Properties

Label 768.4.f.c.383.3
Level $768$
Weight $4$
Character 768.383
Analytic conductor $45.313$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\), degree \(1\), not 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: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.3
Root \(1.40126 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 768.383
Dual form 768.4.f.c.383.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.87298 + 3.46410i) q^{3} -8.94427 q^{5} +7.74597i q^{7} +(3.00000 - 26.8328i) q^{9} +O(q^{10})\) \(q+(-3.87298 + 3.46410i) q^{3} -8.94427 q^{5} +7.74597i q^{7} +(3.00000 - 26.8328i) q^{9} +34.6410i q^{11} -10.0000i q^{13} +(34.6410 - 30.9839i) q^{15} -35.7771i q^{17} -69.7137 q^{19} +(-26.8328 - 30.0000i) q^{21} -96.9948 q^{23} -45.0000 q^{25} +(81.3327 + 114.315i) q^{27} -152.053 q^{29} -224.633i q^{31} +(-120.000 - 134.164i) q^{33} -69.2820i q^{35} +130.000i q^{37} +(34.6410 + 38.7298i) q^{39} -125.220i q^{41} +224.633 q^{43} +(-26.8328 + 240.000i) q^{45} +193.990 q^{47} +283.000 q^{49} +(123.935 + 138.564i) q^{51} -545.601 q^{53} -309.839i q^{55} +(270.000 - 241.495i) q^{57} +173.205i q^{59} -442.000i q^{61} +(207.846 + 23.2379i) q^{63} +89.4427i q^{65} +735.867 q^{67} +(375.659 - 336.000i) q^{69} +1039.23 q^{71} -410.000 q^{73} +(174.284 - 155.885i) q^{75} -268.328 q^{77} +85.2056i q^{79} +(-711.000 - 160.997i) q^{81} +1254.00i q^{83} +320.000i q^{85} +(588.897 - 526.726i) q^{87} +840.762i q^{89} +77.4597 q^{91} +(778.152 + 870.000i) q^{93} +623.538 q^{95} +770.000 q^{97} +(929.516 + 103.923i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 360 q^{25} - 960 q^{33} + 2264 q^{49} + 2160 q^{57} - 3280 q^{73} - 5688 q^{81} + 6160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) −3.87298 + 3.46410i −0.745356 + 0.666667i
\(4\) 0 0
\(5\) −8.94427 −0.800000 −0.400000 0.916515i \(-0.630990\pi\)
−0.400000 + 0.916515i \(0.630990\pi\)
\(6\) 0 0
\(7\) 7.74597i 0.418243i 0.977890 + 0.209121i \(0.0670604\pi\)
−0.977890 + 0.209121i \(0.932940\pi\)
\(8\) 0 0
\(9\) 3.00000 26.8328i 0.111111 0.993808i
\(10\) 0 0
\(11\) 34.6410i 0.949514i 0.880117 + 0.474757i \(0.157464\pi\)
−0.880117 + 0.474757i \(0.842536\pi\)
\(12\) 0 0
\(13\) 10.0000i 0.213346i −0.994294 0.106673i \(-0.965980\pi\)
0.994294 0.106673i \(-0.0340198\pi\)
\(14\) 0 0
\(15\) 34.6410 30.9839i 0.596285 0.533333i
\(16\) 0 0
\(17\) 35.7771i 0.510425i −0.966885 0.255212i \(-0.917855\pi\)
0.966885 0.255212i \(-0.0821454\pi\)
\(18\) 0 0
\(19\) −69.7137 −0.841759 −0.420879 0.907117i \(-0.638278\pi\)
−0.420879 + 0.907117i \(0.638278\pi\)
\(20\) 0 0
\(21\) −26.8328 30.0000i −0.278829 0.311740i
\(22\) 0 0
\(23\) −96.9948 −0.879340 −0.439670 0.898159i \(-0.644905\pi\)
−0.439670 + 0.898159i \(0.644905\pi\)
\(24\) 0 0
\(25\) −45.0000 −0.360000
\(26\) 0 0
\(27\) 81.3327 + 114.315i 0.579721 + 0.814815i
\(28\) 0 0
\(29\) −152.053 −0.973637 −0.486818 0.873503i \(-0.661843\pi\)
−0.486818 + 0.873503i \(0.661843\pi\)
\(30\) 0 0
\(31\) 224.633i 1.30146i −0.759309 0.650730i \(-0.774463\pi\)
0.759309 0.650730i \(-0.225537\pi\)
\(32\) 0 0
\(33\) −120.000 134.164i −0.633010 0.707726i
\(34\) 0 0
\(35\) 69.2820i 0.334594i
\(36\) 0 0
\(37\) 130.000i 0.577618i 0.957387 + 0.288809i \(0.0932593\pi\)
−0.957387 + 0.288809i \(0.906741\pi\)
\(38\) 0 0
\(39\) 34.6410 + 38.7298i 0.142231 + 0.159019i
\(40\) 0 0
\(41\) 125.220i 0.476977i −0.971145 0.238488i \(-0.923348\pi\)
0.971145 0.238488i \(-0.0766519\pi\)
\(42\) 0 0
\(43\) 224.633 0.796656 0.398328 0.917243i \(-0.369591\pi\)
0.398328 + 0.917243i \(0.369591\pi\)
\(44\) 0 0
\(45\) −26.8328 + 240.000i −0.0888889 + 0.795046i
\(46\) 0 0
\(47\) 193.990 0.602049 0.301025 0.953616i \(-0.402671\pi\)
0.301025 + 0.953616i \(0.402671\pi\)
\(48\) 0 0
\(49\) 283.000 0.825073
\(50\) 0 0
\(51\) 123.935 + 138.564i 0.340283 + 0.380448i
\(52\) 0 0
\(53\) −545.601 −1.41404 −0.707019 0.707195i \(-0.749960\pi\)
−0.707019 + 0.707195i \(0.749960\pi\)
\(54\) 0 0
\(55\) 309.839i 0.759612i
\(56\) 0 0
\(57\) 270.000 241.495i 0.627410 0.561173i
\(58\) 0 0
\(59\) 173.205i 0.382193i 0.981571 + 0.191096i \(0.0612043\pi\)
−0.981571 + 0.191096i \(0.938796\pi\)
\(60\) 0 0
\(61\) 442.000i 0.927743i −0.885903 0.463871i \(-0.846460\pi\)
0.885903 0.463871i \(-0.153540\pi\)
\(62\) 0 0
\(63\) 207.846 + 23.2379i 0.415653 + 0.0464714i
\(64\) 0 0
\(65\) 89.4427i 0.170677i
\(66\) 0 0
\(67\) 735.867 1.34180 0.670899 0.741549i \(-0.265908\pi\)
0.670899 + 0.741549i \(0.265908\pi\)
\(68\) 0 0
\(69\) 375.659 336.000i 0.655421 0.586227i
\(70\) 0 0
\(71\) 1039.23 1.73710 0.868549 0.495603i \(-0.165053\pi\)
0.868549 + 0.495603i \(0.165053\pi\)
\(72\) 0 0
\(73\) −410.000 −0.657354 −0.328677 0.944442i \(-0.606603\pi\)
−0.328677 + 0.944442i \(0.606603\pi\)
\(74\) 0 0
