Properties

Label 384.4.c.b.383.11
Level $384$
Weight $4$
Character 384.383
Analytic conductor $22.657$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.11
Root \(2.29679i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.4.c.b.383.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.00172 - 1.40813i) q^{3} +5.86626i q^{5} +5.92149i q^{7} +(23.0343 - 14.0861i) q^{9} +O(q^{10})\) \(q+(5.00172 - 1.40813i) q^{3} +5.86626i q^{5} +5.92149i q^{7} +(23.0343 - 14.0861i) q^{9} +27.9652 q^{11} -0.0653967 q^{13} +(8.26046 + 29.3414i) q^{15} -36.9675i q^{17} +30.7691i q^{19} +(8.33823 + 29.6176i) q^{21} +61.2864 q^{23} +90.5870 q^{25} +(95.3761 - 102.890i) q^{27} +143.566i q^{29} +299.568i q^{31} +(139.874 - 39.3787i) q^{33} -34.7370 q^{35} +340.559 q^{37} +(-0.327096 + 0.0920870i) q^{39} +379.315i q^{41} -470.926i q^{43} +(82.6330 + 135.125i) q^{45} -428.593 q^{47} +307.936 q^{49} +(-52.0551 - 184.901i) q^{51} -505.868i q^{53} +164.051i q^{55} +(43.3269 + 153.898i) q^{57} -207.827 q^{59} +578.221 q^{61} +(83.4109 + 136.398i) q^{63} -0.383634i q^{65} -415.350i q^{67} +(306.537 - 86.2992i) q^{69} -547.669 q^{71} +194.572 q^{73} +(453.090 - 127.558i) q^{75} +165.596i q^{77} +308.374i q^{79} +(332.162 - 648.930i) q^{81} +62.3019 q^{83} +216.861 q^{85} +(202.159 + 718.076i) q^{87} +1065.01i q^{89} -0.387246i q^{91} +(421.831 + 1498.36i) q^{93} -180.500 q^{95} +703.293 q^{97} +(644.161 - 393.922i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{3} - 36 q^{11} + 84 q^{15} - 136 q^{21} - 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} - 528 q^{37} + 620 q^{39} - 440 q^{45} - 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} - 624 q^{61} + 2744 q^{63} + 24 q^{69} - 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} - 480 q^{85} + 4156 q^{87} + 112 q^{93} - 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.00172 1.40813i 0.962581 0.270995i
\(4\) 0 0
\(5\) 5.86626i 0.524694i 0.964974 + 0.262347i \(0.0844966\pi\)
−0.964974 + 0.262347i \(0.915503\pi\)
\(6\) 0 0
\(7\) 5.92149i 0.319730i 0.987139 + 0.159865i \(0.0511060\pi\)
−0.987139 + 0.159865i \(0.948894\pi\)
\(8\) 0 0
\(9\) 23.0343 14.0861i 0.853124 0.521709i
\(10\) 0 0
\(11\) 27.9652 0.766531 0.383265 0.923638i \(-0.374799\pi\)
0.383265 + 0.923638i \(0.374799\pi\)
\(12\) 0 0
\(13\) −0.0653967 −0.00139521 −0.000697607 1.00000i \(-0.500222\pi\)
−0.000697607 1.00000i \(0.500222\pi\)
\(14\) 0 0
\(15\) 8.26046 + 29.3414i 0.142189 + 0.505061i
\(16\) 0 0
\(17\) 36.9675i 0.527408i −0.964604 0.263704i \(-0.915056\pi\)
0.964604 0.263704i \(-0.0849443\pi\)
\(18\) 0 0
\(19\) 30.7691i 0.371522i 0.982595 + 0.185761i \(0.0594751\pi\)
−0.982595 + 0.185761i \(0.940525\pi\)
\(20\) 0 0
\(21\) 8.33823 + 29.6176i 0.0866452 + 0.307766i
\(22\) 0 0
\(23\) 61.2864 0.555613 0.277806 0.960637i \(-0.410393\pi\)
0.277806 + 0.960637i \(0.410393\pi\)
\(24\) 0 0
\(25\) 90.5870 0.724696
\(26\) 0 0
\(27\) 95.3761 102.890i 0.679820 0.733379i
\(28\) 0 0
\(29\) 143.566i 0.919294i 0.888102 + 0.459647i \(0.152024\pi\)
−0.888102 + 0.459647i \(0.847976\pi\)
\(30\) 0 0
\(31\) 299.568i 1.73562i 0.496900 + 0.867808i \(0.334471\pi\)
−0.496900 + 0.867808i \(0.665529\pi\)
\(32\) 0 0
\(33\) 139.874 39.3787i 0.737848 0.207726i
\(34\) 0 0
\(35\) −34.7370 −0.167761
\(36\) 0 0
\(37\) 340.559 1.51318 0.756589 0.653891i \(-0.226865\pi\)
0.756589 + 0.653891i \(0.226865\pi\)
\(38\) 0 0
\(39\) −0.327096 + 0.0920870i −0.00134301 + 0.000378095i
\(40\) 0 0
\(41\) 379.315i 1.44486i 0.691447 + 0.722428i \(0.256974\pi\)
−0.691447 + 0.722428i \(0.743026\pi\)
\(42\) 0 0
\(43\) 470.926i 1.67013i −0.550152 0.835065i \(-0.685430\pi\)
0.550152 0.835065i \(-0.314570\pi\)
\(44\) 0 0
\(45\) 82.6330 + 135.125i 0.273738 + 0.447629i
\(46\) 0 0
\(47\) −428.593 −1.33014 −0.665072 0.746779i \(-0.731599\pi\)
−0.665072 + 0.746779i \(0.731599\pi\)
\(48\) 0 0
\(49\) 307.936 0.897772
\(50\) 0 0
\(51\) −52.0551 184.901i −0.142925 0.507673i
\(52\) 0 0
\(53\) 505.868i 1.31106i −0.755168 0.655531i \(-0.772445\pi\)
0.755168 0.655531i \(-0.227555\pi\)
\(54\) 0 0
\(55\) 164.051i 0.402194i
\(56\) 0 0
\(57\) 43.3269 + 153.898i 0.100681 + 0.357620i
\(58\) 0 0
\(59\) −207.827 −0.458589 −0.229295 0.973357i \(-0.573642\pi\)
−0.229295 + 0.973357i \(0.573642\pi\)
\(60\) 0 0
\(61\) 578.221 1.21367 0.606833 0.794830i \(-0.292440\pi\)
0.606833 + 0.794830i \(0.292440\pi\)
\(62\) 0 0
\(63\) 83.4109 + 136.398i 0.166806 + 0.272770i
\(64\) 0 0
\(65\) 0.383634i 0.000732061i
\(66\) 0 0
\(67\) 415.350i 0.757359i −0.925528 0.378679i \(-0.876378\pi\)
0.925528 0.378679i \(-0.123622\pi\)
\(68\) 0 0
\(69\) 306.537 86.2992i 0.534822 0.150568i
\(70\) 0 0
\(71\) −547.669 −0.915441 −0.457721 0.889096i \(-0.651334\pi\)
−0.457721 + 0.889096i \(0.651334\pi\)
\(72\) 0 0
\(73\) 194.572 0.311957 0.155979 0.987760i \(-0.450147\pi\)
