Properties

Label 1575.4.a.bg.1.4
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(5.87199\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} +O(q^{10})\) \(q+4.87199 q^{2} +15.7363 q^{4} +7.00000 q^{7} +37.6910 q^{8} -36.9922 q^{11} -61.3165 q^{13} +34.1039 q^{14} +57.7401 q^{16} +44.8345 q^{17} -139.701 q^{19} -180.226 q^{22} -217.580 q^{23} -298.733 q^{26} +110.154 q^{28} +33.8226 q^{29} +124.437 q^{31} -20.2192 q^{32} +218.433 q^{34} +237.270 q^{37} -680.620 q^{38} -195.117 q^{41} -343.725 q^{43} -582.119 q^{44} -1060.05 q^{46} -16.8224 q^{47} +49.0000 q^{49} -964.894 q^{52} -346.965 q^{53} +263.837 q^{56} +164.783 q^{58} -135.340 q^{59} +490.414 q^{61} +606.255 q^{62} -560.428 q^{64} +477.969 q^{67} +705.529 q^{68} -45.2557 q^{71} +100.781 q^{73} +1155.98 q^{74} -2198.37 q^{76} -258.945 q^{77} +880.534 q^{79} -950.609 q^{82} +1155.03 q^{83} -1674.62 q^{86} -1394.27 q^{88} -619.374 q^{89} -429.216 q^{91} -3423.90 q^{92} -81.9585 q^{94} -231.195 q^{97} +238.727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32} + 203 q^{34} + 403 q^{37} - 555 q^{38} - 671 q^{41} - 411 q^{43} - 438 q^{44} - 997 q^{46} + 8 q^{47} + 196 q^{49} + 74 q^{52} - 90 q^{53} - 189 q^{56} + 673 q^{58} - 1018 q^{59} + 50 q^{61} + 1626 q^{62} - 2421 q^{64} + 424 q^{67} + 617 q^{68} - 215 q^{71} - 1207 q^{73} - 623 q^{74} - 3257 q^{76} - 700 q^{77} - 951 q^{79} + 1695 q^{82} + 3035 q^{83} + 99 q^{86} + 163 q^{88} - 2819 q^{89} + 308 q^{91} - 3073 q^{92} - 3056 q^{94} - 1100 q^{97} - 196 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.87199 1.72251 0.861254 0.508175i \(-0.169680\pi\)
0.861254 + 0.508175i \(0.169680\pi\)
\(3\) 0 0
\(4\) 15.7363 1.96703
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 37.6910 1.66572
\(9\) 0 0
\(10\) 0 0
\(11\) −36.9922 −1.01396 −0.506980 0.861958i \(-0.669238\pi\)
−0.506980 + 0.861958i \(0.669238\pi\)
\(12\) 0 0
\(13\) −61.3165 −1.30817 −0.654083 0.756423i \(-0.726945\pi\)
−0.654083 + 0.756423i \(0.726945\pi\)
\(14\) 34.1039 0.651047
\(15\) 0 0
\(16\) 57.7401 0.902189
\(17\) 44.8345 0.639646 0.319823 0.947477i \(-0.396377\pi\)
0.319823 + 0.947477i \(0.396377\pi\)
\(18\) 0 0
\(19\) −139.701 −1.68682 −0.843408 0.537273i \(-0.819455\pi\)
−0.843408 + 0.537273i \(0.819455\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −180.226 −1.74656
\(23\) −217.580 −1.97255 −0.986275 0.165110i \(-0.947202\pi\)
−0.986275 + 0.165110i \(0.947202\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −298.733 −2.25333
\(27\) 0 0
\(28\) 110.154 0.743469
\(29\) 33.8226 0.216576 0.108288 0.994120i \(-0.465463\pi\)
0.108288 + 0.994120i \(0.465463\pi\)
\(30\) 0 0
\(31\) 124.437 0.720952 0.360476 0.932769i \(-0.382614\pi\)
0.360476 + 0.932769i \(0.382614\pi\)
\(32\) −20.2192 −0.111697
\(33\) 0 0
\(34\) 218.433 1.10179
\(35\) 0 0
\(36\) 0 0
\(37\) 237.270 1.05424 0.527121 0.849790i \(-0.323271\pi\)
0.527121 + 0.849790i \(0.323271\pi\)
\(38\) −680.620 −2.90556
\(39\) 0 0
\(40\) 0 0
\(41\) −195.117 −0.743224 −0.371612 0.928388i \(-0.621195\pi\)
−0.371612 + 0.928388i \(0.621195\pi\)
\(42\) 0 0
\(43\) −343.725 −1.21901 −0.609506 0.792781i \(-0.708632\pi\)
−0.609506 + 0.792781i \(0.708632\pi\)
\(44\) −582.119 −1.99450
\(45\) 0 0
\(46\) −1060.05 −3.39773
\(47\) −16.8224 −0.0522085 −0.0261042 0.999659i \(-0.508310\pi\)
−0.0261042 + 0.999659i \(0.508310\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −964.894 −2.57321
\(53\) −346.965 −0.899231 −0.449616 0.893222i \(-0.648439\pi\)
−0.449616 + 0.893222i \(0.648439\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 263.837 0.629584
\(57\) 0 0
\(58\) 164.783 0.373054
\(59\) −135.340 −0.298640 −0.149320 0.988789i \(-0.547708\pi\)
−0.149320 + 0.988789i \(0.547708\pi\)
\(60\) 0 0
\(61\) 490.414 1.02936 0.514681 0.857382i \(-0.327911\pi\)
0.514681 + 0.857382i \(0.327911\pi\)
\(62\) 606.255 1.24185
\(63\) 0 0
\(64\) −560.428 −1.09459
\(65\) 0 0
\(66\) 0 0
\(67\) 477.969 0.871540 0.435770 0.900058i \(-0.356476\pi\)
0.435770 + 0.900058i \(0.356476\pi\)
\(68\) 705.529 1.25820
\(69\) 0 0
\(70\) 0 0
\(71\) −45.2557 −0.0756460 −0.0378230 0.999284i \(-0.512042\pi\)
−0.0378230 + 0.999284i \(0.512042\pi\)
\(72\) 0 0
\(73\) 100.781 0.161583 0.0807913 0.996731i \(-0.474255\pi\)
0.0807913 + 0.996731i \(0.474255\pi\)
\(74\) 1155.98 1.81594
\(75\) 0 0
\(76\) −2198.37 −3.31803