\(75\) 174.284 155.885i 0.268328 0.240000i
\(76\) 0 0
\(77\) −268.328 −0.397128
\(78\) 0 0
\(79\) 85.2056i 0.121347i 0.998158 + 0.0606733i \(0.0193248\pi\)
−0.998158 + 0.0606733i \(0.980675\pi\)
\(80\) 0 0
\(81\) −711.000 160.997i −0.975309 0.220846i
\(82\) 0 0
\(83\) 1254.00i 1.65837i 0.558973 + 0.829186i \(0.311196\pi\)
−0.558973 + 0.829186i \(0.688804\pi\)
\(84\) 0 0
\(85\) 320.000i 0.408340i
\(86\) 0 0
\(87\) 588.897 526.726i 0.725706 0.649091i
\(88\) 0 0
\(89\) 840.762i 1.00135i 0.865634 + 0.500677i \(0.166916\pi\)
−0.865634 + 0.500677i \(0.833084\pi\)
\(90\) 0 0
\(91\) 77.4597 0.0892305
\(92\) 0 0
\(93\) 778.152 + 870.000i 0.867641 + 0.970052i
\(94\) 0 0
\(95\) 623.538 0.673407
\(96\) 0 0
\(97\) 770.000 0.805996 0.402998 0.915201i \(-0.367968\pi\)
0.402998 + 0.915201i \(0.367968\pi\)
\(98\) 0 0
\(99\) 929.516 + 103.923i 0.943635 + 0.105502i
\(100\) 0 0
\(101\) 1493.69 1.47156 0.735782 0.677218i \(-0.236815\pi\)
0.735782 + 0.677218i \(0.236815\pi\)
\(102\) 0 0
\(103\) 1355.54i 1.29675i −0.761319 0.648377i \(-0.775448\pi\)
0.761319 0.648377i \(-0.224552\pi\)
\(104\) 0 0
\(105\) 240.000 + 268.328i 0.223063 + 0.249392i
\(106\) 0 0
\(107\) 644.323i 0.582141i 0.956702 + 0.291070i \(0.0940114\pi\)
−0.956702 + 0.291070i \(0.905989\pi\)
\(108\) 0 0
\(109\) 1066.00i 0.936737i −0.883533 0.468368i \(-0.844842\pi\)
0.883533 0.468368i \(-0.155158\pi\)
\(110\) 0 0
\(111\) −450.333 503.488i −0.385079 0.430531i
\(112\) 0 0
\(113\) 1037.54i 0.863745i 0.901935 + 0.431872i \(0.142147\pi\)
−0.901935 + 0.431872i \(0.857853\pi\)
\(114\) 0 0
\(115\) 867.548 0.703472
\(116\) 0 0
\(117\) −268.328 30.0000i −0.212025 0.0237051i
\(118\) 0 0
\(119\) 277.128 0.213481
\(120\) 0 0
\(121\) 131.000 0.0984222
\(122\) 0 0
\(123\) 433.774 + 484.974i 0.317985 + 0.355518i
\(124\) 0 0
\(125\) 1520.53 1.08800
\(126\) 0 0
\(127\) 1835.79i 1.28268i −0.767257 0.641340i \(-0.778379\pi\)
0.767257 0.641340i \(-0.221621\pi\)
\(128\) 0 0
\(129\) −870.000 + 778.152i −0.593792 + 0.531104i
\(130\) 0 0
\(131\) 450.333i 0.300350i 0.988659 + 0.150175i \(0.0479836\pi\)
−0.988659 + 0.150175i \(0.952016\pi\)
\(132\) 0 0
\(133\) 540.000i 0.352060i
\(134\) 0 0
\(135\) −727.461 1022.47i −0.463777 0.651852i
\(136\) 0 0
\(137\) 89.4427i 0.0557782i 0.999611 + 0.0278891i \(0.00887852\pi\)
−0.999611 + 0.0278891i \(0.991121\pi\)
\(138\) 0 0
\(139\) 1959.73 1.19584 0.597921 0.801555i \(-0.295994\pi\)
0.597921 + 0.801555i \(0.295994\pi\)
\(140\) 0 0
\(141\) −751.319 + 672.000i −0.448741 + 0.401366i
\(142\) 0 0
\(143\) 346.410 0.202575
\(144\) 0 0
\(145\) 1360.00 0.778909
\(146\) 0 0
\(147\) −1096.05 + 980.341i −0.614973 + 0.550049i
\(148\) 0 0
\(149\) −1618.91 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(150\) 0 0
\(151\) 565.456i 0.304743i 0.988323 + 0.152371i \(0.0486909\pi\)
−0.988323 + 0.152371i \(0.951309\pi\)
\(152\) 0 0
\(153\) −960.000 107.331i −0.507264 0.0567138i
\(154\) 0 0
\(155\) 2009.18i 1.04117i
\(156\) 0 0
\(157\) 730.000i 0.371085i −0.982636 0.185542i \(-0.940596\pi\)
0.982636 0.185542i \(-0.0594042\pi\)
\(158\) 0 0
\(159\) 2113.10 1890.02i 1.05396 0.942692i
\(160\) 0 0
\(161\) 751.319i 0.367778i
\(162\) 0 0
\(163\) −255.617 −0.122831 −0.0614155 0.998112i \(-0.519561\pi\)
−0.0614155 + 0.998112i \(0.519561\pi\)
\(164\) 0 0
\(165\) 1073.31 + 1200.00i 0.506408 + 0.566181i
\(166\) 0 0
\(167\) −13.8564 −0.00642060 −0.00321030 0.999995i \(-0.501022\pi\)
−0.00321030 + 0.999995i \(0.501022\pi\)
\(168\) 0 0
\(169\) 2097.00 0.954483
\(170\) 0 0
\(171\) −209.141 + 1870.61i −0.0935288 + 0.836547i
\(172\) 0 0
\(173\) −1118.03 −0.491344 −0.245672 0.969353i \(-0.579009\pi\)
−0.245672 + 0.969353i \(0.579009\pi\)
\(174\) 0 0
\(175\) 348.569i 0.150567i
\(176\) 0 0
\(177\) −600.000 670.820i −0.254795 0.284870i
\(178\) 0 0
\(179\) 1351.00i 0.564125i −0.959396 0.282063i \(-0.908981\pi\)
0.959396 0.282063i \(-0.0910186\pi\)
\(180\) 0 0
\(181\) 1262.00i 0.518253i −0.965843 0.259126i \(-0.916565\pi\)
0.965843 0.259126i \(-0.0834346\pi\)
\(182\) 0 0
\(183\) 1531.13 + 1711.86i 0.618495 + 0.691499i
\(184\) 0 0
\(185\) 1162.76i 0.462094i
\(186\) 0 0
\(187\) 1239.35 0.484656
\(188\) 0 0
\(189\) −885.483 + 630.000i −0.340791 + 0.242464i
\(190\) 0 0
\(191\) 2771.28 1.04986 0.524929 0.851146i \(-0.324092\pi\)
0.524929 + 0.851146i \(0.324092\pi\)
\(192\) 0 0
\(193\) −190.000 −0.0708627 −0.0354313 0.999372i \(-0.511281\pi\)
−0.0354313 + 0.999372i \(0.511281\pi\)
\(194\) 0 0
\(195\) −309.839 346.410i −0.113785 0.127215i
\(196\) 0 0
\(197\) 2137.68 0.773114 0.386557 0.922266i \(-0.373664\pi\)
0.386557 + 0.922266i \(0.373664\pi\)
\(198\) 0 0
\(199\) 255.617i 0.0910563i 0.998963 + 0.0455281i \(0.0144971\pi\)
−0.998963 + 0.0455281i \(0.985503\pi\)
\(200\) 0 0
\(201\) −2850.00 + 2549.12i −1.00012 + 0.894532i
