Properties

Label 76.4.i.a.25.5
Level $76$
Weight $4$
Character 76.25
Analytic conductor $4.484$
Analytic rank $0$
Dimension $30$
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] = [76,4,Mod(5,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), degree \(6\), 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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 25.5
Character \(\chi\) \(=\) 76.25
Dual form 76.4.i.a.73.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.26044 + 2.27861i) q^{3} +(2.15308 + 12.2107i) q^{5} +(-4.57544 + 7.92489i) q^{7} +(13.3179 + 11.1750i) q^{9} +O(q^{10})\) \(q+(6.26044 + 2.27861i) q^{3} +(2.15308 + 12.2107i) q^{5} +(-4.57544 + 7.92489i) q^{7} +(13.3179 + 11.1750i) q^{9} +(-7.90249 - 13.6875i) q^{11} +(23.5530 - 8.57261i) q^{13} +(-14.3443 + 81.3505i) q^{15} +(23.2232 - 19.4866i) q^{17} +(82.6404 + 5.43657i) q^{19} +(-46.7020 + 39.1877i) q^{21} +(-2.18772 + 12.4072i) q^{23} +(-27.0040 + 9.82867i) q^{25} +(-32.0279 - 55.4739i) q^{27} +(-78.9982 - 66.2873i) q^{29} +(117.534 - 203.576i) q^{31} +(-18.2845 - 103.697i) q^{33} +(-106.620 - 38.8064i) q^{35} -338.196 q^{37} +166.986 q^{39} +(-361.846 - 131.701i) q^{41} +(-83.3298 - 472.587i) q^{43} +(-107.780 + 186.681i) q^{45} +(412.476 + 346.108i) q^{47} +(129.631 + 224.527i) q^{49} +(189.790 - 69.0779i) q^{51} +(-93.1773 + 528.435i) q^{53} +(150.119 - 125.965i) q^{55} +(504.978 + 222.341i) q^{57} +(305.958 - 256.729i) q^{59} +(-13.1610 + 74.6398i) q^{61} +(-149.496 + 54.4120i) q^{63} +(155.389 + 269.142i) q^{65} +(-160.604 - 134.763i) q^{67} +(-41.9672 + 72.6893i) q^{69} +(134.844 + 764.741i) q^{71} +(-826.398 - 300.784i) q^{73} -191.453 q^{75} +144.629 q^{77} +(539.847 + 196.488i) q^{79} +(-155.616 - 882.540i) q^{81} +(-472.093 + 817.688i) q^{83} +(287.946 + 241.616i) q^{85} +(-343.520 - 594.994i) q^{87} +(127.516 - 46.4121i) q^{89} +(-39.8285 + 225.879i) q^{91} +(1199.69 - 1006.66i) q^{93} +(111.547 + 1020.80i) q^{95} +(525.497 - 440.944i) q^{97} +(47.7139 - 270.599i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9} + 42 q^{11} - 42 q^{13} + 78 q^{15} + 30 q^{17} + 282 q^{19} + 198 q^{21} - 300 q^{23} - 276 q^{25} + 219 q^{27} + 216 q^{29} + 30 q^{31} - 597 q^{33} - 636 q^{35} + 60 q^{37} - 2172 q^{39} - 63 q^{41} - 246 q^{43} - 882 q^{45} + 762 q^{47} - 525 q^{49} + 2613 q^{51} + 882 q^{53} + 1350 q^{55} + 924 q^{57} + 2085 q^{59} + 1530 q^{61} + 2424 q^{63} + 1530 q^{65} - 3609 q^{67} + 756 q^{69} - 4962 q^{71} - 2394 q^{73} - 3516 q^{77} - 630 q^{79} - 3723 q^{81} - 2382 q^{83} + 3228 q^{85} - 1110 q^{87} + 2196 q^{89} + 6036 q^{91} + 5010 q^{93} + 6204 q^{95} + 6459 q^{97} + 6189 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\)

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
\(3\) 6.26044 + 2.27861i 1.20482 + 0.438520i 0.864905 0.501935i \(-0.167379\pi\)
0.339918 + 0.940455i \(0.389601\pi\)
\(4\) 0 0
\(5\) 2.15308 + 12.2107i 0.192577 + 1.09216i 0.915827 + 0.401573i \(0.131536\pi\)
−0.723250 + 0.690586i \(0.757353\pi\)
\(6\) 0 0
\(7\) −4.57544 + 7.92489i −0.247050 + 0.427904i −0.962706 0.270549i \(-0.912795\pi\)
0.715656 + 0.698453i \(0.246128\pi\)
\(8\) 0 0
\(9\) 13.3179 + 11.1750i 0.493254 + 0.413889i
\(10\) 0 0
\(11\) −7.90249 13.6875i −0.216608 0.375176i 0.737161 0.675717i \(-0.236166\pi\)
−0.953769 + 0.300541i \(0.902833\pi\)
\(12\) 0 0
\(13\) 23.5530 8.57261i 0.502495 0.182893i −0.0783207 0.996928i \(-0.524956\pi\)
0.580816 + 0.814035i \(0.302734\pi\)
\(14\) 0 0
\(15\) −14.3443 + 81.3505i −0.246912 + 1.40031i
\(16\) 0 0
\(17\) 23.2232 19.4866i 0.331321 0.278011i −0.461917 0.886923i \(-0.652838\pi\)
0.793238 + 0.608912i \(0.208394\pi\)
\(18\) 0 0
\(19\) 82.6404 + 5.43657i 0.997843 + 0.0656439i
\(20\) 0 0
\(21\) −46.7020 + 39.1877i −0.485296 + 0.407212i
\(22\) 0 0
\(23\) −2.18772 + 12.4072i −0.0198335 + 0.112481i −0.993117 0.117124i \(-0.962632\pi\)
0.973284 + 0.229605i \(0.0737436\pi\)
\(24\) 0 0
\(25\) −27.0040 + 9.82867i −0.216032 + 0.0786294i
\(26\) 0 0
\(27\) −32.0279 55.4739i −0.228288 0.395406i
\(28\) 0 0
\(29\) −78.9982 66.2873i −0.505848 0.424457i 0.353817 0.935315i \(-0.384883\pi\)
−0.859665 + 0.510858i \(0.829328\pi\)
\(30\) 0 0
\(31\) 117.534 203.576i 0.680962 1.17946i −0.293726 0.955890i \(-0.594895\pi\)
0.974688 0.223570i \(-0.0717713\pi\)
\(32\) 0 0
\(33\) −18.2845 103.697i −0.0964522 0.547008i
\(34\) 0 0
\(35\) −106.620 38.8064i −0.514915 0.187414i
\(36\) 0 0
\(37\) −338.196 −1.50268 −0.751340 0.659916i \(-0.770592\pi\)
−0.751340 + 0.659916i \(0.770592\pi\)
\(38\) 0 0
\(39\) 166.986 0.685620
\(40\) 0 0
\(41\) −361.846 131.701i −1.37831 0.501665i −0.456646 0.889648i \(-0.650949\pi\)
−0.921666 + 0.387984i \(0.873172\pi\)
\(42\) 0 0
\(43\) −83.3298 472.587i −0.295527 1.67602i −0.665051 0.746798i \(-0.731590\pi\)
0.369524 0.929221i \(-0.379521\pi\)
\(44\) 0 0
\(45\) −107.780 + 186.681i −0.357044 + 0.618418i
\(46\) 0 0
\(47\) 412.476 + 346.108i 1.28012 + 1.07415i 0.993227 + 0.116190i \(0.0370681\pi\)
0.286897 + 0.957962i \(0.407376\pi\)
\(48\) 0 0
\(49\) 129.631 + 224.527i 0.377932 + 0.654598i
\(50\) 0 0
\(51\) 189.790 69.0779i 0.521096 0.189664i
\(52\) 0 0
\(53\) −93.1773 + 528.435i −0.241488 + 1.36955i 0.587020 + 0.809573i \(0.300301\pi\)
−0.828508 + 0.559977i \(0.810810\pi\)
\(54\) 0 0
\(55\) 150.119 125.965i 0.368038 0.308821i
\(56\) 0 0
\(57\) 504.978 + 222.341i 1.17344 + 0.516663i
\(58\) 0 0
\(59\) 305.958 256.729i 0.675125 0.566497i −0.239452 0.970908i \(-0.576968\pi\)
0.914577 + 0.404411i \(0.132523\pi\)
\(60\) 0 0
