Properties

Label 448.3.s.h.129.5
Level $448$
Weight $3$
Character 448.129
Analytic conductor $12.207$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(129,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (of order \(6\), degree \(2\), 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: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} - 33728 x^{7} - 49760 x^{6} + 203528 x^{5} + 27401 x^{4} - 156928 x^{3} + \cdots + 208849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 129.5
Root \(0.869658 + 0.314400i\) of defining polynomial
Character \(\chi\) \(=\) 448.129
Dual form 448.3.s.h.257.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.729881 + 0.421397i) q^{3} +(1.21685 - 0.702550i) q^{5} +(1.56553 + 6.82269i) q^{7} +(-4.14485 - 7.17909i) q^{9} +O(q^{10})\) \(q+(0.729881 + 0.421397i) q^{3} +(1.21685 - 0.702550i) q^{5} +(1.56553 + 6.82269i) q^{7} +(-4.14485 - 7.17909i) q^{9} +(7.44562 - 12.8962i) q^{11} -2.67477i q^{13} +1.18421 q^{15} +(11.7354 + 6.77544i) q^{17} +(25.2564 - 14.5818i) q^{19} +(-1.73241 + 5.63947i) q^{21} +(11.4367 + 19.8089i) q^{23} +(-11.5128 + 19.9408i) q^{25} -14.5717i q^{27} +3.76543 q^{29} +(11.4937 + 6.63592i) q^{31} +(10.8688 - 6.27513i) q^{33} +(6.69830 + 7.20235i) q^{35} +(1.32272 + 2.29102i) q^{37} +(1.12714 - 1.95227i) q^{39} -45.2712i q^{41} +51.5858 q^{43} +(-10.0873 - 5.82393i) q^{45} +(58.1281 - 33.5603i) q^{47} +(-44.0982 + 21.3623i) q^{49} +(5.71030 + 9.89054i) q^{51} +(19.9616 - 34.5744i) q^{53} -20.9237i q^{55} +24.5789 q^{57} +(11.0804 + 6.39728i) q^{59} +(-67.4084 + 38.9182i) q^{61} +(42.4918 - 39.5181i) q^{63} +(-1.87916 - 3.25480i) q^{65} +(-22.9199 + 39.6985i) q^{67} +19.2775i q^{69} +40.6812 q^{71} +(55.6834 + 32.1488i) q^{73} +(-16.8060 + 9.70296i) q^{75} +(99.6431 + 30.6098i) q^{77} +(-40.2603 - 69.7329i) q^{79} +(-31.1632 + 53.9762i) q^{81} +121.864i q^{83} +19.0403 q^{85} +(2.74832 + 1.58674i) q^{87} +(-39.7614 + 22.9562i) q^{89} +(18.2492 - 4.18744i) q^{91} +(5.59271 + 9.68687i) q^{93} +(20.4889 - 35.4878i) q^{95} -134.268i q^{97} -123.444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{9} - 48 q^{17} + 136 q^{21} + 80 q^{25} + 16 q^{29} - 264 q^{33} - 72 q^{37} - 312 q^{45} + 128 q^{49} - 40 q^{53} + 368 q^{57} - 216 q^{61} - 168 q^{65} - 312 q^{73} - 64 q^{77} - 384 q^{81} + 1072 q^{85} + 24 q^{89} + 168 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) 0.729881 + 0.421397i 0.243294 + 0.140466i 0.616690 0.787206i \(-0.288473\pi\)
−0.373396 + 0.927672i \(0.621807\pi\)
\(4\) 0 0
\(5\) 1.21685 0.702550i 0.243371 0.140510i −0.373354 0.927689i \(-0.621792\pi\)
0.616725 + 0.787179i \(0.288459\pi\)
\(6\) 0 0
\(7\) 1.56553 + 6.82269i 0.223647 + 0.974670i
\(8\) 0 0
\(9\) −4.14485 7.17909i −0.460539 0.797677i
\(10\) 0 0
\(11\) 7.44562 12.8962i 0.676874 1.17238i −0.299043 0.954240i \(-0.596667\pi\)
0.975917 0.218141i \(-0.0699994\pi\)
\(12\) 0 0
\(13\) 2.67477i 0.205752i −0.994694 0.102876i \(-0.967196\pi\)
0.994694 0.102876i \(-0.0328045\pi\)
\(14\) 0 0
\(15\) 1.18421 0.0789474
\(16\) 0 0
\(17\) 11.7354 + 6.77544i 0.690318 + 0.398555i 0.803731 0.594993i \(-0.202845\pi\)
−0.113413 + 0.993548i \(0.536178\pi\)
\(18\) 0 0
\(19\) 25.2564 14.5818i 1.32928 0.767462i 0.344095 0.938935i \(-0.388186\pi\)
0.985189 + 0.171473i \(0.0548526\pi\)
\(20\) 0 0
\(21\) −1.73241 + 5.63947i −0.0824958 + 0.268546i
\(22\) 0 0
\(23\) 11.4367 + 19.8089i 0.497247 + 0.861257i 0.999995 0.00317621i \(-0.00101102\pi\)
−0.502748 + 0.864433i \(0.667678\pi\)
\(24\) 0 0
\(25\) −11.5128 + 19.9408i −0.460514 + 0.797633i
\(26\) 0 0
\(27\) 14.5717i 0.539691i
\(28\) 0 0
\(29\) 3.76543 0.129843 0.0649213 0.997890i \(-0.479320\pi\)
0.0649213 + 0.997890i \(0.479320\pi\)
\(30\) 0 0
\(31\) 11.4937 + 6.63592i 0.370766 + 0.214062i 0.673793 0.738920i \(-0.264664\pi\)
−0.303027 + 0.952982i \(0.597997\pi\)
\(32\) 0 0
\(33\) 10.8688 6.27513i 0.329359 0.190155i
\(34\) 0 0
\(35\) 6.69830 + 7.20235i 0.191380 + 0.205781i
\(36\) 0 0
\(37\) 1.32272 + 2.29102i 0.0357491 + 0.0619193i 0.883346 0.468721i \(-0.155285\pi\)
−0.847597 + 0.530640i \(0.821952\pi\)