\(202\) 0 0
\(203\) 1177.79i 0.407217i
\(204\) 0 0
\(205\) 1120.00i 0.381581i
\(206\) 0 0
\(207\) −290.985 + 2602.64i −0.0977045 + 0.873895i
\(208\) 0 0
\(209\) 2414.95i 0.799262i
\(210\) 0 0
\(211\) 549.964 0.179436 0.0897181 0.995967i \(-0.471403\pi\)
0.0897181 + 0.995967i \(0.471403\pi\)
\(212\) 0 0
\(213\) −4024.92 + 3600.00i −1.29476 + 1.15807i
\(214\) 0 0
\(215\) −2009.18 −0.637325
\(216\) 0 0
\(217\) 1740.00 0.544327
\(218\) 0 0
\(219\) 1587.92 1420.28i 0.489963 0.438236i
\(220\) 0 0
\(221\) −357.771 −0.108897
\(222\) 0 0
\(223\) 472.504i 0.141889i −0.997480 0.0709444i \(-0.977399\pi\)
0.997480 0.0709444i \(-0.0226013\pi\)
\(224\) 0 0
\(225\) −135.000 + 1207.48i −0.0400000 + 0.357771i
\(226\) 0 0
\(227\) 505.759i 0.147878i 0.997263 + 0.0739392i \(0.0235571\pi\)
−0.997263 + 0.0739392i \(0.976443\pi\)
\(228\) 0 0
\(229\) 4094.00i 1.18139i −0.806894 0.590697i \(-0.798853\pi\)
0.806894 0.590697i \(-0.201147\pi\)
\(230\) 0 0
\(231\) 1039.23 929.516i 0.296001 0.264752i
\(232\) 0 0
\(233\) 5277.12i 1.48376i −0.670534 0.741879i \(-0.733935\pi\)
0.670534 0.741879i \(-0.266065\pi\)
\(234\) 0 0
\(235\) −1735.10 −0.481639
\(236\) 0 0
\(237\) −295.161 330.000i −0.0808977 0.0904464i
\(238\) 0 0
\(239\) −5681.13 −1.53758 −0.768790 0.639502i \(-0.779141\pi\)
−0.768790 + 0.639502i \(0.779141\pi\)
\(240\) 0 0
\(241\) −1198.00 −0.320207 −0.160104 0.987100i \(-0.551183\pi\)
−0.160104 + 0.987100i \(0.551183\pi\)
\(242\) 0 0
\(243\) 3311.40 1839.44i 0.874183 0.485597i
\(244\) 0 0
\(245\) −2531.23 −0.660058
\(246\) 0 0
\(247\) 697.137i 0.179586i
\(248\) 0 0
\(249\) −4344.00 4856.74i −1.10558 1.23608i
\(250\) 0 0
\(251\) 4260.84i 1.07148i −0.844382 0.535741i \(-0.820032\pi\)
0.844382 0.535741i \(-0.179968\pi\)
\(252\) 0 0
\(253\) 3360.00i 0.834946i
\(254\) 0 0
\(255\) −1108.51 1239.35i −0.272226 0.304358i
\(256\) 0 0
\(257\) 3148.38i 0.764166i −0.924128 0.382083i \(-0.875207\pi\)
0.924128 0.382083i \(-0.124793\pi\)
\(258\) 0 0
\(259\) −1006.98 −0.241585
\(260\) 0 0
\(261\) −456.158 + 4080.00i −0.108182 + 0.967608i
\(262\) 0 0
\(263\) 4253.92 0.997368 0.498684 0.866784i \(-0.333817\pi\)
0.498684 + 0.866784i \(0.333817\pi\)
\(264\) 0 0
\(265\) 4880.00 1.13123
\(266\) 0 0
\(267\) −2912.48 3256.26i −0.667570 0.746366i
\(268\) 0 0
\(269\) −44.7214 −0.0101365 −0.00506823 0.999987i \(-0.501613\pi\)
−0.00506823 + 0.999987i \(0.501613\pi\)
\(270\) 0 0
\(271\) 8760.69i 1.96374i 0.189552 + 0.981871i \(0.439296\pi\)
−0.189552 + 0.981871i \(0.560704\pi\)
\(272\) 0 0
\(273\) −300.000 + 268.328i −0.0665085 + 0.0594870i
\(274\) 0 0
\(275\) 1558.85i 0.341825i
\(276\) 0 0
\(277\) 6350.00i 1.37738i −0.725055 0.688690i \(-0.758186\pi\)
0.725055 0.688690i \(-0.241814\pi\)
\(278\) 0 0
\(279\) −6027.54 673.899i −1.29340 0.144607i
\(280\) 0 0
\(281\) 5563.34i 1.18107i 0.807012 + 0.590535i \(0.201083\pi\)
−0.807012 + 0.590535i \(0.798917\pi\)
\(282\) 0 0
\(283\) −6777.72 −1.42365 −0.711826 0.702356i \(-0.752132\pi\)
−0.711826 + 0.702356i \(0.752132\pi\)
\(284\) 0 0
\(285\) −2414.95 + 2160.00i −0.501928 + 0.448938i
\(286\) 0 0
\(287\) 969.948 0.199492
\(288\) 0 0
\(289\) 3633.00 0.739467
\(290\) 0 0
\(291\) −2982.20 + 2667.36i −0.600754 + 0.537331i
\(292\) 0 0
\(293\) −652.932 −0.130187 −0.0650933 0.997879i \(-0.520735\pi\)
−0.0650933 + 0.997879i \(0.520735\pi\)
\(294\) 0 0
\(295\) 1549.19i 0.305754i
\(296\) 0 0
\(297\) −3960.00 + 2817.45i −0.773678 + 0.550454i
\(298\) 0 0
\(299\) 969.948i 0.187604i
\(300\) 0 0
\(301\) 1740.00i 0.333196i
\(302\) 0 0
\(303\) −5785.05 + 5174.31i −1.09684 + 0.981043i
\(304\) 0 0
\(305\) 3953.37i 0.742194i
\(306\) 0 0
\(307\) −1556.94 −0.289444 −0.144722 0.989472i \(-0.546229\pi\)
−0.144722 + 0.989472i \(0.546229\pi\)
\(308\) 0 0
\(309\) 4695.74 + 5250.00i 0.864503 + 0.966544i
\(310\) 0 0
\(311\) −3256.26 −0.593715 −0.296857 0.954922i \(-0.595939\pi\)
−0.296857 + 0.954922i \(0.595939\pi\)
\(312\) 0 0
\(313\) 7030.00 1.26952 0.634759 0.772710i \(-0.281099\pi\)
0.634759 + 0.772710i \(0.281099\pi\)
\(314\) 0 0
\(315\) −1859.03 207.846i −0.332522 0.0371771i
\(316\) 0 0
\(317\) 491.935 0.0871603 0.0435802 0.999050i \(-0.486124\pi\)
0.0435802 + 0.999050i \(0.486124\pi\)
\(318\) 0 0
\(319\) 5267.26i 0.924482i
\(320\) 0 0
\(321\) −2232.00 2495.45i −0.388094 0.433902i
\(322\) 0 0
\(323\) 2494.15i 0.429654i
\(324\) 0 0
\(325\) 450.000i 0.0768046i
\(326\) 0 0
\(327\) 3692.73 + 4128.60i 0.624491 + 0.698202i
\(328\) 0 0
\(329\) 1502.64i 0.251803i
\(330\) 0 0
\(331\) −4237.04 −0.703592 −0.351796 0.936077i \(-0.614429\pi\)
−0.351796 + 0.936077i \(0.614429\pi\)
\(332\) 0 0
\(333\) 3488.27 + 390.000i 0.574041 + 0.0641798i
\(334\) 0 0
\(335\) −6581.79 −1.07344
\(336\) 0 0
\(337\) 1490.00 0.240847 0.120424 0.992723i \(-0.461575\pi\)
0.120424 + 0.992723i \(0.461575\pi\)
