Properties

Label 245.4.a.p.1.1
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $6$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.10376\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.51797 q^{2} +3.86039 q^{3} +22.4480 q^{4} -5.00000 q^{5} -21.3015 q^{6} -79.7239 q^{8} -12.0974 q^{9} +O(q^{10})\) \(q-5.51797 q^{2} +3.86039 q^{3} +22.4480 q^{4} -5.00000 q^{5} -21.3015 q^{6} -79.7239 q^{8} -12.0974 q^{9} +27.5899 q^{10} +34.5211 q^{11} +86.6582 q^{12} +68.8935 q^{13} -19.3020 q^{15} +260.330 q^{16} -91.4346 q^{17} +66.7530 q^{18} +11.8278 q^{19} -112.240 q^{20} -190.486 q^{22} -0.104165 q^{23} -307.765 q^{24} +25.0000 q^{25} -380.152 q^{26} -150.931 q^{27} +190.863 q^{29} +106.508 q^{30} +159.802 q^{31} -798.703 q^{32} +133.265 q^{33} +504.534 q^{34} -271.562 q^{36} -177.908 q^{37} -65.2657 q^{38} +265.956 q^{39} +398.619 q^{40} +145.247 q^{41} +8.25729 q^{43} +774.930 q^{44} +60.4869 q^{45} +0.574782 q^{46} +260.529 q^{47} +1004.98 q^{48} -137.949 q^{50} -352.973 q^{51} +1546.52 q^{52} +353.107 q^{53} +832.834 q^{54} -172.605 q^{55} +45.6601 q^{57} -1053.18 q^{58} +240.495 q^{59} -433.291 q^{60} +778.188 q^{61} -881.783 q^{62} +2324.58 q^{64} -344.467 q^{65} -735.352 q^{66} +151.945 q^{67} -2052.53 q^{68} -0.402119 q^{69} -311.449 q^{71} +964.450 q^{72} +639.888 q^{73} +981.690 q^{74} +96.5098 q^{75} +265.512 q^{76} -1467.54 q^{78} +391.186 q^{79} -1301.65 q^{80} -256.024 q^{81} -801.472 q^{82} +493.205 q^{83} +457.173 q^{85} -45.5635 q^{86} +736.807 q^{87} -2752.15 q^{88} +473.850 q^{89} -333.765 q^{90} -2.33831 q^{92} +616.898 q^{93} -1437.59 q^{94} -59.1392 q^{95} -3083.31 q^{96} +839.005 q^{97} -417.615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9} + 10 q^{10} - 16 q^{11} + 160 q^{12} + 168 q^{13} - 80 q^{15} + 298 q^{16} - 4 q^{17} + 354 q^{18} + 308 q^{19} - 70 q^{20} - 236 q^{22} - 336 q^{23} - 92 q^{24} + 150 q^{25} + 56 q^{26} + 964 q^{27} + 176 q^{29} - 120 q^{30} + 392 q^{31} - 770 q^{32} + 188 q^{33} + 812 q^{34} + 230 q^{36} - 140 q^{37} + 20 q^{38} + 140 q^{39} + 330 q^{40} + 656 q^{41} - 388 q^{43} - 160 q^{44} - 350 q^{45} - 388 q^{46} + 628 q^{47} + 1396 q^{48} - 50 q^{50} + 744 q^{51} + 1520 q^{52} - 676 q^{53} + 2284 q^{54} + 80 q^{55} + 1468 q^{57} - 2012 q^{58} + 996 q^{59} - 800 q^{60} + 740 q^{61} - 364 q^{62} + 1426 q^{64} - 840 q^{65} - 3620 q^{66} + 1768 q^{67} - 2940 q^{68} - 1048 q^{69} - 224 q^{71} + 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 400 q^{75} - 1340 q^{76} + 8 q^{78} + 1636 q^{79} - 1490 q^{80} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 20 q^{85} + 1180 q^{86} + 1940 q^{87} - 5652 q^{88} - 1904 q^{89} - 1770 q^{90} - 1952 q^{92} - 1592 q^{93} - 3332 q^{94} - 1540 q^{95} - 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.51797 −1.95090 −0.975449 0.220225i \(-0.929321\pi\)
−0.975449 + 0.220225i \(0.929321\pi\)
\(3\) 3.86039 0.742933 0.371466 0.928446i \(-0.378855\pi\)
0.371466 + 0.928446i \(0.378855\pi\)
\(4\) 22.4480 2.80600
\(5\) −5.00000 −0.447214
\(6\) −21.3015 −1.44939
\(7\) 0 0
\(8\) −79.7239 −3.52333
\(9\) −12.0974 −0.448051
\(10\) 27.5899 0.872468
\(11\) 34.5211 0.946227 0.473113 0.881002i \(-0.343130\pi\)
0.473113 + 0.881002i \(0.343130\pi\)
\(12\) 86.6582 2.08467
\(13\) 68.8935 1.46982 0.734908 0.678167i \(-0.237225\pi\)
0.734908 + 0.678167i \(0.237225\pi\)
\(14\) 0 0
\(15\) −19.3020 −0.332250
\(16\) 260.330 4.06766
\(17\) −91.4346 −1.30448 −0.652239 0.758013i \(-0.726170\pi\)
−0.652239 + 0.758013i \(0.726170\pi\)
\(18\) 66.7530 0.874102
\(19\) 11.8278 0.142815 0.0714077 0.997447i \(-0.477251\pi\)
0.0714077 + 0.997447i \(0.477251\pi\)
\(20\) −112.240 −1.25488
\(21\) 0 0
\(22\) −190.486 −1.84599
\(23\) −0.104165 −0.000944348 0 −0.000472174 1.00000i \(-0.500150\pi\)
−0.000472174 1.00000i \(0.500150\pi\)
\(24\) −307.765 −2.61760
\(25\) 25.0000 0.200000
\(26\) −380.152 −2.86746
\(27\) −150.931 −1.07580
\(28\) 0 0
\(29\) 190.863 1.22215 0.611076 0.791572i \(-0.290737\pi\)
0.611076 + 0.791572i \(0.290737\pi\)
\(30\) 106.508 0.648185
\(31\) 159.802 0.925847 0.462924 0.886398i \(-0.346800\pi\)
0.462924 + 0.886398i \(0.346800\pi\)
\(32\) −798.703 −4.41225
\(33\) 133.265 0.702983
\(34\) 504.534 2.54491
\(35\) 0 0
\(36\) −271.562 −1.25723
\(37\) −177.908 −0.790482 −0.395241 0.918577i \(-0.629339\pi\)
−0.395241 + 0.918577i \(0.629339\pi\)
\(38\) −65.2657 −0.278618
\(39\) 265.956 1.09197
\(40\) 398.619 1.57568
\(41\) 145.247 0.553264 0.276632 0.960976i \(-0.410782\pi\)
0.276632 + 0.960976i \(0.410782\pi\)
\(42\) 0 0
\(43\) 8.25729 0.0292843 0.0146421 0.999893i \(-0.495339\pi\)
0.0146421 + 0.999893i \(0.495339\pi\)
\(44\) 774.930 2.65512
\(45\) 60.4869 0.200375
\(46\) 0.574782 0.00184233
\(47\) 260.529 0.808553 0.404277 0.914637i \(-0.367523\pi\)
0.404277 + 0.914637i \(0.367523\pi\)
\(48\) 1004.98 3.02199
\(49\) 0 0
\(50\) −137.949 −0.390180
\(51\) −352.973 −0.969140
\(52\) 1546.52 4.12431
\(53\) 353.107 0.915151 0.457576 0.889171i \(-0.348718\pi\)