\(38\) 0 0
\(39\) 1.12714 1.95227i 0.0289011 0.0500581i
\(40\) 0 0
\(41\) 45.2712i 1.10417i −0.833786 0.552087i \(-0.813832\pi\)
0.833786 0.552087i \(-0.186168\pi\)
\(42\) 0 0
\(43\) 51.5858 1.19967 0.599834 0.800124i \(-0.295233\pi\)
0.599834 + 0.800124i \(0.295233\pi\)
\(44\) 0 0
\(45\) −10.0873 5.82393i −0.224163 0.129421i
\(46\) 0 0
\(47\) 58.1281 33.5603i 1.23677 0.714049i 0.268336 0.963325i \(-0.413526\pi\)
0.968432 + 0.249276i \(0.0801928\pi\)
\(48\) 0 0
\(49\) −44.0982 + 21.3623i −0.899964 + 0.435964i
\(50\) 0 0
\(51\) 5.71030 + 9.89054i 0.111967 + 0.193932i
\(52\) 0 0
\(53\) 19.9616 34.5744i 0.376633 0.652348i −0.613937 0.789355i \(-0.710415\pi\)
0.990570 + 0.137007i \(0.0437484\pi\)
\(54\) 0 0
\(55\) 20.9237i 0.380431i
\(56\) 0 0
\(57\) 24.5789 0.431209
\(58\) 0 0
\(59\) 11.0804 + 6.39728i 0.187804 + 0.108429i 0.590954 0.806705i \(-0.298751\pi\)
−0.403150 + 0.915134i \(0.632085\pi\)
\(60\) 0 0
\(61\) −67.4084 + 38.9182i −1.10506 + 0.638004i −0.937544 0.347866i \(-0.886906\pi\)
−0.167511 + 0.985870i \(0.553573\pi\)
\(62\) 0 0
\(63\) 42.4918 39.5181i 0.674473 0.627271i
\(64\) 0 0
\(65\) −1.87916 3.25480i −0.0289102 0.0500739i
\(66\) 0 0
\(67\) −22.9199 + 39.6985i −0.342089 + 0.592515i −0.984820 0.173576i \(-0.944468\pi\)
0.642732 + 0.766091i \(0.277801\pi\)
\(68\) 0 0
\(69\) 19.2775i 0.279385i
\(70\) 0 0
\(71\) 40.6812 0.572975 0.286487 0.958084i \(-0.407512\pi\)
0.286487 + 0.958084i \(0.407512\pi\)
\(72\) 0 0
\(73\) 55.6834 + 32.1488i 0.762786 + 0.440395i 0.830295 0.557324i \(-0.188172\pi\)
−0.0675092 + 0.997719i \(0.521505\pi\)
\(74\) 0 0
\(75\) −16.8060 + 9.70296i −0.224080 + 0.129373i
\(76\) 0 0
\(77\) 99.6431 + 30.6098i 1.29407 + 0.397530i
\(78\) 0 0
\(79\) −40.2603 69.7329i −0.509624 0.882695i −0.999938 0.0111488i \(-0.996451\pi\)
0.490314 0.871546i \(-0.336882\pi\)
\(80\) 0 0
\(81\) −31.1632 + 53.9762i −0.384731 + 0.666373i
\(82\) 0 0
\(83\) 121.864i 1.46824i 0.679021 + 0.734118i \(0.262404\pi\)
−0.679021 + 0.734118i \(0.737596\pi\)
\(84\) 0 0
\(85\) 19.0403 0.224004
\(86\) 0 0
\(87\) 2.74832 + 1.58674i 0.0315899 + 0.0182384i
\(88\) 0 0
\(89\) −39.7614 + 22.9562i −0.446757 + 0.257935i −0.706460 0.707753i \(-0.749709\pi\)
0.259703 + 0.965689i \(0.416375\pi\)
\(90\) 0 0
\(91\) 18.2492 4.18744i 0.200540 0.0460158i
\(92\) 0 0
\(93\) 5.59271 + 9.68687i 0.0601367 + 0.104160i
\(94\) 0 0
\(95\) 20.4889 35.4878i 0.215672 0.373555i
\(96\) 0 0
\(97\) 134.268i 1.38421i −0.721798 0.692104i \(-0.756684\pi\)
0.721798 0.692104i \(-0.243316\pi\)
\(98\) 0 0
\(99\) −123.444 −1.24691
\(100\) 0 0
\(101\) −170.810 98.6169i −1.69118 0.976405i −0.953565 0.301188i \(-0.902617\pi\)
−0.737619 0.675218i \(-0.764050\pi\)
\(102\) 0 0
\(103\) −121.371 + 70.0736i −1.17836 + 0.680326i −0.955635 0.294555i \(-0.904829\pi\)
−0.222725 + 0.974881i \(0.571495\pi\)
\(104\) 0 0
\(105\) 1.85392 + 8.07950i 0.0176564 + 0.0769477i
\(106\) 0 0
\(107\) −53.0099 91.8159i −0.495420 0.858093i 0.504566 0.863373i \(-0.331652\pi\)
−0.999986 + 0.00528051i \(0.998319\pi\)
\(108\) 0 0
\(109\) 71.3762 123.627i 0.654828 1.13419i −0.327109 0.944986i \(-0.606075\pi\)
0.981937 0.189208i \(-0.0605921\pi\)
\(110\) 0 0
\(111\) 2.22956i 0.0200861i
\(112\) 0 0
\(113\) 108.519 0.960344 0.480172 0.877174i \(-0.340574\pi\)
0.480172 + 0.877174i \(0.340574\pi\)
\(114\) 0 0
\(115\) 27.8335 + 16.0697i 0.242030 + 0.139736i
\(116\) 0 0
\(117\) −19.2024 + 11.0865i −0.164123 + 0.0947567i
\(118\) 0 0
\(119\) −27.8546 + 90.6742i −0.234072 + 0.761968i
\(120\) 0 0
\(121\) −50.3745 87.2512i −0.416318 0.721084i
\(122\) 0 0
\(123\) 19.0771 33.0426i 0.155099 0.268639i
\(124\) 0 0
\(125\) 67.4809i 0.539847i
\(126\) 0 0
\(127\) −139.789 −1.10070 −0.550351 0.834934i \(-0.685506\pi\)
−0.550351 + 0.834934i \(0.685506\pi\)
\(128\) 0 0
\(129\) 37.6515 + 21.7381i 0.291872 + 0.168512i
\(130\) 0 0
\(131\) −82.2943 + 47.5126i −0.628200 + 0.362692i −0.780055 0.625711i \(-0.784809\pi\)
0.151854 + 0.988403i \(0.451476\pi\)
\(132\) 0 0
\(133\) 139.027 + 149.488i 1.04531 + 1.12397i
\(134\) 0 0
\(135\) −10.2373 17.7316i −0.0758320 0.131345i