\(338\) 0 0
\(339\) −3594.13 4018.36i −0.575830 0.643797i
\(340\) 0 0
\(341\) 7781.52 1.23576
\(342\) 0 0
\(343\) 4848.98i 0.763324i
\(344\) 0 0
\(345\) −3360.00 + 3005.28i −0.524337 + 0.468981i
\(346\) 0 0
\(347\) 1988.39i 0.307616i −0.988101 0.153808i \(-0.950846\pi\)
0.988101 0.153808i \(-0.0491537\pi\)
\(348\) 0 0
\(349\) 2074.00i 0.318105i −0.987270 0.159053i \(-0.949156\pi\)
0.987270 0.159053i \(-0.0508439\pi\)
\(350\) 0 0
\(351\) 1143.15 813.327i 0.173838 0.123681i
\(352\) 0 0
\(353\) 8658.06i 1.30544i 0.757597 + 0.652722i \(0.226373\pi\)
−0.757597 + 0.652722i \(0.773627\pi\)
\(354\) 0 0
\(355\) −9295.16 −1.38968
\(356\) 0 0
\(357\) −1073.31 + 960.000i −0.159120 + 0.142321i
\(358\) 0 0
\(359\) −8106.00 −1.19169 −0.595847 0.803098i \(-0.703184\pi\)
−0.595847 + 0.803098i \(0.703184\pi\)
\(360\) 0 0
\(361\) −1999.00 −0.291442
\(362\) 0 0
\(363\) −507.361 + 453.797i −0.0733596 + 0.0656148i
\(364\) 0 0
\(365\) 3667.15 0.525884
\(366\) 0 0
\(367\) 7893.14i 1.12267i 0.827590 + 0.561333i \(0.189711\pi\)
−0.827590 + 0.561333i \(0.810289\pi\)
\(368\) 0 0
\(369\) −3360.00 375.659i −0.474023 0.0529974i
\(370\) 0 0
\(371\) 4226.20i 0.591411i
\(372\) 0 0
\(373\) 4910.00i 0.681582i −0.940139 0.340791i \(-0.889305\pi\)
0.940139 0.340791i \(-0.110695\pi\)
\(374\) 0 0
\(375\) −5888.97 + 5267.26i −0.810947 + 0.725333i
\(376\) 0 0
\(377\) 1520.53i 0.207722i
\(378\) 0 0
\(379\) 3137.12 0.425179 0.212590 0.977142i \(-0.431810\pi\)
0.212590 + 0.977142i \(0.431810\pi\)
\(380\) 0 0
\(381\) 6359.38 + 7110.00i 0.855120 + 0.956053i
\(382\) 0 0
\(383\) 6207.67 0.828191 0.414095 0.910233i \(-0.364098\pi\)
0.414095 + 0.910233i \(0.364098\pi\)
\(384\) 0 0
\(385\) 2400.00 0.317702
\(386\) 0 0
\(387\) 673.899 6027.54i 0.0885174 0.791723i
\(388\) 0 0
\(389\) −9454.10 −1.23224 −0.616120 0.787652i \(-0.711297\pi\)
−0.616120 + 0.787652i \(0.711297\pi\)
\(390\) 0 0
\(391\) 3470.19i 0.448837i
\(392\) 0 0
\(393\) −1560.00 1744.13i −0.200233 0.223867i
\(394\) 0 0
\(395\) 762.102i 0.0970773i
\(396\) 0 0
\(397\) 10570.0i 1.33625i −0.744047 0.668127i \(-0.767096\pi\)
0.744047 0.668127i \(-0.232904\pi\)
\(398\) 0 0
\(399\) 1870.61 + 2091.41i 0.234706 + 0.262410i
\(400\) 0 0
\(401\) 1681.52i 0.209405i 0.994504 + 0.104702i \(0.0333890\pi\)
−0.994504 + 0.104702i \(0.966611\pi\)
\(402\) 0 0
\(403\) −2246.33 −0.277662
\(404\) 0 0
\(405\) 6359.38 + 1440.00i 0.780247 + 0.176677i
\(406\) 0 0
\(407\) −4503.33 −0.548457
\(408\) 0 0
\(409\) 3574.00 0.432085 0.216043 0.976384i \(-0.430685\pi\)
0.216043 + 0.976384i \(0.430685\pi\)
\(410\) 0 0
\(411\) −309.839 346.410i −0.0371854 0.0415746i
\(412\) 0 0
\(413\) −1341.64 −0.159849
\(414\) 0 0
\(415\) 11216.2i 1.32670i
\(416\) 0 0
\(417\) −7590.00 + 6788.70i −0.891328 + 0.797228i
\(418\) 0 0
\(419\) 15346.0i 1.78926i −0.446808 0.894630i \(-0.647439\pi\)
0.446808 0.894630i \(-0.352561\pi\)
\(420\) 0 0
\(421\) 3518.00i 0.407261i −0.979048 0.203630i \(-0.934726\pi\)
0.979048 0.203630i \(-0.0652741\pi\)
\(422\) 0 0
\(423\) 581.969 5205.29i 0.0668943 0.598321i
\(424\) 0 0
\(425\) 1609.97i 0.183753i
\(426\) 0 0
\(427\) 3423.72 0.388022
\(428\) 0 0
\(429\) −1341.64 + 1200.00i −0.150991 + 0.135050i
\(430\) 0 0
\(431\) −12886.5 −1.44018 −0.720091 0.693879i \(-0.755900\pi\)
−0.720091 + 0.693879i \(0.755900\pi\)
\(432\) 0 0
\(433\) 14450.0 1.60375 0.801874 0.597493i \(-0.203837\pi\)
0.801874 + 0.597493i \(0.203837\pi\)
\(434\) 0 0
\(435\) −5267.26 + 4711.18i −0.580565 + 0.519273i
\(436\) 0 0
\(437\) 6761.87 0.740192
\(438\) 0 0
\(439\) 15065.9i 1.63794i 0.573835 + 0.818971i \(0.305455\pi\)
−0.573835 + 0.818971i \(0.694545\pi\)
\(440\) 0 0
\(441\) 849.000 7593.69i 0.0916748 0.819964i
\(442\) 0 0
\(443\) 3041.48i 0.326197i −0.986610 0.163098i \(-0.947851\pi\)
0.986610 0.163098i \(-0.0521488\pi\)
\(444\) 0 0
\(445\) 7520.00i 0.801084i
\(446\) 0 0
\(447\) 6270.02 5608.08i 0.663450 0.593407i
\(448\) 0 0
\(449\) 14310.8i 1.50416i −0.659069 0.752082i \(-0.729049\pi\)
0.659069 0.752082i \(-0.270951\pi\)
\(450\) 0 0
\(451\) 4337.74 0.452896
\(452\) 0 0
\(453\) −1958.80 2190.00i −0.203162 0.227142i
\(454\) 0 0
\(455\) −692.820 −0.0713844
\(456\) 0 0
\(457\) 3430.00 0.351091 0.175546 0.984471i \(-0.443831\pi\)
0.175546 + 0.984471i \(0.443831\pi\)
\(458\) 0 0
\(459\) 4089.87 2909.85i 0.415902 0.295904i
\(460\) 0 0
\(461\) −3908.65 −0.394889 −0.197445 0.980314i \(-0.563264\pi\)
−0.197445 + 0.980314i \(0.563264\pi\)
\(462\) 0 0
\(463\) 18179.8i 1.82481i 0.409291 + 0.912404i \(0.365776\pi\)
−0.409291 + 0.912404i \(0.634224\pi\)
\(464\) 0 0
\(465\) −6960.00 7781.52i −0.694112 0.776041i
\(466\) 0 0
\(467\) 1849.83i 0.183298i −0.995791 0.0916488i \(-0.970786\pi\)
0.995791 0.0916488i \(-0.0292137\pi\)
\(468\) 0 0
\(469\) 5700.00i 0.561197i
\(470\) 0 0