\(61\) −13.1610 + 74.6398i −0.0276245 + 0.156666i −0.995500 0.0947648i \(-0.969790\pi\)
0.967875 + 0.251431i \(0.0809012\pi\)
\(62\) 0 0
\(63\) −149.496 + 54.4120i −0.298963 + 0.108814i
\(64\) 0 0
\(65\) 155.389 + 269.142i 0.296518 + 0.513584i
\(66\) 0 0
\(67\) −160.604 134.763i −0.292849 0.245730i 0.484511 0.874785i \(-0.338997\pi\)
−0.777361 + 0.629055i \(0.783442\pi\)
\(68\) 0 0
\(69\) −41.9672 + 72.6893i −0.0732211 + 0.126823i
\(70\) 0 0
\(71\) 134.844 + 764.741i 0.225396 + 1.27828i 0.861928 + 0.507031i \(0.169257\pi\)
−0.636532 + 0.771250i \(0.719632\pi\)
\(72\) 0 0
\(73\) −826.398 300.784i −1.32497 0.482249i −0.419921 0.907561i \(-0.637942\pi\)
−0.905047 + 0.425312i \(0.860164\pi\)
\(74\) 0 0
\(75\) −191.453 −0.294761
\(76\) 0 0
\(77\) 144.629 0.214052
\(78\) 0 0
\(79\) 539.847 + 196.488i 0.768830 + 0.279831i 0.696507 0.717550i \(-0.254737\pi\)
0.0723228 + 0.997381i \(0.476959\pi\)
\(80\) 0 0
\(81\) −155.616 882.540i −0.213465 1.21062i
\(82\) 0 0
\(83\) −472.093 + 817.688i −0.624324 + 1.08136i 0.364347 + 0.931263i \(0.381292\pi\)
−0.988671 + 0.150098i \(0.952041\pi\)
\(84\) 0 0
\(85\) 287.946 + 241.616i 0.367437 + 0.308316i
\(86\) 0 0
\(87\) −343.520 594.994i −0.423325 0.733220i
\(88\) 0 0
\(89\) 127.516 46.4121i 0.151873 0.0552772i −0.264965 0.964258i \(-0.585360\pi\)
0.416838 + 0.908981i \(0.363138\pi\)
\(90\) 0 0
\(91\) −39.8285 + 225.879i −0.0458809 + 0.260203i
\(92\) 0 0
\(93\) 1199.69 1006.66i 1.33765 1.12243i
\(94\) 0 0
\(95\) 111.547 + 1020.80i 0.120468 + 1.10244i
\(96\) 0 0
\(97\) 525.497 440.944i 0.550063 0.461558i −0.324899 0.945749i \(-0.605330\pi\)
0.874962 + 0.484191i \(0.160886\pi\)
\(98\) 0 0
\(99\) 47.7139 270.599i 0.0484386 0.274709i
\(100\) 0 0
\(101\) 226.806 82.5507i 0.223446 0.0813277i −0.227871 0.973691i \(-0.573176\pi\)
0.451317 + 0.892364i \(0.350954\pi\)
\(102\) 0 0
\(103\) −336.162 582.249i −0.321583 0.556997i 0.659232 0.751940i \(-0.270881\pi\)
−0.980815 + 0.194942i \(0.937548\pi\)
\(104\) 0 0
\(105\) −579.062 485.891i −0.538197 0.451601i
\(106\) 0 0
\(107\) −53.3019 + 92.3216i −0.0481579 + 0.0834119i −0.889100 0.457714i \(-0.848668\pi\)
0.840942 + 0.541126i \(0.182002\pi\)
\(108\) 0 0
\(109\) 94.3891 + 535.307i 0.0829435 + 0.470396i 0.997781 + 0.0665747i \(0.0212071\pi\)
−0.914838 + 0.403821i \(0.867682\pi\)
\(110\) 0 0
\(111\) −2117.26 770.619i −1.81046 0.658954i
\(112\) 0 0
\(113\) −2000.81 −1.66567 −0.832835 0.553522i \(-0.813284\pi\)
−0.832835 + 0.553522i \(0.813284\pi\)
\(114\) 0 0
\(115\) −156.210 −0.126667
\(116\) 0 0
\(117\) 409.475 + 149.037i 0.323556 + 0.117765i
\(118\) 0 0
\(119\) 48.1727 + 273.201i 0.0371091 + 0.210456i
\(120\) 0 0
\(121\) 540.601 936.349i 0.406162 0.703493i
\(122\) 0 0
\(123\) −1965.22 1649.01i −1.44063 1.20883i
\(124\) 0 0
\(125\) 596.785 + 1033.66i 0.427025 + 0.739629i
\(126\) 0 0
\(127\) 1260.82 458.902i 0.880944 0.320637i 0.138353 0.990383i \(-0.455819\pi\)
0.742591 + 0.669746i \(0.233597\pi\)
\(128\) 0 0
\(129\) 555.162 3148.48i 0.378909 2.14890i
\(130\) 0 0
\(131\) −402.594 + 337.816i −0.268510 + 0.225307i −0.767094 0.641535i \(-0.778298\pi\)
0.498584 + 0.866841i \(0.333854\pi\)
\(132\) 0 0
\(133\) −421.200 + 630.042i −0.274607 + 0.410763i
\(134\) 0 0
\(135\) 608.417 510.522i 0.387883 0.325472i
\(136\) 0 0
\(137\) −441.677 + 2504.87i −0.275438 + 1.56209i 0.462129 + 0.886813i \(0.347086\pi\)
−0.737567 + 0.675274i \(0.764025\pi\)
\(138\) 0 0
\(139\) 666.214 242.482i 0.406529 0.147964i −0.130658 0.991427i \(-0.541709\pi\)
0.537187 + 0.843463i \(0.319487\pi\)
\(140\) 0 0
\(141\) 1793.63 + 3106.67i 1.07129 + 1.85552i
\(142\) 0 0
\(143\) −303.465 254.638i −0.177462 0.148908i
\(144\) 0 0
\(145\) 639.326 1107.35i 0.366160 0.634207i
\(146\) 0 0
\(147\) 299.935 + 1701.02i 0.168287 + 0.954405i
\(148\) 0 0
\(149\) −66.1491 24.0763i −0.0363701 0.0132376i 0.323771 0.946135i \(-0.395049\pi\)
−0.360141 + 0.932898i \(0.617271\pi\)
\(150\) 0 0
\(151\) −533.022 −0.287263 −0.143631 0.989631i \(-0.545878\pi\)
−0.143631 + 0.989631i \(0.545878\pi\)
\(152\) 0 0
\(153\) 527.046 0.278491
\(154\) 0 0
\(155\) 2738.86 + 996.865i 1.41930 + 0.516581i
\(156\) 0 0
\(157\) 250.432 + 1420.27i 0.127303 + 0.721974i 0.979913 + 0.199425i \(0.0639075\pi\)
−0.852610 + 0.522548i \(0.824981\pi\)
\(158\) 0 0
\(159\) −1787.43 + 3095.92i −0.891525 + 1.54417i
\(160\) 0 0
\(161\) −88.3156 74.1056i −0.0432313 0.0362754i
\(162\) 0 0
\(163\) 470.642 + 815.175i 0.226156 + 0.391715i 0.956666 0.291188i \(-0.0940506\pi\)
−0.730509 + 0.682903i \(0.760717\pi\)
\(164\) 0 0
\(165\) 1226.84 446.533i 0.578845 0.210682i
\(166\) 0 0
\(167\) 252.301 1430.87i 0.116908 0.663019i −0.868880 0.495023i \(-0.835159\pi\)
0.985788 0.167995i \(-0.0537294\pi\)
\(168\) 0 0
\(169\) −1201.74 + 1008.38i −0.546993 + 0.458981i
\(170\) 0 0
\(171\) 1039.84 + 995.912i 0.465021 + 0.445376i
\(172\) 0 0
\(173\) 228.128 191.422i 0.100256 0.0841246i −0.591282 0.806465i \(-0.701378\pi\)
0.691538 + 0.722340i \(0.256934\pi\)
\(174\) 0 0
\(175\) 45.6642 258.975i 0.0197251 0.111866i
\(176\) 0 0
\(177\) 2500.42 910.079i 1.06183 0.386473i
\(178\) 0 0
\(179\) 228.176 + 395.212i 0.0952774 + 0.165025i 0.909724 0.415213i \(-0.136293\pi\)
−0.814447 + 0.580238i \(0.802960\pi\)
\(180\) 0 0
\(181\) −2602.24 2183.54i −1.06864 0.896692i −0.0737081 0.997280i \(-0.523483\pi\)
−0.994928 + 0.100588i \(0.967928\pi\)
\(182\) 0 0
\(183\) −252.469 + 437.289i −0.101984 + 0.176641i
\(184\) 0 0
\(185\) −728.163 4129.62i −0.289382 1.64116i
\(186\) 0 0
\(187\) −450.244 163.875i −0.176070 0.0640842i
\(188\) 0 0