\(136\) 0 0
\(137\) −76.7845 + 132.995i −0.560471 + 0.970764i 0.436984 + 0.899469i \(0.356046\pi\)
−0.997455 + 0.0712949i \(0.977287\pi\)
\(138\) 0 0
\(139\) 217.529i 1.56496i −0.622676 0.782480i \(-0.713954\pi\)
0.622676 0.782480i \(-0.286046\pi\)
\(140\) 0 0
\(141\) 56.5689 0.401198
\(142\) 0 0
\(143\) −34.4944 19.9153i −0.241219 0.139268i
\(144\) 0 0
\(145\) 4.58198 2.64541i 0.0315998 0.0182442i
\(146\) 0 0
\(147\) −41.1885 2.99096i −0.280194 0.0203467i
\(148\) 0 0
\(149\) 32.0451 + 55.5037i 0.215068 + 0.372508i 0.953294 0.302045i \(-0.0976694\pi\)
−0.738226 + 0.674554i \(0.764336\pi\)
\(150\) 0 0
\(151\) −136.023 + 235.598i −0.900813 + 1.56025i −0.0743729 + 0.997230i \(0.523696\pi\)
−0.826440 + 0.563024i \(0.809638\pi\)
\(152\) 0 0
\(153\) 112.333i 0.734201i
\(154\) 0 0
\(155\) 18.6483 0.120311
\(156\) 0 0
\(157\) −187.295 108.135i −1.19296 0.688757i −0.233984 0.972240i \(-0.575176\pi\)
−0.958977 + 0.283484i \(0.908510\pi\)
\(158\) 0 0
\(159\) 29.1391 16.8235i 0.183265 0.105808i
\(160\) 0 0
\(161\) −117.246 + 109.040i −0.728233 + 0.677269i
\(162\) 0 0
\(163\) 27.3465 + 47.3655i 0.167770 + 0.290586i 0.937635 0.347620i \(-0.113010\pi\)
−0.769866 + 0.638206i \(0.779677\pi\)
\(164\) 0 0
\(165\) 8.81718 15.2718i 0.0534375 0.0925564i
\(166\) 0 0
\(167\) 248.098i 1.48562i 0.669504 + 0.742809i \(0.266507\pi\)
−0.669504 + 0.742809i \(0.733493\pi\)
\(168\) 0 0
\(169\) 161.846 0.957666
\(170\) 0 0
\(171\) −209.368 120.879i −1.22437 0.706892i
\(172\) 0 0
\(173\) −224.883 + 129.836i −1.29990 + 0.750499i −0.980387 0.197082i \(-0.936853\pi\)
−0.319515 + 0.947581i \(0.603520\pi\)
\(174\) 0 0
\(175\) −154.074 47.3306i −0.880422 0.270461i
\(176\) 0 0
\(177\) 5.39159 + 9.33851i 0.0304610 + 0.0527600i
\(178\) 0 0
\(179\) −69.0883 + 119.664i −0.385968 + 0.668516i −0.991903 0.126998i \(-0.959466\pi\)
0.605935 + 0.795514i \(0.292799\pi\)
\(180\) 0 0
\(181\) 114.497i 0.632579i −0.948663 0.316290i \(-0.897563\pi\)
0.948663 0.316290i \(-0.102437\pi\)
\(182\) 0 0
\(183\) −65.6001 −0.358471
\(184\) 0 0
\(185\) 3.21911 + 1.85855i 0.0174006 + 0.0100462i
\(186\) 0 0
\(187\) 174.755 100.895i 0.934517 0.539544i
\(188\) 0 0
\(189\) 99.4180 22.8124i 0.526021 0.120700i
\(190\) 0 0
\(191\) 76.2145 + 132.007i 0.399029 + 0.691138i 0.993606 0.112900i \(-0.0360141\pi\)
−0.594578 + 0.804038i \(0.702681\pi\)
\(192\) 0 0
\(193\) −121.156 + 209.848i −0.627749 + 1.08729i 0.360253 + 0.932855i \(0.382690\pi\)
−0.988002 + 0.154439i \(0.950643\pi\)
\(194\) 0 0
\(195\) 3.16750i 0.0162436i
\(196\) 0 0
\(197\) 197.518 1.00263 0.501314 0.865265i \(-0.332850\pi\)
0.501314 + 0.865265i \(0.332850\pi\)
\(198\) 0 0
\(199\) −142.172 82.0833i −0.714435 0.412479i 0.0982663 0.995160i \(-0.468670\pi\)
−0.812701 + 0.582681i \(0.802004\pi\)
\(200\) 0 0
\(201\) −33.4577 + 19.3168i −0.166456 + 0.0961035i
\(202\) 0 0
\(203\) 5.89490 + 25.6904i 0.0290389 + 0.126554i
\(204\) 0 0
\(205\) −31.8053 55.0883i −0.155148 0.268724i
\(206\) 0 0
\(207\) 94.8066 164.210i 0.458003 0.793284i
\(208\) 0 0
\(209\) 434.282i 2.07790i
\(210\) 0 0
\(211\) −53.1445 −0.251870 −0.125935 0.992039i \(-0.540193\pi\)
−0.125935 + 0.992039i \(0.540193\pi\)
\(212\) 0 0
\(213\) 29.6925 + 17.1429i 0.139401 + 0.0804833i
\(214\) 0 0
\(215\) 62.7723 36.2416i 0.291964 0.168565i
\(216\) 0 0
\(217\) −27.2810 + 88.8070i −0.125719 + 0.409249i
\(218\) 0 0
\(219\) 27.0948 + 46.9296i 0.123721 + 0.214291i
\(220\) 0 0
\(221\) 18.1228 31.3896i 0.0820035 0.142034i
\(222\) 0 0
\(223\) 310.755i 1.39352i −0.717303 0.696761i \(-0.754624\pi\)
0.717303 0.696761i \(-0.245376\pi\)
\(224\) 0 0
\(225\) 190.876 0.848338
\(226\) 0 0
\(227\) 43.2131 + 24.9491i 0.190366 + 0.109908i 0.592154 0.805825i \(-0.298278\pi\)
−0.401788 + 0.915733i \(0.631611\pi\)
\(228\) 0 0
\(229\) −182.833 + 105.559i −0.798398 + 0.460955i −0.842911 0.538054i \(-0.819160\pi\)
0.0445127 + 0.999009i \(0.485826\pi\)
\(230\) 0 0
\(231\) 59.8287 + 64.3308i 0.258999 + 0.278488i
\(232\) 0 0
\(233\) −37.7274 65.3457i −0.161920 0.280454i 0.773637 0.633629i \(-0.218435\pi\)
−0.935557 + 0.353175i \(0.885102\pi\)