\(471\) 2528.79 + 2827.28i 0.247390 + 0.276590i
\(472\) 0 0
\(473\) 7781.52i 0.756437i
\(474\) 0 0
\(475\) 3137.12 0.303033
\(476\) 0 0
\(477\) −1636.80 + 14640.0i −0.157115 + 1.40528i
\(478\) 0 0
\(479\) 15242.0 1.45392 0.726959 0.686681i \(-0.240933\pi\)
0.726959 + 0.686681i \(0.240933\pi\)
\(480\) 0 0
\(481\) 1300.00 0.123233
\(482\) 0 0
\(483\) 2602.64 + 2909.85i 0.245185 + 0.274125i
\(484\) 0 0
\(485\) −6887.09 −0.644797
\(486\) 0 0
\(487\) 9783.16i 0.910302i −0.890414 0.455151i \(-0.849585\pi\)
0.890414 0.455151i \(-0.150415\pi\)
\(488\) 0 0
\(489\) 990.000 885.483i 0.0915529 0.0818874i
\(490\) 0 0
\(491\) 6893.56i 0.633609i −0.948491 0.316805i \(-0.897390\pi\)
0.948491 0.316805i \(-0.102610\pi\)
\(492\) 0 0
\(493\) 5440.00i 0.496968i
\(494\) 0 0
\(495\) −8313.84 929.516i −0.754908 0.0844013i
\(496\) 0 0
\(497\) 8049.84i 0.726529i
\(498\) 0 0
\(499\) −1309.07 −0.117439 −0.0587194 0.998275i \(-0.518702\pi\)
−0.0587194 + 0.998275i \(0.518702\pi\)
\(500\) 0 0
\(501\) 53.6656 48.0000i 0.00478564 0.00428040i
\(502\) 0 0
\(503\) 7939.72 0.703806 0.351903 0.936036i \(-0.385535\pi\)
0.351903 + 0.936036i \(0.385535\pi\)
\(504\) 0 0
\(505\) −13360.0 −1.17725
\(506\) 0 0
\(507\) −8121.65 + 7264.22i −0.711430 + 0.636322i
\(508\) 0 0
\(509\) −14534.4 −1.26567 −0.632837 0.774285i \(-0.718110\pi\)
−0.632837 + 0.774285i \(0.718110\pi\)
\(510\) 0 0
\(511\) 3175.85i 0.274934i
\(512\) 0 0
\(513\) −5670.00 7969.35i −0.487986 0.685878i
\(514\) 0 0
\(515\) 12124.4i 1.03740i
\(516\) 0 0
\(517\) 6720.00i 0.571654i
\(518\) 0 0
\(519\) 4330.13 3872.98i 0.366226 0.327563i
\(520\) 0 0
\(521\) 9355.71i 0.786720i −0.919385 0.393360i \(-0.871313\pi\)
0.919385 0.393360i \(-0.128687\pi\)
\(522\) 0 0
\(523\) −10062.0 −0.841264 −0.420632 0.907231i \(-0.638192\pi\)
−0.420632 + 0.907231i \(0.638192\pi\)
\(524\) 0 0
\(525\) 1207.48 + 1350.00i 0.100378 + 0.112226i
\(526\) 0 0
\(527\) −8036.72 −0.664298
\(528\) 0 0
\(529\) −2759.00 −0.226761
\(530\) 0 0
\(531\) 4647.58 + 519.615i 0.379826 + 0.0424659i
\(532\) 0 0
\(533\) −1252.20 −0.101761
\(534\) 0 0
\(535\) 5763.00i 0.465712i
\(536\) 0 0
\(537\) 4680.00 + 5232.40i 0.376084 + 0.420474i
\(538\) 0 0
\(539\) 9803.41i 0.783419i
\(540\) 0 0
\(541\) 23962.0i 1.90426i −0.305687 0.952132i \(-0.598886\pi\)
0.305687 0.952132i \(-0.401114\pi\)
\(542\) 0 0
\(543\) 4371.70 + 4887.70i 0.345502 + 0.386283i
\(544\) 0 0
\(545\) 9534.59i 0.749389i
\(546\) 0 0
\(547\) 15112.4 1.18128 0.590639 0.806936i \(-0.298876\pi\)
0.590639 + 0.806936i \(0.298876\pi\)
\(548\) 0 0
\(549\) −11860.1 1326.00i −0.921998 0.103083i
\(550\) 0 0
\(551\) 10600.2 0.819567
\(552\) 0 0
\(553\) −660.000 −0.0507524
\(554\) 0 0
\(555\) 4027.90 + 4503.33i 0.308063 + 0.344425i
\(556\) 0 0
\(557\) 16055.0 1.22131 0.610656 0.791896i \(-0.290906\pi\)
0.610656 + 0.791896i \(0.290906\pi\)
\(558\) 0 0
\(559\) 2246.33i 0.169964i
\(560\) 0 0
\(561\) −4800.00 + 4293.25i −0.361241 + 0.323104i
\(562\) 0 0
\(563\) 25142.4i 1.88211i 0.338254 + 0.941055i \(0.390164\pi\)
−0.338254 + 0.941055i \(0.609836\pi\)
\(564\) 0 0
\(565\) 9280.00i 0.690996i
\(566\) 0 0
\(567\) 1247.08 5507.38i 0.0923674 0.407916i
\(568\) 0 0
\(569\) 23416.1i 1.72523i −0.505864 0.862613i \(-0.668826\pi\)
0.505864 0.862613i \(-0.331174\pi\)
\(570\) 0 0
\(571\) −4918.69 −0.360492 −0.180246 0.983622i \(-0.557689\pi\)
−0.180246 + 0.983622i \(0.557689\pi\)
\(572\) 0 0
\(573\) −10733.1 + 9600.00i −0.782518 + 0.699905i
\(574\) 0 0
\(575\) 4364.77 0.316562
\(576\) 0 0
\(577\) 19490.0 1.40620 0.703102 0.711089i \(-0.251798\pi\)
0.703102 + 0.711089i \(0.251798\pi\)
\(578\) 0 0
\(579\) 735.867 658.179i 0.0528179 0.0472418i
\(580\) 0 0
\(581\) −9713.48 −0.693602
\(582\) 0 0
\(583\) 18900.2i 1.34265i
\(584\) 0 0
\(585\) 2400.00 + 268.328i 0.169620 + 0.0189641i
\(586\) 0 0
\(587\) 1364.86i 0.0959687i 0.998848 + 0.0479844i \(0.0152798\pi\)
−0.998848 + 0.0479844i \(0.984720\pi\)
\(588\) 0 0
\(589\) 15660.0i 1.09552i
\(590\) 0 0
\(591\) −8279.20 + 7405.14i −0.576245 + 0.515409i
\(592\) 0 0
\(593\) 25795.3i 1.78632i −0.449743 0.893158i \(-0.648485\pi\)
0.449743 0.893158i \(-0.351515\pi\)
\(594\) 0 0
\(595\) −2478.71 −0.170785
\(596\) 0 0
\(597\) −885.483 990.000i −0.0607042 0.0678694i
\(598\) 0 0
\(599\) −2424.87 −0.165405 −0.0827025 0.996574i \(-0.526355\pi\)
−0.0827025 + 0.996574i \(0.526355\pi\)
\(600\) 0 0
\(601\) 8758.00 0.594420 0.297210 0.954812i \(-0.403944\pi\)
0.297210 + 0.954812i \(0.403944\pi\)
\(602\) 0 0
\(603\) 2207.60 19745.4i 0.149089 1.33349i
\(604\) 0 0
\(605\) −1171.70 −0.0787378
\(606\) 0 0
\(607\) 19558.6i 1.30784i −0.756565 0.653919i \(-0.773124\pi\)
0.756565 0.653919i \(-0.226876\pi\)
\(608\) 0 0
\(609\) 4080.00 + 4561.58i 0.271478 + 0.303521i
\(610\) 0 0
\(611\) 1939.90i 0.128445i
\(612\) 0 0