\(189\) 586.166 0.225594
\(190\) 0 0
\(191\) −2645.80 −1.00232 −0.501161 0.865354i \(-0.667094\pi\)
−0.501161 + 0.865354i \(0.667094\pi\)
\(192\) 0 0
\(193\) 1981.71 + 721.282i 0.739100 + 0.269011i 0.684012 0.729471i \(-0.260234\pi\)
0.0550885 + 0.998481i \(0.482456\pi\)
\(194\) 0 0
\(195\) 359.534 + 2039.02i 0.132035 + 0.748806i
\(196\) 0 0
\(197\) −2118.61 + 3669.55i −0.766219 + 1.32713i 0.173381 + 0.984855i \(0.444531\pi\)
−0.939600 + 0.342275i \(0.888803\pi\)
\(198\) 0 0
\(199\) −3983.16 3342.27i −1.41889 1.19059i −0.951932 0.306309i \(-0.900906\pi\)
−0.466957 0.884280i \(-0.654650\pi\)
\(200\) 0 0
\(201\) −698.379 1209.63i −0.245074 0.424481i
\(202\) 0 0
\(203\) 886.771 322.758i 0.306597 0.111592i
\(204\) 0 0
\(205\) 829.081 4701.95i 0.282466 1.60195i
\(206\) 0 0
\(207\) −167.786 + 140.789i −0.0563378 + 0.0472730i
\(208\) 0 0
\(209\) −578.652 1174.10i −0.191513 0.388586i
\(210\) 0 0
\(211\) −1392.21 + 1168.21i −0.454237 + 0.381150i −0.841005 0.541027i \(-0.818036\pi\)
0.386768 + 0.922177i \(0.373591\pi\)
\(212\) 0 0
\(213\) −898.364 + 5094.87i −0.288990 + 1.63894i
\(214\) 0 0
\(215\) 5591.20 2035.03i 1.77357 0.645526i
\(216\) 0 0
\(217\) 1075.54 + 1862.89i 0.336464 + 0.582772i
\(218\) 0 0
\(219\) −4488.25 3766.09i −1.38488 1.16205i
\(220\) 0 0
\(221\) 379.926 658.052i 0.115641 0.200296i
\(222\) 0 0
\(223\) −110.607 627.282i −0.0332142 0.188367i 0.963687 0.267036i \(-0.0860442\pi\)
−0.996901 + 0.0786685i \(0.974933\pi\)
\(224\) 0 0
\(225\) −469.472 170.874i −0.139103 0.0506293i
\(226\) 0 0
\(227\) 4874.69 1.42531 0.712653 0.701516i \(-0.247493\pi\)
0.712653 + 0.701516i \(0.247493\pi\)
\(228\) 0 0
\(229\) −3195.65 −0.922158 −0.461079 0.887359i \(-0.652538\pi\)
−0.461079 + 0.887359i \(0.652538\pi\)
\(230\) 0 0
\(231\) 905.443 + 329.554i 0.257895 + 0.0938662i
\(232\) 0 0
\(233\) −966.384 5480.63i −0.271716 1.54098i −0.749203 0.662340i \(-0.769563\pi\)
0.477487 0.878639i \(-0.341548\pi\)
\(234\) 0 0
\(235\) −3338.14 + 5781.82i −0.926621 + 1.60496i
\(236\) 0 0
\(237\) 2931.96 + 2460.21i 0.803592 + 0.674294i
\(238\) 0 0
\(239\) −2643.59 4578.84i −0.715480 1.23925i −0.962774 0.270308i \(-0.912875\pi\)
0.247294 0.968941i \(-0.420459\pi\)
\(240\) 0 0
\(241\) 6315.36 2298.60i 1.68800 0.614382i 0.693630 0.720332i \(-0.256010\pi\)
0.994372 + 0.105949i \(0.0337881\pi\)
\(242\) 0 0
\(243\) 736.421 4176.45i 0.194409 1.10255i
\(244\) 0 0
\(245\) −2462.53 + 2066.31i −0.642144 + 0.538823i
\(246\) 0 0
\(247\) 1993.04 580.396i 0.513417 0.149513i
\(248\) 0 0
\(249\) −4818.71 + 4043.37i −1.22640 + 1.02907i
\(250\) 0 0
\(251\) −114.775 + 650.924i −0.0288628 + 0.163689i −0.995832 0.0912030i \(-0.970929\pi\)
0.966970 + 0.254892i \(0.0820399\pi\)
\(252\) 0 0
\(253\) 187.111 68.1030i 0.0464964 0.0169233i
\(254\) 0 0
\(255\) 1252.12 + 2168.74i 0.307494 + 0.532595i
\(256\) 0 0
\(257\) −1324.92 1111.74i −0.321582 0.269839i 0.467678 0.883899i \(-0.345091\pi\)
−0.789259 + 0.614060i \(0.789535\pi\)
\(258\) 0 0
\(259\) 1547.40 2680.17i 0.371237 0.643002i
\(260\) 0 0
\(261\) −311.325 1765.61i −0.0738334 0.418730i
\(262\) 0 0
\(263\) 647.332 + 235.610i 0.151773 + 0.0552407i 0.416789 0.909003i \(-0.363155\pi\)
−0.265017 + 0.964244i \(0.585377\pi\)
\(264\) 0 0
\(265\) −6653.18 −1.54227
\(266\) 0 0
\(267\) 904.063 0.207220
\(268\) 0 0
\(269\) −6231.45 2268.06i −1.41241 0.514075i −0.480573 0.876955i \(-0.659571\pi\)
−0.931836 + 0.362880i \(0.881794\pi\)
\(270\) 0 0
\(271\) 713.033 + 4043.81i 0.159829 + 0.906435i 0.954237 + 0.299051i \(0.0966701\pi\)
−0.794408 + 0.607384i \(0.792219\pi\)
\(272\) 0 0
\(273\) −764.035 + 1323.35i −0.169383 + 0.293379i
\(274\) 0 0
\(275\) 347.929 + 291.947i 0.0762942 + 0.0640185i
\(276\) 0 0
\(277\) 3816.25 + 6609.93i 0.827784 + 1.43376i 0.899773 + 0.436358i \(0.143732\pi\)
−0.0719893 + 0.997405i \(0.522935\pi\)
\(278\) 0 0
\(279\) 3840.27 1397.74i 0.824053 0.299931i
\(280\) 0 0
\(281\) 605.923 3436.36i 0.128635 0.729523i −0.850448 0.526060i \(-0.823669\pi\)
0.979082 0.203464i \(-0.0652200\pi\)
\(282\) 0 0
\(283\) 40.7950 34.2310i 0.00856894 0.00719019i −0.638493 0.769628i \(-0.720442\pi\)
0.647062 + 0.762437i \(0.275997\pi\)
\(284\) 0 0
\(285\) −1627.69 + 6644.86i −0.338301 + 1.38108i
\(286\) 0 0
\(287\) 2699.32 2265.00i 0.555177 0.465849i
\(288\) 0 0
\(289\) −693.543 + 3933.28i −0.141165 + 0.800586i
\(290\) 0 0
\(291\) 4294.58 1563.10i 0.865130 0.314882i
\(292\) 0 0
\(293\) 2814.07 + 4874.11i 0.561091 + 0.971838i 0.997402 + 0.0720418i \(0.0229515\pi\)
−0.436311 + 0.899796i \(0.643715\pi\)
\(294\) 0 0
\(295\) 3793.60 + 3183.21i 0.748718 + 0.628249i
\(296\) 0 0
\(297\) −506.199 + 876.763i −0.0988979 + 0.171296i
\(298\) 0 0
\(299\) 54.8343 + 310.981i 0.0106058 + 0.0601488i
\(300\) 0 0
\(301\) 4126.47 + 1501.91i 0.790185 + 0.287604i
\(302\) 0 0
\(303\) 1608.01 0.304877
\(304\) 0 0
\(305\) −939.741 −0.176424
\(306\) 0 0
\(307\) 9833.35 + 3579.05i 1.82808 + 0.665365i 0.993411 + 0.114606i \(0.0365605\pi\)
0.834664 + 0.550759i \(0.185662\pi\)
\(308\) 0 0
\(309\) −777.800 4411.12i −0.143196 0.812104i
\(310\) 0 0
\(311\) −2428.55 + 4206.37i −0.442799 + 0.766950i −0.997896 0.0648353i \(-0.979348\pi\)
0.555097 + 0.831786i \(0.312681\pi\)
\(312\) 0 0
\(313\) −254.377 213.447i −0.0459368 0.0385456i 0.619530 0.784973i \(-0.287323\pi\)
−0.665467 + 0.746427i \(0.731768\pi\)
\(314\) 0 0
\(315\) −986.285 1708.30i −0.176415 0.305561i
\(316\) 0 0
\(317\) 5714.93 2080.06i 1.01256 0.368543i 0.218146 0.975916i \(-0.429999\pi\)
0.794416 + 0.607373i \(0.207777\pi\)
\(318\) 0 0