\(77\) −258.945 −0.383241
\(78\) 0 0
\(79\) 880.534 1.25402 0.627012 0.779010i \(-0.284278\pi\)
0.627012 + 0.779010i \(0.284278\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −950.609 −1.28021
\(83\) 1155.03 1.52748 0.763739 0.645525i \(-0.223362\pi\)
0.763739 + 0.645525i \(0.223362\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1674.62 −2.09976
\(87\) 0 0
\(88\) −1394.27 −1.68898
\(89\) −619.374 −0.737680 −0.368840 0.929493i \(-0.620245\pi\)
−0.368840 + 0.929493i \(0.620245\pi\)
\(90\) 0 0
\(91\) −429.216 −0.494440
\(92\) −3423.90 −3.88007
\(93\) 0 0
\(94\) −81.9585 −0.0899295
\(95\) 0 0
\(96\) 0 0
\(97\) −231.195 −0.242003 −0.121001 0.992652i \(-0.538611\pi\)
−0.121001 + 0.992652i \(0.538611\pi\)
\(98\) 238.727 0.246073
\(99\) 0 0
\(100\) 0 0
\(101\) −1875.99 −1.84820 −0.924101 0.382149i \(-0.875184\pi\)
−0.924101 + 0.382149i \(0.875184\pi\)
\(102\) 0 0
\(103\) 1855.94 1.77545 0.887723 0.460377i \(-0.152286\pi\)
0.887723 + 0.460377i \(0.152286\pi\)
\(104\) −2311.08 −2.17904
\(105\) 0 0
\(106\) −1690.41 −1.54893
\(107\) 218.016 0.196976 0.0984879 0.995138i \(-0.468599\pi\)
0.0984879 + 0.995138i \(0.468599\pi\)
\(108\) 0 0
\(109\) 847.217 0.744484 0.372242 0.928136i \(-0.378589\pi\)
0.372242 + 0.928136i \(0.378589\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 404.181 0.340995
\(113\) −1430.05 −1.19051 −0.595254 0.803537i \(-0.702949\pi\)
−0.595254 + 0.803537i \(0.702949\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 532.242 0.426012
\(117\) 0 0
\(118\) −659.374 −0.514409
\(119\) 313.842 0.241763
\(120\) 0 0
\(121\) 37.4229 0.0281164
\(122\) 2389.29 1.77308
\(123\) 0 0
\(124\) 1958.17 1.41814
\(125\) 0 0
\(126\) 0 0
\(127\) −1732.84 −1.21075 −0.605374 0.795941i \(-0.706976\pi\)
−0.605374 + 0.795941i \(0.706976\pi\)
\(128\) −2568.65 −1.77374
\(129\) 0 0
\(130\) 0 0
\(131\) 2429.76 1.62052 0.810262 0.586068i \(-0.199325\pi\)
0.810262 + 0.586068i \(0.199325\pi\)
\(132\) 0 0
\(133\) −977.904 −0.637557
\(134\) 2328.66 1.50123
\(135\) 0 0
\(136\) 1689.86 1.06547
\(137\) −1452.84 −0.906018 −0.453009 0.891506i \(-0.649649\pi\)
−0.453009 + 0.891506i \(0.649649\pi\)
\(138\) 0 0
\(139\) 2927.47 1.78637 0.893183 0.449693i \(-0.148467\pi\)
0.893183 + 0.449693i \(0.148467\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −220.485 −0.130301
\(143\) 2268.23 1.32643
\(144\) 0 0
\(145\) 0 0
\(146\) 491.004 0.278327
\(147\) 0 0
\(148\) 3733.75 2.07373
\(149\) −901.724 −0.495786 −0.247893 0.968787i \(-0.579738\pi\)
−0.247893 + 0.968787i \(0.579738\pi\)
\(150\) 0 0
\(151\) −1357.63 −0.731669 −0.365834 0.930680i \(-0.619216\pi\)
−0.365834 + 0.930680i \(0.619216\pi\)
\(152\) −5265.46 −2.80977
\(153\) 0 0
\(154\) −1261.58 −0.660136
\(155\) 0 0
\(156\) 0 0
\(157\) −2318.72 −1.17869 −0.589343 0.807883i \(-0.700613\pi\)
−0.589343 + 0.807883i \(0.700613\pi\)
\(158\) 4289.95 2.16006
\(159\) 0 0
\(160\) 0 0
\(161\) −1523.06 −0.745554
\(162\) 0 0
\(163\) −1577.42 −0.757996 −0.378998 0.925397i \(-0.623731\pi\)
−0.378998 + 0.925397i \(0.623731\pi\)
\(164\) −3070.42 −1.46195
\(165\) 0 0
\(166\) 5627.28 2.63109
\(167\) −1038.02 −0.480982 −0.240491 0.970651i \(-0.577309\pi\)
−0.240491 + 0.970651i \(0.577309\pi\)
\(168\) 0 0
\(169\) 1562.72 0.711296
\(170\) 0 0
\(171\) 0 0
\(172\) −5408.95 −2.39784
\(173\) 482.641 0.212107 0.106054 0.994360i \(-0.466178\pi\)
0.106054 + 0.994360i \(0.466178\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2135.93 −0.914784
\(177\) 0 0
\(178\) −3017.58 −1.27066
\(179\) −2407.19 −1.00515 −0.502575 0.864534i \(-0.667614\pi\)
−0.502575 + 0.864534i \(0.667614\pi\)
\(180\) 0 0
\(181\) 532.600 0.218717 0.109359 0.994002i \(-0.465120\pi\)
0.109359 + 0.994002i \(0.465120\pi\)
\(182\) −2091.13 −0.851677
\(183\) 0 0
\(184\) −8200.83 −3.28572
\(185\) 0 0
\(186\) 0 0
\(187\) −1658.53 −0.648575
\(188\) −264.722 −0.102696
\(189\) 0 0
\(190\) 0 0
\(191\) −336.675 −0.127544 −0.0637721 0.997964i \(-0.520313\pi\)
−0.0637721 + 0.997964i \(0.520313\pi\)
\(192\) 0 0
\(193\) 22.7366 0.00847988 0.00423994 0.999991i \(-0.498650\pi\)
0.00423994 + 0.999991i \(0.498650\pi\)
\(194\) −1126.38 −0.416852
\(195\) 0 0
\(196\) 771.077 0.281005
\(197\) 1085.33 0.392520 0.196260 0.980552i \(-0.437120\pi\)
0.196260 + 0.980552i \(0.437120\pi\)
\(198\) 0 0
\(199\) −2630.28 −0.936963 −0.468482 0.883473i \(-0.655199\pi\)