\(613\) 16450.0i 1.08386i 0.840422 + 0.541932i \(0.182307\pi\)
−0.840422 + 0.541932i \(0.817693\pi\)
\(614\) 0 0
\(615\) −3879.79 4337.74i −0.254388 0.284414i
\(616\) 0 0
\(617\) 8461.28i 0.552088i 0.961145 + 0.276044i \(0.0890236\pi\)
−0.961145 + 0.276044i \(0.910976\pi\)
\(618\) 0 0
\(619\) 19930.4 1.29413 0.647067 0.762433i \(-0.275995\pi\)
0.647067 + 0.762433i \(0.275995\pi\)
\(620\) 0 0
\(621\) −7888.85 11088.0i −0.509772 0.716499i
\(622\) 0 0
\(623\) −6512.51 −0.418809
\(624\) 0 0
\(625\) −7975.00 −0.510400
\(626\) 0 0
\(627\) 8365.64 + 9353.07i 0.532842 + 0.595735i
\(628\) 0 0
\(629\) 4651.02 0.294830
\(630\) 0 0
\(631\) 12199.9i 0.769683i −0.922983 0.384842i \(-0.874256\pi\)
0.922983 0.384842i \(-0.125744\pi\)
\(632\) 0 0
\(633\) −2130.00 + 1905.13i −0.133744 + 0.119624i
\(634\) 0 0
\(635\) 16419.8i 1.02614i
\(636\) 0 0
\(637\) 2830.00i 0.176026i
\(638\) 0 0
\(639\) 3117.69 27885.5i 0.193011 1.72634i
\(640\) 0 0
\(641\) 7012.31i 0.432090i −0.976383 0.216045i \(-0.930684\pi\)
0.976383 0.216045i \(-0.0693158\pi\)
\(642\) 0 0
\(643\) 15979.9 0.980073 0.490036 0.871702i \(-0.336983\pi\)
0.490036 + 0.871702i \(0.336983\pi\)
\(644\) 0 0
\(645\) 7781.52 6960.00i 0.475034 0.424883i
\(646\) 0 0
\(647\) 17999.5 1.09371 0.546856 0.837226i \(-0.315824\pi\)
0.546856 + 0.837226i \(0.315824\pi\)
\(648\) 0 0
\(649\) −6000.00 −0.362898
\(650\) 0 0
\(651\) −6738.99 + 6027.54i −0.405717 + 0.362884i
\(652\) 0 0
\(653\) −5196.62 −0.311423 −0.155712 0.987803i \(-0.549767\pi\)
−0.155712 + 0.987803i \(0.549767\pi\)
\(654\) 0 0
\(655\) 4027.90i 0.240280i
\(656\) 0 0
\(657\) −1230.00 + 11001.5i −0.0730394 + 0.653284i
\(658\) 0 0
\(659\) 6062.18i 0.358344i −0.983818 0.179172i \(-0.942658\pi\)
0.983818 0.179172i \(-0.0573419\pi\)
\(660\) 0 0
\(661\) 9422.00i 0.554423i −0.960809 0.277211i \(-0.910590\pi\)
0.960809 0.277211i \(-0.0894102\pi\)
\(662\) 0 0
\(663\) 1385.64 1239.35i 0.0811672 0.0725981i
\(664\) 0 0
\(665\) 4829.91i 0.281648i
\(666\) 0 0
\(667\) 14748.3 0.856158
\(668\) 0 0
\(669\) 1636.80 + 1830.00i 0.0945925 + 0.105758i
\(670\) 0 0
\(671\) 15311.3 0.880905
\(672\) 0 0
\(673\) −17470.0 −1.00062 −0.500311 0.865846i \(-0.666781\pi\)
−0.500311 + 0.865846i \(0.666781\pi\)
\(674\) 0 0
\(675\) −3659.97 5144.19i −0.208700 0.293333i
\(676\) 0 0
\(677\) 20813.3 1.18157 0.590784 0.806830i \(-0.298819\pi\)
0.590784 + 0.806830i \(0.298819\pi\)
\(678\) 0 0
\(679\) 5964.39i 0.337102i
\(680\) 0 0
\(681\) −1752.00 1958.80i −0.0985856 0.110222i
\(682\) 0 0
\(683\) 12616.3i 0.706805i 0.935471 + 0.353402i \(0.114975\pi\)
−0.935471 + 0.353402i \(0.885025\pi\)
\(684\) 0 0
\(685\) 800.000i 0.0446225i
\(686\) 0 0
\(687\) 14182.0 + 15856.0i 0.787596 + 0.880559i
\(688\) 0 0
\(689\) 5456.01i 0.301680i
\(690\) 0 0
\(691\) 3028.67 0.166738 0.0833691 0.996519i \(-0.473432\pi\)
0.0833691 + 0.996519i \(0.473432\pi\)
\(692\) 0 0
\(693\) −804.984 + 7200.00i −0.0441253 + 0.394669i
\(694\) 0 0
\(695\) −17528.4 −0.956674
\(696\) 0 0
\(697\) −4480.00 −0.243461
\(698\) 0 0
\(699\) 18280.5 + 20438.2i 0.989172 + 1.10593i
\(700\) 0 0
\(701\) 17664.9 0.951777 0.475888 0.879506i \(-0.342127\pi\)
0.475888 + 0.879506i \(0.342127\pi\)
\(702\) 0 0
\(703\) 9062.78i 0.486215i
\(704\) 0 0
\(705\) 6720.00 6010.55i 0.358993 0.321093i
\(706\) 0 0
\(707\) 11570.1i 0.615472i
\(708\) 0 0
\(709\) 14174.0i 0.750798i −0.926863 0.375399i \(-0.877506\pi\)
0.926863 0.375399i \(-0.122494\pi\)
\(710\) 0 0
\(711\) 2286.31 + 255.617i 0.120595 + 0.0134830i
\(712\) 0 0
\(713\) 21788.2i 1.14443i
\(714\) 0 0
\(715\) −3098.39 −0.162060
\(716\) 0 0
\(717\) 22002.9 19680.0i 1.14604 1.02505i
\(718\) 0 0
\(719\) 32839.7 1.70336 0.851678 0.524065i \(-0.175585\pi\)
0.851678 + 0.524065i \(0.175585\pi\)
\(720\) 0 0
\(721\) 10500.0 0.542358
\(722\) 0 0
\(723\) 4639.83 4149.99i 0.238668 0.213472i
\(724\) 0 0
\(725\) 6842.37 0.350509
\(726\) 0 0
\(727\) 8001.58i 0.408201i 0.978950 + 0.204101i \(0.0654270\pi\)
−0.978950 + 0.204101i \(0.934573\pi\)
\(728\) 0 0
\(729\) −6453.00 + 18595.1i −0.327846 + 0.944731i
\(730\) 0 0
\(731\) 8036.72i 0.406633i
\(732\) 0 0
\(733\) 11750.0i 0.592082i 0.955175 + 0.296041i \(0.0956665\pi\)
−0.955175 + 0.296041i \(0.904333\pi\)
\(734\) 0 0
\(735\) 9803.41 8768.43i 0.491978 0.440039i
\(736\) 0 0
\(737\) 25491.2i 1.27406i
\(738\) 0 0
\(739\) −19961.4 −0.993627 −0.496814 0.867857i \(-0.665497\pi\)
−0.496814 + 0.867857i \(0.665497\pi\)
\(740\) 0 0
\(741\) −2414.95 2700.00i −0.119724 0.133856i
\(742\) 0 0
\(743\) 25592.8 1.26367 0.631836 0.775102i \(-0.282302\pi\)
0.631836 + 0.775102i \(0.282302\pi\)
\(744\) 0 0
\(745\) 14480.0 0.712089
\(746\) 0 0
\(747\) 33648.5 + 3762.01i 1.64810 + 0.184264i
\(748\) 0 0
\(749\) −4990.90 −0.243476
\(750\) 0 0
\(751\) 5244.02i 0.254803i −0.991851 0.127401i \(-0.959336\pi\)