0.155979 + 0.987760i \(0.450147\pi\)
\(74\) 0 0
\(75\) 453.090 127.558i 0.697578 0.196389i
\(76\) 0 0
\(77\) 165.596i 0.245083i
\(78\) 0 0
\(79\) 308.374i 0.439175i 0.975593 + 0.219587i \(0.0704711\pi\)
−0.975593 + 0.219587i \(0.929529\pi\)
\(80\) 0 0
\(81\) 332.162 648.930i 0.455640 0.890164i
\(82\) 0 0
\(83\) 62.3019 0.0823918 0.0411959 0.999151i \(-0.486883\pi\)
0.0411959 + 0.999151i \(0.486883\pi\)
\(84\) 0 0
\(85\) 216.861 0.276728
\(86\) 0 0
\(87\) 202.159 + 718.076i 0.249124 + 0.884895i
\(88\) 0 0
\(89\) 1065.01i 1.26844i 0.773153 + 0.634220i \(0.218679\pi\)
−0.773153 + 0.634220i \(0.781321\pi\)
\(90\) 0 0
\(91\) 0.387246i 0.000446092i
\(92\) 0 0
\(93\) 421.831 + 1498.36i 0.470343 + 1.67067i
\(94\) 0 0
\(95\) −180.500 −0.194936
\(96\) 0 0
\(97\) 703.293 0.736171 0.368085 0.929792i \(-0.380013\pi\)
0.368085 + 0.929792i \(0.380013\pi\)
\(98\) 0 0
\(99\) 644.161 393.922i 0.653946 0.399906i
\(100\) 0 0
\(101\) 942.553i 0.928589i −0.885681 0.464295i \(-0.846308\pi\)
0.885681 0.464295i \(-0.153692\pi\)
\(102\) 0 0
\(103\) 86.0316i 0.0823005i 0.999153 + 0.0411502i \(0.0131022\pi\)
−0.999153 + 0.0411502i \(0.986898\pi\)
\(104\) 0 0
\(105\) −173.745 + 48.9142i −0.161483 + 0.0454623i
\(106\) 0 0
\(107\) −688.082 −0.621676 −0.310838 0.950463i \(-0.600610\pi\)
−0.310838 + 0.950463i \(0.600610\pi\)
\(108\) 0 0
\(109\) −1849.94 −1.62561 −0.812807 0.582534i \(-0.802061\pi\)
−0.812807 + 0.582534i \(0.802061\pi\)
\(110\) 0 0
\(111\) 1703.38 479.551i 1.45656 0.410063i
\(112\) 0 0
\(113\) 1054.54i 0.877902i −0.898511 0.438951i \(-0.855350\pi\)
0.898511 0.438951i \(-0.144650\pi\)
\(114\) 0 0
\(115\) 359.522i 0.291527i
\(116\) 0 0
\(117\) −1.50637 + 0.921186i −0.00119029 + 0.000727895i
\(118\) 0 0
\(119\) 218.903 0.168628
\(120\) 0 0
\(121\) −548.945 −0.412431
\(122\) 0 0
\(123\) 534.125 + 1897.23i 0.391548 + 1.39079i
\(124\) 0 0
\(125\) 1264.69i 0.904938i
\(126\) 0 0
\(127\) 957.493i 0.669006i −0.942395 0.334503i \(-0.891432\pi\)
0.942395 0.334503i \(-0.108568\pi\)
\(128\) 0 0
\(129\) −663.125 2355.44i −0.452596 1.60763i
\(130\) 0 0
\(131\) −550.306 −0.367026 −0.183513 0.983017i \(-0.558747\pi\)
−0.183513 + 0.983017i \(0.558747\pi\)
\(132\) 0 0
\(133\) −182.199 −0.118787
\(134\) 0 0
\(135\) 603.581 + 559.501i 0.384800 + 0.356698i
\(136\) 0 0
\(137\) 1991.37i 1.24185i 0.783869 + 0.620926i \(0.213243\pi\)
−0.783869 + 0.620926i \(0.786757\pi\)
\(138\) 0 0
\(139\) 746.850i 0.455733i 0.973692 + 0.227867i \(0.0731751\pi\)
−0.973692 + 0.227867i \(0.926825\pi\)
\(140\) 0 0
\(141\) −2143.70 + 603.515i −1.28037 + 0.360462i
\(142\) 0 0
\(143\) −1.82883 −0.00106947
\(144\) 0 0
\(145\) −842.195 −0.482348
\(146\) 0 0
\(147\) 1540.21 433.614i 0.864179 0.243292i
\(148\) 0 0
\(149\) 3198.80i 1.75877i −0.476114 0.879383i \(-0.657955\pi\)
0.476114 0.879383i \(-0.342045\pi\)
\(150\) 0 0
\(151\) 1475.50i 0.795195i −0.917560 0.397598i \(-0.869844\pi\)
0.917560 0.397598i \(-0.130156\pi\)
\(152\) 0 0
\(153\) −520.730 851.523i −0.275154 0.449945i
\(154\) 0 0
\(155\) −1757.35 −0.910668
\(156\) 0 0
\(157\) 713.140 0.362514 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(158\) 0 0
\(159\) −712.328 2530.21i −0.355291 1.26200i
\(160\) 0 0
\(161\) 362.907i 0.177646i
\(162\) 0 0
\(163\) 1986.44i 0.954538i 0.878757 + 0.477269i \(0.158373\pi\)
−0.878757 + 0.477269i \(0.841627\pi\)
\(164\) 0 0
\(165\) 231.006 + 820.539i 0.108993 + 0.387145i
\(166\) 0 0
\(167\) −3376.74 −1.56467 −0.782336 0.622857i \(-0.785972\pi\)
−0.782336 + 0.622857i \(0.785972\pi\)
\(168\) 0 0
\(169\) −2197.00 −0.999998
\(170\) 0 0
\(171\) 433.418 + 708.747i 0.193826 + 0.316954i
\(172\) 0 0
\(173\) 2696.12i 1.18487i −0.805619 0.592435i \(-0.798167\pi\)
0.805619 0.592435i \(-0.201833\pi\)
\(174\) 0 0
\(175\) 536.410i 0.231707i
\(176\) 0 0
\(177\) −1039.49 + 292.647i −0.441429 + 0.124275i
\(178\) 0 0
\(179\) −3347.25 −1.39768 −0.698842 0.715276i \(-0.746301\pi\)
−0.698842 + 0.715276i \(0.746301\pi\)
\(180\) 0 0
\(181\) −1843.89 −0.757210 −0.378605 0.925558i \(-0.623596\pi\)
−0.378605 + 0.925558i \(0.623596\pi\)
\(182\) 0 0
\(183\) 2892.10 814.210i 1.16825 0.328897i
\(184\) 0 0
\(185\) 1997.81i 0.793956i
\(186\) 0 0
\(187\) 1033.81i 0.404275i
\(188\) 0 0
\(189\) 609.263 + 564.769i 0.234483 + 0.217359i
\(190\) 0 0
\(191\) −2382.31 −0.902501 −0.451251 0.892397i \(-0.649022\pi\)
−0.451251 + 0.892397i \(0.649022\pi\)
\(192\) 0 0
\(193\) −2546.25 −0.949654 −0.474827 0.880079i \(-0.657489\pi\)
−0.474827 + 0.880079i \(0.657489\pi\)
\(194\) 0 0
\(195\) −0.540207 1.91883i −0.000198385 0.000704668i
\(196\) 0 0
\(197\) 2189.58i 0.791885i −0.918275 0.395943i \(-0.870418\pi\)
0.918275 0.395943i \(-0.129582\pi\)
\(198\) 0 0
\(199\) 5379.16i 1.91617i −0.286476 0.958087i \(-0.592484\pi\)
0.286476 0.958087i \(-0.407516\pi\)