\(234\) 0 0
\(235\) 47.1556 81.6759i 0.200662 0.347557i
\(236\) 0 0
\(237\) 67.8623i 0.286339i
\(238\) 0 0
\(239\) 310.396 1.29873 0.649365 0.760477i \(-0.275035\pi\)
0.649365 + 0.760477i \(0.275035\pi\)
\(240\) 0 0
\(241\) 93.7200 + 54.1093i 0.388880 + 0.224520i 0.681675 0.731656i \(-0.261252\pi\)
−0.292795 + 0.956175i \(0.594585\pi\)
\(242\) 0 0
\(243\) −159.066 + 91.8366i −0.654591 + 0.377929i
\(244\) 0 0
\(245\) −38.6530 + 56.9759i −0.157767 + 0.232555i
\(246\) 0 0
\(247\) −39.0030 67.5551i −0.157907 0.273502i
\(248\) 0 0
\(249\) −51.3530 + 88.9460i −0.206237 + 0.357213i
\(250\) 0 0
\(251\) 302.023i 1.20328i 0.798768 + 0.601639i \(0.205485\pi\)
−0.798768 + 0.601639i \(0.794515\pi\)
\(252\) 0 0
\(253\) 340.613 1.34629
\(254\) 0 0
\(255\) 13.8972 + 8.02355i 0.0544988 + 0.0314649i
\(256\) 0 0
\(257\) −89.8068 + 51.8500i −0.349443 + 0.201751i −0.664440 0.747342i \(-0.731330\pi\)
0.314997 + 0.949093i \(0.397996\pi\)
\(258\) 0 0
\(259\) −13.5601 + 12.6112i −0.0523557 + 0.0486917i
\(260\) 0 0
\(261\) −15.6072 27.0324i −0.0597975 0.103572i
\(262\) 0 0
\(263\) 128.972 223.386i 0.490387 0.849375i −0.509552 0.860440i \(-0.670189\pi\)
0.999939 + 0.0110651i \(0.00352220\pi\)
\(264\) 0 0
\(265\) 56.0960i 0.211683i
\(266\) 0 0
\(267\) −38.6948 −0.144924
\(268\) 0 0
\(269\) −140.523 81.1310i −0.522390 0.301602i 0.215522 0.976499i \(-0.430855\pi\)
−0.737912 + 0.674897i \(0.764188\pi\)
\(270\) 0 0
\(271\) 54.3429 31.3749i 0.200527 0.115775i −0.396374 0.918089i \(-0.629732\pi\)
0.596901 + 0.802315i \(0.296398\pi\)
\(272\) 0 0
\(273\) 15.0843 + 4.63381i 0.0552538 + 0.0169737i
\(274\) 0 0
\(275\) 171.441 + 296.944i 0.623420 + 1.07980i
\(276\) 0 0
\(277\) −128.576 + 222.701i −0.464175 + 0.803974i −0.999164 0.0408848i \(-0.986982\pi\)
0.534989 + 0.844859i \(0.320316\pi\)
\(278\) 0 0
\(279\) 110.019i 0.394335i
\(280\) 0 0
\(281\) 230.243 0.819369 0.409685 0.912227i \(-0.365639\pi\)
0.409685 + 0.912227i \(0.365639\pi\)
\(282\) 0 0
\(283\) 354.637 + 204.750i 1.25313 + 0.723497i 0.971730 0.236093i \(-0.0758671\pi\)
0.281402 + 0.959590i \(0.409200\pi\)
\(284\) 0 0
\(285\) 29.9089 17.2679i 0.104943 0.0605891i
\(286\) 0 0
\(287\) 308.871 70.8733i 1.07621 0.246945i
\(288\) 0 0
\(289\) −52.6868 91.2562i −0.182307 0.315765i
\(290\) 0 0
\(291\) 56.5803 97.9999i 0.194434 0.336769i
\(292\) 0 0
\(293\) 438.106i 1.49524i −0.664126 0.747621i \(-0.731196\pi\)
0.664126 0.747621i \(-0.268804\pi\)
\(294\) 0 0
\(295\) 17.9776 0.0609412
\(296\) 0 0
\(297\) −187.919 108.495i −0.632724 0.365303i
\(298\) 0 0
\(299\) 52.9843 30.5905i 0.177205 0.102309i
\(300\) 0 0
\(301\) 80.7590 + 351.954i 0.268302 + 1.16928i
\(302\) 0 0
\(303\) −83.1138 143.957i −0.274303 0.475107i
\(304\) 0 0
\(305\) −54.6840 + 94.7155i −0.179292 + 0.310543i
\(306\) 0 0
\(307\) 136.514i 0.444672i −0.974970 0.222336i \(-0.928632\pi\)
0.974970 0.222336i \(-0.0713682\pi\)
\(308\) 0 0
\(309\) −118.115 −0.382250
\(310\) 0 0
\(311\) 503.092 + 290.461i 1.61766 + 0.933957i 0.987523 + 0.157473i \(0.0503348\pi\)
0.630138 + 0.776484i \(0.282999\pi\)
\(312\) 0 0
\(313\) −162.118 + 93.5987i −0.517948 + 0.299037i −0.736095 0.676879i \(-0.763332\pi\)
0.218147 + 0.975916i \(0.429999\pi\)
\(314\) 0 0
\(315\) 23.9428 77.9403i 0.0760090 0.247430i
\(316\) 0 0
\(317\) 82.9440 + 143.663i 0.261653 + 0.453196i 0.966681 0.255983i \(-0.0823991\pi\)
−0.705028 + 0.709179i \(0.749066\pi\)
\(318\) 0 0
\(319\) 28.0360 48.5597i 0.0878871 0.152225i
\(320\) 0 0
\(321\) 89.3530i 0.278358i
\(322\) 0 0
\(323\) 395.192 1.22350
\(324\) 0 0
\(325\) 53.3372 + 30.7943i 0.164115 + 0.0947516i
\(326\) 0 0
\(327\) 104.192 60.1555i 0.318631 0.183962i
\(328\) 0 0
\(329\) 319.973 + 344.051i 0.972562 + 1.04575i
\(330\) 0 0
\(331\) −209.099 362.170i −0.631719 1.09417i −0.987200 0.159486i \(-0.949016\pi\)
0.355481 0.934683i \(-0.384317\pi\)
\(332\) 0 0
\(333\) 10.9649 18.9918i 0.0329277 0.0570325i
\(334\) 0 0
\(335\) 64.4096i 0.192268i
\(336\) 0 0
\(337\) 104.000 0.308607 0.154303 0.988024i \(-0.450687\pi\)
0.154303 + 0.988024i \(0.450687\pi\)
\(338\) 0 0