0.991851 0.127401i \(-0.0406637\pi\)
\(752\) 0 0
\(753\) 14760.0 + 16502.2i 0.714322 + 0.798636i
\(754\) 0 0
\(755\) 5057.59i 0.243794i
\(756\) 0 0
\(757\) 14290.0i 0.686102i 0.939317 + 0.343051i \(0.111460\pi\)
−0.939317 + 0.343051i \(0.888540\pi\)
\(758\) 0 0
\(759\) 11639.4 + 13013.2i 0.556631 + 0.622332i
\(760\) 0 0
\(761\) 16976.2i 0.808657i −0.914614 0.404328i \(-0.867505\pi\)
0.914614 0.404328i \(-0.132495\pi\)
\(762\) 0 0
\(763\) 8257.20 0.391783
\(764\) 0 0
\(765\) 8586.50 + 960.000i 0.405811 + 0.0453711i
\(766\) 0 0
\(767\) 1732.05 0.0815394
\(768\) 0 0
\(769\) −29566.0 −1.38645 −0.693223 0.720723i \(-0.743810\pi\)
−0.693223 + 0.720723i \(0.743810\pi\)
\(770\) 0 0
\(771\) 10906.3 + 12193.6i 0.509444 + 0.569576i
\(772\) 0 0
\(773\) 21457.3 0.998403 0.499202 0.866486i \(-0.333627\pi\)
0.499202 + 0.866486i \(0.333627\pi\)
\(774\) 0 0
\(775\) 10108.5i 0.468526i
\(776\) 0 0
\(777\) 3900.00 3488.27i 0.180067 0.161056i
\(778\) 0 0
\(779\) 8729.54i 0.401499i
\(780\) 0 0
\(781\) 36000.0i 1.64940i
\(782\) 0 0
\(783\) −12366.8 17381.9i −0.564438 0.793334i
\(784\) 0 0
\(785\) 6529.32i 0.296868i
\(786\) 0 0
\(787\) 3896.22 0.176474 0.0882372 0.996099i \(-0.471877\pi\)
0.0882372 + 0.996099i \(0.471877\pi\)
\(788\) 0 0
\(789\) −16475.3 + 14736.0i −0.743394 + 0.664912i
\(790\) 0 0
\(791\) −8036.72 −0.361255
\(792\) 0 0
\(793\) −4420.00 −0.197930
\(794\) 0 0
\(795\) −18900.2 + 16904.8i −0.843169 + 0.754154i
\(796\) 0 0
\(797\) 30759.4 1.36707 0.683533 0.729919i \(-0.260442\pi\)
0.683533 + 0.729919i \(0.260442\pi\)
\(798\) 0 0
\(799\) 6940.39i 0.307301i
\(800\) 0 0
\(801\) 22560.0 + 2522.28i 0.995154 + 0.111262i
\(802\) 0 0
\(803\) 14202.8i 0.624168i
\(804\) 0 0
\(805\) 6720.00i 0.294222i
\(806\) 0 0
\(807\) 173.205 154.919i 0.00755528 0.00675764i
\(808\) 0 0
\(809\) 10429.0i 0.453232i −0.973984 0.226616i \(-0.927234\pi\)
0.973984 0.226616i \(-0.0727663\pi\)
\(810\) 0 0
\(811\) 8156.50 0.353161 0.176580 0.984286i \(-0.443496\pi\)
0.176580 + 0.984286i \(0.443496\pi\)
\(812\) 0 0
\(813\) −30347.9 33930.0i −1.30916 1.46369i
\(814\) 0 0
\(815\) 2286.31 0.0982648
\(816\) 0 0
\(817\) −15660.0 −0.670592
\(818\) 0 0
\(819\) 232.379 2078.46i 0.00991450 0.0886780i
\(820\) 0 0
\(821\) −35750.3 −1.51972 −0.759861 0.650085i \(-0.774733\pi\)
−0.759861 + 0.650085i \(0.774733\pi\)
\(822\) 0 0
\(823\) 20875.4i 0.884168i −0.896974 0.442084i \(-0.854239\pi\)
0.896974 0.442084i \(-0.145761\pi\)
\(824\) 0 0
\(825\) 5400.00 + 6037.38i 0.227883 + 0.254781i
\(826\) 0 0
\(827\) 12907.2i 0.542719i −0.962478 0.271360i \(-0.912527\pi\)
0.962478 0.271360i \(-0.0874733\pi\)
\(828\) 0 0
\(829\) 14074.0i 0.589638i −0.955553 0.294819i \(-0.904741\pi\)
0.955553 0.294819i \(-0.0952594\pi\)
\(830\) 0 0
\(831\) 21997.0 + 24593.4i 0.918254 + 1.02664i
\(832\) 0 0
\(833\) 10124.9i 0.421138i
\(834\) 0 0
\(835\) 123.935 0.00513648
\(836\) 0 0
\(837\) 25679.0 18270.0i 1.06045 0.754485i
\(838\) 0 0
\(839\) −16697.0 −0.687060 −0.343530 0.939142i \(-0.611623\pi\)
−0.343530 + 0.939142i \(0.611623\pi\)
\(840\) 0 0
\(841\) −1269.00 −0.0520317
\(842\) 0 0
\(843\) −19272.0 21546.7i −0.787380 0.880318i
\(844\) 0 0
\(845\) −18756.1 −0.763587
\(846\) 0 0
\(847\) 1014.72i 0.0411644i
\(848\) 0 0
\(849\) 26250.0 23478.7i 1.06113 0.949102i
\(850\) 0 0
\(851\) 12609.3i 0.507923i
\(852\) 0 0
\(853\) 23630.0i 0.948506i −0.880389 0.474253i \(-0.842718\pi\)
0.880389 0.474253i \(-0.157282\pi\)
\(854\) 0 0
\(855\) 1870.61 16731.3i 0.0748230 0.669237i
\(856\) 0 0
\(857\) 31322.8i 1.24850i 0.781223 + 0.624252i \(0.214596\pi\)
−0.781223 + 0.624252i \(0.785404\pi\)
\(858\) 0 0
\(859\) 13671.6 0.543038 0.271519 0.962433i \(-0.412474\pi\)
0.271519 + 0.962433i \(0.412474\pi\)
\(860\) 0 0
\(861\) −3756.59 + 3360.00i −0.148693 + 0.132995i
\(862\) 0 0
\(863\) −25107.8 −0.990359 −0.495179 0.868791i \(-0.664898\pi\)
−0.495179 + 0.868791i \(0.664898\pi\)
\(864\) 0 0
\(865\) 10000.0 0.393075
\(866\) 0 0
\(867\) −14070.5 + 12585.1i −0.551166 + 0.492978i
\(868\) 0 0
\(869\) −2951.61 −0.115220
\(870\) 0 0
\(871\) 7358.67i 0.286267i
\(872\) 0 0
\(873\) 2310.00 20661.3i 0.0895552 0.801006i
\(874\) 0 0
\(875\) 11777.9i 0.455048i
\(876\) 0 0
\(877\) 41750.0i 1.60752i 0.594952 + 0.803761i \(0.297171\pi\)
−0.594952 + 0.803761i \(0.702829\pi\)
\(878\) 0 0
\(879\) 2528.79 2261.82i 0.0970354 0.0867911i
\(880\) 0 0
\(881\) 2397.06i 0.0916676i −0.998949 0.0458338i \(-0.985406\pi\)
0.998949 0.0458338i \(-0.0145945\pi\)
\(882\) 0 0
\(883\) 43431.6 1.65526 0.827628 0.561277i \(-0.189690\pi\)
0.827628 + 0.561277i \(0.189690\pi\)
\(884\) 0 0
\(885\) 5366.56 + 6000.00i 0.203836 + 0.227896i
\(886\) 0 0
\(887\) 6387.80 0.241805 0.120903 0.992664i \(-0.461421\pi\)
0.120903 + 0.992664i \(0.461421\pi\)
\(888\) 0 0