\(200\) 0 0
\(201\) −584.866 2077.46i −0.205240 0.729019i
\(202\) 0 0
\(203\) −850.124 −0.293926
\(204\) 0 0
\(205\) −2225.16 −0.758107
\(206\) 0 0
\(207\) 1411.69 863.288i 0.474006 0.289868i
\(208\) 0 0
\(209\) 860.466i 0.284783i
\(210\) 0 0
\(211\) 5320.28i 1.73584i 0.496701 + 0.867922i \(0.334544\pi\)
−0.496701 + 0.867922i \(0.665456\pi\)
\(212\) 0 0
\(213\) −2739.28 + 771.189i −0.881186 + 0.248080i
\(214\) 0 0
\(215\) 2762.58 0.876307
\(216\) 0 0
\(217\) −1773.89 −0.554929
\(218\) 0 0
\(219\) 973.192 273.982i 0.300284 0.0845388i
\(220\) 0 0
\(221\) 2.41755i 0.000735847i
\(222\) 0 0
\(223\) 2708.41i 0.813310i 0.913582 + 0.406655i \(0.133305\pi\)
−0.913582 + 0.406655i \(0.866695\pi\)
\(224\) 0 0
\(225\) 2086.61 1276.02i 0.618255 0.378080i
\(226\) 0 0
\(227\) −2335.26 −0.682806 −0.341403 0.939917i \(-0.610902\pi\)
−0.341403 + 0.939917i \(0.610902\pi\)
\(228\) 0 0
\(229\) 139.302 0.0401980 0.0200990 0.999798i \(-0.493602\pi\)
0.0200990 + 0.999798i \(0.493602\pi\)
\(230\) 0 0
\(231\) 233.181 + 828.264i 0.0664162 + 0.235912i
\(232\) 0 0
\(233\) 7.07987i 0.00199064i 1.00000 0.000995318i \(0.000316819\pi\)
−1.00000 0.000995318i \(0.999683\pi\)
\(234\) 0 0
\(235\) 2514.24i 0.697919i
\(236\) 0 0
\(237\) 434.231 + 1542.40i 0.119014 + 0.422741i
\(238\) 0 0
\(239\) 890.272 0.240949 0.120475 0.992716i \(-0.461558\pi\)
0.120475 + 0.992716i \(0.461558\pi\)
\(240\) 0 0
\(241\) 1273.98 0.340515 0.170258 0.985400i \(-0.445540\pi\)
0.170258 + 0.985400i \(0.445540\pi\)
\(242\) 0 0
\(243\) 747.601 3713.49i 0.197361 0.980331i
\(244\) 0 0
\(245\) 1806.43i 0.471056i
\(246\) 0 0
\(247\) 2.01220i 0.000518353i
\(248\) 0 0
\(249\) 311.616 87.7291i 0.0793088 0.0223277i
\(250\) 0 0
\(251\) 3032.85 0.762678 0.381339 0.924435i \(-0.375463\pi\)
0.381339 + 0.924435i \(0.375463\pi\)
\(252\) 0 0
\(253\) 1713.89 0.425894
\(254\) 0 0
\(255\) 1084.68 305.369i 0.266373 0.0749919i
\(256\) 0 0
\(257\) 2585.22i 0.627478i 0.949509 + 0.313739i \(0.101582\pi\)
−0.949509 + 0.313739i \(0.898418\pi\)
\(258\) 0 0
\(259\) 2016.62i 0.483809i
\(260\) 0 0
\(261\) 2022.29 + 3306.95i 0.479604 + 0.784271i
\(262\) 0 0
\(263\) −5444.20 −1.27644 −0.638220 0.769854i \(-0.720329\pi\)
−0.638220 + 0.769854i \(0.720329\pi\)
\(264\) 0 0
\(265\) 2967.55 0.687907
\(266\) 0 0
\(267\) 1499.68 + 5326.89i 0.343740 + 1.22098i
\(268\) 0 0
\(269\) 2061.38i 0.467229i −0.972329 0.233614i \(-0.924945\pi\)
0.972329 0.233614i \(-0.0750553\pi\)
\(270\) 0 0
\(271\) 5846.72i 1.31056i 0.755384 + 0.655282i \(0.227450\pi\)
−0.755384 + 0.655282i \(0.772550\pi\)
\(272\) 0 0
\(273\) −0.545292 1.93689i −0.000120889 0.000429400i
\(274\) 0 0
\(275\) 2533.29 0.555502
\(276\) 0 0
\(277\) 1488.28 0.322823 0.161412 0.986887i \(-0.448395\pi\)
0.161412 + 0.986887i \(0.448395\pi\)
\(278\) 0 0
\(279\) 4219.76 + 6900.36i 0.905485 + 1.48069i
\(280\) 0 0
\(281\) 6456.97i 1.37079i −0.728174 0.685393i \(-0.759631\pi\)
0.728174 0.685393i \(-0.240369\pi\)
\(282\) 0 0
\(283\) 5390.67i 1.13231i −0.824301 0.566153i \(-0.808431\pi\)
0.824301 0.566153i \(-0.191569\pi\)
\(284\) 0 0
\(285\) −902.809 + 254.167i −0.187641 + 0.0528265i
\(286\) 0 0
\(287\) −2246.11 −0.461964
\(288\) 0 0
\(289\) 3546.40 0.721840
\(290\) 0 0
\(291\) 3517.67 990.328i 0.708624 0.199498i
\(292\) 0 0
\(293\) 5400.85i 1.07686i 0.842669 + 0.538432i \(0.180983\pi\)
−0.842669 + 0.538432i \(0.819017\pi\)
\(294\) 0 0
\(295\) 1219.17i 0.240619i
\(296\) 0 0
\(297\) 2667.22 2877.35i 0.521103 0.562157i
\(298\) 0 0
\(299\) −4.00793 −0.000775198
\(300\) 0 0
\(301\) 2788.58 0.533991
\(302\) 0 0
\(303\) −1327.24 4714.38i −0.251643 0.893842i
\(304\) 0 0
\(305\) 3391.99i 0.636803i
\(306\) 0 0
\(307\) 3598.12i 0.668910i −0.942412 0.334455i \(-0.891448\pi\)
0.942412 0.334455i \(-0.108552\pi\)
\(308\) 0 0
\(309\) 121.144 + 430.306i 0.0223030 + 0.0792209i
\(310\) 0 0
\(311\) 7894.55 1.43942 0.719709 0.694276i \(-0.244275\pi\)
0.719709 + 0.694276i \(0.244275\pi\)
\(312\) 0 0
\(313\) 8500.57 1.53508 0.767541 0.641000i \(-0.221480\pi\)
0.767541 + 0.641000i \(0.221480\pi\)
\(314\) 0 0
\(315\) −800.144 + 489.310i −0.143121 + 0.0875222i
\(316\) 0 0
\(317\) 7352.01i 1.30262i −0.758813 0.651309i \(-0.774220\pi\)
0.758813 0.651309i \(-0.225780\pi\)
\(318\) 0 0
\(319\) 4014.86i 0.704667i
\(320\) 0 0
\(321\) −3441.59 + 968.908i −0.598414 + 0.168471i
\(322\) 0 0
\(323\) 1137.46 0.195944
\(324\) 0 0
\(325\) −5.92409 −0.00101111
\(326\) 0 0
\(327\) −9252.86 + 2604.95i −1.56478 + 0.440533i
\(328\) 0 0
\(329\) 2537.91i 0.425287i
\(330\) 0 0
\(331\) 4348.48i 0.722097i −0.932547 0.361048i \(-0.882419\pi\)
0.932547 0.361048i \(-0.117581\pi\)
\(332\) 0 0
\(333\) 7844.55 4797.16i 1.29093 0.789438i
\(334\) 0 0
\(335\) 2436.55 0.397382
\(336\) 0 0
\(337\) −2451.75 −0.396307 −0.198153 0.980171i \(-0.563494\pi\)