0.457576 + 0.889171i \(0.348718\pi\)
\(54\) 832.834 2.09879
\(55\) −172.605 −0.423165
\(56\) 0 0
\(57\) 45.6601 0.106102
\(58\) −1053.18 −2.38429
\(59\) 240.495 0.530673 0.265337 0.964156i \(-0.414517\pi\)
0.265337 + 0.964156i \(0.414517\pi\)
\(60\) −433.291 −0.932294
\(61\) 778.188 1.63339 0.816695 0.577069i \(-0.195804\pi\)
0.816695 + 0.577069i \(0.195804\pi\)
\(62\) −881.783 −1.80623
\(63\) 0 0
\(64\) 2324.58 4.54020
\(65\) −344.467 −0.657322
\(66\) −735.352 −1.37145
\(67\) 151.945 0.277059 0.138530 0.990358i \(-0.455762\pi\)
0.138530 + 0.990358i \(0.455762\pi\)
\(68\) −2052.53 −3.66037
\(69\) −0.402119 −0.000701587 0
\(70\) 0 0
\(71\) −311.449 −0.520594 −0.260297 0.965529i \(-0.583820\pi\)
−0.260297 + 0.965529i \(0.583820\pi\)
\(72\) 964.450 1.57863
\(73\) 639.888 1.02593 0.512967 0.858408i \(-0.328546\pi\)
0.512967 + 0.858408i \(0.328546\pi\)
\(74\) 981.690 1.54215
\(75\) 96.5098 0.148587
\(76\) 265.512 0.400740
\(77\) 0 0
\(78\) −1467.54 −2.13033
\(79\) 391.186 0.557112 0.278556 0.960420i \(-0.410144\pi\)
0.278556 + 0.960420i \(0.410144\pi\)
\(80\) −1301.65 −1.81911
\(81\) −256.024 −0.351199
\(82\) −801.472 −1.07936
\(83\) 493.205 0.652245 0.326122 0.945328i \(-0.394258\pi\)
0.326122 + 0.945328i \(0.394258\pi\)
\(84\) 0 0
\(85\) 457.173 0.583381
\(86\) −45.5635 −0.0571307
\(87\) 736.807 0.907977
\(88\) −2752.15 −3.33387
\(89\) 473.850 0.564359 0.282180 0.959362i \(-0.408943\pi\)
0.282180 + 0.959362i \(0.408943\pi\)
\(90\) −333.765 −0.390910
\(91\) 0 0
\(92\) −2.33831 −0.00264984
\(93\) 616.898 0.687842
\(94\) −1437.59 −1.57741
\(95\) −59.1392 −0.0638690
\(96\) −3083.31 −3.27801
\(97\) 839.005 0.878227 0.439114 0.898431i \(-0.355293\pi\)
0.439114 + 0.898431i \(0.355293\pi\)
\(98\) 0 0
\(99\) −417.615 −0.423958
\(100\) 561.201 0.561201
\(101\) 887.440 0.874293 0.437146 0.899390i \(-0.355989\pi\)
0.437146 + 0.899390i \(0.355989\pi\)
\(102\) 1947.70 1.89069
\(103\) 619.087 0.592238 0.296119 0.955151i \(-0.404308\pi\)
0.296119 + 0.955151i \(0.404308\pi\)
\(104\) −5492.46 −5.17865
\(105\) 0 0
\(106\) −1948.44 −1.78537
\(107\) −2151.30 −1.94368 −0.971841 0.235639i \(-0.924282\pi\)
−0.971841 + 0.235639i \(0.924282\pi\)
\(108\) −3388.11 −3.01871
\(109\) −407.076 −0.357714 −0.178857 0.983875i \(-0.557240\pi\)
−0.178857 + 0.983875i \(0.557240\pi\)
\(110\) 952.432 0.825553
\(111\) −686.793 −0.587275
\(112\) 0 0
\(113\) −349.581 −0.291025 −0.145513 0.989356i \(-0.546483\pi\)
−0.145513 + 0.989356i \(0.546483\pi\)
\(114\) −251.951 −0.206995
\(115\) 0.520827 0.000422325 0
\(116\) 4284.50 3.42936
\(117\) −833.431 −0.658553
\(118\) −1327.04 −1.03529
\(119\) 0 0
\(120\) 1538.83 1.17063
\(121\) −139.296 −0.104655
\(122\) −4294.02 −3.18658
\(123\) 560.712 0.411038
\(124\) 3587.24 2.59793
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1183.78 0.827114 0.413557 0.910478i \(-0.364286\pi\)
0.413557 + 0.910478i \(0.364286\pi\)
\(128\) −6437.36 −4.44522
\(129\) 31.8764 0.0217563
\(130\) 1900.76 1.28237
\(131\) 223.357 0.148968 0.0744840 0.997222i \(-0.476269\pi\)
0.0744840 + 0.997222i \(0.476269\pi\)
\(132\) 2991.53 1.97257
\(133\) 0 0
\(134\) −838.426 −0.540515
\(135\) 754.656 0.481114
\(136\) 7289.52 4.59611
\(137\) 2036.66 1.27010 0.635050 0.772471i \(-0.280979\pi\)
0.635050 + 0.772471i \(0.280979\pi\)
\(138\) 2.21888 0.00136872
\(139\) 2687.00 1.63963 0.819815 0.572629i \(-0.194076\pi\)
0.819815 + 0.572629i \(0.194076\pi\)
\(140\) 0 0
\(141\) 1005.74 0.600701
\(142\) 1718.57 1.01563
\(143\) 2378.28 1.39078
\(144\) −3149.31 −1.82252
\(145\) −954.316 −0.546563
\(146\) −3530.88 −2.00149
\(147\) 0 0
\(148\) −3993.68 −2.21810
\(149\) 673.500 0.370304 0.185152 0.982710i \(-0.440722\pi\)
0.185152 + 0.982710i \(0.440722\pi\)
\(150\) −532.538 −0.289877
\(151\) −2125.18 −1.14533 −0.572664 0.819790i \(-0.694090\pi\)
−0.572664 + 0.819790i \(0.694090\pi\)
\(152\) −942.961 −0.503186
\(153\) 1106.12 0.584473
\(154\) 0 0
\(155\) −799.010 −0.414052
\(156\) 5970.18 3.06409
\(157\) −2813.03 −1.42997 −0.714983 0.699142i \(-0.753565\pi\)
−0.714983 + 0.699142i \(0.753565\pi\)
\(158\) −2158.55 −1.08687
\(159\) 1363.13 0.679896
\(160\) 3993.52 1.97322
\(161\) 0 0
\(162\) 1412.73 0.685153
\(163\) −1344.42 −0.646032 −0.323016 0.946394i \(-0.604697\pi\)
−0.323016 + 0.946394i \(0.604697\pi\)
\(164\) 3260.52 1.55246
\(165\) −666.324 −0.314383
\(166\) −2721.49 −1.27246
\(167\) 1451.24 0.672456 0.336228 0.941781i \(-0.390849\pi\)
0.336228 + 0.941781i \(0.390849\pi\)
\(168\) 0 0
\(169\) 2549.31 1.16036
\(170\) −2522.67 −1.13812
\(171\) −143.086 −0.0639886
\(172\) 185.360 0.0821718
\(173\) −1979.16 −0.869784 −0.434892 0.900483i \(-0.643213\pi\)
−0.434892 + 0.900483i \(0.643213\pi\)
\(174\) −4065.68 −1.77137
\(175\) 0 0
\(176\) 8986.87 3.84893
\(177\) 928.403 0.394254
\(178\) −2614.69 −1.10101
\(179\) −4358.66 −1.82001 −0.910005 0.414598i \(-0.863922\pi\)
−0.910005 + 0.414598i \(0.863922\pi\)
\(180\) 1357.81 0.562252
\(181\) −377.923 −0.155198 −0.0775988 0.996985i \(-0.524725\pi\)