\(889\) 14220.0 0.536472
\(890\) 0 0
\(891\) 5577.10 24629.8i 0.209697 0.926070i
\(892\) 0 0
\(893\) −13523.7 −0.506780
\(894\) 0 0
\(895\) 12083.7i 0.451300i
\(896\) 0 0
\(897\) −3360.00 3756.59i −0.125069 0.139832i
\(898\) 0 0
\(899\) 34156.0i 1.26715i
\(900\) 0 0
\(901\) 19520.0i 0.721760i
\(902\) 0 0
\(903\) −6027.54 6738.99i −0.222131 0.248349i
\(904\) 0 0
\(905\) 11287.7i 0.414602i
\(906\) 0 0
\(907\) −37327.8 −1.36654 −0.683269 0.730167i \(-0.739442\pi\)
−0.683269 + 0.730167i \(0.739442\pi\)
\(908\) 0 0
\(909\) 4481.08 40080.0i 0.163507 1.46245i
\(910\) 0 0
\(911\) −50853.0 −1.84944 −0.924718 0.380654i \(-0.875699\pi\)
−0.924718 + 0.380654i \(0.875699\pi\)
\(912\) 0 0
\(913\) −43440.0 −1.57465
\(914\) 0 0
\(915\) −13694.9 15311.3i −0.494796 0.553199i
\(916\) 0 0
\(917\) −3488.27 −0.125619
\(918\) 0 0
\(919\) 14198.4i 0.509642i 0.966988 + 0.254821i \(0.0820165\pi\)
−0.966988 + 0.254821i \(0.917984\pi\)
\(920\) 0 0
\(921\) 6030.00 5393.40i 0.215739 0.192962i
\(922\) 0 0
\(923\) 10392.3i 0.370603i
\(924\) 0 0
\(925\) 5850.00i 0.207943i
\(926\) 0 0
\(927\) −36373.1 4066.63i −1.28873 0.144084i
\(928\) 0 0
\(929\) 10232.2i 0.361366i −0.983541 0.180683i \(-0.942169\pi\)
0.983541 0.180683i \(-0.0578308\pi\)
\(930\) 0 0
\(931\) −19729.0 −0.694512
\(932\) 0 0
\(933\) 12611.4 11280.0i 0.442529 0.395810i
\(934\) 0 0
\(935\) −11085.1 −0.387724
\(936\) 0 0
\(937\) −37850.0 −1.31964 −0.659822 0.751422i \(-0.729368\pi\)
−0.659822 + 0.751422i \(0.729368\pi\)
\(938\) 0 0
\(939\) −27227.1 + 24352.6i −0.946243 + 0.846345i
\(940\) 0 0
\(941\) −13997.8 −0.484925 −0.242463 0.970161i \(-0.577955\pi\)
−0.242463 + 0.970161i \(0.577955\pi\)
\(942\) 0 0
\(943\) 12145.7i 0.419425i
\(944\) 0 0
\(945\) 7920.00 5634.89i 0.272632 0.193971i
\(946\) 0 0
\(947\) 7738.80i 0.265552i 0.991146 + 0.132776i \(0.0423890\pi\)
−0.991146 + 0.132776i \(0.957611\pi\)
\(948\) 0 0
\(949\) 4100.00i 0.140244i
\(950\) 0 0
\(951\) −1905.26 + 1704.11i −0.0649655 + 0.0581069i
\(952\) 0 0
\(953\) 44882.4i 1.52558i −0.646644 0.762792i \(-0.723828\pi\)
0.646644 0.762792i \(-0.276172\pi\)
\(954\) 0 0
\(955\) −24787.1 −0.839886
\(956\) 0 0
\(957\) 18246.3 + 20400.0i 0.616321 + 0.689068i
\(958\) 0 0
\(959\) −692.820 −0.0233288
\(960\) 0 0
\(961\) −20669.0 −0.693800
\(962\) 0 0
\(963\) 17289.0 + 1932.97i 0.578536 + 0.0646823i
\(964\) 0 0
\(965\) 1699.41 0.0566902
\(966\) 0 0
\(967\) 15856.0i 0.527295i −0.964619 0.263648i \(-0.915074\pi\)
0.964619 0.263648i \(-0.0849256\pi\)
\(968\) 0 0
\(969\) −8640.00 9659.81i −0.286436 0.320246i
\(970\) 0 0
\(971\) 40426.1i 1.33608i −0.744125 0.668040i \(-0.767133\pi\)
0.744125 0.668040i \(-0.232867\pi\)
\(972\) 0 0
\(973\) 15180.0i 0.500153i
\(974\) 0 0
\(975\) −1558.85 1742.84i −0.0512031 0.0572468i
\(976\) 0 0
\(977\) 6404.10i 0.209709i 0.994488 + 0.104854i \(0.0334376\pi\)
−0.994488 + 0.104854i \(0.966562\pi\)
\(978\) 0 0
\(979\) −29124.8 −0.950801
\(980\) 0 0
\(981\) −28603.8 3198.00i −0.930936 0.104082i
\(982\) 0 0
\(983\) 25953.0 0.842089 0.421045 0.907040i \(-0.361664\pi\)
0.421045 + 0.907040i \(0.361664\pi\)
\(984\) 0 0
\(985\) −19120.0 −0.618491
\(986\) 0 0
\(987\) −5205.29 5819.69i −0.167868 0.187683i
\(988\) 0 0
\(989\) −21788.2 −0.700532
\(990\) 0 0
\(991\) 48358.1i 1.55010i 0.631903 + 0.775048i \(0.282274\pi\)
−0.631903 + 0.775048i \(0.717726\pi\)
\(992\) 0 0
\(993\) 16410.0 14677.6i 0.524427 0.469061i
\(994\) 0 0
\(995\) 2286.31i 0.0728450i
\(996\) 0 0
\(997\) 18370.0i 0.583534i 0.956489 + 0.291767i \(0.0942432\pi\)
−0.956489 + 0.291767i \(0.905757\pi\)
\(998\) 0 0
\(999\) −14861.0 + 10573.2i −0.470652 + 0.334858i
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 768.4.f.c.383.3 8
3.2 odd 2 inner 768.4.f.c.383.2 8
4.3 odd 2 inner 768.4.f.c.383.5 8
8.3 odd 2 inner 768.4.f.c.383.4 8
8.5 even 2 inner 768.4.f.c.383.6 8
12.11 even 2 inner 768.4.f.c.383.8 8
16.3 odd 4 192.4.c.b.191.4 4
16.5 even 4 12.4.b.a.11.2 yes 4
16.11 odd 4 12.4.b.a.11.4 yes 4
16.13 even 4 192.4.c.b.191.1 4
24.5 odd 2 inner 768.4.f.c.383.7 8
24.11 even 2 inner 768.4.f.c.383.1 8
48.5 odd 4 12.4.b.a.11.3 yes 4
48.11 even 4 12.4.b.a.11.1 4
48.29 odd 4 192.4.c.b.191.3 4
48.35 even 4 192.4.c.b.191.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.b.a.11.1 4 48.11 even 4
12.4.b.a.11.2 yes 4 16.5 even 4
12.4.b.a.11.3 yes 4 48.5 odd 4
12.4.b.a.11.4 yes 4 16.11 odd 4
192.4.c.b.191.1 4 16.13 even 4
192.4.c.b.191.2 4 48.35 even 4
192.4.c.b.191.3 4 48.29 odd 4
192.4.c.b.191.4 4 16.3 odd 4
768.4.f.c.383.1 8 24.11 even 2 inner
768.4.f.c.383.2 8 3.2 odd 2 inner
768.4.f.c.383.3 8 1.1 even 1 trivial
768.4.f.c.383.4 8 8.3 odd 2 inner
768.4.f.c.383.5 8 4.3 odd 2 inner
768.4.f.c.383.6 8 8.5 even 2 inner
768.4.f.c.383.7 8 24.5 odd 2 inner
768.4.f.c.383.8 8 12.11 even 2 inner