−0.198153 + 0.980171i \(0.563494\pi\)
\(338\) 0 0
\(339\) −1484.93 5274.52i −0.237907 0.845051i
\(340\) 0 0
\(341\) 8377.50i 1.33040i
\(342\) 0 0
\(343\) 3854.51i 0.606776i
\(344\) 0 0
\(345\) 506.254 + 1798.23i 0.0790023 + 0.280618i
\(346\) 0 0
\(347\) −3671.15 −0.567947 −0.283974 0.958832i \(-0.591653\pi\)
−0.283974 + 0.958832i \(0.591653\pi\)
\(348\) 0 0
\(349\) 2543.31 0.390086 0.195043 0.980795i \(-0.437515\pi\)
0.195043 + 0.980795i \(0.437515\pi\)
\(350\) 0 0
\(351\) −6.23728 + 6.72868i −0.000948494 + 0.00102322i
\(352\) 0 0
\(353\) 12015.9i 1.81173i 0.423562 + 0.905867i \(0.360779\pi\)
−0.423562 + 0.905867i \(0.639221\pi\)
\(354\) 0 0
\(355\) 3212.77i 0.480327i
\(356\) 0 0
\(357\) 1094.89 308.244i 0.162319 0.0456974i
\(358\) 0 0
\(359\) −11569.3 −1.70085 −0.850424 0.526097i \(-0.823655\pi\)
−0.850424 + 0.526097i \(0.823655\pi\)
\(360\) 0 0
\(361\) 5912.26 0.861971
\(362\) 0 0
\(363\) −2745.67 + 772.986i −0.396998 + 0.111767i
\(364\) 0 0
\(365\) 1141.41i 0.163682i
\(366\) 0 0
\(367\) 675.208i 0.0960370i −0.998846 0.0480185i \(-0.984709\pi\)
0.998846 0.0480185i \(-0.0152906\pi\)
\(368\) 0 0
\(369\) 5343.08 + 8737.27i 0.753793 + 1.23264i
\(370\) 0 0
\(371\) 2995.49 0.419186
\(372\) 0 0
\(373\) 9000.61 1.24942 0.624710 0.780857i \(-0.285217\pi\)
0.624710 + 0.780857i \(0.285217\pi\)
\(374\) 0 0
\(375\) 1780.85 + 6325.62i 0.245234 + 0.871076i
\(376\) 0 0
\(377\) 9.38873i 0.00128261i
\(378\) 0 0
\(379\) 6453.84i 0.874700i −0.899291 0.437350i \(-0.855917\pi\)
0.899291 0.437350i \(-0.144083\pi\)
\(380\) 0 0
\(381\) −1348.28 4789.11i −0.181297 0.643973i
\(382\) 0 0
\(383\) −5211.10 −0.695234 −0.347617 0.937637i \(-0.613009\pi\)
−0.347617 + 0.937637i \(0.613009\pi\)
\(384\) 0 0
\(385\) −971.429 −0.128594
\(386\) 0 0
\(387\) −6633.53 10847.5i −0.871321 1.42483i
\(388\) 0 0
\(389\) 1399.05i 0.182351i 0.995835 + 0.0911756i \(0.0290625\pi\)
−0.995835 + 0.0911756i \(0.970938\pi\)
\(390\) 0 0
\(391\) 2265.61i 0.293035i
\(392\) 0 0
\(393\) −2752.47 + 774.902i −0.353292 + 0.0994622i
\(394\) 0 0
\(395\) −1809.00 −0.230433
\(396\) 0 0
\(397\) 6624.77 0.837500 0.418750 0.908101i \(-0.362468\pi\)
0.418750 + 0.908101i \(0.362468\pi\)
\(398\) 0 0
\(399\) −911.308 + 256.560i −0.114342 + 0.0321906i
\(400\) 0 0
\(401\) 5283.09i 0.657918i −0.944344 0.328959i \(-0.893302\pi\)
0.944344 0.328959i \(-0.106698\pi\)
\(402\) 0 0
\(403\) 19.5908i 0.00242155i
\(404\) 0 0
\(405\) 3806.79 + 1948.55i 0.467064 + 0.239072i
\(406\) 0 0
\(407\) 9523.81 1.15990
\(408\) 0 0
\(409\) −7555.96 −0.913492 −0.456746 0.889597i \(-0.650985\pi\)
−0.456746 + 0.889597i \(0.650985\pi\)
\(410\) 0 0
\(411\) 2804.10 + 9960.25i 0.336536 + 1.19538i
\(412\) 0 0
\(413\) 1230.65i 0.146625i
\(414\) 0 0
\(415\) 365.479i 0.0432305i
\(416\) 0 0
\(417\) 1051.66 + 3735.53i 0.123501 + 0.438680i
\(418\) 0 0
\(419\) 6784.17 0.790999 0.395500 0.918466i \(-0.370571\pi\)
0.395500 + 0.918466i \(0.370571\pi\)
\(420\) 0 0
\(421\) −12740.9 −1.47495 −0.737476 0.675374i \(-0.763982\pi\)
−0.737476 + 0.675374i \(0.763982\pi\)
\(422\) 0 0
\(423\) −9872.36 + 6037.22i −1.13478 + 0.693947i
\(424\) 0 0
\(425\) 3348.78i 0.382211i
\(426\) 0 0
\(427\) 3423.93i 0.388046i
\(428\) 0 0
\(429\) −9.14731 + 2.57524i −0.00102946 + 0.000289822i
\(430\) 0 0
\(431\) 16239.7 1.81493 0.907467 0.420123i \(-0.138013\pi\)
0.907467 + 0.420123i \(0.138013\pi\)
\(432\) 0 0
\(433\) 4832.73 0.536365 0.268183 0.963368i \(-0.413577\pi\)
0.268183 + 0.963368i \(0.413577\pi\)
\(434\) 0 0
\(435\) −4212.42 + 1185.92i −0.464299 + 0.130714i
\(436\) 0 0
\(437\) 1885.73i 0.206422i
\(438\) 0 0
\(439\) 4286.58i 0.466031i −0.972473 0.233015i \(-0.925141\pi\)
0.972473 0.233015i \(-0.0748592\pi\)
\(440\) 0 0
\(441\) 7093.10 4337.63i 0.765911 0.468376i
\(442\) 0 0
\(443\) 2140.54 0.229572 0.114786 0.993390i \(-0.463382\pi\)
0.114786 + 0.993390i \(0.463382\pi\)
\(444\) 0 0
\(445\) −6247.64 −0.665543
\(446\) 0 0
\(447\) −4504.33 15999.5i −0.476617 1.69296i
\(448\) 0 0
\(449\) 9598.09i 1.00882i −0.863463 0.504412i \(-0.831709\pi\)
0.863463 0.504412i \(-0.168291\pi\)
\(450\) 0 0
\(451\) 10607.6i 1.10753i
\(452\) 0 0
\(453\) −2077.70 7380.03i −0.215494 0.765440i
\(454\) 0 0
\(455\) 2.27168 0.000234062
\(456\) 0 0
\(457\) −9542.53 −0.976763 −0.488382 0.872630i \(-0.662413\pi\)
−0.488382 + 0.872630i \(0.662413\pi\)
\(458\) 0 0
\(459\) −3803.60 3525.82i −0.386790 0.358543i
\(460\) 0 0
\(461\) 8017.02i 0.809956i 0.914326 + 0.404978i \(0.132721\pi\)
−0.914326 + 0.404978i \(0.867279\pi\)
\(462\) 0 0
\(463\) 10933.1i 1.09742i 0.836014 + 0.548709i \(0.184880\pi\)
−0.836014 + 0.548709i \(0.815120\pi\)
\(464\) 0 0
\(465\) −8789.75 + 2474.57i −0.876591 + 0.246786i
\(466\) 0 0
\(467\) 3991.55 0.395518 0.197759 0.980251i \(-0.436634\pi\)
0.197759 + 0.980251i \(0.436634\pi\)
\(468\) 0 0
\(469\) 2459.49 0.242151