−0.0775988 + 0.996985i \(0.524725\pi\)
\(182\) 0 0
\(183\) 3004.11 1.21350
\(184\) 8.30447 0.00332725
\(185\) 889.538 0.353514
\(186\) −3404.03 −1.34191
\(187\) −3156.42 −1.23433
\(188\) 5848.36 2.26880
\(189\) 0 0
\(190\) 326.328 0.124602
\(191\) −2425.95 −0.919033 −0.459517 0.888169i \(-0.651977\pi\)
−0.459517 + 0.888169i \(0.651977\pi\)
\(192\) 8973.80 3.37306
\(193\) −622.923 −0.232326 −0.116163 0.993230i \(-0.537060\pi\)
−0.116163 + 0.993230i \(0.537060\pi\)
\(194\) −4629.61 −1.71333
\(195\) −1329.78 −0.488346
\(196\) 0 0
\(197\) 2842.29 1.02794 0.513971 0.857807i \(-0.328174\pi\)
0.513971 + 0.857807i \(0.328174\pi\)
\(198\) 2304.39 0.827099
\(199\) −867.364 −0.308974 −0.154487 0.987995i \(-0.549372\pi\)
−0.154487 + 0.987995i \(0.549372\pi\)
\(200\) −1993.10 −0.704666
\(201\) 586.565 0.205836
\(202\) −4896.87 −1.70566
\(203\) 0 0
\(204\) −7923.55 −2.71941
\(205\) −726.237 −0.247427
\(206\) −3416.11 −1.15540
\(207\) 1.26013 0.000423116 0
\(208\) 17935.0 5.97871
\(209\) 408.309 0.135136
\(210\) 0 0
\(211\) −5975.92 −1.94976 −0.974880 0.222730i \(-0.928503\pi\)
−0.974880 + 0.222730i \(0.928503\pi\)
\(212\) 7926.57 2.56792
\(213\) −1202.31 −0.386766
\(214\) 11870.8 3.79192
\(215\) −41.2864 −0.0130963
\(216\) 12032.8 3.79042
\(217\) 0 0
\(218\) 2246.24 0.697864
\(219\) 2470.22 0.762200
\(220\) −3874.65 −1.18740
\(221\) −6299.25 −1.91734
\(222\) 3789.71 1.14571
\(223\) −5181.58 −1.55598 −0.777992 0.628274i \(-0.783762\pi\)
−0.777992 + 0.628274i \(0.783762\pi\)
\(224\) 0 0
\(225\) −302.435 −0.0896102
\(226\) 1928.98 0.567761
\(227\) 3753.06 1.09735 0.548677 0.836035i \(-0.315132\pi\)
0.548677 + 0.836035i \(0.315132\pi\)
\(228\) 1024.98 0.297723
\(229\) −6258.14 −1.80589 −0.902947 0.429752i \(-0.858601\pi\)
−0.902947 + 0.429752i \(0.858601\pi\)
\(230\) −2.87391 −0.000823913 0
\(231\) 0 0
\(232\) −15216.4 −4.30605
\(233\) 1779.96 0.500469 0.250234 0.968185i \(-0.419492\pi\)
0.250234 + 0.968185i \(0.419492\pi\)
\(234\) 4598.85 1.28477
\(235\) −1302.64 −0.361596
\(236\) 5398.63 1.48907
\(237\) 1510.13 0.413897
\(238\) 0 0
\(239\) 3519.46 0.952532 0.476266 0.879301i \(-0.341990\pi\)
0.476266 + 0.879301i \(0.341990\pi\)
\(240\) −5024.88 −1.35148
\(241\) 362.930 0.0970058 0.0485029 0.998823i \(-0.484555\pi\)
0.0485029 + 0.998823i \(0.484555\pi\)
\(242\) 768.629 0.204171
\(243\) 3086.79 0.814887
\(244\) 17468.8 4.58330
\(245\) 0 0
\(246\) −3093.99 −0.801894
\(247\) 814.861 0.209912
\(248\) −12740.0 −3.26207
\(249\) 1903.97 0.484574
\(250\) 689.747 0.174494
\(251\) −5333.85 −1.34131 −0.670656 0.741768i \(-0.733988\pi\)
−0.670656 + 0.741768i \(0.733988\pi\)
\(252\) 0 0
\(253\) −3.59590 −0.000893567 0
\(254\) −6532.06 −1.61361
\(255\) 1764.87 0.433412
\(256\) 16924.5 4.13197
\(257\) −2438.78 −0.591933 −0.295966 0.955198i \(-0.595642\pi\)
−0.295966 + 0.955198i \(0.595642\pi\)
\(258\) −175.893 −0.0424442
\(259\) 0 0
\(260\) −7732.62 −1.84445
\(261\) −2308.95 −0.547587
\(262\) −1232.48 −0.290621
\(263\) 1526.25 0.357843 0.178921 0.983863i \(-0.442739\pi\)
0.178921 + 0.983863i \(0.442739\pi\)
\(264\) −10624.4 −2.47684
\(265\) −1765.54 −0.409268
\(266\) 0 0
\(267\) 1829.25 0.419281
\(268\) 3410.86 0.777430
\(269\) −7564.12 −1.71447 −0.857235 0.514925i \(-0.827820\pi\)
−0.857235 + 0.514925i \(0.827820\pi\)
\(270\) −4164.17 −0.938605
\(271\) 4282.68 0.959980 0.479990 0.877274i \(-0.340640\pi\)
0.479990 + 0.877274i \(0.340640\pi\)
\(272\) −23803.2 −5.30617
\(273\) 0 0
\(274\) −11238.2 −2.47784
\(275\) 863.027 0.189245
\(276\) −9.02679 −0.00196866
\(277\) −4008.41 −0.869465 −0.434732 0.900560i \(-0.643157\pi\)
−0.434732 + 0.900560i \(0.643157\pi\)
\(278\) −14826.8 −3.19875
\(279\) −1933.18 −0.414827
\(280\) 0 0
\(281\) 6935.44 1.47236 0.736181 0.676785i \(-0.236627\pi\)
0.736181 + 0.676785i \(0.236627\pi\)
\(282\) −5549.66 −1.17191
\(283\) −2666.64 −0.560124 −0.280062 0.959982i \(-0.590355\pi\)
−0.280062 + 0.959982i \(0.590355\pi\)
\(284\) −6991.41 −1.46079
\(285\) −228.300 −0.0474503
\(286\) −13123.3 −2.71327
\(287\) 0 0
\(288\) 9662.22 1.97692
\(289\) 3447.28 0.701665
\(290\) 5265.89 1.06629
\(291\) 3238.89 0.652464
\(292\) 14364.2 2.87878
\(293\) 5939.10 1.18418 0.592092 0.805870i \(-0.298302\pi\)
0.592092 + 0.805870i \(0.298302\pi\)
\(294\) 0 0
\(295\) −1202.47 −0.237324
\(296\) 14183.5 2.78513
\(297\) −5210.31 −1.01795
\(298\) −3716.36 −0.722425
\(299\) −7.17632 −0.00138802
\(300\) 2166.45 0.416934
\(301\) 0 0
\(302\) 11726.7 2.23442
\(303\) 3425.87 0.649541
\(304\) 3079.14 0.580924
\(305\) −3890.94 −0.730474
\(306\) −6103.53 −1.14025
\(307\) 10381.5 1.92998 0.964992 0.262278i \(-0.0844737\pi\)
0.964992 + 0.262278i \(0.0844737\pi\)
\(308\) 0 0
\(309\) 2389.92 0.439993
\(310\) 4408.91 0.807772
\(311\) 4240.54 0.773181 0.386590 0.922252i \(-0.373653\pi\)
0.386590 + 0.922252i \(0.373653\pi\)
\(312\) −21203.0 −3.84739
\(313\) −283.903 −0.0512688 −0.0256344 0.999671i \(-0.508161\pi\)
−0.0256344 + 0.999671i \(0.508161\pi\)