\(319\) −283.026 + 1605.12i −0.0496753 + 0.281723i
\(320\) 0 0
\(321\) −544.059 + 456.520i −0.0945994 + 0.0793784i
\(322\) 0 0
\(323\) 2025.12 1484.13i 0.348856 0.255662i
\(324\) 0 0
\(325\) −551.770 + 462.990i −0.0941745 + 0.0790218i
\(326\) 0 0
\(327\) −628.841 + 3566.34i −0.106346 + 0.603116i
\(328\) 0 0
\(329\) −4630.13 + 1685.23i −0.775888 + 0.282400i
\(330\) 0 0
\(331\) −2721.96 4714.58i −0.452002 0.782891i 0.546508 0.837454i \(-0.315957\pi\)
−0.998510 + 0.0545631i \(0.982623\pi\)
\(332\) 0 0
\(333\) −4504.05 3779.35i −0.741203 0.621943i
\(334\) 0 0
\(335\) 1299.76 2251.24i 0.211980 0.367160i
\(336\) 0 0
\(337\) 299.208 + 1696.90i 0.0483648 + 0.274290i 0.999394 0.0348110i \(-0.0110829\pi\)
−0.951029 + 0.309101i \(0.899972\pi\)
\(338\) 0 0
\(339\) −12526.0 4559.08i −2.00684 0.730429i
\(340\) 0 0
\(341\) −3715.26 −0.590007
\(342\) 0 0
\(343\) −5511.22 −0.867574
\(344\) 0 0
\(345\) −977.947 355.943i −0.152611 0.0555460i
\(346\) 0 0
\(347\) −568.366 3223.37i −0.0879294 0.498672i −0.996686 0.0813458i \(-0.974078\pi\)
0.908757 0.417327i \(-0.137033\pi\)
\(348\) 0 0
\(349\) 1000.80 1733.44i 0.153501 0.265871i −0.779011 0.627010i \(-0.784279\pi\)
0.932512 + 0.361139i \(0.117612\pi\)
\(350\) 0 0
\(351\) −1229.91 1032.02i −0.187031 0.156937i
\(352\) 0 0
\(353\) 240.987 + 417.402i 0.0363355 + 0.0629350i 0.883621 0.468202i \(-0.155098\pi\)
−0.847286 + 0.531137i \(0.821765\pi\)
\(354\) 0 0
\(355\) −9047.69 + 3293.09i −1.35268 + 0.492336i
\(356\) 0 0
\(357\) −320.937 + 1820.13i −0.0475793 + 0.269836i
\(358\) 0 0
\(359\) 8141.14 6831.23i 1.19686 1.00429i 0.197147 0.980374i \(-0.436832\pi\)
0.999714 0.0239119i \(-0.00761212\pi\)
\(360\) 0 0
\(361\) 6799.89 + 898.561i 0.991382 + 0.131005i
\(362\) 0 0
\(363\) 5517.98 4630.14i 0.797849 0.669474i
\(364\) 0 0
\(365\) 1893.49 10738.5i 0.271534 1.53994i
\(366\) 0 0
\(367\) −8677.98 + 3158.52i −1.23430 + 0.449247i −0.875067 0.484002i \(-0.839183\pi\)
−0.359229 + 0.933249i \(0.616960\pi\)
\(368\) 0 0
\(369\) −3347.25 5797.61i −0.472225 0.817917i
\(370\) 0 0
\(371\) −3761.46 3156.24i −0.526375 0.441681i
\(372\) 0 0
\(373\) 2181.71 3778.84i 0.302855 0.524560i −0.673927 0.738798i \(-0.735394\pi\)
0.976781 + 0.214238i \(0.0687270\pi\)
\(374\) 0 0
\(375\) 1380.82 + 7831.03i 0.190148 + 1.07838i
\(376\) 0 0
\(377\) −2428.90 884.048i −0.331817 0.120771i
\(378\) 0 0
\(379\) −10524.7 −1.42644 −0.713218 0.700942i \(-0.752763\pi\)
−0.713218 + 0.700942i \(0.752763\pi\)
\(380\) 0 0
\(381\) 8938.97 1.20199
\(382\) 0 0
\(383\) 5547.15 + 2019.00i 0.740069 + 0.269363i 0.684421 0.729087i \(-0.260055\pi\)
0.0556482 + 0.998450i \(0.482277\pi\)
\(384\) 0 0
\(385\) 311.398 + 1766.03i 0.0412216 + 0.233779i
\(386\) 0 0
\(387\) 4171.39 7225.06i 0.547916 0.949019i
\(388\) 0 0
\(389\) −2731.85 2292.29i −0.356067 0.298776i 0.447154 0.894457i \(-0.352438\pi\)
−0.803221 + 0.595681i \(0.796882\pi\)
\(390\) 0 0
\(391\) 190.967 + 330.765i 0.0246998 + 0.0427814i
\(392\) 0 0
\(393\) −3290.17 + 1197.52i −0.422308 + 0.153708i
\(394\) 0 0
\(395\) −1236.93 + 7014.97i −0.157561 + 0.893573i
\(396\) 0 0
\(397\) 1115.03 935.617i 0.140961 0.118280i −0.569581 0.821935i \(-0.692894\pi\)
0.710542 + 0.703655i \(0.248450\pi\)
\(398\) 0 0
\(399\) −4072.52 + 2984.59i −0.510980 + 0.374477i
\(400\) 0 0
\(401\) 5830.05 4891.99i 0.726032 0.609213i −0.203015 0.979176i \(-0.565074\pi\)
0.929047 + 0.369963i \(0.120630\pi\)
\(402\) 0 0
\(403\) 1023.12 5802.40i 0.126465 0.717217i
\(404\) 0 0
\(405\) 10441.4 3800.35i 1.28108 0.466274i
\(406\) 0 0
\(407\) 2672.59 + 4629.06i 0.325492 + 0.563769i
\(408\) 0 0
\(409\) 7178.29 + 6023.30i 0.867832 + 0.728198i 0.963641 0.267202i \(-0.0860991\pi\)
−0.0958082 + 0.995400i \(0.530544\pi\)
\(410\) 0 0
\(411\) −8472.74 + 14675.2i −1.01686 + 1.76125i
\(412\) 0 0
\(413\) 634.660 + 3599.33i 0.0756164 + 0.428842i
\(414\) 0 0
\(415\) −11001.0 4004.04i −1.30125 0.473616i
\(416\) 0 0
\(417\) 4723.32 0.554681
\(418\) 0 0
\(419\) 5260.54 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(420\) 0 0
\(421\) −9246.90 3365.60i −1.07047 0.389618i −0.254115 0.967174i \(-0.581784\pi\)
−0.816351 + 0.577556i \(0.804007\pi\)
\(422\) 0 0
\(423\) 1625.53 + 9218.85i 0.186847 + 1.05966i
\(424\) 0 0
\(425\) −435.593 + 754.470i −0.0497162 + 0.0861110i
\(426\) 0 0
\(427\) −531.294 445.809i −0.0602134 0.0505251i
\(428\) 0 0
\(429\) −1319.61 2285.62i −0.148511 0.257228i
\(430\) 0 0
\(431\) −8194.59 + 2982.59i −0.915823 + 0.333332i −0.756575 0.653906i \(-0.773129\pi\)
−0.159247 + 0.987239i \(0.550907\pi\)
\(432\) 0 0
\(433\) −514.972 + 2920.55i −0.0571546 + 0.324140i −0.999958 0.00921057i \(-0.997068\pi\)
0.942803 + 0.333351i \(0.108179\pi\)
\(434\) 0 0
\(435\) 6525.68 5475.69i 0.719270 0.603539i
\(436\) 0 0
\(437\) −248.246 + 1013.44i −0.0271744 + 0.110937i
\(438\) 0 0
\(439\) 6639.59 5571.28i 0.721846 0.605701i −0.206049 0.978542i \(-0.566061\pi\)
0.927895 + 0.372841i \(0.121616\pi\)
\(440\) 0 0
\(441\) −782.688 + 4438.85i −0.0845145 + 0.479305i
\(442\) 0 0
\(443\) −8523.60 + 3102.34i −0.914150 + 0.332723i −0.755909 0.654677i \(-0.772805\pi\)
−0.158241 + 0.987401i \(0.550582\pi\)
\(444\) 0 0
\(445\) 841.276 + 1457.13i 0.0896187 + 0.155224i
\(446\) 0 0
\(447\) −359.262 301.457i −0.0380146 0.0318980i
\(448\) 0 0
\(449\) 3757.00 6507.31i 0.394886 0.683962i −0.598201 0.801346i \(-0.704118\pi\)
0.993087 + 0.117384i \(0.0374508\pi\)
\(450\) 0 0
\(451\) 1056.82 + 5993.53i 0.110341 + 0.625775i
\(452\) 0 0
\(453\) −3336.95 1214.55i −0.346101 0.125970i
\(454\) 0 0
\(455\) −2843.89 −0.293019
\(456\) 0 0