−0.468482 + 0.883473i \(0.655199\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −9139.82 −3.18354
\(203\) 236.758 0.0818580
\(204\) 0 0
\(205\) 0 0
\(206\) 9042.11 3.05822
\(207\) 0 0
\(208\) −3540.42 −1.18021
\(209\) 5167.83 1.71037
\(210\) 0 0
\(211\) 614.178 0.200388 0.100194 0.994968i \(-0.468054\pi\)
0.100194 + 0.994968i \(0.468054\pi\)
\(212\) −5459.93 −1.76882
\(213\) 0 0
\(214\) 1062.17 0.339292
\(215\) 0 0
\(216\) 0 0
\(217\) 871.057 0.272494
\(218\) 4127.63 1.28238
\(219\) 0 0
\(220\) 0 0
\(221\) −2749.10 −0.836762
\(222\) 0 0
\(223\) 1930.30 0.579651 0.289826 0.957079i \(-0.406403\pi\)
0.289826 + 0.957079i \(0.406403\pi\)
\(224\) −141.535 −0.0422173
\(225\) 0 0
\(226\) −6967.17 −2.05066
\(227\) −765.365 −0.223784 −0.111892 0.993720i \(-0.535691\pi\)
−0.111892 + 0.993720i \(0.535691\pi\)
\(228\) 0 0
\(229\) −5712.66 −1.64849 −0.824243 0.566237i \(-0.808399\pi\)
−0.824243 + 0.566237i \(0.808399\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1274.81 0.360756
\(233\) 864.418 0.243047 0.121523 0.992589i \(-0.461222\pi\)
0.121523 + 0.992589i \(0.461222\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2129.74 −0.587435
\(237\) 0 0
\(238\) 1529.03 0.416439
\(239\) 1816.64 0.491669 0.245834 0.969312i \(-0.420938\pi\)
0.245834 + 0.969312i \(0.420938\pi\)
\(240\) 0 0
\(241\) −2354.53 −0.629330 −0.314665 0.949203i \(-0.601892\pi\)
−0.314665 + 0.949203i \(0.601892\pi\)
\(242\) 182.324 0.0484307
\(243\) 0 0
\(244\) 7717.28 2.02479
\(245\) 0 0
\(246\) 0 0
\(247\) 8565.96 2.20664
\(248\) 4690.15 1.20091
\(249\) 0 0
\(250\) 0 0
\(251\) 2983.66 0.750306 0.375153 0.926963i \(-0.377590\pi\)
0.375153 + 0.926963i \(0.377590\pi\)
\(252\) 0 0
\(253\) 8048.78 2.00009
\(254\) −8442.39 −2.08552
\(255\) 0 0
\(256\) −8030.99 −1.96069
\(257\) 5121.50 1.24307 0.621537 0.783385i \(-0.286508\pi\)
0.621537 + 0.783385i \(0.286508\pi\)
\(258\) 0 0
\(259\) 1660.89 0.398466
\(260\) 0 0
\(261\) 0 0
\(262\) 11837.7 2.79137
\(263\) −4663.66 −1.09344 −0.546718 0.837317i \(-0.684123\pi\)
−0.546718 + 0.837317i \(0.684123\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4764.34 −1.09820
\(267\) 0 0
\(268\) 7521.45 1.71435
\(269\) −5018.86 −1.13757 −0.568783 0.822488i \(-0.692586\pi\)
−0.568783 + 0.822488i \(0.692586\pi\)
\(270\) 0 0
\(271\) −2512.66 −0.563222 −0.281611 0.959529i \(-0.590869\pi\)
−0.281611 + 0.959529i \(0.590869\pi\)
\(272\) 2588.75 0.577081
\(273\) 0 0
\(274\) −7078.21 −1.56062
\(275\) 0 0
\(276\) 0 0
\(277\) 6286.82 1.36368 0.681839 0.731503i \(-0.261181\pi\)
0.681839 + 0.731503i \(0.261181\pi\)
\(278\) 14262.6 3.07703
\(279\) 0 0
\(280\) 0 0
\(281\) −8804.33 −1.86912 −0.934560 0.355807i \(-0.884206\pi\)
−0.934560 + 0.355807i \(0.884206\pi\)
\(282\) 0 0
\(283\) 485.298 0.101936 0.0509681 0.998700i \(-0.483769\pi\)
0.0509681 + 0.998700i \(0.483769\pi\)
\(284\) −712.157 −0.148798
\(285\) 0 0
\(286\) 11050.8 2.28478
\(287\) −1365.82 −0.280912
\(288\) 0 0
\(289\) −2902.86 −0.590854
\(290\) 0 0
\(291\) 0 0
\(292\) 1585.92 0.317838
\(293\) −4004.48 −0.798444 −0.399222 0.916854i \(-0.630720\pi\)
−0.399222 + 0.916854i \(0.630720\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8942.96 1.75608
\(297\) 0 0
\(298\) −4393.19 −0.853995
\(299\) 13341.3 2.58042
\(300\) 0 0
\(301\) −2406.07 −0.460743
\(302\) −6614.33 −1.26031
\(303\) 0 0
\(304\) −8066.32 −1.52183
\(305\) 0 0
\(306\) 0 0
\(307\) 3154.13 0.586370 0.293185 0.956056i \(-0.405285\pi\)
0.293185 + 0.956056i \(0.405285\pi\)
\(308\) −4074.84 −0.753848
\(309\) 0 0
\(310\) 0 0
\(311\) 1738.29 0.316944 0.158472 0.987363i \(-0.449343\pi\)
0.158472 + 0.987363i \(0.449343\pi\)
\(312\) 0 0
\(313\) 5092.30 0.919597 0.459799 0.888023i \(-0.347922\pi\)
0.459799 + 0.888023i \(0.347922\pi\)
\(314\) −11296.8 −2.03030
\(315\) 0 0
\(316\) 13856.3 2.46671
\(317\) 4874.67 0.863686 0.431843 0.901949i \(-0.357863\pi\)
0.431843 + 0.901949i \(0.357863\pi\)
\(318\) 0 0
\(319\) −1251.17 −0.219600
\(320\) 0 0
\(321\) 0 0
\(322\) −7420.35 −1.28422
\(323\) −6263.41 −1.07896
\(324\) 0 0
\(325\) 0 0
\(326\) −7685.19 −1.30565
\(327\) 0 0
\(328\) −7354.17 −1.23801
\(329\) −117.757 −0.0197329
\(330\) 0 0
\(331\) −2448.46 −0.406585 −0.203292 0.979118i \(-0.565164\pi\)
−0.203292 + 0.979118i \(0.565164\pi\)
\(332\) 18175.8 3.00460
\(333\) 0 0