\(339\) 79.2059 + 45.7295i 0.233646 + 0.134895i
\(340\) 0 0
\(341\) 171.156 98.8170i 0.501924 0.289786i
\(342\) 0 0
\(343\) −214.785 267.425i −0.626196 0.779666i
\(344\) 0 0
\(345\) 13.5434 + 23.4579i 0.0392563 + 0.0679940i
\(346\) 0 0
\(347\) 48.9796 84.8352i 0.141152 0.244482i −0.786779 0.617235i \(-0.788253\pi\)
0.927931 + 0.372753i \(0.121586\pi\)
\(348\) 0 0
\(349\) 248.606i 0.712339i −0.934421 0.356169i \(-0.884083\pi\)
0.934421 0.356169i \(-0.115917\pi\)
\(350\) 0 0
\(351\) −38.9759 −0.111042
\(352\) 0 0
\(353\) −113.804 65.7047i −0.322390 0.186132i 0.330067 0.943957i \(-0.392929\pi\)
−0.652458 + 0.757825i \(0.726262\pi\)
\(354\) 0 0
\(355\) 49.5030 28.5806i 0.139445 0.0805087i
\(356\) 0 0
\(357\) −58.5404 + 54.4436i −0.163979 + 0.152503i
\(358\) 0 0
\(359\) 176.634 + 305.939i 0.492016 + 0.852196i 0.999958 0.00919491i \(-0.00292687\pi\)
−0.507942 + 0.861391i \(0.669594\pi\)
\(360\) 0 0
\(361\) 244.757 423.931i 0.677996 1.17432i
\(362\) 0 0
\(363\) 84.9107i 0.233914i
\(364\) 0 0
\(365\) 90.3446 0.247519
\(366\) 0 0
\(367\) 105.306 + 60.7983i 0.286937 + 0.165663i 0.636560 0.771228i \(-0.280357\pi\)
−0.349623 + 0.936891i \(0.613690\pi\)
\(368\) 0 0
\(369\) −325.006 + 187.642i −0.880774 + 0.508515i
\(370\) 0 0
\(371\) 267.141 + 82.0642i 0.720057 + 0.221197i
\(372\) 0 0
\(373\) −245.458 425.145i −0.658063 1.13980i −0.981116 0.193418i \(-0.938043\pi\)
0.323053 0.946381i \(-0.395291\pi\)
\(374\) 0 0
\(375\) −28.4363 + 49.2531i −0.0758301 + 0.131342i
\(376\) 0 0
\(377\) 10.0717i 0.0267153i
\(378\) 0 0
\(379\) 325.094 0.857768 0.428884 0.903360i \(-0.358907\pi\)
0.428884 + 0.903360i \(0.358907\pi\)
\(380\) 0 0
\(381\) −102.029 58.9067i −0.267794 0.154611i
\(382\) 0 0
\(383\) −385.028 + 222.296i −1.00529 + 0.580407i −0.909810 0.415024i \(-0.863773\pi\)
−0.0954837 + 0.995431i \(0.530440\pi\)
\(384\) 0 0
\(385\) 142.756 32.7567i 0.370794 0.0850822i
\(386\) 0 0
\(387\) −213.815 370.339i −0.552494 0.956948i
\(388\) 0 0
\(389\) −209.219 + 362.379i −0.537839 + 0.931565i 0.461181 + 0.887306i \(0.347426\pi\)
−0.999020 + 0.0442585i \(0.985907\pi\)
\(390\) 0 0
\(391\) 309.954i 0.792721i
\(392\) 0 0
\(393\) −80.0867 −0.203783
\(394\) 0 0
\(395\) −97.9817 56.5698i −0.248055 0.143215i
\(396\) 0 0
\(397\) −378.814 + 218.708i −0.954192 + 0.550903i −0.894381 0.447307i \(-0.852383\pi\)
−0.0598112 + 0.998210i \(0.519050\pi\)
\(398\) 0 0
\(399\) 38.4790 + 167.694i 0.0964385 + 0.420286i
\(400\) 0 0
\(401\) 45.2362 + 78.3513i 0.112808 + 0.195390i 0.916902 0.399113i \(-0.130682\pi\)
−0.804093 + 0.594503i \(0.797349\pi\)
\(402\) 0 0
\(403\) 17.7496 30.7432i 0.0440436 0.0762858i
\(404\) 0 0
\(405\) 87.5748i 0.216234i
\(406\) 0 0
\(407\) 39.3938 0.0967907
\(408\) 0 0
\(409\) −607.126 350.524i −1.48442 0.857028i −0.484573 0.874751i \(-0.661025\pi\)
−0.999843 + 0.0177233i \(0.994358\pi\)
\(410\) 0 0
\(411\) −112.087 + 64.7136i −0.272718 + 0.157454i
\(412\) 0 0
\(413\) −26.3000 + 85.6134i −0.0636803 + 0.207296i
\(414\) 0 0
\(415\) 85.6153 + 148.290i 0.206302 + 0.357326i
\(416\) 0 0
\(417\) 91.6663 158.771i 0.219823 0.380745i
\(418\) 0 0
\(419\) 374.252i 0.893202i −0.894733 0.446601i \(-0.852634\pi\)
0.894733 0.446601i \(-0.147366\pi\)
\(420\) 0 0
\(421\) 733.801 1.74299 0.871497 0.490400i \(-0.163149\pi\)
0.871497 + 0.490400i \(0.163149\pi\)
\(422\) 0 0
\(423\) −481.865 278.205i −1.13916 0.657694i
\(424\) 0 0
\(425\) −270.216 + 156.009i −0.635802 + 0.367081i
\(426\) 0 0
\(427\) −371.057 398.979i −0.868986 0.934376i
\(428\) 0 0
\(429\) −16.7845 29.0717i −0.0391248 0.0677662i
\(430\) 0 0
\(431\) −416.503 + 721.405i −0.966365 + 1.67379i −0.260462 + 0.965484i \(0.583875\pi\)
−0.705903 + 0.708308i \(0.749459\pi\)
\(432\) 0 0
\(433\) 414.085i 0.956316i 0.878274 + 0.478158i \(0.158695\pi\)
−0.878274 + 0.478158i \(0.841305\pi\)
\(434\) 0 0
\(435\) 4.45907 0.0102507
\(436\) 0 0
\(437\) 577.698 + 333.534i 1.32196 + 0.763236i
\(438\) 0 0
\(439\) 61.9802 35.7843i 0.141185 0.0815132i −0.427744 0.903900i \(-0.640691\pi\)
0.568929 + 0.822387i \(0.307358\pi\)
\(440\) 0 0
\(441\) 336.142 + 228.042i 0.762227 + 0.517102i