\(457\) 15863.4 1.62376 0.811882 0.583822i \(-0.198443\pi\)
0.811882 + 0.583822i \(0.198443\pi\)
\(458\) 0 0
\(459\) −1824.79 664.168i −0.185564 0.0675397i
\(460\) 0 0
\(461\) −3070.27 17412.3i −0.310188 1.75916i −0.598020 0.801481i \(-0.704046\pi\)
0.287833 0.957681i \(-0.407065\pi\)
\(462\) 0 0
\(463\) 7521.42 13027.5i 0.754968 1.30764i −0.190422 0.981702i \(-0.560986\pi\)
0.945390 0.325940i \(-0.105681\pi\)
\(464\) 0 0
\(465\) 14875.0 + 12481.6i 1.48347 + 1.24478i
\(466\) 0 0
\(467\) 5537.24 + 9590.79i 0.548679 + 0.950340i 0.998365 + 0.0571535i \(0.0182024\pi\)
−0.449686 + 0.893187i \(0.648464\pi\)
\(468\) 0 0
\(469\) 1802.81 656.170i 0.177497 0.0646037i
\(470\) 0 0
\(471\) −1668.43 + 9462.15i −0.163222 + 0.925675i
\(472\) 0 0
\(473\) −5810.02 + 4875.19i −0.564789 + 0.473914i
\(474\) 0 0
\(475\) −2285.06 + 665.436i −0.220728 + 0.0642785i
\(476\) 0 0
\(477\) −7146.19 + 5996.37i −0.685957 + 0.575586i
\(478\) 0 0
\(479\) −1248.65 + 7081.45i −0.119107 + 0.675490i 0.865527 + 0.500862i \(0.166983\pi\)
−0.984634 + 0.174628i \(0.944128\pi\)
\(480\) 0 0
\(481\) −7965.55 + 2899.22i −0.755089 + 0.274830i
\(482\) 0 0
\(483\) −384.037 665.171i −0.0361786 0.0626632i
\(484\) 0 0
\(485\) 6515.67 + 5467.30i 0.610024 + 0.511871i
\(486\) 0 0
\(487\) 395.052 684.249i 0.0367587 0.0636680i −0.847061 0.531496i \(-0.821630\pi\)
0.883819 + 0.467828i \(0.154963\pi\)
\(488\) 0 0
\(489\) 1088.95 + 6175.77i 0.100704 + 0.571121i
\(490\) 0 0
\(491\) 12995.0 + 4729.80i 1.19441 + 0.434731i 0.861272 0.508145i \(-0.169669\pi\)
0.333143 + 0.942876i \(0.391891\pi\)
\(492\) 0 0
\(493\) −3126.30 −0.285602
\(494\) 0 0
\(495\) 3406.93 0.309354
\(496\) 0 0
\(497\) −6677.45 2430.39i −0.602666 0.219352i
\(498\) 0 0
\(499\) −24.6955 140.055i −0.00221548 0.0125646i 0.983680 0.179927i \(-0.0575861\pi\)
−0.985895 + 0.167362i \(0.946475\pi\)
\(500\) 0 0
\(501\) 4839.92 8382.99i 0.431600 0.747554i
\(502\) 0 0
\(503\) −7413.09 6220.32i −0.657124 0.551393i 0.252099 0.967701i \(-0.418879\pi\)
−0.909223 + 0.416309i \(0.863324\pi\)
\(504\) 0 0
\(505\) 1496.33 + 2591.73i 0.131853 + 0.228377i
\(506\) 0 0
\(507\) −9821.16 + 3574.61i −0.860302 + 0.313124i
\(508\) 0 0
\(509\) −3636.15 + 20621.6i −0.316639 + 1.79575i 0.246238 + 0.969209i \(0.420806\pi\)
−0.562877 + 0.826541i \(0.690306\pi\)
\(510\) 0 0
\(511\) 6164.82 5172.89i 0.533690 0.447819i
\(512\) 0 0
\(513\) −2345.21 4758.51i −0.201839 0.409539i
\(514\) 0 0
\(515\) 6385.89 5358.40i 0.546400 0.458484i
\(516\) 0 0
\(517\) 1477.78 8380.89i 0.125711 0.712942i
\(518\) 0 0
\(519\) 1864.36 678.571i 0.157681 0.0573911i
\(520\) 0 0
\(521\) −1757.60 3044.25i −0.147796 0.255991i 0.782616 0.622504i \(-0.213885\pi\)
−0.930413 + 0.366514i \(0.880551\pi\)
\(522\) 0 0
\(523\) 6716.93 + 5636.17i 0.561589 + 0.471229i 0.878843 0.477112i \(-0.158316\pi\)
−0.317254 + 0.948341i \(0.602761\pi\)
\(524\) 0 0
\(525\) 875.981 1517.24i 0.0728209 0.126129i
\(526\) 0 0
\(527\) −1237.47 7018.02i −0.102286 0.580095i
\(528\) 0 0
\(529\) 11284.1 + 4107.07i 0.927434 + 0.337558i
\(530\) 0 0
\(531\) 6943.67 0.567475
\(532\) 0 0
\(533\) −9651.59 −0.784347
\(534\) 0 0
\(535\) −1242.08 452.079i −0.100373 0.0365328i
\(536\) 0 0
\(537\) 527.945 + 2994.13i 0.0424255 + 0.240607i
\(538\) 0 0
\(539\) 2048.81 3548.64i 0.163726 0.283582i
\(540\) 0 0
\(541\) −2253.01 1890.50i −0.179047 0.150238i 0.548860 0.835914i \(-0.315062\pi\)
−0.727907 + 0.685676i \(0.759507\pi\)
\(542\) 0 0
\(543\) −11315.7 19599.4i −0.894300 1.54897i
\(544\) 0 0
\(545\) −6333.25 + 2305.12i −0.497774 + 0.181175i
\(546\) 0 0
\(547\) 1971.24 11179.5i 0.154085 0.873857i −0.805533 0.592550i \(-0.798121\pi\)
0.959618 0.281307i \(-0.0907678\pi\)
\(548\) 0 0
\(549\) −1009.38 + 846.968i −0.0784684 + 0.0658428i
\(550\) 0 0
\(551\) −6168.07 5907.49i −0.476894 0.456747i
\(552\) 0 0
\(553\) −4027.18 + 3379.21i −0.309680 + 0.259853i
\(554\) 0 0
\(555\) 4851.18 27512.4i 0.371029 2.10421i
\(556\) 0 0
\(557\) −8365.06 + 3044.63i −0.636336 + 0.231607i −0.639987 0.768386i \(-0.721060\pi\)
0.00365087 + 0.999993i \(0.498838\pi\)
\(558\) 0 0
\(559\) −6013.97 10416.5i −0.455034 0.788142i
\(560\) 0 0
\(561\) −2445.32 2051.86i −0.184031 0.154420i
\(562\) 0 0
\(563\) −8827.77 + 15290.1i −0.660828 + 1.14459i 0.319571 + 0.947562i \(0.396461\pi\)
−0.980398 + 0.197025i \(0.936872\pi\)
\(564\) 0 0
\(565\) −4307.90 24431.3i −0.320770 1.81918i
\(566\) 0 0
\(567\) 7706.04 + 2804.77i 0.570764 + 0.207741i
\(568\) 0 0
\(569\) 12764.0 0.940412 0.470206 0.882557i \(-0.344180\pi\)
0.470206 + 0.882557i \(0.344180\pi\)
\(570\) 0 0
\(571\) −26572.6 −1.94751 −0.973755 0.227597i \(-0.926913\pi\)
−0.973755 + 0.227597i \(0.926913\pi\)
\(572\) 0 0
\(573\) −16563.9 6028.76i −1.20762 0.439538i
\(574\) 0 0
\(575\) −62.8686 356.546i −0.00455966 0.0258591i
\(576\) 0 0
\(577\) −11865.9 + 20552.3i −0.856121 + 1.48285i 0.0194801 + 0.999810i \(0.493799\pi\)
−0.875601 + 0.483035i \(0.839534\pi\)
\(578\) 0 0
\(579\) 10762.8 + 9031.09i 0.772519 + 0.648220i
\(580\) 0 0
\(581\) −4320.06 7482.56i −0.308479 0.534301i
\(582\) 0 0
\(583\) 7969.29 2900.58i 0.566131 0.206055i
\(584\) 0 0
\(585\) −938.213 + 5320.87i −0.0663083 + 0.376053i
\(586\) 0 0
\(587\) −10707.0 + 8984.23i −0.752853 + 0.631719i −0.936256 0.351319i \(-0.885733\pi\)
0.183403 + 0.983038i \(0.441289\pi\)
\(588\) 0 0
\(589\) 10819.9 16184.6i 0.756917 1.13222i
\(590\) 0 0
\(591\) −21625.0 + 18145.5i −1.50513 + 1.26295i
\(592\) 0 0
\(593\) −1694.21 + 9608.33i −0.117323 + 0.665374i 0.868250 + 0.496127i \(0.165245\pi\)
−0.985574 + 0.169248i \(0.945866\pi\)