\(314\) 15522.2 2.78972
\(315\) 0 0
\(316\) 8781.36 1.56326
\(317\) 1739.49 0.308201 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(318\) −7521.73 −1.32641
\(319\) 6588.80 1.15643
\(320\) −11622.9 −2.03044
\(321\) −8304.85 −1.44402
\(322\) 0 0
\(323\) −1081.47 −0.186300
\(324\) −5747.24 −0.985466
\(325\) 1722.34 0.293963
\(326\) 7418.48 1.26034
\(327\) −1571.47 −0.265758
\(328\) −11579.7 −1.94933
\(329\) 0 0
\(330\) 3676.76 0.613330
\(331\) 5606.96 0.931077 0.465538 0.885028i \(-0.345861\pi\)
0.465538 + 0.885028i \(0.345861\pi\)
\(332\) 11071.5 1.83020
\(333\) 2152.22 0.354176
\(334\) −8007.89 −1.31189
\(335\) −759.723 −0.123905
\(336\) 0 0
\(337\) −9427.44 −1.52387 −0.761937 0.647652i \(-0.775751\pi\)
−0.761937 + 0.647652i \(0.775751\pi\)
\(338\) −14067.0 −2.26375
\(339\) −1349.52 −0.216212
\(340\) 10262.6 1.63697
\(341\) 5516.53 0.876062
\(342\) 789.544 0.124835
\(343\) 0 0
\(344\) −658.303 −0.103178
\(345\) 2.01060 0.000313759 0
\(346\) 10920.9 1.69686
\(347\) 11634.2 1.79988 0.899939 0.436017i \(-0.143611\pi\)
0.899939 + 0.436017i \(0.143611\pi\)
\(348\) 16539.9 2.54779
\(349\) −1317.10 −0.202013 −0.101006 0.994886i \(-0.532206\pi\)
−0.101006 + 0.994886i \(0.532206\pi\)
\(350\) 0 0
\(351\) −10398.2 −1.58124
\(352\) −27572.1 −4.17499
\(353\) 5848.82 0.881874 0.440937 0.897538i \(-0.354646\pi\)
0.440937 + 0.897538i \(0.354646\pi\)
\(354\) −5122.90 −0.769150
\(355\) 1557.24 0.232817
\(356\) 10637.0 1.58359
\(357\) 0 0
\(358\) 24051.0 3.55065
\(359\) −12422.0 −1.82621 −0.913104 0.407727i \(-0.866321\pi\)
−0.913104 + 0.407727i \(0.866321\pi\)
\(360\) −4822.25 −0.705986
\(361\) −6719.10 −0.979604
\(362\) 2085.37 0.302775
\(363\) −537.735 −0.0777515
\(364\) 0 0
\(365\) −3199.44 −0.458812
\(366\) −16576.6 −2.36741
\(367\) −6590.78 −0.937427 −0.468713 0.883350i \(-0.655282\pi\)
−0.468713 + 0.883350i \(0.655282\pi\)
\(368\) −27.1174 −0.00384128
\(369\) −1757.11 −0.247891
\(370\) −4908.45 −0.689670
\(371\) 0 0
\(372\) 13848.1 1.93009
\(373\) 344.278 0.0477910 0.0238955 0.999714i \(-0.492393\pi\)
0.0238955 + 0.999714i \(0.492393\pi\)
\(374\) 17417.0 2.40806
\(375\) −482.549 −0.0664499
\(376\) −20770.4 −2.84880
\(377\) 13149.2 1.79634
\(378\) 0 0
\(379\) 5241.23 0.710353 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(380\) −1327.56 −0.179217
\(381\) 4569.85 0.614490
\(382\) 13386.3 1.79294
\(383\) 7597.37 1.01360 0.506798 0.862065i \(-0.330829\pi\)
0.506798 + 0.862065i \(0.330829\pi\)
\(384\) −24850.7 −3.30250
\(385\) 0 0
\(386\) 3437.27 0.453245
\(387\) −99.8916 −0.0131209
\(388\) 18834.0 2.46431
\(389\) 3101.84 0.404291 0.202146 0.979355i \(-0.435209\pi\)
0.202146 + 0.979355i \(0.435209\pi\)
\(390\) 7337.69 0.952713
\(391\) 9.52432 0.00123188
\(392\) 0 0
\(393\) 862.246 0.110673
\(394\) −15683.7 −2.00541
\(395\) −1955.93 −0.249148
\(396\) −9374.63 −1.18963
\(397\) −2932.06 −0.370669 −0.185335 0.982675i \(-0.559337\pi\)
−0.185335 + 0.982675i \(0.559337\pi\)
\(398\) 4786.09 0.602777
\(399\) 0 0
\(400\) 6508.25 0.813531
\(401\) 89.2375 0.0111130 0.00555649 0.999985i \(-0.498231\pi\)
0.00555649 + 0.999985i \(0.498231\pi\)
\(402\) −3236.65 −0.401566
\(403\) 11009.3 1.36083
\(404\) 19921.3 2.45327
\(405\) 1280.12 0.157061
\(406\) 0 0
\(407\) −6141.56 −0.747975
\(408\) 28140.4 3.41460
\(409\) 147.388 0.0178188 0.00890940 0.999960i \(-0.497164\pi\)
0.00890940 + 0.999960i \(0.497164\pi\)
\(410\) 4007.36 0.482706
\(411\) 7862.31 0.943599
\(412\) 13897.3 1.66182
\(413\) 0 0
\(414\) −6.95336 −0.000825457 0
\(415\) −2466.03 −0.291693
\(416\) −55025.4 −6.48520
\(417\) 10372.9 1.21813
\(418\) −2253.04 −0.263636
\(419\) −3781.67 −0.440923 −0.220462 0.975396i \(-0.570756\pi\)
−0.220462 + 0.975396i \(0.570756\pi\)
\(420\) 0 0
\(421\) −10899.2 −1.26175 −0.630874 0.775885i \(-0.717304\pi\)
−0.630874 + 0.775885i \(0.717304\pi\)
\(422\) 32975.0 3.80378
\(423\) −3151.71 −0.362273
\(424\) −28151.1 −3.22438
\(425\) −2285.86 −0.260896
\(426\) 6634.33 0.754541
\(427\) 0 0
\(428\) −48292.4 −5.45398
\(429\) 9181.08 1.03326
\(430\) 227.817 0.0255496
\(431\) −11283.4 −1.26102 −0.630512 0.776180i \(-0.717155\pi\)
−0.630512 + 0.776180i \(0.717155\pi\)
\(432\) −39291.9 −4.37600
\(433\) −8906.19 −0.988462 −0.494231 0.869331i \(-0.664550\pi\)
−0.494231 + 0.869331i \(0.664550\pi\)
\(434\) 0 0
\(435\) −3684.03 −0.406059
\(436\) −9138.07 −1.00375
\(437\) −1.23205 −0.000134867 0
\(438\) −13630.6 −1.48697
\(439\) 8149.49 0.886000 0.443000 0.896522i \(-0.353914\pi\)
0.443000 + 0.896522i \(0.353914\pi\)
\(440\) 13760.8 1.49095
\(441\) 0 0
\(442\) 34759.1 3.74054
\(443\) −11472.8 −1.23045 −0.615223 0.788353i \(-0.710934\pi\)
−0.615223 + 0.788353i \(0.710934\pi\)
\(444\) −15417.2 −1.64790
\(445\) −2369.25 −0.252389
\(446\) 28591.8 3.03557
\(447\) 2599.97 0.275111
\(448\) 0 0
\(449\) 1963.80 0.206409 0.103204 0.994660i \(-0.467090\pi\)
0.103204 + 0.994660i \(0.467090\pi\)
\(450\) 1668.83 0.174820
\(451\) 5014.10 0.523514
\(452\) −7847.42 −0.816618
\(453\) −8204.02 −0.850902