\(442\) 0 0
\(443\) 414.628 + 718.157i 0.935956 + 1.62112i 0.772921 + 0.634503i \(0.218795\pi\)
0.163035 + 0.986620i \(0.447872\pi\)
\(444\) 0 0
\(445\) −32.2558 + 55.8687i −0.0724850 + 0.125548i
\(446\) 0 0
\(447\) 54.0149i 0.120839i
\(448\) 0 0
\(449\) 247.153 0.550452 0.275226 0.961380i \(-0.411247\pi\)
0.275226 + 0.961380i \(0.411247\pi\)
\(450\) 0 0
\(451\) −583.825 337.072i −1.29451 0.747388i
\(452\) 0 0
\(453\) −198.561 + 114.639i −0.438325 + 0.253067i
\(454\) 0 0
\(455\) 19.2646 17.9164i 0.0423399 0.0393768i
\(456\) 0 0
\(457\) 240.746 + 416.984i 0.526796 + 0.912437i 0.999512 + 0.0312226i \(0.00994009\pi\)
−0.472717 + 0.881214i \(0.656727\pi\)
\(458\) 0 0
\(459\) 98.7294 171.004i 0.215097 0.372559i
\(460\) 0 0
\(461\) 256.190i 0.555727i −0.960621 0.277863i \(-0.910374\pi\)
0.960621 0.277863i \(-0.0896263\pi\)
\(462\) 0 0
\(463\) −142.004 −0.306705 −0.153353 0.988172i \(-0.549007\pi\)
−0.153353 + 0.988172i \(0.549007\pi\)
\(464\) 0 0
\(465\) 13.6110 + 7.85832i 0.0292710 + 0.0168996i
\(466\) 0 0
\(467\) 375.153 216.594i 0.803324 0.463800i −0.0413078 0.999146i \(-0.513152\pi\)
0.844632 + 0.535347i \(0.179819\pi\)
\(468\) 0 0
\(469\) −306.732 94.2265i −0.654014 0.200909i
\(470\) 0 0
\(471\) −91.1354 157.851i −0.193493 0.335140i
\(472\) 0 0
\(473\) 384.088 665.260i 0.812025 1.40647i
\(474\) 0 0
\(475\) 671.511i 1.41371i
\(476\) 0 0
\(477\) −330.950 −0.693816
\(478\) 0 0
\(479\) −227.894 131.575i −0.475770 0.274686i 0.242882 0.970056i \(-0.421907\pi\)
−0.718652 + 0.695370i \(0.755241\pi\)
\(480\) 0 0
\(481\) 6.12795 3.53797i 0.0127400 0.00735545i
\(482\) 0 0
\(483\) −131.525 + 30.1796i −0.272308 + 0.0624836i
\(484\) 0 0
\(485\) −94.3302 163.385i −0.194495 0.336876i
\(486\) 0 0
\(487\) 338.504 586.307i 0.695081 1.20392i −0.275072 0.961424i \(-0.588702\pi\)
0.970153 0.242492i \(-0.0779649\pi\)
\(488\) 0 0
\(489\) 46.0950i 0.0942637i
\(490\) 0 0
\(491\) −807.748 −1.64511 −0.822554 0.568686i \(-0.807452\pi\)
−0.822554 + 0.568686i \(0.807452\pi\)
\(492\) 0 0
\(493\) 44.1889 + 25.5125i 0.0896326 + 0.0517494i
\(494\) 0 0
\(495\) −150.213 + 86.7255i −0.303461 + 0.175203i
\(496\) 0 0
\(497\) 63.6876 + 277.555i 0.128144 + 0.558461i
\(498\) 0 0
\(499\) −78.2519 135.536i −0.156817 0.271616i 0.776902 0.629622i \(-0.216790\pi\)
−0.933719 + 0.358006i \(0.883457\pi\)
\(500\) 0 0
\(501\) −104.548 + 181.082i −0.208678 + 0.361441i
\(502\) 0 0
\(503\) 499.753i 0.993545i −0.867881 0.496773i \(-0.834518\pi\)
0.867881 0.496773i \(-0.165482\pi\)
\(504\) 0 0
\(505\) −277.133 −0.548779
\(506\) 0 0
\(507\) 118.128 + 68.2013i 0.232994 + 0.134519i
\(508\) 0 0
\(509\) 462.478 267.012i 0.908602 0.524582i 0.0286209 0.999590i \(-0.490888\pi\)
0.879981 + 0.475009i \(0.157555\pi\)
\(510\) 0 0
\(511\) −132.167 + 430.240i −0.258645 + 0.841958i
\(512\) 0 0
\(513\) −212.481 368.027i −0.414193 0.717403i
\(514\) 0 0
\(515\) −98.4604 + 170.538i −0.191185 + 0.331143i
\(516\) 0 0
\(517\) 999.509i 1.93329i
\(518\) 0 0
\(519\) −218.851 −0.421677
\(520\) 0 0
\(521\) −623.856 360.184i −1.19742 0.691331i −0.237442 0.971402i \(-0.576309\pi\)
−0.959979 + 0.280070i \(0.909642\pi\)
\(522\) 0 0
\(523\) −523.499 + 302.243i −1.00095 + 0.577902i −0.908531 0.417818i \(-0.862795\pi\)
−0.0924242 + 0.995720i \(0.529462\pi\)
\(524\) 0 0
\(525\) −92.5107 99.4721i −0.176211 0.189471i
\(526\) 0 0
\(527\) 89.9225 + 155.750i 0.170631 + 0.295542i
\(528\) 0 0
\(529\) 2.90487 5.03138i 0.00549125 0.00951112i
\(530\) 0 0
\(531\) 106.063i 0.199742i
\(532\) 0 0
\(533\) −121.090 −0.227186
\(534\) 0 0
\(535\) −129.011 74.4843i −0.241141 0.139223i
\(536\) 0 0
\(537\) −100.853 + 58.2272i −0.187807 + 0.108431i
\(538\) 0 0
\(539\) −52.8470 + 727.754i −0.0980463 + 1.35019i
\(540\) 0 0
\(541\) 177.344 + 307.168i 0.327807 + 0.567779i 0.982077 0.188483i \(-0.0603570\pi\)
−0.654269 + 0.756262i \(0.727024\pi\)
\(542\) 0 0
\(543\) 48.2487 83.5692i 0.0888557 0.153903i
\(544\) 0 0
\(545\) 200.581i 0.368039i
\(546\) 0 0
\(547\) −867.407 −1.58575 −0.792877 0.609382i \(-0.791417\pi\)
−0.792877 + 0.609382i \(0.791417\pi\)
\(548\) 0 0