\(594\) 0 0
\(595\) −3232.26 + 1176.45i −0.222705 + 0.0810581i
\(596\) 0 0
\(597\) −17320.6 30000.2i −1.18741 2.05666i
\(598\) 0 0
\(599\) 15432.1 + 12949.1i 1.05265 + 0.883281i 0.993370 0.114961i \(-0.0366742\pi\)
0.0592828 + 0.998241i \(0.481119\pi\)
\(600\) 0 0
\(601\) 13140.4 22759.9i 0.891862 1.54475i 0.0542207 0.998529i \(-0.482733\pi\)
0.837641 0.546221i \(-0.183934\pi\)
\(602\) 0 0
\(603\) −632.926 3589.50i −0.0427442 0.242414i
\(604\) 0 0
\(605\) 12597.4 + 4585.09i 0.846544 + 0.308117i
\(606\) 0 0
\(607\) 2188.00 0.146307 0.0731533 0.997321i \(-0.476694\pi\)
0.0731533 + 0.997321i \(0.476694\pi\)
\(608\) 0 0
\(609\) 6287.02 0.418330
\(610\) 0 0
\(611\) 12682.1 + 4615.91i 0.839711 + 0.305630i
\(612\) 0 0
\(613\) 2331.96 + 13225.2i 0.153649 + 0.871390i 0.960010 + 0.279965i \(0.0903230\pi\)
−0.806361 + 0.591424i \(0.798566\pi\)
\(614\) 0 0
\(615\) 15904.4 27547.2i 1.04281 1.80619i
\(616\) 0 0
\(617\) 14320.4 + 12016.3i 0.934389 + 0.784046i 0.976600 0.215063i \(-0.0689957\pi\)
−0.0422109 + 0.999109i \(0.513440\pi\)
\(618\) 0 0
\(619\) 13935.1 + 24136.3i 0.904846 + 1.56724i 0.821123 + 0.570751i \(0.193348\pi\)
0.0837230 + 0.996489i \(0.473319\pi\)
\(620\) 0 0
\(621\) 758.341 276.014i 0.0490035 0.0178358i
\(622\) 0 0
\(623\) −215.631 + 1222.91i −0.0138669 + 0.0786432i
\(624\) 0 0
\(625\) −14088.6 + 11821.7i −0.901668 + 0.756589i
\(626\) 0 0
\(627\) −947.286 8668.94i −0.0603364 0.552159i
\(628\) 0 0
\(629\) −7854.00 + 6590.29i −0.497869 + 0.417762i
\(630\) 0 0
\(631\) 386.405 2191.41i 0.0243780 0.138255i −0.970190 0.242347i \(-0.922083\pi\)
0.994568 + 0.104093i \(0.0331938\pi\)
\(632\) 0 0
\(633\) −11377.8 + 4141.17i −0.714417 + 0.260027i
\(634\) 0 0
\(635\) 8318.16 + 14407.5i 0.519837 + 0.900383i
\(636\) 0 0
\(637\) 4977.98 + 4177.02i 0.309631 + 0.259811i
\(638\) 0 0
\(639\) −6750.15 + 11691.6i −0.417890 + 0.723807i
\(640\) 0 0
\(641\) −392.851 2227.97i −0.0242070 0.137285i 0.970309 0.241868i \(-0.0777602\pi\)
−0.994516 + 0.104584i \(0.966649\pi\)
\(642\) 0 0
\(643\) 19158.6 + 6973.16i 1.17503 + 0.427674i 0.854442 0.519546i \(-0.173899\pi\)
0.320583 + 0.947220i \(0.396121\pi\)
\(644\) 0 0
\(645\) 39640.5 2.41991
\(646\) 0 0
\(647\) 8354.21 0.507632 0.253816 0.967252i \(-0.418314\pi\)
0.253816 + 0.967252i \(0.418314\pi\)
\(648\) 0 0
\(649\) −5931.82 2159.00i −0.358774 0.130583i
\(650\) 0 0
\(651\) 2488.55 + 14113.3i 0.149822 + 0.849683i
\(652\) 0 0
\(653\) −746.125 + 1292.33i −0.0447138 + 0.0774466i −0.887516 0.460777i \(-0.847571\pi\)
0.842802 + 0.538223i \(0.180904\pi\)
\(654\) 0 0
\(655\) −4991.79 4188.61i −0.297779 0.249867i
\(656\) 0 0
\(657\) −7644.59 13240.8i −0.453948 0.786261i
\(658\) 0 0
\(659\) −5538.44 + 2015.83i −0.327385 + 0.119159i −0.500484 0.865746i \(-0.666844\pi\)
0.173098 + 0.984905i \(0.444622\pi\)
\(660\) 0 0
\(661\) −2332.97 + 13230.9i −0.137280 + 0.778554i 0.835965 + 0.548783i \(0.184909\pi\)
−0.973245 + 0.229771i \(0.926202\pi\)
\(662\) 0 0
\(663\) 3877.95 3253.99i 0.227160 0.190610i
\(664\) 0 0
\(665\) −8600.13 3786.63i −0.501502 0.220811i
\(666\) 0 0
\(667\) 995.263 835.125i 0.0577762 0.0484800i
\(668\) 0 0
\(669\) 736.887 4179.09i 0.0425855 0.241514i
\(670\) 0 0
\(671\) 1125.64 409.698i 0.0647611 0.0235711i
\(672\) 0 0
\(673\) −3402.96 5894.10i −0.194910 0.337594i 0.751961 0.659208i \(-0.229108\pi\)
−0.946871 + 0.321613i \(0.895775\pi\)
\(674\) 0 0
\(675\) 1410.12 + 1183.23i 0.0804080 + 0.0674703i
\(676\) 0 0
\(677\) −9972.07 + 17272.1i −0.566112 + 0.980534i 0.430834 + 0.902431i \(0.358220\pi\)
−0.996945 + 0.0781028i \(0.975114\pi\)
\(678\) 0 0
\(679\) 1090.06 + 6182.01i 0.0616090 + 0.349402i
\(680\) 0 0
\(681\) 30517.7 + 11107.5i 1.71724 + 0.625025i
\(682\) 0 0
\(683\) 5186.63 0.290572 0.145286 0.989390i \(-0.453590\pi\)
0.145286 + 0.989390i \(0.453590\pi\)
\(684\) 0 0
\(685\) −31537.3 −1.75909
\(686\) 0 0
\(687\) −20006.2 7281.65i −1.11104 0.404385i
\(688\) 0 0
\(689\) 2335.45 + 13245.0i 0.129135 + 0.732359i
\(690\) 0 0
\(691\) 13242.4 22936.5i 0.729038 1.26273i −0.228253 0.973602i \(-0.573301\pi\)
0.957290 0.289128i \(-0.0933654\pi\)
\(692\) 0 0
\(693\) 1926.15 + 1616.23i 0.105582 + 0.0885940i
\(694\) 0 0
\(695\) 4395.29 + 7612.86i 0.239889 + 0.415500i
\(696\) 0 0
\(697\) −10969.6 + 3992.62i −0.596132 + 0.216974i
\(698\) 0 0
\(699\) 6438.26 36513.2i 0.348380 1.97576i
\(700\) 0 0
\(701\) −22689.8 + 19039.0i −1.22251 + 1.02581i −0.223824 + 0.974630i \(0.571854\pi\)
−0.998689 + 0.0511810i \(0.983701\pi\)
\(702\) 0 0
\(703\) −27948.7 1838.63i −1.49944 0.0986418i
\(704\) 0 0
\(705\) −34072.8 + 28590.4i −1.82022 + 1.52735i
\(706\) 0 0
\(707\) −383.532 + 2175.12i −0.0204020 + 0.115705i
\(708\) 0 0
\(709\) 24656.2 8974.11i 1.30604 0.475359i 0.407080 0.913392i \(-0.366547\pi\)
0.898959 + 0.438033i \(0.144325\pi\)
\(710\) 0 0
\(711\) 4993.85 + 8649.60i 0.263409 + 0.456238i
\(712\) 0 0
\(713\) 2268.66 + 1903.63i 0.119161 + 0.0999883i
\(714\) 0 0
\(715\) 2455.92 4253.78i 0.128456 0.222493i
\(716\) 0 0
\(717\) −6116.66 34689.3i −0.318592 1.80683i
\(718\) 0 0
\(719\) 20946.6 + 7623.93i 1.08648 + 0.395445i 0.822314 0.569034i \(-0.192683\pi\)
0.264161 + 0.964479i \(0.414905\pi\)
\(720\) 0 0
\(721\) 6152.35 0.317788
\(722\) 0 0
\(723\) 44774.6 2.30316
\(724\) 0 0
\(725\) 2784.79 + 1013.58i 0.142654 + 0.0519219i
\(726\) 0 0
\(727\) −906.627 5141.74i −0.0462516 0.262306i 0.952910 0.303254i \(-0.0980732\pi\)
−0.999161 + 0.0409484i \(0.986962\pi\)
\(728\) 0 0
\(729\) 2028.76 3513.91i 0.103072 0.178525i
\(730\) 0 0
\(731\) −11144.3 9351.17i −0.563866 0.473140i