\(454\) −20709.3 −2.14083
\(455\) 0 0
\(456\) −3640.20 −0.373833
\(457\) 5589.85 0.572171 0.286086 0.958204i \(-0.407646\pi\)
0.286086 + 0.958204i \(0.407646\pi\)
\(458\) 34532.3 3.52311
\(459\) 13800.3 1.40336
\(460\) 11.6915 0.00118505
\(461\) −18790.9 −1.89844 −0.949219 0.314618i \(-0.898124\pi\)
−0.949219 + 0.314618i \(0.898124\pi\)
\(462\) 0 0
\(463\) −7892.22 −0.792187 −0.396094 0.918210i \(-0.629634\pi\)
−0.396094 + 0.918210i \(0.629634\pi\)
\(464\) 49687.4 4.97130
\(465\) −3084.49 −0.307612
\(466\) −9821.78 −0.976363
\(467\) 385.511 0.0381998 0.0190999 0.999818i \(-0.493920\pi\)
0.0190999 + 0.999818i \(0.493920\pi\)
\(468\) −18708.9 −1.84790
\(469\) 0 0
\(470\) 7187.95 0.705437
\(471\) −10859.4 −1.06237
\(472\) −19173.2 −1.86974
\(473\) 285.050 0.0277096
\(474\) −8332.86 −0.807471
\(475\) 295.696 0.0285631
\(476\) 0 0
\(477\) −4271.67 −0.410035
\(478\) −19420.3 −1.85829
\(479\) 1694.46 0.161632 0.0808160 0.996729i \(-0.474247\pi\)
0.0808160 + 0.996729i \(0.474247\pi\)
\(480\) 15416.5 1.46597
\(481\) −12256.7 −1.16186
\(482\) −2002.64 −0.189248
\(483\) 0 0
\(484\) −3126.91 −0.293662
\(485\) −4195.03 −0.392755
\(486\) −17032.8 −1.58976
\(487\) 15711.2 1.46189 0.730945 0.682436i \(-0.239079\pi\)
0.730945 + 0.682436i \(0.239079\pi\)
\(488\) −62040.2 −5.75498
\(489\) −5189.99 −0.479958
\(490\) 0 0
\(491\) −2716.34 −0.249667 −0.124834 0.992178i \(-0.539840\pi\)
−0.124834 + 0.992178i \(0.539840\pi\)
\(492\) 12586.9 1.15337
\(493\) −17451.5 −1.59427
\(494\) −4496.38 −0.409518
\(495\) 2088.07 0.189600
\(496\) 41601.2 3.76603
\(497\) 0 0
\(498\) −10506.0 −0.945355
\(499\) 4295.34 0.385342 0.192671 0.981263i \(-0.438285\pi\)
0.192671 + 0.981263i \(0.438285\pi\)
\(500\) −2806.00 −0.250977
\(501\) 5602.34 0.499589
\(502\) 29432.0 2.61677
\(503\) 6515.26 0.577537 0.288769 0.957399i \(-0.406754\pi\)
0.288769 + 0.957399i \(0.406754\pi\)
\(504\) 0 0
\(505\) −4437.20 −0.390996
\(506\) 19.8421 0.00174326
\(507\) 9841.34 0.862070
\(508\) 26573.5 2.32088
\(509\) 14391.8 1.25325 0.626624 0.779321i \(-0.284436\pi\)
0.626624 + 0.779321i \(0.284436\pi\)
\(510\) −9738.48 −0.845544
\(511\) 0 0
\(512\) −41890.2 −3.61583
\(513\) −1785.19 −0.153641
\(514\) 13457.1 1.15480
\(515\) −3095.44 −0.264857
\(516\) 715.562 0.0610481
\(517\) 8993.73 0.765075
\(518\) 0 0
\(519\) −7640.32 −0.646191
\(520\) 27462.3 2.31596
\(521\) 2913.36 0.244984 0.122492 0.992469i \(-0.460911\pi\)
0.122492 + 0.992469i \(0.460911\pi\)
\(522\) 12740.7 1.06829
\(523\) 16870.2 1.41048 0.705242 0.708967i \(-0.250838\pi\)
0.705242 + 0.708967i \(0.250838\pi\)
\(524\) 5013.93 0.418005
\(525\) 0 0
\(526\) −8421.82 −0.698115
\(527\) −14611.4 −1.20775
\(528\) 34692.8 2.85949
\(529\) −12167.0 −0.999999
\(530\) 9742.18 0.798441
\(531\) −2909.35 −0.237769
\(532\) 0 0
\(533\) 10006.6 0.813197
\(534\) −10093.7 −0.817974
\(535\) 10756.5 0.869241
\(536\) −12113.6 −0.976172
\(537\) −16826.1 −1.35214
\(538\) 41738.6 3.34476
\(539\) 0 0
\(540\) 16940.5 1.35001
\(541\) −2229.59 −0.177186 −0.0885929 0.996068i \(-0.528237\pi\)
−0.0885929 + 0.996068i \(0.528237\pi\)
\(542\) −23631.7 −1.87282
\(543\) −1458.93 −0.115301
\(544\) 73029.1 5.75569
\(545\) 2035.38 0.159975
\(546\) 0 0
\(547\) −1218.73 −0.0952639 −0.0476319 0.998865i \(-0.515167\pi\)
−0.0476319 + 0.998865i \(0.515167\pi\)
\(548\) 45719.1 3.56391
\(549\) −9414.04 −0.731843
\(550\) −4762.16 −0.369198
\(551\) 2257.50 0.174542
\(552\) 32.0585 0.00247192
\(553\) 0 0
\(554\) 22118.3 1.69624
\(555\) 3433.96 0.262637
\(556\) 60317.9 4.60081
\(557\) −22734.5 −1.72943 −0.864714 0.502264i \(-0.832501\pi\)
−0.864714 + 0.502264i \(0.832501\pi\)
\(558\) 10667.3 0.809285
\(559\) 568.873 0.0430425
\(560\) 0 0
\(561\) −12185.0 −0.917026
\(562\) −38269.6 −2.87243
\(563\) 4302.11 0.322047 0.161023 0.986951i \(-0.448521\pi\)
0.161023 + 0.986951i \(0.448521\pi\)
\(564\) 22576.9 1.68557
\(565\) 1747.91 0.130150
\(566\) 14714.4 1.09275
\(567\) 0 0
\(568\) 24829.9 1.83422
\(569\) 14866.9 1.09535 0.547675 0.836691i \(-0.315513\pi\)
0.547675 + 0.836691i \(0.315513\pi\)
\(570\) 1259.76 0.0925708
\(571\) 16514.5 1.21035 0.605174 0.796093i \(-0.293103\pi\)
0.605174 + 0.796093i \(0.293103\pi\)
\(572\) 53387.6 3.90253
\(573\) −9365.10 −0.682780
\(574\) 0 0
\(575\) −2.60414 −0.000188870 0
\(576\) −28121.4 −2.03424
\(577\) 1867.97 0.134774 0.0673870 0.997727i \(-0.478534\pi\)
0.0673870 + 0.997727i \(0.478534\pi\)
\(578\) −19022.0 −1.36888
\(579\) −2404.73 −0.172603
\(580\) −21422.5 −1.53366
\(581\) 0 0
\(582\) −17872.1 −1.27289
\(583\) 12189.6 0.865941
\(584\) −51014.3 −3.61471
\(585\) 4167.15 0.294514
\(586\) −32771.8 −2.31022
\(587\) 3926.15 0.276064 0.138032 0.990428i \(-0.455922\pi\)
0.138032 + 0.990428i \(0.455922\pi\)
\(588\) 0 0
\(589\) 1890.11 0.132225
\(590\) 6635.21 0.462996
\(591\) 10972.3 0.763692
\(592\) −46314.7 −3.21541
\(593\) −7554.18 −0.523125 −0.261562 0.965187i \(-0.584238\pi\)
−0.261562 + 0.965187i \(0.584238\pi\)
\(594\) 28750.3 1.98593