\(549\) 558.795 + 322.620i 1.01784 + 0.587651i
\(550\) 0 0
\(551\) 95.1012 54.9067i 0.172597 0.0996492i
\(552\) 0 0
\(553\) 412.737 383.853i 0.746360 0.694128i
\(554\) 0 0
\(555\) 1.56638 + 2.71304i 0.00282230 + 0.00488837i
\(556\) 0 0
\(557\) −67.3674 + 116.684i −0.120947 + 0.209486i −0.920141 0.391586i \(-0.871926\pi\)
0.799194 + 0.601073i \(0.205260\pi\)
\(558\) 0 0
\(559\) 137.980i 0.246834i
\(560\) 0 0
\(561\) 170.067 0.303150
\(562\) 0 0
\(563\) 413.611 + 238.798i 0.734655 + 0.424153i 0.820123 0.572188i \(-0.193905\pi\)
−0.0854680 + 0.996341i \(0.527239\pi\)
\(564\) 0 0
\(565\) 132.051 76.2399i 0.233719 0.134938i
\(566\) 0 0
\(567\) −417.050 128.115i −0.735538 0.225953i
\(568\) 0 0
\(569\) 166.341 + 288.110i 0.292338 + 0.506345i 0.974362 0.224985i \(-0.0722333\pi\)
−0.682024 + 0.731330i \(0.738900\pi\)
\(570\) 0 0
\(571\) −525.564 + 910.304i −0.920427 + 1.59423i −0.121672 + 0.992570i \(0.538826\pi\)
−0.798755 + 0.601657i \(0.794508\pi\)
\(572\) 0 0
\(573\) 128.466i 0.224199i
\(574\) 0 0
\(575\) −526.675 −0.915956
\(576\) 0 0
\(577\) 350.366 + 202.284i 0.607221 + 0.350579i 0.771877 0.635772i \(-0.219318\pi\)
−0.164656 + 0.986351i \(0.552651\pi\)
\(578\) 0 0
\(579\) −176.858 + 102.109i −0.305455 + 0.176355i
\(580\) 0 0
\(581\) −831.438 + 190.781i −1.43105 + 0.328367i
\(582\) 0 0
\(583\) −297.252 514.856i −0.509867 0.883115i
\(584\) 0 0
\(585\) −15.5777 + 26.9813i −0.0266285 + 0.0461220i
\(586\) 0 0
\(587\) 64.6639i 0.110160i −0.998482 0.0550800i \(-0.982459\pi\)
0.998482 0.0550800i \(-0.0175414\pi\)
\(588\) 0 0
\(589\) 387.054 0.657137
\(590\) 0 0
\(591\) 144.165 + 83.2335i 0.243933 + 0.140835i
\(592\) 0 0
\(593\) 36.0843 20.8333i 0.0608504 0.0351320i −0.469266 0.883057i \(-0.655482\pi\)
0.530117 + 0.847925i \(0.322148\pi\)
\(594\) 0 0
\(595\) 29.8082 + 129.906i 0.0500979 + 0.218330i
\(596\) 0 0
\(597\) −69.1794 119.822i −0.115878 0.200707i
\(598\) 0 0
\(599\) 60.0025 103.927i 0.100171 0.173502i −0.811584 0.584236i \(-0.801394\pi\)
0.911755 + 0.410734i \(0.134728\pi\)
\(600\) 0 0
\(601\) 1120.08i 1.86369i 0.362858 + 0.931844i \(0.381801\pi\)
−0.362858 + 0.931844i \(0.618199\pi\)
\(602\) 0 0
\(603\) 379.999 0.630180
\(604\) 0 0
\(605\) −122.597 70.7812i −0.202639 0.116994i
\(606\) 0 0
\(607\) 383.831 221.605i 0.632342 0.365083i −0.149317 0.988789i \(-0.547707\pi\)
0.781658 + 0.623707i \(0.214374\pi\)
\(608\) 0 0
\(609\) −6.52328 + 21.2350i −0.0107115 + 0.0348687i
\(610\) 0 0
\(611\) −89.7662 155.480i −0.146917 0.254467i
\(612\) 0 0
\(613\) −568.642 + 984.916i −0.927637 + 1.60672i −0.140374 + 0.990099i \(0.544830\pi\)
−0.787264 + 0.616617i \(0.788503\pi\)
\(614\) 0 0
\(615\) 53.6106i 0.0871717i
\(616\) 0 0
\(617\) −357.454 −0.579343 −0.289671 0.957126i \(-0.593546\pi\)
−0.289671 + 0.957126i \(0.593546\pi\)
\(618\) 0 0
\(619\) −686.616 396.418i −1.10923 0.640416i −0.170603 0.985340i \(-0.554572\pi\)
−0.938631 + 0.344923i \(0.887905\pi\)
\(620\) 0 0
\(621\) 288.649 166.651i 0.464813 0.268360i
\(622\) 0 0
\(623\) −218.871 235.341i −0.351318 0.377754i
\(624\) 0 0
\(625\) −240.412 416.407i −0.384660 0.666250i
\(626\) 0 0
\(627\) 183.005 316.974i 0.291874 0.505541i
\(628\) 0 0
\(629\) 35.8480i 0.0569920i
\(630\) 0 0
\(631\) 587.326 0.930786 0.465393 0.885104i \(-0.345913\pi\)
0.465393 + 0.885104i \(0.345913\pi\)
\(632\) 0 0
\(633\) −38.7892 22.3950i −0.0612783 0.0353791i
\(634\) 0 0
\(635\) −170.103 + 98.2088i −0.267878 + 0.154660i
\(636\) 0 0
\(637\) 57.1392 + 117.953i 0.0897004 + 0.185169i
\(638\) 0 0
\(639\) −168.617 292.054i −0.263877 0.457048i
\(640\) 0 0
\(641\) −229.349 + 397.244i −0.357798 + 0.619725i −0.987593 0.157037i \(-0.949806\pi\)
0.629794 + 0.776762i \(0.283139\pi\)
\(642\) 0 0
\(643\) 242.009i 0.376374i −0.982133 0.188187i \(-0.939739\pi\)
0.982133 0.188187i \(-0.0602611\pi\)
\(644\) 0 0
\(645\) 61.0884 0.0947107
\(646\) 0 0
\(647\) 68.6970 + 39.6622i 0.106178 + 0.0613017i 0.552148 0.833746i \(-0.313808\pi\)
−0.445971 + 0.895048i \(0.647141\pi\)
\(648\) 0 0
\(649\) 165.001 95.2634i 0.254239 0.146785i
\(650\) 0 0
\(651\) −57.3349 + 53.3224i −0.0880721 + 0.0819085i