\(732\) 0 0
\(733\) −9.61916 16.6609i −0.000484709 0.000839541i 0.865783 0.500420i \(-0.166821\pi\)
−0.866268 + 0.499580i \(0.833488\pi\)
\(734\) 0 0
\(735\) −20124.8 + 7324.84i −1.00995 + 0.367593i
\(736\) 0 0
\(737\) −575.395 + 3263.23i −0.0287584 + 0.163097i
\(738\) 0 0
\(739\) −7023.94 + 5893.79i −0.349634 + 0.293378i −0.800643 0.599141i \(-0.795509\pi\)
0.451009 + 0.892519i \(0.351064\pi\)
\(740\) 0 0
\(741\) 13799.8 + 907.832i 0.684141 + 0.0450068i
\(742\) 0 0
\(743\) 25389.8 21304.6i 1.25365 1.05194i 0.257320 0.966326i \(-0.417161\pi\)
0.996329 0.0856096i \(-0.0272838\pi\)
\(744\) 0 0
\(745\) 151.565 859.566i 0.00745356 0.0422712i
\(746\) 0 0
\(747\) −15424.9 + 5614.22i −0.755514 + 0.274985i
\(748\) 0 0
\(749\) −487.759 844.824i −0.0237948 0.0412139i
\(750\) 0 0
\(751\) 14825.3 + 12439.9i 0.720348 + 0.604443i 0.927481 0.373869i \(-0.121969\pi\)
−0.207134 + 0.978313i \(0.566414\pi\)
\(752\) 0 0
\(753\) −2201.75 + 3813.54i −0.106555 + 0.184559i
\(754\) 0 0
\(755\) −1147.64 6508.57i −0.0553203 0.313737i
\(756\) 0 0
\(757\) 8679.37 + 3159.03i 0.416720 + 0.151674i 0.541866 0.840465i \(-0.317718\pi\)
−0.125146 + 0.992138i \(0.539940\pi\)
\(758\) 0 0
\(759\) 1326.58 0.0634411
\(760\) 0 0
\(761\) −18921.9 −0.901336 −0.450668 0.892692i \(-0.648814\pi\)
−0.450668 + 0.892692i \(0.648814\pi\)
\(762\) 0 0
\(763\) −4674.12 1701.24i −0.221775 0.0807196i
\(764\) 0 0
\(765\) 1134.77 + 6435.61i 0.0536310 + 0.304157i
\(766\) 0 0
\(767\) 5005.41 8669.62i 0.235639 0.408138i
\(768\) 0 0
\(769\) 1729.72 + 1451.41i 0.0811123 + 0.0680613i 0.682442 0.730939i \(-0.260918\pi\)
−0.601330 + 0.799001i \(0.705362\pi\)
\(770\) 0 0
\(771\) −5761.38 9979.00i −0.269119 0.466128i
\(772\) 0 0
\(773\) 7074.05 2574.74i 0.329154 0.119802i −0.172157 0.985070i \(-0.555074\pi\)
0.501311 + 0.865267i \(0.332851\pi\)
\(774\) 0 0
\(775\) −1173.03 + 6652.57i −0.0543696 + 0.308345i
\(776\) 0 0
\(777\) 15794.5 13253.1i 0.729244 0.611909i
\(778\) 0 0
\(779\) −29187.1 12851.0i −1.34241 0.591060i
\(780\) 0 0
\(781\) 9401.78 7889.03i 0.430758 0.361449i
\(782\) 0 0
\(783\) −1147.07 + 6505.38i −0.0523538 + 0.296913i
\(784\) 0 0
\(785\) −16803.3 + 6115.90i −0.763994 + 0.278071i
\(786\) 0 0
\(787\) −4288.31 7427.58i −0.194234 0.336423i 0.752415 0.658689i \(-0.228889\pi\)
−0.946649 + 0.322266i \(0.895555\pi\)
\(788\) 0 0
\(789\) 3515.72 + 2950.04i 0.158635 + 0.133111i
\(790\) 0 0
\(791\) 9154.59 15856.2i 0.411504 0.712746i
\(792\) 0 0
\(793\) 329.876 + 1870.82i 0.0147720 + 0.0837764i
\(794\) 0 0
\(795\) −41651.9 15160.0i −1.85816 0.676316i
\(796\) 0 0
\(797\) −3324.77 −0.147766 −0.0738828 0.997267i \(-0.523539\pi\)
−0.0738828 + 0.997267i \(0.523539\pi\)
\(798\) 0 0
\(799\) 16323.5 0.722758
\(800\) 0 0
\(801\) 2216.90 + 806.885i 0.0977906 + 0.0355929i
\(802\) 0 0
\(803\) 2413.61 + 13688.3i 0.106070 + 0.601555i
\(804\) 0 0
\(805\) 714.731 1237.95i 0.0312931 0.0542013i
\(806\) 0 0
\(807\) −33843.6 28398.1i −1.47627 1.23874i
\(808\) 0 0
\(809\) −15652.3 27110.6i −0.680229 1.17819i −0.974911 0.222596i \(-0.928547\pi\)
0.294682 0.955595i \(-0.404786\pi\)
\(810\) 0 0
\(811\) −33322.3 + 12128.3i −1.44279 + 0.525133i −0.940568 0.339606i \(-0.889706\pi\)
−0.502223 + 0.864738i \(0.667484\pi\)
\(812\) 0 0
\(813\) −4750.39 + 26940.8i −0.204924 + 1.16218i
\(814\) 0 0
\(815\) −8940.54 + 7502.00i −0.384262 + 0.322434i
\(816\) 0 0
\(817\) −4317.16 39507.8i −0.184869 1.69180i
\(818\) 0 0
\(819\) −3054.63 + 2563.14i −0.130326 + 0.109357i
\(820\) 0 0
\(821\) 7023.01 39829.5i 0.298544 1.69313i −0.353894 0.935286i \(-0.615143\pi\)
0.652438 0.757842i \(-0.273746\pi\)
\(822\) 0 0
\(823\) 11727.4 4268.41i 0.496707 0.180787i −0.0815050 0.996673i \(-0.525973\pi\)
0.578213 + 0.815886i \(0.303750\pi\)
\(824\) 0 0
\(825\) 1512.96 + 2620.51i 0.0638477 + 0.110587i
\(826\) 0 0
\(827\) −8391.47 7041.28i −0.352842 0.296069i 0.449088 0.893487i \(-0.351749\pi\)
−0.801930 + 0.597418i \(0.796193\pi\)
\(828\) 0 0
\(829\) −17615.7 + 30511.3i −0.738020 + 1.27829i 0.215365 + 0.976534i \(0.430906\pi\)
−0.953386 + 0.301755i \(0.902428\pi\)
\(830\) 0 0
\(831\) 8829.91 + 50076.9i 0.368599 + 2.09043i
\(832\) 0 0
\(833\) 7385.71 + 2688.18i 0.307202 + 0.111813i
\(834\) 0 0
\(835\) 18015.2 0.746636
\(836\) 0 0
\(837\) −15057.5 −0.621820
\(838\) 0 0
\(839\) 22899.3 + 8334.68i 0.942280 + 0.342962i 0.767066 0.641569i \(-0.221716\pi\)
0.175214 + 0.984530i \(0.443938\pi\)
\(840\) 0 0
\(841\) −2388.41 13545.3i −0.0979296 0.555387i
\(842\) 0 0
\(843\) 11623.5 20132.5i 0.474892 0.822538i
\(844\) 0 0
\(845\) −14900.5 12503.0i −0.606619 0.509014i
\(846\) 0 0
\(847\) 4946.97 + 8568.41i 0.200685 + 0.347596i
\(848\) 0 0
\(849\) 333.394 121.345i 0.0134771 0.00490526i
\(850\) 0 0
\(851\) 739.877 4196.05i 0.0298034 0.169023i
\(852\) 0 0
\(853\) 25463.0 21366.0i 1.02208 0.857630i 0.0321960 0.999482i \(-0.489750\pi\)
0.989888 + 0.141851i \(0.0453055\pi\)
\(854\) 0 0
\(855\) −9921.93 + 14841.5i −0.396869 + 0.593646i
\(856\) 0 0
\(857\) 32182.4 27004.2i 1.28276 1.07637i 0.289907 0.957055i \(-0.406376\pi\)
0.992857 0.119311i \(-0.0380687\pi\)
\(858\) 0 0
\(859\) −1929.88 + 10944.9i −0.0766552 + 0.434733i 0.922192 + 0.386732i \(0.126396\pi\)
−0.998847 + 0.0480010i \(0.984715\pi\)
\(860\) 0 0
\(861\) 22060.0 8029.18i 0.873173 0.317809i
\(862\) 0 0
\(863\) −5088.42 8813.40i −0.200709 0.347638i 0.748048 0.663645i \(-0.230991\pi\)
−0.948757 + 0.316006i \(0.897658\pi\)
\(864\) 0 0
\(865\) 2828.58 + 2373.46i 0.111184 + 0.0932948i
\(866\) 0 0
\(867\) −13304.3 + 23043.7i −0.521151 + 0.902661i