\(334\) −5057.20 −0.828496
\(335\) 0 0
\(336\) 0 0
\(337\) −6271.50 −1.01374 −0.506870 0.862023i \(-0.669197\pi\)
−0.506870 + 0.862023i \(0.669197\pi\)
\(338\) 7613.54 1.22521
\(339\) 0 0
\(340\) 0 0
\(341\) −4603.19 −0.731017
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −12955.3 −2.03054
\(345\) 0 0
\(346\) 2351.42 0.365356
\(347\) −417.338 −0.0645645 −0.0322822 0.999479i \(-0.510278\pi\)
−0.0322822 + 0.999479i \(0.510278\pi\)
\(348\) 0 0
\(349\) 5971.08 0.915829 0.457915 0.888996i \(-0.348597\pi\)
0.457915 + 0.888996i \(0.348597\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 747.954 0.113256
\(353\) 12012.6 1.81124 0.905620 0.424089i \(-0.139406\pi\)
0.905620 + 0.424089i \(0.139406\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9746.64 −1.45104
\(357\) 0 0
\(358\) −11727.8 −1.73138
\(359\) −6312.46 −0.928020 −0.464010 0.885830i \(-0.653590\pi\)
−0.464010 + 0.885830i \(0.653590\pi\)
\(360\) 0 0
\(361\) 12657.3 1.84535
\(362\) 2594.82 0.376743
\(363\) 0 0
\(364\) −6754.26 −0.972580
\(365\) 0 0
\(366\) 0 0
\(367\) 12228.6 1.73932 0.869659 0.493652i \(-0.164338\pi\)
0.869659 + 0.493652i \(0.164338\pi\)
\(368\) −12563.1 −1.77961
\(369\) 0 0
\(370\) 0 0
\(371\) −2428.75 −0.339877
\(372\) 0 0
\(373\) 7876.24 1.09334 0.546671 0.837348i \(-0.315895\pi\)
0.546671 + 0.837348i \(0.315895\pi\)
\(374\) −8080.33 −1.11718
\(375\) 0 0
\(376\) −634.053 −0.0869649
\(377\) −2073.89 −0.283317
\(378\) 0 0
\(379\) 5992.79 0.812214 0.406107 0.913826i \(-0.366886\pi\)
0.406107 + 0.913826i \(0.366886\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1640.28 −0.219696
\(383\) −7373.61 −0.983743 −0.491872 0.870668i \(-0.663687\pi\)
−0.491872 + 0.870668i \(0.663687\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 110.772 0.0146067
\(387\) 0 0
\(388\) −3638.14 −0.476028
\(389\) −1896.88 −0.247238 −0.123619 0.992330i \(-0.539450\pi\)
−0.123619 + 0.992330i \(0.539450\pi\)
\(390\) 0 0
\(391\) −9755.12 −1.26173
\(392\) 1846.86 0.237961
\(393\) 0 0
\(394\) 5287.71 0.676119
\(395\) 0 0
\(396\) 0 0
\(397\) −791.156 −0.100018 −0.0500088 0.998749i \(-0.515925\pi\)
−0.0500088 + 0.998749i \(0.515925\pi\)
\(398\) −12814.7 −1.61393
\(399\) 0 0
\(400\) 0 0
\(401\) −4410.55 −0.549258 −0.274629 0.961550i \(-0.588555\pi\)
−0.274629 + 0.961550i \(0.588555\pi\)
\(402\) 0 0
\(403\) −7630.03 −0.943124
\(404\) −29521.1 −3.63548
\(405\) 0 0
\(406\) 1153.48 0.141001
\(407\) −8777.14 −1.06896
\(408\) 0 0
\(409\) −4403.73 −0.532397 −0.266199 0.963918i \(-0.585768\pi\)
−0.266199 + 0.963918i \(0.585768\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 29205.5 3.49236
\(413\) −947.379 −0.112875
\(414\) 0 0
\(415\) 0 0
\(416\) 1239.77 0.146117
\(417\) 0 0
\(418\) 25177.6 2.94612
\(419\) 14272.6 1.66411 0.832057 0.554691i \(-0.187163\pi\)
0.832057 + 0.554691i \(0.187163\pi\)
\(420\) 0 0
\(421\) −15530.4 −1.79788 −0.898939 0.438074i \(-0.855661\pi\)
−0.898939 + 0.438074i \(0.855661\pi\)
\(422\) 2992.27 0.345169
\(423\) 0 0
\(424\) −13077.5 −1.49787
\(425\) 0 0
\(426\) 0 0
\(427\) 3432.90 0.389062
\(428\) 3430.76 0.387458
\(429\) 0 0
\(430\) 0 0
\(431\) 5260.75 0.587938 0.293969 0.955815i \(-0.405024\pi\)
0.293969 + 0.955815i \(0.405024\pi\)
\(432\) 0 0
\(433\) 6610.65 0.733690 0.366845 0.930282i \(-0.380438\pi\)
0.366845 + 0.930282i \(0.380438\pi\)
\(434\) 4243.78 0.469373
\(435\) 0 0
\(436\) 13332.0 1.46442
\(437\) 30396.1 3.32733
\(438\) 0 0
\(439\) −8870.46 −0.964383 −0.482191 0.876066i \(-0.660159\pi\)
−0.482191 + 0.876066i \(0.660159\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −13393.6 −1.44133
\(443\) 13221.8 1.41802 0.709012 0.705196i \(-0.249141\pi\)
0.709012 + 0.705196i \(0.249141\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9404.38 0.998454
\(447\) 0 0
\(448\) −3923.00 −0.413715
\(449\) −4192.70 −0.440681 −0.220341 0.975423i \(-0.570717\pi\)
−0.220341 + 0.975423i \(0.570717\pi\)
\(450\) 0 0
\(451\) 7217.82 0.753600
\(452\) −22503.6 −2.34177
\(453\) 0 0
\(454\) −3728.85 −0.385470
\(455\) 0 0
\(456\) 0 0
\(457\) −4701.78 −0.481270 −0.240635 0.970616i \(-0.577356\pi\)
−0.240635 + 0.970616i \(0.577356\pi\)
\(458\) −27832.0 −2.83953
\(459\) 0 0
\(460\) 0 0
\(461\) 1949.82 0.196989 0.0984946 0.995138i \(-0.468597\pi\)
0.0984946 + 0.995138i \(0.468597\pi\)
\(462\) 0 0
\(463\) 7315.85 0.734333 0.367166 0.930155i \(-0.380328\pi\)