\(470\) 0 0
\(471\) 3566.92 1004.19i 0.348949 0.0982394i
\(472\) 0 0
\(473\) 13169.6i 1.28021i
\(474\) 0 0
\(475\) 2787.28i 0.269241i
\(476\) 0 0
\(477\) −7125.72 11652.3i −0.683993 1.11850i
\(478\) 0 0
\(479\) −10524.5 −1.00392 −0.501958 0.864892i \(-0.667387\pi\)
−0.501958 + 0.864892i \(0.667387\pi\)
\(480\) 0 0
\(481\) −22.2714 −0.00211120
\(482\) 0 0
\(483\) 511.020 + 1815.16i 0.0481412 + 0.170999i
\(484\) 0 0
\(485\) 4125.70i 0.386265i
\(486\) 0 0
\(487\) 10750.1i 1.00028i −0.865946 0.500138i \(-0.833283\pi\)
0.865946 0.500138i \(-0.166717\pi\)
\(488\) 0 0
\(489\) 2797.16 + 9935.60i 0.258675 + 0.918820i
\(490\) 0 0
\(491\) 18059.8 1.65994 0.829968 0.557811i \(-0.188359\pi\)
0.829968 + 0.557811i \(0.188359\pi\)
\(492\) 0 0
\(493\) 5307.28 0.484843
\(494\) 0 0
\(495\) 2310.85 + 3778.82i 0.209828 + 0.343122i
\(496\) 0 0
\(497\) 3243.02i 0.292694i
\(498\) 0 0
\(499\) 15164.6i 1.36044i 0.733007 + 0.680221i \(0.238116\pi\)
−0.733007 + 0.680221i \(0.761884\pi\)
\(500\) 0 0
\(501\) −16889.5 + 4754.89i −1.50612 + 0.424018i
\(502\) 0 0
\(503\) 21362.1 1.89362 0.946808 0.321799i \(-0.104288\pi\)
0.946808 + 0.321799i \(0.104288\pi\)
\(504\) 0 0
\(505\) 5529.26 0.487226
\(506\) 0 0
\(507\) −10988.8 + 3093.66i −0.962579 + 0.270994i
\(508\) 0 0
\(509\) 4921.82i 0.428597i 0.976768 + 0.214299i \(0.0687466\pi\)
−0.976768 + 0.214299i \(0.931253\pi\)
\(510\) 0 0
\(511\) 1152.15i 0.0997422i
\(512\) 0 0
\(513\) 3165.84 + 2934.64i 0.272467 + 0.252568i
\(514\) 0 0
\(515\) −504.684 −0.0431826
\(516\) 0 0
\(517\) −11985.7 −1.01960
\(518\) 0 0
\(519\) −3796.49 13485.2i −0.321093 1.14053i
\(520\) 0 0
\(521\) 9816.25i 0.825446i −0.910857 0.412723i \(-0.864578\pi\)
0.910857 0.412723i \(-0.135422\pi\)
\(522\) 0 0
\(523\) 11120.7i 0.929783i −0.885368 0.464891i \(-0.846093\pi\)
0.885368 0.464891i \(-0.153907\pi\)
\(524\) 0 0
\(525\) 755.335 + 2682.97i 0.0627914 + 0.223037i
\(526\) 0 0
\(527\) 11074.3 0.915378
\(528\) 0 0
\(529\) −8410.98 −0.691295
\(530\) 0 0
\(531\) −4787.16 + 2927.48i −0.391233 + 0.239250i
\(532\) 0 0
\(533\) 24.8059i 0.00201588i
\(534\) 0 0
\(535\) 4036.47i 0.326190i
\(536\) 0 0
\(537\) −16742.0 + 4713.37i −1.34538 + 0.378765i
\(538\) 0 0
\(539\) 8611.50 0.688170
\(540\) 0 0
\(541\) −15949.1 −1.26748 −0.633740 0.773546i \(-0.718481\pi\)
−0.633740 + 0.773546i \(0.718481\pi\)
\(542\) 0 0
\(543\) −9222.60 + 2596.43i −0.728876 + 0.205200i
\(544\) 0 0
\(545\) 10852.2i 0.852950i
\(546\) 0 0
\(547\) 12775.8i 0.998637i 0.866419 + 0.499318i \(0.166416\pi\)
−0.866419 + 0.499318i \(0.833584\pi\)
\(548\) 0 0
\(549\) 13318.9 8144.89i 1.03541 0.633180i
\(550\) 0 0
\(551\) −4417.40 −0.341538
\(552\) 0 0
\(553\) −1826.04 −0.140418
\(554\) 0 0
\(555\) 2813.17 + 9992.47i 0.215158 + 0.764246i
\(556\) 0 0
\(557\) 9120.09i 0.693772i 0.937907 + 0.346886i \(0.112761\pi\)
−0.937907 + 0.346886i \(0.887239\pi\)
\(558\) 0 0
\(559\) 30.7970i 0.00233019i
\(560\) 0 0
\(561\) −1455.73 5170.80i −0.109556 0.389147i
\(562\) 0 0
\(563\) −4530.65 −0.339155 −0.169577 0.985517i \(-0.554240\pi\)
−0.169577 + 0.985517i \(0.554240\pi\)
\(564\) 0 0
\(565\) 6186.21 0.460630
\(566\) 0 0
\(567\) 3842.63 + 1966.89i 0.284612 + 0.145682i
\(568\) 0 0
\(569\) 17896.0i 1.31853i 0.751913 + 0.659263i \(0.229132\pi\)
−0.751913 + 0.659263i \(0.770868\pi\)
\(570\) 0 0
\(571\) 11840.1i 0.867760i 0.900971 + 0.433880i \(0.142856\pi\)
−0.900971 + 0.433880i \(0.857144\pi\)
\(572\) 0 0
\(573\) −11915.6 + 3354.60i −0.868731 + 0.244573i
\(574\) 0 0
\(575\) 5551.75 0.402650
\(576\) 0 0
\(577\) −18102.9 −1.30612 −0.653060 0.757306i \(-0.726515\pi\)
−0.653060 + 0.757306i \(0.726515\pi\)
\(578\) 0 0
\(579\) −12735.6 + 3585.45i −0.914119 + 0.257351i
\(580\) 0 0
\(581\) 368.920i 0.0263432i
\(582\) 0 0
\(583\) 14146.7i 1.00497i
\(584\) 0 0
\(585\) −5.40392 8.83676i −0.000381922 0.000624538i
\(586\) 0 0
\(587\) −21946.7 −1.54317 −0.771583 0.636129i \(-0.780535\pi\)
−0.771583 + 0.636129i \(0.780535\pi\)
\(588\) 0 0
\(589\) −9217.46 −0.644820
\(590\) 0 0
\(591\) −3083.22 10951.7i −0.214597 0.762253i
\(592\) 0 0
\(593\) 8554.64i 0.592406i −0.955125 0.296203i \(-0.904279\pi\)
0.955125 0.296203i \(-0.0957206\pi\)
\(594\) 0 0
\(595\) 1284.14i 0.0884784i
\(596\) 0 0
\(597\) −7574.56 26905.0i −0.519273 1.84447i
\(598\) 0 0
\(599\) 19321.4 1.31795 0.658973 0.752167i \(-0.270991\pi\)
0.658973 + 0.752167i \(0.270991\pi\)
\(600\) 0 0
\(601\) −18248.6 −1.23856 −0.619281 0.785170i \(-0.712576\pi\)
−0.619281 + 0.785170i \(0.712576\pi\)
\(602\) 0 0
\(603\) −5850.67 9567.30i −0.395121 0.646121i
\(604\) 0 0
\(605\) 3220.26i 0.216400i
\(606\) 0 0
\(607\) 18473.4i 1.23528i −0.786462 0.617639i \(-0.788089\pi\)
0.786462 0.617639i \(-0.211911\pi\)
\(608\) 0 0
\(609\) −4252.08 + 1197.08i −0.282928 + 0.0796524i
\(610\) 0 0
\(611\) 28.0286 0.00185583