\(595\) 0 0
\(596\) 15118.8 1.03907
\(597\) −3348.36 −0.229547
\(598\) 39.5988 0.00270788
\(599\) 28210.9 1.92432 0.962160 0.272487i \(-0.0878461\pi\)
0.962160 + 0.272487i \(0.0878461\pi\)
\(600\) −7694.13 −0.523519
\(601\) −23181.4 −1.57336 −0.786679 0.617363i \(-0.788201\pi\)
−0.786679 + 0.617363i \(0.788201\pi\)
\(602\) 0 0
\(603\) −1838.13 −0.124137
\(604\) −47706.1 −3.21380
\(605\) 696.478 0.0468031
\(606\) −18903.8 −1.26719
\(607\) −24863.6 −1.66257 −0.831287 0.555844i \(-0.812395\pi\)
−0.831287 + 0.555844i \(0.812395\pi\)
\(608\) −9446.93 −0.630137
\(609\) 0 0
\(610\) 21470.1 1.42508
\(611\) 17948.7 1.18843
\(612\) 24830.2 1.64003
\(613\) −12792.7 −0.842893 −0.421446 0.906853i \(-0.638477\pi\)
−0.421446 + 0.906853i \(0.638477\pi\)
\(614\) −57285.0 −3.76520
\(615\) −2803.56 −0.183822
\(616\) 0 0
\(617\) 19793.3 1.29149 0.645744 0.763554i \(-0.276547\pi\)
0.645744 + 0.763554i \(0.276547\pi\)
\(618\) −13187.5 −0.858381
\(619\) −20951.7 −1.36045 −0.680226 0.733002i \(-0.738118\pi\)
−0.680226 + 0.733002i \(0.738118\pi\)
\(620\) −17936.2 −1.16183
\(621\) 15.7218 0.00101593
\(622\) −23399.2 −1.50840
\(623\) 0 0
\(624\) 69236.3 4.44178
\(625\) 625.000 0.0400000
\(626\) 1566.57 0.100020
\(627\) 1576.23 0.100397
\(628\) −63147.1 −4.01249
\(629\) 16266.9 1.03117
\(630\) 0 0
\(631\) 23273.5 1.46831 0.734156 0.678981i \(-0.237578\pi\)
0.734156 + 0.678981i \(0.237578\pi\)
\(632\) −31186.9 −1.96289
\(633\) −23069.4 −1.44854
\(634\) −9598.47 −0.601268
\(635\) −5918.90 −0.369896
\(636\) 30599.6 1.90779
\(637\) 0 0
\(638\) −36356.8 −2.25608
\(639\) 3767.71 0.233253
\(640\) 32186.8 1.98796
\(641\) −22117.7 −1.36287 −0.681434 0.731880i \(-0.738643\pi\)
−0.681434 + 0.731880i \(0.738643\pi\)
\(642\) 45826.0 2.81714
\(643\) 20269.9 1.24318 0.621591 0.783342i \(-0.286487\pi\)
0.621591 + 0.783342i \(0.286487\pi\)
\(644\) 0 0
\(645\) −159.382 −0.00972969
\(646\) 5967.54 0.363451
\(647\) −3145.25 −0.191117 −0.0955583 0.995424i \(-0.530464\pi\)
−0.0955583 + 0.995424i \(0.530464\pi\)
\(648\) 20411.2 1.23739
\(649\) 8302.13 0.502137
\(650\) −9503.81 −0.573493
\(651\) 0 0
\(652\) −30179.6 −1.81277
\(653\) −5953.35 −0.356773 −0.178386 0.983961i \(-0.557088\pi\)
−0.178386 + 0.983961i \(0.557088\pi\)
\(654\) 8671.35 0.518466
\(655\) −1116.79 −0.0666205
\(656\) 37812.3 2.25049
\(657\) −7740.96 −0.459671
\(658\) 0 0
\(659\) 26277.5 1.55330 0.776652 0.629930i \(-0.216916\pi\)
0.776652 + 0.629930i \(0.216916\pi\)
\(660\) −14957.7 −0.882161
\(661\) −24004.3 −1.41250 −0.706248 0.707964i \(-0.749614\pi\)
−0.706248 + 0.707964i \(0.749614\pi\)
\(662\) −30939.1 −1.81644
\(663\) −24317.6 −1.42446
\(664\) −39320.3 −2.29808
\(665\) 0 0
\(666\) −11875.9 −0.690962
\(667\) −19.8814 −0.00115414
\(668\) 32577.4 1.88691
\(669\) −20002.9 −1.15599
\(670\) 4192.13 0.241725
\(671\) 26863.9 1.54556
\(672\) 0 0
\(673\) 9205.36 0.527252 0.263626 0.964625i \(-0.415082\pi\)
0.263626 + 0.964625i \(0.415082\pi\)
\(674\) 52020.4 2.97292
\(675\) −3773.28 −0.215161
\(676\) 57227.1 3.25598
\(677\) −18773.1 −1.06575 −0.532873 0.846195i \(-0.678888\pi\)
−0.532873 + 0.846195i \(0.678888\pi\)
\(678\) 7446.62 0.421808
\(679\) 0 0
\(680\) −36447.6 −2.05544
\(681\) 14488.3 0.815260
\(682\) −30440.1 −1.70911
\(683\) −10222.2 −0.572684 −0.286342 0.958127i \(-0.592439\pi\)
−0.286342 + 0.958127i \(0.592439\pi\)
\(684\) −3212.00 −0.179552
\(685\) −10183.3 −0.568006
\(686\) 0 0
\(687\) −24158.9 −1.34166
\(688\) 2149.62 0.119118
\(689\) 24326.8 1.34510
\(690\) −11.0944 −0.000612112 0
\(691\) 22355.1 1.23072 0.615361 0.788246i \(-0.289010\pi\)
0.615361 + 0.788246i \(0.289010\pi\)
\(692\) −44428.2 −2.44062
\(693\) 0 0
\(694\) −64197.3 −3.51138
\(695\) −13435.0 −0.733264
\(696\) −58741.1 −3.19910
\(697\) −13280.6 −0.721722
\(698\) 7267.70 0.394107
\(699\) 6871.35 0.371814
\(700\) 0 0
\(701\) 16217.3 0.873779 0.436890 0.899515i \(-0.356080\pi\)
0.436890 + 0.899515i \(0.356080\pi\)
\(702\) 57376.9 3.08483
\(703\) −2104.26 −0.112893
\(704\) 80247.1 4.29606
\(705\) −5028.71 −0.268642
\(706\) −32273.7 −1.72045
\(707\) 0 0
\(708\) 20840.8 1.10628
\(709\) 3922.92 0.207798 0.103899 0.994588i \(-0.466868\pi\)
0.103899 + 0.994588i \(0.466868\pi\)
\(710\) −8592.83 −0.454202
\(711\) −4732.33 −0.249615
\(712\) −37777.1 −1.98842
\(713\) −16.6458 −0.000874322 0
\(714\) 0 0
\(715\) −11891.4 −0.621976
\(716\) −97843.4 −5.10695
\(717\) 13586.5 0.707667
\(718\) 68544.3 3.56275
\(719\) 23908.6 1.24011 0.620055 0.784558i \(-0.287110\pi\)
0.620055 + 0.784558i \(0.287110\pi\)
\(720\) 15746.6 0.815055
\(721\) 0 0
\(722\) 37075.8 1.91111
\(723\) 1401.05 0.0720688
\(724\) −8483.62 −0.435485
\(725\) 4771.58 0.244430
\(726\) 2967.21 0.151685
\(727\) −26906.4 −1.37263 −0.686315 0.727305i \(-0.740773\pi\)
−0.686315 + 0.727305i \(0.740773\pi\)
\(728\) 0 0
\(729\) 18828.9 0.956605
\(730\) 17654.4 0.895095
\(731\) −755.001 −0.0382007
\(732\) 67436.4 3.40508
\(733\) 27637.5 1.39265 0.696325 0.717726i \(-0.254817\pi\)