0.367166 + 0.930155i \(0.380328\pi\)
\(464\) 1952.92 0.195392
\(465\) 0 0
\(466\) 4211.43 0.418650
\(467\) 1897.54 0.188025 0.0940126 0.995571i \(-0.470031\pi\)
0.0940126 + 0.995571i \(0.470031\pi\)
\(468\) 0 0
\(469\) 3345.78 0.329411
\(470\) 0 0
\(471\) 0 0
\(472\) −5101.10 −0.497451
\(473\) 12715.1 1.23603
\(474\) 0 0
\(475\) 0 0
\(476\) 4938.70 0.475557
\(477\) 0 0
\(478\) 8850.66 0.846903
\(479\) −10534.4 −1.00486 −0.502429 0.864619i \(-0.667560\pi\)
−0.502429 + 0.864619i \(0.667560\pi\)
\(480\) 0 0
\(481\) −14548.6 −1.37912
\(482\) −11471.2 −1.08403
\(483\) 0 0
\(484\) 588.897 0.0553058
\(485\) 0 0
\(486\) 0 0
\(487\) 11937.0 1.11071 0.555355 0.831614i \(-0.312583\pi\)
0.555355 + 0.831614i \(0.312583\pi\)
\(488\) 18484.2 1.71463
\(489\) 0 0
\(490\) 0 0
\(491\) −100.060 −0.00919683 −0.00459841 0.999989i \(-0.501464\pi\)
−0.00459841 + 0.999989i \(0.501464\pi\)
\(492\) 0 0
\(493\) 1516.42 0.138532
\(494\) 41733.3 3.80095
\(495\) 0 0
\(496\) 7184.99 0.650435
\(497\) −316.790 −0.0285915
\(498\) 0 0
\(499\) −15611.7 −1.40055 −0.700275 0.713874i \(-0.746939\pi\)
−0.700275 + 0.713874i \(0.746939\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14536.3 1.29241
\(503\) −2270.39 −0.201256 −0.100628 0.994924i \(-0.532085\pi\)
−0.100628 + 0.994924i \(0.532085\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 39213.6 3.44517
\(507\) 0 0
\(508\) −27268.5 −2.38158
\(509\) 2579.79 0.224651 0.112325 0.993671i \(-0.464170\pi\)
0.112325 + 0.993671i \(0.464170\pi\)
\(510\) 0 0
\(511\) 705.467 0.0610725
\(512\) −18577.7 −1.60357
\(513\) 0 0
\(514\) 24951.9 2.14121
\(515\) 0 0
\(516\) 0 0
\(517\) 622.297 0.0529373
\(518\) 8091.84 0.686361
\(519\) 0 0
\(520\) 0 0
\(521\) −12793.9 −1.07584 −0.537918 0.842997i \(-0.680789\pi\)
−0.537918 + 0.842997i \(0.680789\pi\)
\(522\) 0 0
\(523\) −11273.4 −0.942548 −0.471274 0.881987i \(-0.656206\pi\)
−0.471274 + 0.881987i \(0.656206\pi\)
\(524\) 38235.3 3.18763
\(525\) 0 0
\(526\) −22721.3 −1.88345
\(527\) 5579.07 0.461154
\(528\) 0 0
\(529\) 35174.2 2.89095
\(530\) 0 0
\(531\) 0 0
\(532\) −15388.6 −1.25410
\(533\) 11963.9 0.972260
\(534\) 0 0
\(535\) 0 0
\(536\) 18015.1 1.45175
\(537\) 0 0
\(538\) −24451.8 −1.95947
\(539\) −1812.62 −0.144852
\(540\) 0 0
\(541\) 10282.9 0.817183 0.408591 0.912717i \(-0.366020\pi\)
0.408591 + 0.912717i \(0.366020\pi\)
\(542\) −12241.6 −0.970154
\(543\) 0 0
\(544\) −906.520 −0.0714462
\(545\) 0 0
\(546\) 0 0
\(547\) −2160.94 −0.168912 −0.0844562 0.996427i \(-0.526915\pi\)
−0.0844562 + 0.996427i \(0.526915\pi\)
\(548\) −22862.3 −1.78217
\(549\) 0 0
\(550\) 0 0
\(551\) −4725.04 −0.365324
\(552\) 0 0
\(553\) 6163.74 0.473976
\(554\) 30629.3 2.34895
\(555\) 0 0
\(556\) 46067.5 3.51384
\(557\) −15648.4 −1.19038 −0.595190 0.803585i \(-0.702923\pi\)
−0.595190 + 0.803585i \(0.702923\pi\)
\(558\) 0 0
\(559\) 21076.0 1.59467
\(560\) 0 0
\(561\) 0 0
\(562\) −42894.6 −3.21957
\(563\) 21951.6 1.64325 0.821623 0.570031i \(-0.193069\pi\)
0.821623 + 0.570031i \(0.193069\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2364.36 0.175586
\(567\) 0 0
\(568\) −1705.74 −0.126005
\(569\) −14019.2 −1.03289 −0.516445 0.856321i \(-0.672745\pi\)
−0.516445 + 0.856321i \(0.672745\pi\)
\(570\) 0 0
\(571\) −21706.3 −1.59086 −0.795430 0.606045i \(-0.792755\pi\)
−0.795430 + 0.606045i \(0.792755\pi\)
\(572\) 35693.5 2.60913
\(573\) 0 0
\(574\) −6654.26 −0.483874
\(575\) 0 0
\(576\) 0 0
\(577\) 12569.0 0.906853 0.453427 0.891294i \(-0.350201\pi\)
0.453427 + 0.891294i \(0.350201\pi\)
\(578\) −14142.7 −1.01775
\(579\) 0 0
\(580\) 0 0
\(581\) 8085.19 0.577332
\(582\) 0 0
\(583\) 12835.0 0.911785
\(584\) 3798.54 0.269152
\(585\) 0 0
\(586\) −19509.8 −1.37533
\(587\) 1330.20 0.0935322 0.0467661 0.998906i \(-0.485108\pi\)
0.0467661 + 0.998906i \(0.485108\pi\)
\(588\) 0 0
\(589\) −17383.9 −1.21611
\(590\) 0 0
\(591\) 0 0
\(592\) 13700.0 0.951126
\(593\) −9866.91 −0.683281 −0.341640 0.939831i \(-0.610982\pi\)
−0.341640 + 0.939831i \(0.610982\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14189.8 −0.975228
\(597\) 0 0
\(598\) 64998.6 4.44480
\(599\) −4185.17 −0.285478 −0.142739 0.989760i \(-0.545591\pi\)
−0.142739 + 0.989760i \(0.545591\pi\)
\(600\) 0 0
\(601\) 10599.8 0.719426 0.359713 0.933063i \(-0.382875\pi\)
0.359713 + 0.933063i \(0.382875\pi\)