\(868\) 0 0
\(869\) −1576.70 8941.90i −0.0615487 0.349060i
\(870\) 0 0
\(871\) −4937.98 1797.28i −0.192098 0.0699178i
\(872\) 0 0
\(873\) 11926.0 0.462355
\(874\) 0 0
\(875\) −10922.2 −0.421987
\(876\) 0 0
\(877\) −832.509 303.008i −0.0320545 0.0116669i 0.325943 0.945389i \(-0.394318\pi\)
−0.357998 + 0.933722i \(0.616540\pi\)
\(878\) 0 0
\(879\) 6511.10 + 36926.3i 0.249845 + 1.41694i
\(880\) 0 0
\(881\) −974.955 + 1688.67i −0.0372839 + 0.0645775i −0.884065 0.467364i \(-0.845204\pi\)
0.846781 + 0.531941i \(0.178537\pi\)
\(882\) 0 0
\(883\) 9964.07 + 8360.84i 0.379748 + 0.318646i 0.812604 0.582817i \(-0.198050\pi\)
−0.432855 + 0.901463i \(0.642494\pi\)
\(884\) 0 0
\(885\) 16496.3 + 28572.4i 0.626573 + 1.08526i
\(886\) 0 0
\(887\) −4138.11 + 1506.15i −0.156645 + 0.0570141i −0.419153 0.907916i \(-0.637673\pi\)
0.262508 + 0.964930i \(0.415451\pi\)
\(888\) 0 0
\(889\) −2132.07 + 12091.6i −0.0804356 + 0.456173i
\(890\) 0 0
\(891\) −10850.0 + 9104.25i −0.407957 + 0.342316i
\(892\) 0 0
\(893\) 32205.6 + 30845.0i 1.20685 + 1.15587i
\(894\) 0 0
\(895\) −4334.54 + 3637.11i −0.161886 + 0.135838i
\(896\) 0 0
\(897\) −365.318 + 2071.82i −0.0135982 + 0.0771195i
\(898\) 0 0
\(899\) −22779.5 + 8291.06i −0.845093 + 0.307589i
\(900\) 0 0
\(901\) 8133.51 + 14087.7i 0.300740 + 0.520897i
\(902\) 0 0
\(903\) 22411.2 + 18805.3i 0.825913 + 0.693023i
\(904\) 0 0
\(905\) 21059.7 36476.6i 0.773536 1.33980i
\(906\) 0 0
\(907\) 4100.04 + 23252.5i 0.150099 + 0.851252i 0.963131 + 0.269033i \(0.0867041\pi\)
−0.813032 + 0.582219i \(0.802185\pi\)
\(908\) 0 0
\(909\) 3943.08 + 1435.16i 0.143876 + 0.0523667i
\(910\) 0 0
\(911\) −46961.0 −1.70789 −0.853944 0.520365i \(-0.825796\pi\)
−0.853944 + 0.520365i \(0.825796\pi\)
\(912\) 0 0
\(913\) 14922.8 0.540935
\(914\) 0 0
\(915\) −5883.19 2141.31i −0.212560 0.0773655i
\(916\) 0 0
\(917\) −835.114 4736.17i −0.0300741 0.170558i
\(918\) 0 0
\(919\) −7643.95 + 13239.7i −0.274375 + 0.475231i −0.969977 0.243196i \(-0.921804\pi\)
0.695602 + 0.718427i \(0.255138\pi\)
\(920\) 0 0
\(921\) 53405.9 + 44812.9i 1.91073 + 1.60329i
\(922\) 0 0
\(923\) 9731.82 + 16856.0i 0.347049 + 0.601107i
\(924\) 0 0
\(925\) 9132.67 3324.02i 0.324627 0.118155i
\(926\) 0 0
\(927\) 2029.69 11510.9i 0.0719134 0.407841i
\(928\) 0 0
\(929\) 32830.3 27547.9i 1.15945 0.972893i 0.159551 0.987190i \(-0.448995\pi\)
0.999898 + 0.0142963i \(0.00455080\pi\)
\(930\) 0 0
\(931\) 9492.09 + 19259.8i 0.334147 + 0.677995i
\(932\) 0 0
\(933\) −24788.5 + 20800.0i −0.869817 + 0.729863i
\(934\) 0 0
\(935\) 1031.62 5850.63i 0.0360831 0.204638i
\(936\) 0 0
\(937\) −18292.3 + 6657.86i −0.637763 + 0.232127i −0.640607 0.767869i \(-0.721317\pi\)
0.00284361 + 0.999996i \(0.499095\pi\)
\(938\) 0 0
\(939\) −1106.15 1915.90i −0.0384427 0.0665848i
\(940\) 0 0
\(941\) −8172.65 6857.67i −0.283125 0.237570i 0.490154 0.871636i \(-0.336941\pi\)
−0.773279 + 0.634065i \(0.781385\pi\)
\(942\) 0 0
\(943\) 2425.65 4201.35i 0.0837647 0.145085i
\(944\) 0 0
\(945\) 1262.06 + 7157.50i 0.0434442 + 0.246385i
\(946\) 0 0
\(947\) −19411.4 7065.16i −0.666087 0.242436i −0.0132248 0.999913i \(-0.504210\pi\)
−0.652862 + 0.757477i \(0.726432\pi\)
\(948\) 0 0
\(949\) −22042.7 −0.753990
\(950\) 0 0
\(951\) 40517.7 1.38157
\(952\) 0 0
\(953\) 33465.4 + 12180.4i 1.13751 + 0.414021i 0.841015 0.541012i \(-0.181959\pi\)
0.296499 + 0.955033i \(0.404181\pi\)
\(954\) 0 0
\(955\) −5696.61 32307.1i −0.193024 1.09469i
\(956\) 0 0
\(957\) −5429.33 + 9403.87i −0.183391 + 0.317643i
\(958\) 0 0
\(959\) −17830.0 14961.1i −0.600376 0.503775i
\(960\) 0 0
\(961\) −12733.2 22054.5i −0.427417 0.740309i
\(962\) 0 0
\(963\) −1741.56 + 633.877i −0.0582774 + 0.0212112i
\(964\) 0 0
\(965\) −4540.60 + 25751.0i −0.151468 + 0.859020i
\(966\) 0 0
\(967\) −11682.1 + 9802.44i −0.388491 + 0.325983i −0.816025 0.578017i \(-0.803827\pi\)
0.427534 + 0.903999i \(0.359382\pi\)
\(968\) 0 0
\(969\) 16059.9 4676.82i 0.532423 0.155048i
\(970\) 0 0
\(971\) 39659.7 33278.4i 1.31075 1.09985i 0.322573 0.946545i \(-0.395452\pi\)
0.988179 0.153307i \(-0.0489922\pi\)
\(972\) 0 0
\(973\) −1126.58 + 6389.13i −0.0371186 + 0.210510i
\(974\) 0 0
\(975\) −4509.30 + 1641.25i −0.148116 + 0.0539099i
\(976\) 0 0
\(977\) −14233.2 24652.6i −0.466080 0.807275i 0.533169 0.846009i \(-0.321001\pi\)
−0.999250 + 0.0387336i \(0.987668\pi\)
\(978\) 0 0
\(979\) −1642.96 1378.61i −0.0536356 0.0450056i
\(980\) 0 0
\(981\) −4725.01 + 8183.95i −0.153780 + 0.266354i
\(982\) 0 0
\(983\) −4070.02 23082.2i −0.132059 0.748941i −0.976863 0.213866i \(-0.931394\pi\)
0.844804 0.535075i \(-0.179717\pi\)
\(984\) 0 0
\(985\) −49369.3 17969.0i −1.59699 0.581258i
\(986\) 0 0
\(987\) −32826.6 −1.05865
\(988\) 0 0
\(989\) 6045.76 0.194382
\(990\) 0 0
\(991\) −30107.7 10958.3i −0.965089 0.351264i −0.189063 0.981965i \(-0.560545\pi\)
−0.776026 + 0.630701i \(0.782767\pi\)
\(992\) 0 0
\(993\) −6297.99 35717.7i −0.201270 1.14146i
\(994\) 0 0
\(995\) 32235.4 55833.4i 1.02707 1.77893i
\(996\) 0 0
\(997\) 15963.6 + 13395.1i 0.507094 + 0.425503i 0.860105 0.510117i \(-0.170398\pi\)
−0.353011 + 0.935619i \(0.614842\pi\)
\(998\) 0 0
\(999\) 10831.7 + 18761.1i 0.343043 + 0.594168i
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 76.4.i.a.25.5 30
19.4 even 9 1444.4.a.j.1.3 15
19.15 odd 18 1444.4.a.k.1.13 15
19.16 even 9 inner 76.4.i.a.73.5 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.i.a.25.5 30 1.1 even 1 trivial
76.4.i.a.73.5 yes 30 19.16 even 9 inner
1444.4.a.j.1.3 15 19.4 even 9
1444.4.a.k.1.13 15 19.15 odd 18