0.696325 + 0.717726i \(0.254817\pi\)
\(734\) 36367.7 1.82882
\(735\) 0 0
\(736\) 83.1973 0.00416670
\(737\) 5245.29 0.262161
\(738\) 9695.71 0.483610
\(739\) −14038.9 −0.698821 −0.349410 0.936970i \(-0.613618\pi\)
−0.349410 + 0.936970i \(0.613618\pi\)
\(740\) 19968.4 0.991963
\(741\) 3145.68 0.155951
\(742\) 0 0
\(743\) 17698.7 0.873893 0.436946 0.899488i \(-0.356060\pi\)
0.436946 + 0.899488i \(0.356060\pi\)
\(744\) −49181.5 −2.42350
\(745\) −3367.50 −0.165605
\(746\) −1899.72 −0.0932354
\(747\) −5966.50 −0.292239
\(748\) −70855.4 −3.46354
\(749\) 0 0
\(750\) 2662.69 0.129637
\(751\) 17199.7 0.835719 0.417859 0.908512i \(-0.362780\pi\)
0.417859 + 0.908512i \(0.362780\pi\)
\(752\) 67823.4 3.28892
\(753\) −20590.7 −0.996505
\(754\) −72557.1 −3.50448
\(755\) 10625.9 0.512206
\(756\) 0 0
\(757\) −19200.7 −0.921879 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(758\) −28921.0 −1.38583
\(759\) −13.8816 −0.000663860 0
\(760\) 4714.80 0.225031
\(761\) −27918.7 −1.32990 −0.664949 0.746889i \(-0.731547\pi\)
−0.664949 + 0.746889i \(0.731547\pi\)
\(762\) −25216.3 −1.19881
\(763\) 0 0
\(764\) −54457.7 −2.57881
\(765\) −5530.59 −0.261384
\(766\) −41922.1 −1.97742
\(767\) 16568.5 0.779992
\(768\) 65335.3 3.06977
\(769\) −7436.29 −0.348712 −0.174356 0.984683i \(-0.555784\pi\)
−0.174356 + 0.984683i \(0.555784\pi\)
\(770\) 0 0
\(771\) −9414.64 −0.439766
\(772\) −13983.4 −0.651908
\(773\) 4989.84 0.232176 0.116088 0.993239i \(-0.462965\pi\)
0.116088 + 0.993239i \(0.462965\pi\)
\(774\) 551.199 0.0255975
\(775\) 3995.05 0.185169
\(776\) −66888.7 −3.09429
\(777\) 0 0
\(778\) −17115.9 −0.788731
\(779\) 1717.96 0.0790146
\(780\) −29850.9 −1.37030
\(781\) −10751.5 −0.492600
\(782\) −52.5550 −0.00240328
\(783\) −28807.2 −1.31480
\(784\) 0 0
\(785\) 14065.2 0.639500
\(786\) −4757.85 −0.215912
\(787\) −2870.69 −0.130024 −0.0650122 0.997884i \(-0.520709\pi\)
−0.0650122 + 0.997884i \(0.520709\pi\)
\(788\) 63803.8 2.88441
\(789\) 5891.93 0.265853
\(790\) 10792.8 0.486063
\(791\) 0 0
\(792\) 33293.9 1.49374
\(793\) 53612.1 2.40078
\(794\) 16179.0 0.723138
\(795\) −6815.66 −0.304059
\(796\) −19470.6 −0.866982
\(797\) 4676.61 0.207847 0.103923 0.994585i \(-0.466860\pi\)
0.103923 + 0.994585i \(0.466860\pi\)
\(798\) 0 0
\(799\) −23821.3 −1.05474
\(800\) −19967.6 −0.882451
\(801\) −5732.34 −0.252862
\(802\) −492.410 −0.0216803
\(803\) 22089.6 0.970766
\(804\) 13167.2 0.577578
\(805\) 0 0
\(806\) −60749.1 −2.65483
\(807\) −29200.5 −1.27374
\(808\) −70750.2 −3.08042
\(809\) −15376.9 −0.668261 −0.334131 0.942527i \(-0.608443\pi\)
−0.334131 + 0.942527i \(0.608443\pi\)
\(810\) −7063.67 −0.306410
\(811\) 36422.9 1.57704 0.788522 0.615007i \(-0.210847\pi\)
0.788522 + 0.615007i \(0.210847\pi\)
\(812\) 0 0
\(813\) 16532.8 0.713200
\(814\) 33889.0 1.45922
\(815\) 6722.10 0.288914
\(816\) −91889.5 −3.94213
\(817\) 97.6658 0.00418225
\(818\) −813.285 −0.0347627
\(819\) 0 0
\(820\) −16302.6 −0.694282
\(821\) −25479.0 −1.08310 −0.541550 0.840669i \(-0.682162\pi\)
−0.541550 + 0.840669i \(0.682162\pi\)
\(822\) −43384.0 −1.84087
\(823\) −17933.3 −0.759557 −0.379779 0.925077i \(-0.624000\pi\)
−0.379779 + 0.925077i \(0.624000\pi\)
\(824\) −49356.1 −2.08665
\(825\) 3331.62 0.140597
\(826\) 0 0
\(827\) 13021.6 0.547529 0.273764 0.961797i \(-0.411731\pi\)
0.273764 + 0.961797i \(0.411731\pi\)
\(828\) 28.2874 0.00118727
\(829\) −11397.6 −0.477509 −0.238754 0.971080i \(-0.576739\pi\)
−0.238754 + 0.971080i \(0.576739\pi\)
\(830\) 13607.5 0.569063
\(831\) −15474.0 −0.645954
\(832\) 160149. 6.67326
\(833\) 0 0
\(834\) −57237.2 −2.37646
\(835\) −7256.19 −0.300731
\(836\) 9165.75 0.379191
\(837\) −24119.1 −0.996031
\(838\) 20867.2 0.860196
\(839\) −37681.2 −1.55053 −0.775267 0.631633i \(-0.782385\pi\)
−0.775267 + 0.631633i \(0.782385\pi\)
\(840\) 0 0
\(841\) 12039.8 0.493656
\(842\) 60141.7 2.46154
\(843\) 26773.5 1.09386
\(844\) −134148. −5.47104
\(845\) −12746.6 −0.518929
\(846\) 17391.1 0.706758
\(847\) 0 0
\(848\) 91924.4 3.72252
\(849\) −10294.3 −0.416135
\(850\) 12613.3 0.508981
\(851\) 18.5318 0.000746490 0
\(852\) −26989.6 −1.08527
\(853\) 21771.6 0.873911 0.436956 0.899483i \(-0.356057\pi\)
0.436956 + 0.899483i \(0.356057\pi\)
\(854\) 0 0
\(855\) 715.429 0.0286166
\(856\) 171510. 6.84823
\(857\) 29860.2 1.19021 0.595103 0.803649i \(-0.297111\pi\)
0.595103 + 0.803649i \(0.297111\pi\)
\(858\) −50661.0 −2.01578
\(859\) 18530.6 0.736039 0.368019 0.929818i \(-0.380036\pi\)
0.368019 + 0.929818i \(0.380036\pi\)
\(860\) −926.799 −0.0367484
\(861\) 0 0
\(862\) 62261.4 2.46013
\(863\) −21296.5 −0.840024 −0.420012 0.907519i \(-0.637974\pi\)
−0.420012 + 0.907519i \(0.637974\pi\)
\(864\) 120549. 4.74672
\(865\) 9895.79 0.388979
\(866\) 49144.1 1.92839
\(867\) 13307.8 0.521290
\(868\) 0 0
\(869\) 13504.2 0.527155
\(870\) 20328.4 0.792181
\(871\) 10468.0 0.407226
\(872\) 32453.7 1.26035
\(873\) −10149.8 −0.393491
\(874\) 6.79843 0.000263112 0
\(875\) 0 0
\(876\) 55451.5 2.13874