\(602\) −11722.4 −0.793634
\(603\) 0 0
\(604\) −21364.0 −1.43922
\(605\) 0 0
\(606\) 0 0
\(607\) −13290.1 −0.888679 −0.444340 0.895858i \(-0.646562\pi\)
−0.444340 + 0.895858i \(0.646562\pi\)
\(608\) 2824.64 0.188412
\(609\) 0 0
\(610\) 0 0
\(611\) 1031.49 0.0682973
\(612\) 0 0
\(613\) 21327.1 1.40521 0.702606 0.711579i \(-0.252020\pi\)
0.702606 + 0.711579i \(0.252020\pi\)
\(614\) 15366.9 1.01003
\(615\) 0 0
\(616\) −9759.92 −0.638374
\(617\) −13741.6 −0.896622 −0.448311 0.893878i \(-0.647974\pi\)
−0.448311 + 0.893878i \(0.647974\pi\)
\(618\) 0 0
\(619\) 19444.3 1.26257 0.631285 0.775551i \(-0.282528\pi\)
0.631285 + 0.775551i \(0.282528\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8468.94 0.545938
\(623\) −4335.62 −0.278817
\(624\) 0 0
\(625\) 0 0
\(626\) 24809.6 1.58401
\(627\) 0 0
\(628\) −36488.0 −2.31852
\(629\) 10637.9 0.674341
\(630\) 0 0
\(631\) −27425.6 −1.73026 −0.865130 0.501547i \(-0.832764\pi\)
−0.865130 + 0.501547i \(0.832764\pi\)
\(632\) 33188.2 2.08886
\(633\) 0 0
\(634\) 23749.3 1.48771
\(635\) 0 0
\(636\) 0 0
\(637\) −3004.51 −0.186881
\(638\) −6095.70 −0.378262
\(639\) 0 0
\(640\) 0 0
\(641\) 10053.8 0.619502 0.309751 0.950818i \(-0.399754\pi\)
0.309751 + 0.950818i \(0.399754\pi\)
\(642\) 0 0
\(643\) −4044.18 −0.248036 −0.124018 0.992280i \(-0.539578\pi\)
−0.124018 + 0.992280i \(0.539578\pi\)
\(644\) −23967.3 −1.46653
\(645\) 0 0
\(646\) −30515.3 −1.85853
\(647\) −1486.90 −0.0903491 −0.0451746 0.998979i \(-0.514384\pi\)
−0.0451746 + 0.998979i \(0.514384\pi\)
\(648\) 0 0
\(649\) 5006.52 0.302809
\(650\) 0 0
\(651\) 0 0
\(652\) −24822.8 −1.49100
\(653\) −4269.30 −0.255851 −0.127925 0.991784i \(-0.540832\pi\)
−0.127925 + 0.991784i \(0.540832\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −11266.1 −0.670528
\(657\) 0 0
\(658\) −573.710 −0.0339902
\(659\) −1607.29 −0.0950094 −0.0475047 0.998871i \(-0.515127\pi\)
−0.0475047 + 0.998871i \(0.515127\pi\)
\(660\) 0 0
\(661\) 8001.31 0.470824 0.235412 0.971896i \(-0.424356\pi\)
0.235412 + 0.971896i \(0.424356\pi\)
\(662\) −11928.9 −0.700345
\(663\) 0 0
\(664\) 43534.1 2.54436
\(665\) 0 0
\(666\) 0 0
\(667\) −7359.14 −0.427207
\(668\) −16334.5 −0.946109
\(669\) 0 0
\(670\) 0 0
\(671\) −18141.5 −1.04373
\(672\) 0 0
\(673\) 3347.96 0.191760 0.0958800 0.995393i \(-0.469433\pi\)
0.0958800 + 0.995393i \(0.469433\pi\)
\(674\) −30554.7 −1.74617
\(675\) 0 0
\(676\) 24591.4 1.39914
\(677\) 7.91810 0.000449508 0 0.000224754 1.00000i \(-0.499928\pi\)
0.000224754 1.00000i \(0.499928\pi\)
\(678\) 0 0
\(679\) −1618.36 −0.0914684
\(680\) 0 0
\(681\) 0 0
\(682\) −22426.7 −1.25918
\(683\) −32136.0 −1.80036 −0.900182 0.435513i \(-0.856567\pi\)
−0.900182 + 0.435513i \(0.856567\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1671.09 0.0930067
\(687\) 0 0
\(688\) −19846.7 −1.09978
\(689\) 21274.7 1.17634
\(690\) 0 0
\(691\) −6196.84 −0.341156 −0.170578 0.985344i \(-0.554563\pi\)
−0.170578 + 0.985344i \(0.554563\pi\)
\(692\) 7594.98 0.417222
\(693\) 0 0
\(694\) −2033.27 −0.111213
\(695\) 0 0
\(696\) 0 0
\(697\) −8747.99 −0.475400
\(698\) 29091.0 1.57752
\(699\) 0 0
\(700\) 0 0
\(701\) −1651.38 −0.0889755 −0.0444878 0.999010i \(-0.514166\pi\)
−0.0444878 + 0.999010i \(0.514166\pi\)
\(702\) 0 0
\(703\) −33146.8 −1.77831
\(704\) 20731.5 1.10987
\(705\) 0 0
\(706\) 58525.4 3.11988
\(707\) −13132.0 −0.698555
\(708\) 0 0
\(709\) 15399.8 0.815728 0.407864 0.913043i \(-0.366274\pi\)
0.407864 + 0.913043i \(0.366274\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −23344.8 −1.22877
\(713\) −27075.0 −1.42211
\(714\) 0 0
\(715\) 0 0
\(716\) −37880.2 −1.97716
\(717\) 0 0
\(718\) −30754.3 −1.59852
\(719\) −3507.31 −0.181920 −0.0909600 0.995855i \(-0.528994\pi\)
−0.0909600 + 0.995855i \(0.528994\pi\)
\(720\) 0 0
\(721\) 12991.6 0.671056
\(722\) 61666.0 3.17863
\(723\) 0 0
\(724\) 8381.14 0.430225
\(725\) 0 0
\(726\) 0 0
\(727\) 22516.7 1.14869 0.574346 0.818613i \(-0.305256\pi\)
0.574346 + 0.818613i \(0.305256\pi\)
\(728\) −16177.6 −0.823600
\(729\) 0 0
\(730\) 0 0
\(731\) −15410.7 −0.779736
\(732\) 0 0
\(733\) −10964.7 −0.552509 −0.276255 0.961084i \(-0.589093\pi\)
−0.276255 + 0.961084i \(0.589093\pi\)
\(734\) 59577.8 2.99599
\(735\) 0 0
\(736\) 4399.31 0.220327
\(737\) −17681.1 −0.883707
\(738\) 0 0
\(739\) 3029.33 0.150793 0.0753963 0.997154i \(-0.475978\pi\)