\(612\) 0 0
\(613\) −1007.68 −0.0663946 −0.0331973 0.999449i \(-0.510569\pi\)
−0.0331973 + 0.999449i \(0.510569\pi\)
\(614\) 0 0
\(615\) −11129.6 + 3133.32i −0.729740 + 0.205443i
\(616\) 0 0
\(617\) 8342.82i 0.544359i 0.962247 + 0.272179i \(0.0877444\pi\)
−0.962247 + 0.272179i \(0.912256\pi\)
\(618\) 0 0
\(619\) 5153.14i 0.334608i −0.985905 0.167304i \(-0.946494\pi\)
0.985905 0.167304i \(-0.0535061\pi\)
\(620\) 0 0
\(621\) 5845.26 6305.77i 0.377717 0.407475i
\(622\) 0 0
\(623\) −6306.46 −0.405559
\(624\) 0 0
\(625\) 3904.37 0.249880
\(626\) 0 0
\(627\) 1211.65 + 4303.81i 0.0771748 + 0.274127i
\(628\) 0 0
\(629\) 12589.6i 0.798062i
\(630\) 0 0
\(631\) 3224.67i 0.203442i −0.994813 0.101721i \(-0.967565\pi\)
0.994813 0.101721i \(-0.0324350\pi\)
\(632\) 0 0
\(633\) 7491.64 + 26610.5i 0.470404 + 1.67089i
\(634\) 0 0
\(635\) 5616.91 0.351024
\(636\) 0 0
\(637\) −20.1380 −0.00125258
\(638\) 0 0
\(639\) −12615.2 + 7714.54i −0.780985 + 0.477594i
\(640\) 0 0
\(641\) 13243.1i 0.816025i −0.912976 0.408013i \(-0.866222\pi\)
0.912976 0.408013i \(-0.133778\pi\)
\(642\) 0 0
\(643\) 17030.3i 1.04449i −0.852795 0.522246i \(-0.825094\pi\)
0.852795 0.522246i \(-0.174906\pi\)
\(644\) 0 0
\(645\) 13817.6 3890.07i 0.843517 0.237475i
\(646\) 0 0
\(647\) −29077.9 −1.76688 −0.883438 0.468548i \(-0.844777\pi\)
−0.883438 + 0.468548i \(0.844777\pi\)
\(648\) 0 0
\(649\) −5811.93 −0.351523
\(650\) 0 0
\(651\) −8872.50 + 2497.87i −0.534164 + 0.150383i
\(652\) 0 0
\(653\) 14617.7i 0.876008i 0.898973 + 0.438004i \(0.144315\pi\)
−0.898973 + 0.438004i \(0.855685\pi\)
\(654\) 0 0
\(655\) 3228.24i 0.192577i
\(656\) 0 0
\(657\) 4481.83 2740.76i 0.266138 0.162751i
\(658\) 0 0
\(659\) −3426.04 −0.202518 −0.101259 0.994860i \(-0.532287\pi\)
−0.101259 + 0.994860i \(0.532287\pi\)
\(660\) 0 0
\(661\) 9256.76 0.544699 0.272350 0.962198i \(-0.412199\pi\)
0.272350 + 0.962198i \(0.412199\pi\)
\(662\) 0 0
\(663\) 3.40423 + 12.0919i 0.000199411 + 0.000708312i
\(664\) 0 0
\(665\) 1068.83i 0.0623269i
\(666\) 0 0
\(667\) 8798.63i 0.510771i
\(668\) 0 0
\(669\) 3813.79 + 13546.7i 0.220403 + 0.782877i
\(670\) 0 0
\(671\) 16170.1 0.930312
\(672\) 0 0
\(673\) 26622.1 1.52482 0.762412 0.647092i \(-0.224015\pi\)
0.762412 + 0.647092i \(0.224015\pi\)
\(674\) 0 0
\(675\) 8639.84 9320.51i 0.492663 0.531476i
\(676\) 0 0
\(677\) 23048.4i 1.30845i −0.756300 0.654225i \(-0.772995\pi\)
0.756300 0.654225i \(-0.227005\pi\)
\(678\) 0 0
\(679\) 4164.54i 0.235376i
\(680\) 0 0
\(681\) −11680.3 + 3288.36i −0.657256 + 0.185037i
\(682\) 0 0
\(683\) 27884.8 1.56220 0.781100 0.624406i \(-0.214659\pi\)
0.781100 + 0.624406i \(0.214659\pi\)
\(684\) 0 0
\(685\) −11681.9 −0.651593
\(686\) 0 0
\(687\) 696.749 196.155i 0.0386938 0.0108934i
\(688\) 0 0
\(689\) 33.0821i 0.00182921i
\(690\) 0 0
\(691\) 8029.81i 0.442067i −0.975266 0.221034i \(-0.929057\pi\)
0.975266 0.221034i \(-0.0709431\pi\)
\(692\) 0 0
\(693\) 2332.61 + 3814.39i 0.127862 + 0.209086i
\(694\) 0 0
\(695\) −4381.22 −0.239121
\(696\) 0 0
\(697\) 14022.3 0.762029
\(698\) 0 0
\(699\) 9.96938 + 35.4115i 0.000539452 + 0.00191615i
\(700\) 0 0
\(701\) 19827.7i 1.06830i 0.845388 + 0.534152i \(0.179369\pi\)
−0.845388 + 0.534152i \(0.820631\pi\)
\(702\) 0 0
\(703\) 10478.7i 0.562179i
\(704\) 0 0
\(705\) −3540.38 12575.5i −0.189132 0.671803i
\(706\) 0 0
\(707\) 5581.32 0.296898
\(708\) 0 0
\(709\) −5444.68 −0.288405 −0.144202 0.989548i \(-0.546062\pi\)
−0.144202 + 0.989548i \(0.546062\pi\)
\(710\) 0 0
\(711\) 4343.80 + 7103.20i 0.229121 + 0.374671i
\(712\) 0 0
\(713\) 18359.5i 0.964330i
\(714\) 0 0
\(715\) 10.7284i 0.000561147i
\(716\) 0 0
\(717\) 4452.89 1253.62i 0.231933 0.0652960i
\(718\) 0 0
\(719\) 4118.06 0.213599 0.106799 0.994281i \(-0.465940\pi\)
0.106799 + 0.994281i \(0.465940\pi\)
\(720\) 0 0
\(721\) −509.435 −0.0263140
\(722\) 0 0
\(723\) 6372.08 1793.93i 0.327774 0.0922779i
\(724\) 0 0
\(725\) 13005.2i 0.666208i
\(726\) 0 0
\(727\) 26169.4i 1.33504i 0.744594 + 0.667518i \(0.232643\pi\)
−0.744594 + 0.667518i \(0.767357\pi\)
\(728\) 0 0
\(729\) −1489.78 19626.5i −0.0756889 0.997131i
\(730\) 0 0
\(731\) −17409.0 −0.880840
\(732\) 0 0
\(733\) 28634.0 1.44286 0.721432 0.692485i \(-0.243484\pi\)
0.721432 + 0.692485i \(0.243484\pi\)
\(734\) 0 0
\(735\) 2543.69 + 9035.27i 0.127654 + 0.453430i
\(736\) 0 0
\(737\) 11615.4i 0.580539i
\(738\) 0 0
\(739\) 1607.43i 0.0800139i −0.999199 0.0400070i \(-0.987262\pi\)
0.999199 0.0400070i \(-0.0127380\pi\)
\(740\) 0 0
\(741\) −2.83344 10.0644i −0.000140471 0.000498957i
\(742\) 0 0
\(743\) 600.872 0.0296687 0.0148344 0.999890i \(-0.495278\pi\)
0.0148344 + 0.999890i \(0.495278\pi\)
\(744\) 0 0
\(745\) 18765.0 0.922815
\(746\) 0 0
\(747\) 1435.08 877.593i 0.0702904 0.0429845i
\(748\) 0 0
\(749\) 4074.47i 0.198769i
\(750\) 0 0