\(877\) −18793.4 −0.723614 −0.361807 0.932253i \(-0.617840\pi\)
−0.361807 + 0.932253i \(0.617840\pi\)
\(878\) −44968.7 −1.72850
\(879\) 22927.3 0.879769
\(880\) −44934.4 −1.72129
\(881\) 1638.62 0.0626634 0.0313317 0.999509i \(-0.490025\pi\)
0.0313317 + 0.999509i \(0.490025\pi\)
\(882\) 0 0
\(883\) −35424.1 −1.35008 −0.675038 0.737783i \(-0.735873\pi\)
−0.675038 + 0.737783i \(0.735873\pi\)
\(884\) −141406. −5.38008
\(885\) −4642.01 −0.176316
\(886\) 63306.5 2.40048
\(887\) −5131.41 −0.194246 −0.0971229 0.995272i \(-0.530964\pi\)
−0.0971229 + 0.995272i \(0.530964\pi\)
\(888\) 54753.8 2.06916
\(889\) 0 0
\(890\) 13073.5 0.492386
\(891\) −8838.22 −0.332314
\(892\) −116316. −4.36610
\(893\) 3081.49 0.115474
\(894\) −14346.6 −0.536713
\(895\) 21793.3 0.813933
\(896\) 0 0
\(897\) −27.7034 −0.00103120
\(898\) −10836.2 −0.402682
\(899\) 30500.3 1.13153
\(900\) −6789.06 −0.251447
\(901\) −32286.2 −1.19380
\(902\) −27667.7 −1.02132
\(903\) 0 0
\(904\) 27870.0 1.02538
\(905\) 1889.61 0.0694065
\(906\) 45269.6 1.66002
\(907\) −19934.6 −0.729787 −0.364893 0.931049i \(-0.618895\pi\)
−0.364893 + 0.931049i \(0.618895\pi\)
\(908\) 84248.8 3.07918
\(909\) −10735.7 −0.391728
\(910\) 0 0
\(911\) 48387.4 1.75976 0.879882 0.475193i \(-0.157622\pi\)
0.879882 + 0.475193i \(0.157622\pi\)
\(912\) 11886.7 0.431587
\(913\) 17026.0 0.617172
\(914\) −30844.7 −1.11625
\(915\) −15020.6 −0.542693
\(916\) −140483. −5.06735
\(917\) 0 0
\(918\) −76149.8 −2.73782
\(919\) −14431.7 −0.518019 −0.259009 0.965875i \(-0.583396\pi\)
−0.259009 + 0.965875i \(0.583396\pi\)
\(920\) −41.5224 −0.00148799
\(921\) 40076.8 1.43385
\(922\) 103688. 3.70366
\(923\) −21456.8 −0.765177
\(924\) 0 0
\(925\) −4447.69 −0.158096
\(926\) 43549.1 1.54548
\(927\) −7489.34 −0.265353
\(928\) −152443. −5.39244
\(929\) 5549.82 0.196000 0.0979998 0.995186i \(-0.468756\pi\)
0.0979998 + 0.995186i \(0.468756\pi\)
\(930\) 17020.1 0.600120
\(931\) 0 0
\(932\) 39956.7 1.40432
\(933\) 16370.2 0.574421
\(934\) −2127.24 −0.0745239
\(935\) 15782.1 0.552010
\(936\) 66444.3 2.32030
\(937\) 26231.9 0.914576 0.457288 0.889319i \(-0.348821\pi\)
0.457288 + 0.889319i \(0.348821\pi\)
\(938\) 0 0
\(939\) −1095.98 −0.0380893
\(940\) −29241.8 −1.01464
\(941\) 2836.80 0.0982753 0.0491376 0.998792i \(-0.484353\pi\)
0.0491376 + 0.998792i \(0.484353\pi\)
\(942\) 59921.9 2.07257
\(943\) −15.1298 −0.000522474 0
\(944\) 62607.9 2.15860
\(945\) 0 0
\(946\) −1572.90 −0.0540586
\(947\) −40272.9 −1.38193 −0.690967 0.722886i \(-0.742815\pi\)
−0.690967 + 0.722886i \(0.742815\pi\)
\(948\) 33899.5 1.16140
\(949\) 44084.1 1.50793
\(950\) −1631.64 −0.0557236
\(951\) 6715.12 0.228972
\(952\) 0 0
\(953\) 17770.1 0.604019 0.302010 0.953305i \(-0.402343\pi\)
0.302010 + 0.953305i \(0.402343\pi\)
\(954\) 23571.0 0.799936
\(955\) 12129.7 0.411004
\(956\) 79005.0 2.67281
\(957\) 25435.4 0.859152
\(958\) −9349.97 −0.315328
\(959\) 0 0
\(960\) −44869.0 −1.50848
\(961\) −4254.35 −0.142807
\(962\) 67632.0 2.26668
\(963\) 26025.1 0.870869
\(964\) 8147.07 0.272199
\(965\) 3114.61 0.103899
\(966\) 0 0
\(967\) −3530.14 −0.117396 −0.0586978 0.998276i \(-0.518695\pi\)
−0.0586978 + 0.998276i \(0.518695\pi\)
\(968\) 11105.2 0.368734
\(969\) −4174.91 −0.138408
\(970\) 23148.0 0.766226
\(971\) 17650.7 0.583356 0.291678 0.956517i \(-0.405786\pi\)
0.291678 + 0.956517i \(0.405786\pi\)
\(972\) 69292.3 2.28658
\(973\) 0 0
\(974\) −86693.8 −2.85200
\(975\) 6648.89 0.218395
\(976\) 202586. 6.64407
\(977\) −14477.6 −0.474085 −0.237042 0.971499i \(-0.576178\pi\)
−0.237042 + 0.971499i \(0.576178\pi\)
\(978\) 28638.2 0.936349
\(979\) 16357.8 0.534012
\(980\) 0 0
\(981\) 4924.56 0.160274
\(982\) 14988.7 0.487075
\(983\) −8764.09 −0.284365 −0.142183 0.989840i \(-0.545412\pi\)
−0.142183 + 0.989840i \(0.545412\pi\)
\(984\) −44702.1 −1.44822
\(985\) −14211.4 −0.459710
\(986\) 96296.9 3.11026
\(987\) 0 0
\(988\) 18292.0 0.589015
\(989\) −0.860124 −2.76546e−5 0
\(990\) −11521.9 −0.369890
\(991\) 33624.1 1.07781 0.538903 0.842368i \(-0.318839\pi\)
0.538903 + 0.842368i \(0.318839\pi\)
\(992\) −127634. −4.08507
\(993\) 21645.1 0.691727
\(994\) 0 0
\(995\) 4336.82 0.138177
\(996\) 42740.3 1.35972
\(997\) 16631.3 0.528302 0.264151 0.964481i \(-0.414908\pi\)
0.264151 + 0.964481i \(0.414908\pi\)
\(998\) −23701.6 −0.751764
\(999\) 26851.8 0.850404
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 245.4.a.p.1.1 yes 6
3.2 odd 2 2205.4.a.ca.1.6 6
5.4 even 2 1225.4.a.bi.1.6 6
7.2 even 3 245.4.e.p.116.6 12
7.3 odd 6 245.4.e.q.226.6 12
7.4 even 3 245.4.e.p.226.6 12
7.5 odd 6 245.4.e.q.116.6 12
7.6 odd 2 245.4.a.o.1.1 6
21.20 even 2 2205.4.a.bz.1.6 6
35.34 odd 2 1225.4.a.bj.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.1 6 7.6 odd 2
245.4.a.p.1.1 yes 6 1.1 even 1 trivial
245.4.e.p.116.6 12 7.2 even 3
245.4.e.p.226.6 12 7.4 even 3
245.4.e.q.116.6 12 7.5 odd 6
245.4.e.q.226.6 12 7.3 odd 6
1225.4.a.bi.1.6 6 5.4 even 2
1225.4.a.bj.1.6 6 35.34 odd 2
2205.4.a.bz.1.6 6 21.20 even 2
2205.4.a.ca.1.6 6 3.2 odd 2