0.0753963 + 0.997154i \(0.475978\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11832.9 −0.585442
\(743\) 8921.04 0.440486 0.220243 0.975445i \(-0.429315\pi\)
0.220243 + 0.975445i \(0.429315\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 38373.0 1.88329
\(747\) 0 0
\(748\) −26099.1 −1.27577
\(749\) 1526.11 0.0744498
\(750\) 0 0
\(751\) −6202.64 −0.301382 −0.150691 0.988581i \(-0.548150\pi\)
−0.150691 + 0.988581i \(0.548150\pi\)
\(752\) −971.326 −0.0471019
\(753\) 0 0
\(754\) −10104.0 −0.488016
\(755\) 0 0
\(756\) 0 0
\(757\) −12099.3 −0.580922 −0.290461 0.956887i \(-0.593809\pi\)
−0.290461 + 0.956887i \(0.593809\pi\)
\(758\) 29196.8 1.39905
\(759\) 0 0
\(760\) 0 0
\(761\) −29332.7 −1.39725 −0.698626 0.715487i \(-0.746205\pi\)
−0.698626 + 0.715487i \(0.746205\pi\)
\(762\) 0 0
\(763\) 5930.52 0.281388
\(764\) −5298.01 −0.250884
\(765\) 0 0
\(766\) −35924.1 −1.69451
\(767\) 8298.57 0.390670
\(768\) 0 0
\(769\) −13280.1 −0.622748 −0.311374 0.950287i \(-0.600789\pi\)
−0.311374 + 0.950287i \(0.600789\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 357.789 0.0166802
\(773\) −11248.1 −0.523370 −0.261685 0.965153i \(-0.584278\pi\)
−0.261685 + 0.965153i \(0.584278\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8713.96 −0.403110
\(777\) 0 0
\(778\) −9241.57 −0.425869
\(779\) 27258.0 1.25368
\(780\) 0 0
\(781\) 1674.11 0.0767021
\(782\) −47526.8 −2.17335
\(783\) 0 0
\(784\) 2829.26 0.128884
\(785\) 0 0
\(786\) 0 0
\(787\) −436.193 −0.0197568 −0.00987839 0.999951i \(-0.503144\pi\)
−0.00987839 + 0.999951i \(0.503144\pi\)
\(788\) 17079.0 0.772100
\(789\) 0 0
\(790\) 0 0
\(791\) −10010.3 −0.449970
\(792\) 0 0
\(793\) −30070.5 −1.34657
\(794\) −3854.50 −0.172281
\(795\) 0 0
\(796\) −41390.8 −1.84304
\(797\) 24034.0 1.06816 0.534082 0.845433i \(-0.320657\pi\)
0.534082 + 0.845433i \(0.320657\pi\)
\(798\) 0 0
\(799\) −754.224 −0.0333949
\(800\) 0 0
\(801\) 0 0
\(802\) −21488.1 −0.946101
\(803\) −3728.11 −0.163838
\(804\) 0 0
\(805\) 0 0
\(806\) −37173.4 −1.62454
\(807\) 0 0
\(808\) −70708.1 −3.07859
\(809\) −16188.0 −0.703509 −0.351754 0.936092i \(-0.614415\pi\)
−0.351754 + 0.936092i \(0.614415\pi\)
\(810\) 0 0
\(811\) 3732.42 0.161607 0.0808033 0.996730i \(-0.474251\pi\)
0.0808033 + 0.996730i \(0.474251\pi\)
\(812\) 3725.69 0.161018
\(813\) 0 0
\(814\) −42762.1 −1.84129
\(815\) 0 0
\(816\) 0 0
\(817\) 48018.6 2.05625
\(818\) −21454.9 −0.917058
\(819\) 0 0
\(820\) 0 0
\(821\) −37226.4 −1.58247 −0.791237 0.611510i \(-0.790562\pi\)
−0.791237 + 0.611510i \(0.790562\pi\)
\(822\) 0 0
\(823\) 21355.7 0.904511 0.452256 0.891888i \(-0.350620\pi\)
0.452256 + 0.891888i \(0.350620\pi\)
\(824\) 69952.2 2.95740
\(825\) 0 0
\(826\) −4615.62 −0.194428
\(827\) 18852.9 0.792718 0.396359 0.918096i \(-0.370274\pi\)
0.396359 + 0.918096i \(0.370274\pi\)
\(828\) 0 0
\(829\) −43227.9 −1.81106 −0.905528 0.424286i \(-0.860525\pi\)
−0.905528 + 0.424286i \(0.860525\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 34363.5 1.43190
\(833\) 2196.89 0.0913779
\(834\) 0 0
\(835\) 0 0
\(836\) 81322.4 3.36435
\(837\) 0 0
\(838\) 69536.1 2.86645
\(839\) −30311.3 −1.24727 −0.623637 0.781714i \(-0.714346\pi\)
−0.623637 + 0.781714i \(0.714346\pi\)
\(840\) 0 0
\(841\) −23245.0 −0.953095
\(842\) −75664.0 −3.09686
\(843\) 0 0
\(844\) 9664.88 0.394169
\(845\) 0 0
\(846\) 0 0
\(847\) 261.960 0.0106270
\(848\) −20033.8 −0.811276
\(849\) 0 0
\(850\) 0 0
\(851\) −51625.3 −2.07955
\(852\) 0 0
\(853\) −22797.0 −0.915071 −0.457535 0.889191i \(-0.651268\pi\)
−0.457535 + 0.889191i \(0.651268\pi\)
\(854\) 16725.0 0.670162
\(855\) 0 0
\(856\) 8217.25 0.328107
\(857\) 11280.3 0.449625 0.224813 0.974402i \(-0.427823\pi\)
0.224813 + 0.974402i \(0.427823\pi\)
\(858\) 0 0
\(859\) 43369.3 1.72263 0.861315 0.508071i \(-0.169641\pi\)
0.861315 + 0.508071i \(0.169641\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 25630.3 1.01273
\(863\) −9846.22 −0.388377 −0.194189 0.980964i \(-0.562207\pi\)
−0.194189 + 0.980964i \(0.562207\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 32207.0 1.26379
\(867\) 0 0
\(868\) 13707.2 0.536005
\(869\) −32572.9 −1.27153
\(870\) 0 0
\(871\) −29307.4 −1.14012
\(872\) 31932.5 1.24010
\(873\) 0 0
\(874\) 148090. 5.73136
\(875\) 0 0
\(876\) 0 0
\(877\) −15229.8 −0.586400 −0.293200 0.956051i \(-0.594720\pi\)