\(751\) 16005.7i 0.777707i −0.921300 0.388854i \(-0.872871\pi\)
0.921300 0.388854i \(-0.127129\pi\)
\(752\) 0 0
\(753\) 15169.5 4270.65i 0.734139 0.206682i
\(754\) 0 0
\(755\) 8655.67 0.417235
\(756\) 0 0
\(757\) 16731.0 0.803300 0.401650 0.915793i \(-0.368437\pi\)
0.401650 + 0.915793i \(0.368437\pi\)
\(758\) 0 0
\(759\) 8572.38 2413.38i 0.409958 0.115415i
\(760\) 0 0
\(761\) 2885.92i 0.137470i −0.997635 0.0687349i \(-0.978104\pi\)
0.997635 0.0687349i \(-0.0218963\pi\)
\(762\) 0 0
\(763\) 10954.4i 0.519758i
\(764\) 0 0
\(765\) 4995.26 3054.74i 0.236083 0.144372i
\(766\) 0 0
\(767\) 13.5912 0.000639830
\(768\) 0 0
\(769\) 5456.60 0.255878 0.127939 0.991782i \(-0.459164\pi\)
0.127939 + 0.991782i \(0.459164\pi\)
\(770\) 0 0
\(771\) 3640.33 + 12930.6i 0.170043 + 0.603998i
\(772\) 0 0
\(773\) 1995.30i 0.0928407i 0.998922 + 0.0464204i \(0.0147814\pi\)
−0.998922 + 0.0464204i \(0.985219\pi\)
\(774\) 0 0
\(775\) 27137.0i 1.25779i
\(776\) 0 0
\(777\) 2839.66 + 10086.5i 0.131110 + 0.465705i
\(778\) 0 0
\(779\) −11671.2 −0.536796
\(780\) 0 0
\(781\) −15315.7 −0.701714
\(782\) 0 0
\(783\) 14771.5 + 13692.8i 0.674191 + 0.624954i
\(784\) 0 0
\(785\) 4183.46i 0.190209i
\(786\) 0 0
\(787\) 38385.1i 1.73860i −0.494282 0.869302i \(-0.664569\pi\)
0.494282 0.869302i \(-0.335431\pi\)
\(788\) 0 0
\(789\) −27230.4 + 7666.14i −1.22868 + 0.345909i
\(790\) 0 0
\(791\) 6244.45 0.280692
\(792\) 0 0
\(793\) −37.8137 −0.00169332
\(794\) 0 0
\(795\) 14842.9 4178.70i 0.662166 0.186419i
\(796\) 0 0
\(797\) 27449.4i 1.21996i 0.792417 + 0.609980i \(0.208822\pi\)
−0.792417 + 0.609980i \(0.791178\pi\)
\(798\) 0 0
\(799\) 15844.0i 0.701529i
\(800\) 0 0
\(801\) 15001.9 + 24531.9i 0.661756 + 1.08214i
\(802\) 0 0
\(803\) 5441.24 0.239125
\(804\) 0 0
\(805\) −2128.91 −0.0932100
\(806\) 0 0
\(807\) −2902.69 10310.4i −0.126617 0.449745i
\(808\) 0 0
\(809\) 430.948i 0.0187285i 0.999956 + 0.00936424i \(0.00298077\pi\)
−0.999956 + 0.00936424i \(0.997019\pi\)
\(810\) 0 0
\(811\) 30047.5i 1.30100i 0.759506 + 0.650500i \(0.225441\pi\)
−0.759506 + 0.650500i \(0.774559\pi\)
\(812\) 0 0
\(813\) 8232.94 + 29243.6i 0.355156 + 1.26152i
\(814\) 0 0
\(815\) −11653.0 −0.500841
\(816\) 0 0
\(817\) 14490.0 0.620490
\(818\) 0 0
\(819\) −5.45480 8.91995i −0.000232730 0.000380572i
\(820\) 0 0
\(821\) 27225.8i 1.15736i −0.815556 0.578678i \(-0.803569\pi\)
0.815556 0.578678i \(-0.196431\pi\)
\(822\) 0 0
\(823\) 41196.3i 1.74485i 0.488748 + 0.872425i \(0.337454\pi\)
−0.488748 + 0.872425i \(0.662546\pi\)
\(824\) 0 0
\(825\) 12670.8 3567.20i 0.534715 0.150538i
\(826\) 0 0
\(827\) 9322.06 0.391971 0.195985 0.980607i \(-0.437209\pi\)
0.195985 + 0.980607i \(0.437209\pi\)
\(828\) 0 0
\(829\) 39860.8 1.66999 0.834996 0.550256i \(-0.185470\pi\)
0.834996 + 0.550256i \(0.185470\pi\)
\(830\) 0 0
\(831\) 7443.95 2095.69i 0.310743 0.0874833i
\(832\) 0 0
\(833\) 11383.6i 0.473493i
\(834\) 0 0
\(835\) 19808.9i 0.820975i
\(836\) 0 0
\(837\) 30822.6 + 28571.7i 1.27286 + 1.17991i
\(838\) 0 0
\(839\) −8509.74 −0.350165 −0.175083 0.984554i \(-0.556019\pi\)
−0.175083 + 0.984554i \(0.556019\pi\)
\(840\) 0 0
\(841\) 3777.83 0.154899
\(842\) 0 0
\(843\) −9092.26 32295.9i −0.371476 1.31949i
\(844\) 0 0
\(845\) 12888.2i 0.524693i
\(846\) 0 0
\(847\) 3250.57i 0.131867i
\(848\) 0 0
\(849\) −7590.77 26962.6i −0.306849 1.08994i
\(850\) 0 0
\(851\) 20871.6 0.840740
\(852\) 0 0
\(853\) 37505.9 1.50548 0.752741 0.658317i \(-0.228731\pi\)
0.752741 + 0.658317i \(0.228731\pi\)
\(854\) 0 0
\(855\) −4157.69 + 2542.54i −0.166304 + 0.101700i
\(856\) 0 0
\(857\) 40647.9i 1.62019i 0.586296 + 0.810097i \(0.300585\pi\)
−0.586296 + 0.810097i \(0.699415\pi\)
\(858\) 0 0
\(859\) 14040.1i 0.557673i −0.960339 0.278836i \(-0.910051\pi\)
0.960339 0.278836i \(-0.0899487\pi\)
\(860\) 0 0
\(861\) −11234.4 + 3162.81i −0.444678 + 0.125190i
\(862\) 0 0
\(863\) −12519.6 −0.493826 −0.246913 0.969038i \(-0.579416\pi\)
−0.246913 + 0.969038i \(0.579416\pi\)
\(864\) 0 0
\(865\) 15816.2 0.621694
\(866\) 0 0
\(867\) 17738.1 4993.79i 0.694830 0.195615i
\(868\) 0 0
\(869\) 8623.76i 0.336641i
\(870\) 0 0
\(871\) 27.1625i 0.00105668i
\(872\) 0 0
\(873\) 16199.9 9906.68i 0.628045 0.384067i
\(874\) 0 0
\(875\) −7488.85 −0.289336
\(876\) 0 0
\(877\) −27995.4 −1.07792 −0.538962 0.842330i \(-0.681183\pi\)
−0.538962 + 0.842330i \(0.681183\pi\)
\(878\) 0 0
\(879\) 7605.11 + 27013.5i 0.291825 + 1.03657i
\(880\) 0 0
\(881\) 16934.0i 0.647583i 0.946128 + 0.323792i \(0.104958\pi\)
−0.946128 + 0.323792i \(0.895042\pi\)
\(882\) 0 0
\(883\) 5835.41i 0.222398i −0.993798 0.111199i \(-0.964531\pi\)
0.993798 0.111199i \(-0.0354691\pi\)
\(884\) 0 0
\(885\) −1716.75 6097.93i −0.0652066 0.231615i
\(886\) 0 0
\(887\) −12966.0 −0.490818 −0.245409 0.969420i \(-0.578922\pi\)
−0.245409 + 0.969420i \(0.578922\pi\)
\(888\) 0 0
\(889\) 5669.79 0.213902
\(890\) 0 0
\(891\) 9288.98 18147.5i 0.349262 0.682338i
\(892\) 0 0
\(893\) 13187.4i 0.494178i
\(894\) 0 0
\(895\) 19635.9i 0.733357i
\(896\) 0 0
\(897\) −20.0465 + 5.64368i −0.000746191 + 0.000210075i
\(898\) 0 0
\(899\) −43007.8 −1.59554
\(900\) 0 0
\(901\) −18700.7 −0.691465
\(902\) 0 0
\(903\) 13947.7 3926.69i 0.514009 0.144709i
\(904\) 0 0
\(905\) 10816.7i 0.397304i
\(906\) 0 0
\(907\) 14718.0i 0.538812i 0.963027 + 0.269406i \(0.0868273\pi\)
−0.963027 + 0.269406i \(0.913173\pi\)
\(908\) 0 0
\(909\) −13276.9 21711.1i −0.484453 0.792202i
\(910\) 0 0
\(911\) −38859.8 −1.41326 −0.706632 0.707581i \(-0.749786\pi\)
−0.706632 + 0.707581i \(0.749786\pi\)
\(912\) 0 0
\(913\) 1742.29 0.0631559
\(914\) 0 0
\(915\) 4776.37 + 16965.8i 0.172570 + 0.612975i
\(916\) 0 0
\(917\) 3258.63i 0.117349i
\(918\) 0 0
\(919\) 38295.3i 1.37459i −0.726380 0.687294i \(-0.758799\pi\)
0.726380 0.687294i \(-0.241201\pi\)
\(920\) 0 0
\(921\) −5066.61 17996.8i −0.181271 0.643880i
\(922\) 0 0
\(923\) 35.8157 0.00127724
\(924\) 0 0
\(925\) 30850.2 1.09659
\(926\) 0 0
\(927\) 1211.85 + 1981.68i 0.0429369 + 0.0702125i
\(928\) 0 0
\(929\) 43777.6i 1.54607i 0.634364 + 0.773034i \(0.281262\pi\)
−0.634364 + 0.773034i \(0.718738\pi\)
\(930\) 0 0
\(931\) 9474.92i 0.333542i
\(932\) 0 0
\(933\) 39486.3 11116.6i 1.38556 0.390075i
\(934\) 0 0
\(935\) 6064.58 0.212121
\(936\) 0 0
\(937\) −28384.6 −0.989631 −0.494816 0.868998i \(-0.664764\pi\)
−0.494816 + 0.868998i \(0.664764\pi\)
\(938\) 0 0
\(939\) 42517.4 11969.9i 1.47764 0.415999i
\(940\) 0 0
\(941\) 54057.6i 1.87272i −0.351044 0.936359i \(-0.614173\pi\)
0.351044 0.936359i \(-0.385827\pi\)
\(942\) 0 0
\(943\) 23246.8i 0.802780i
\(944\) 0 0
\(945\) −3313.08 + 3574.10i −0.114047 + 0.123032i
\(946\) 0 0
\(947\) −54179.3 −1.85912 −0.929562 0.368665i \(-0.879815\pi\)
−0.929562 + 0.368665i \(0.879815\pi\)
\(948\) 0 0
\(949\) −12.7243 −0.000435247
\(950\) 0 0
\(951\) −10352.6 36772.7i −0.353003 1.25388i
\(952\) 0 0
\(953\) 2964.61i 0.100769i −0.998730 0.0503846i \(-0.983955\pi\)
0.998730 0.0503846i \(-0.0160447\pi\)
\(954\) 0 0
\(955\) 13975.2i 0.473537i
\(956\) 0 0
\(957\) 5653.44 + 20081.2i 0.190961 + 0.678299i
\(958\) 0 0
\(959\) −11791.8 −0.397058
\(960\) 0 0
\(961\) −59950.2 −2.01236
\(962\) 0 0
\(963\) −15849.5 + 9692.41i −0.530367 + 0.324334i
\(964\) 0 0
\(965\) 14937.0i 0.498278i
\(966\) 0 0
\(967\) 12087.0i 0.401956i 0.979596 + 0.200978i \(0.0644119\pi\)
−0.979596 + 0.200978i \(0.935588\pi\)
\(968\) 0 0
\(969\) 5689.25 1601.69i 0.188612 0.0530998i
\(970\) 0 0
\(971\) −34336.2 −1.13481 −0.567405 0.823439i \(-0.692053\pi\)
−0.567405 + 0.823439i \(0.692053\pi\)
\(972\) 0 0
\(973\) −4422.46 −0.145712
\(974\) 0 0
\(975\) −29.6306 + 8.34188i −0.000973271 + 0.000274004i
\(976\) 0 0
\(977\) 12445.9i 0.407555i 0.979017 + 0.203777i \(0.0653218\pi\)
−0.979017 + 0.203777i \(0.934678\pi\)
\(978\) 0 0
\(979\) 29783.3i 0.972298i
\(980\) 0 0
\(981\) −42612.1 + 26058.5i −1.38685 + 0.848097i
\(982\) 0 0
\(983\) −50850.7 −1.64994 −0.824968 0.565180i \(-0.808807\pi\)
−0.824968 + 0.565180i \(0.808807\pi\)
\(984\) 0 0
\(985\) 12844.7 0.415498
\(986\) 0 0
\(987\) −3573.71 12693.9i −0.115251 0.409373i
\(988\) 0 0
\(989\) 28861.3i 0.927945i
\(990\) 0 0
\(991\) 16973.3i 0.544071i −0.962287 0.272035i \(-0.912303\pi\)
0.962287 0.272035i \(-0.0876968\pi\)
\(992\) 0 0
\(993\) −6123.23 21749.9i −0.195684 0.695077i
\(994\) 0 0
\(995\) 31555.6 1.00541
\(996\) 0 0
\(997\) −20434.2 −0.649104 −0.324552 0.945868i \(-0.605213\pi\)
−0.324552 + 0.945868i \(0.605213\pi\)
\(998\) 0 0
\(999\) 32481.2 35040.2i 1.02869 1.10973i
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 384.4.c.b.383.11 yes 12
3.2 odd 2 384.4.c.c.383.1 yes 12
4.3 odd 2 384.4.c.c.383.2 yes 12
8.3 odd 2 384.4.c.a.383.11 12
8.5 even 2 384.4.c.d.383.2 yes 12
12.11 even 2 inner 384.4.c.b.383.12 yes 12
16.3 odd 4 768.4.f.h.383.5 12
16.5 even 4 768.4.f.g.383.5 12
16.11 odd 4 768.4.f.e.383.8 12
16.13 even 4 768.4.f.f.383.8 12
24.5 odd 2 384.4.c.a.383.12 yes 12
24.11 even 2 384.4.c.d.383.1 yes 12
48.5 odd 4 768.4.f.h.383.6 12
48.11 even 4 768.4.f.f.383.7 12
48.29 odd 4 768.4.f.e.383.7 12
48.35 even 4 768.4.f.g.383.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.c.a.383.11 12 8.3 odd 2
384.4.c.a.383.12 yes 12 24.5 odd 2
384.4.c.b.383.11 yes 12 1.1 even 1 trivial
384.4.c.b.383.12 yes 12 12.11 even 2 inner
384.4.c.c.383.1 yes 12 3.2 odd 2
384.4.c.c.383.2 yes 12 4.3 odd 2
384.4.c.d.383.1 yes 12 24.11 even 2
384.4.c.d.383.2 yes 12 8.5 even 2
768.4.f.e.383.7 12 48.29 odd 4
768.4.f.e.383.8 12 16.11 odd 4
768.4.f.f.383.7 12 48.11 even 4
768.4.f.f.383.8 12 16.13 even 4
768.4.f.g.383.5 12 16.5 even 4
768.4.f.g.383.6 12 48.35 even 4
768.4.f.h.383.5 12 16.3 odd 4
768.4.f.h.383.6 12 48.5 odd 4