−0.293200 + 0.956051i \(0.594720\pi\)
\(878\) −43216.8 −1.66116
\(879\) 0 0
\(880\) 0 0
\(881\) −27891.7 −1.06662 −0.533311 0.845919i \(-0.679052\pi\)
−0.533311 + 0.845919i \(0.679052\pi\)
\(882\) 0 0
\(883\) 18300.3 0.697456 0.348728 0.937224i \(-0.386614\pi\)
0.348728 + 0.937224i \(0.386614\pi\)
\(884\) −43260.6 −1.64594
\(885\) 0 0
\(886\) 64416.3 2.44256
\(887\) 19104.8 0.723197 0.361598 0.932334i \(-0.382231\pi\)
0.361598 + 0.932334i \(0.382231\pi\)
\(888\) 0 0
\(889\) −12129.9 −0.457620
\(890\) 0 0
\(891\) 0 0
\(892\) 30375.7 1.14019
\(893\) 2350.10 0.0880661
\(894\) 0 0
\(895\) 0 0
\(896\) −17980.5 −0.670410
\(897\) 0 0
\(898\) −20426.8 −0.759077
\(899\) 4208.78 0.156141
\(900\) 0 0
\(901\) −15556.0 −0.575189
\(902\) 35165.1 1.29808
\(903\) 0 0
\(904\) −53899.9 −1.98306
\(905\) 0 0
\(906\) 0 0
\(907\) −29518.8 −1.08066 −0.540328 0.841455i \(-0.681700\pi\)
−0.540328 + 0.841455i \(0.681700\pi\)
\(908\) −12044.0 −0.440191
\(909\) 0 0
\(910\) 0 0
\(911\) 30322.2 1.10277 0.551383 0.834253i \(-0.314100\pi\)
0.551383 + 0.834253i \(0.314100\pi\)
\(912\) 0 0
\(913\) −42727.0 −1.54880
\(914\) −22907.0 −0.828991
\(915\) 0 0
\(916\) −89895.9 −3.24263
\(917\) 17008.3 0.612501
\(918\) 0 0
\(919\) −37780.9 −1.35612 −0.678062 0.735005i \(-0.737180\pi\)
−0.678062 + 0.735005i \(0.737180\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9499.48 0.339315
\(923\) 2774.93 0.0989575
\(924\) 0 0
\(925\) 0 0
\(926\) 35642.7 1.26489
\(927\) 0 0
\(928\) −683.867 −0.0241908
\(929\) −48493.3 −1.71261 −0.856305 0.516470i \(-0.827246\pi\)
−0.856305 + 0.516470i \(0.827246\pi\)
\(930\) 0 0
\(931\) −6845.33 −0.240974
\(932\) 13602.7 0.478081
\(933\) 0 0
\(934\) 9244.81 0.323875
\(935\) 0 0
\(936\) 0 0
\(937\) 18661.7 0.650641 0.325320 0.945604i \(-0.394528\pi\)
0.325320 + 0.945604i \(0.394528\pi\)
\(938\) 16300.6 0.567413
\(939\) 0 0
\(940\) 0 0
\(941\) 31270.1 1.08329 0.541644 0.840608i \(-0.317802\pi\)
0.541644 + 0.840608i \(0.317802\pi\)
\(942\) 0 0
\(943\) 42453.7 1.46605
\(944\) −7814.53 −0.269429
\(945\) 0 0
\(946\) 61948.0 2.12907
\(947\) 22405.3 0.768823 0.384412 0.923162i \(-0.374404\pi\)
0.384412 + 0.923162i \(0.374404\pi\)
\(948\) 0 0
\(949\) −6179.55 −0.211377
\(950\) 0 0
\(951\) 0 0
\(952\) 11829.0 0.402711
\(953\) −21777.9 −0.740247 −0.370123 0.928983i \(-0.620685\pi\)
−0.370123 + 0.928983i \(0.620685\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 28587.2 0.967129
\(957\) 0 0
\(958\) −51323.3 −1.73088
\(959\) −10169.9 −0.342442
\(960\) 0 0
\(961\) −14306.5 −0.480229
\(962\) −70880.5 −2.37555
\(963\) 0 0
\(964\) −37051.5 −1.23791
\(965\) 0 0
\(966\) 0 0
\(967\) −14432.2 −0.479945 −0.239973 0.970780i \(-0.577138\pi\)
−0.239973 + 0.970780i \(0.577138\pi\)
\(968\) 1410.51 0.0468341
\(969\) 0 0
\(970\) 0 0
\(971\) −6672.57 −0.220528 −0.110264 0.993902i \(-0.535170\pi\)
−0.110264 + 0.993902i \(0.535170\pi\)
\(972\) 0 0
\(973\) 20492.3 0.675183
\(974\) 58156.7 1.91321
\(975\) 0 0
\(976\) 28316.5 0.928678
\(977\) 33300.2 1.09045 0.545224 0.838290i \(-0.316444\pi\)
0.545224 + 0.838290i \(0.316444\pi\)
\(978\) 0 0
\(979\) 22912.0 0.747979
\(980\) 0 0
\(981\) 0 0
\(982\) −487.491 −0.0158416
\(983\) −17243.3 −0.559486 −0.279743 0.960075i \(-0.590249\pi\)
−0.279743 + 0.960075i \(0.590249\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7387.99 0.238622
\(987\) 0 0
\(988\) 134796. 4.34053
\(989\) 74787.8 2.40456
\(990\) 0 0
\(991\) −32266.7 −1.03429 −0.517147 0.855897i \(-0.673006\pi\)
−0.517147 + 0.855897i \(0.673006\pi\)
\(992\) −2516.02 −0.0805278
\(993\) 0 0
\(994\) −1543.40 −0.0492491
\(995\) 0 0
\(996\) 0 0
\(997\) −14339.8 −0.455514 −0.227757 0.973718i \(-0.573139\pi\)
−0.227757 + 0.973718i \(0.573139\pi\)
\(998\) −76059.8 −2.41246
\(999\) 0 0
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 1575.4.a.bg.1.4 4
3.2 odd 2 175.4.a.h.1.1 yes 4
5.4 even 2 1575.4.a.bl.1.1 4
15.2 even 4 175.4.b.f.99.1 8
15.8 even 4 175.4.b.f.99.8 8
15.14 odd 2 175.4.a.g.1.4 4
21.20 even 2 1225.4.a.bd.1.1 4
105.104 even 2 1225.4.a.z.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.4 4 15.14 odd 2
175.4.a.h.1.1 yes 4 3.2 odd 2
175.4.b.f.99.1 8 15.2 even 4
175.4.b.f.99.8 8 15.8 even 4
1225.4.a.z.1.4 4 105.104 even 2
1225.4.a.bd.1.1 4 21.20 even 2
1575.4.a.bg.1.4 4 1.1 even 1 trivial
1575.4.a.bl.1.1 4 5.4 even 2