\(652\) 0 0
\(653\) −56.1079 97.1817i −0.0859232 0.148823i 0.819861 0.572563i \(-0.194051\pi\)
−0.905784 + 0.423739i \(0.860717\pi\)
\(654\) 0 0
\(655\) −66.7600 + 115.632i −0.101924 + 0.176537i
\(656\) 0 0
\(657\) 533.008i 0.811275i
\(658\) 0 0
\(659\) 811.930 1.23206 0.616032 0.787721i \(-0.288739\pi\)
0.616032 + 0.787721i \(0.288739\pi\)
\(660\) 0 0
\(661\) 317.842 + 183.506i 0.480850 + 0.277619i 0.720771 0.693174i \(-0.243788\pi\)
−0.239921 + 0.970792i \(0.577121\pi\)
\(662\) 0 0
\(663\) 26.4549 15.2738i 0.0399019 0.0230374i
\(664\) 0 0
\(665\) 274.198 + 84.2321i 0.412328 + 0.126665i
\(666\) 0 0
\(667\) 43.0640 + 74.5891i 0.0645638 + 0.111828i
\(668\) 0 0
\(669\) 130.952 226.815i 0.195742 0.339035i
\(670\) 0 0
\(671\) 1159.08i 1.72739i
\(672\) 0 0
\(673\) −504.526 −0.749668 −0.374834 0.927092i \(-0.622300\pi\)
−0.374834 + 0.927092i \(0.622300\pi\)
\(674\) 0 0
\(675\) 290.571 + 167.761i 0.430476 + 0.248535i
\(676\) 0 0
\(677\) 519.740 300.072i 0.767710 0.443238i −0.0643468 0.997928i \(-0.520496\pi\)
0.832057 + 0.554690i \(0.187163\pi\)
\(678\) 0 0
\(679\) 916.071 210.201i 1.34915 0.309574i
\(680\) 0 0
\(681\) 21.0270 + 36.4198i 0.0308766 + 0.0534798i
\(682\) 0 0
\(683\) −105.717 + 183.107i −0.154783 + 0.268093i −0.932980 0.359928i \(-0.882801\pi\)
0.778197 + 0.628020i \(0.216135\pi\)
\(684\) 0 0
\(685\) 215.780i 0.315007i
\(686\) 0 0
\(687\) −177.929 −0.258994
\(688\) 0 0
\(689\) −92.4787 53.3926i −0.134222 0.0774929i
\(690\) 0 0
\(691\) −16.6678 + 9.62318i −0.0241213 + 0.0139265i −0.512012 0.858978i \(-0.671100\pi\)
0.487891 + 0.872905i \(0.337766\pi\)
\(692\) 0 0
\(693\) −193.255 842.219i −0.278867 1.21532i
\(694\) 0 0
\(695\) −152.825 264.701i −0.219893 0.380865i
\(696\) 0 0
\(697\) 306.732 531.275i 0.440075 0.762232i
\(698\) 0 0
\(699\) 63.5928i 0.0909768i
\(700\) 0 0
\(701\) −1073.36 −1.53119 −0.765595 0.643323i \(-0.777555\pi\)
−0.765595 + 0.643323i \(0.777555\pi\)
\(702\) 0 0
\(703\) 66.8142 + 38.5752i 0.0950415 + 0.0548722i
\(704\) 0 0
\(705\) 68.8360 39.7425i 0.0976397 0.0563723i
\(706\) 0 0
\(707\) 405.425 1319.77i 0.573445 1.86672i
\(708\) 0 0
\(709\) −128.799 223.086i −0.181663 0.314649i 0.760784 0.649005i \(-0.224815\pi\)
−0.942447 + 0.334356i \(0.891481\pi\)
\(710\) 0 0
\(711\) −333.746 + 578.065i −0.469403 + 0.813030i
\(712\) 0 0
\(713\) 303.571i 0.425766i
\(714\) 0 0
\(715\) −55.9661 −0.0782743
\(716\) 0 0
\(717\) 226.553 + 130.800i 0.315973 + 0.182427i
\(718\) 0 0
\(719\) −278.533 + 160.811i −0.387389 + 0.223659i −0.681028 0.732257i \(-0.738467\pi\)
0.293639 + 0.955916i \(0.405134\pi\)
\(720\) 0 0
\(721\) −668.101 718.375i −0.926630 0.996359i
\(722\) 0 0
\(723\) 45.6030 + 78.9867i 0.0630747 + 0.109249i
\(724\) 0 0
\(725\) −43.3509 + 75.0859i −0.0597943 + 0.103567i
\(726\) 0 0
\(727\) 19.9115i 0.0273886i −0.999906 0.0136943i \(-0.995641\pi\)
0.999906 0.0136943i \(-0.00435917\pi\)
\(728\) 0 0
\(729\) 406.138 0.557117
\(730\) 0 0
\(731\) 605.380 + 349.516i 0.828153 + 0.478134i
\(732\) 0 0
\(733\) −514.080 + 296.804i −0.701337 + 0.404917i −0.807845 0.589395i \(-0.799366\pi\)
0.106508 + 0.994312i \(0.466033\pi\)
\(734\) 0 0
\(735\) −52.2216 + 25.2974i −0.0710498 + 0.0344182i
\(736\) 0 0
\(737\) 341.306 + 591.160i 0.463102 + 0.802116i
\(738\) 0 0
\(739\) 505.499 875.550i 0.684031 1.18478i −0.289709 0.957115i \(-0.593559\pi\)
0.973740 0.227662i \(-0.0731080\pi\)
\(740\) 0 0
\(741\) 65.7430i 0.0887219i
\(742\) 0 0
\(743\) 581.910 0.783189 0.391595 0.920138i \(-0.371924\pi\)
0.391595 + 0.920138i \(0.371924\pi\)
\(744\) 0 0
\(745\) 77.9883 + 45.0266i 0.104682 + 0.0604383i
\(746\) 0 0
\(747\) 874.870 505.106i 1.17118 0.676180i
\(748\) 0 0
\(749\) 543.443 505.411i 0.725558 0.674781i
\(750\) 0 0
\(751\) 605.277 + 1048.37i 0.805962 + 1.39597i 0.915640 + 0.402000i \(0.131685\pi\)
−0.109678 + 0.993967i \(0.534982\pi\)
\(752\) 0 0
\(753\) −127.272 + 220.441i −0.169019 + 0.292750i
\(754\) 0 0
\(755\) 382.251i 0.506293i
\(756\) 0 0
\(757\) 245.833 0.324746 0.162373 0.986729i \(-0.448085\pi\)
0.162373 + 0.986729i \(0.448085\pi\)
\(758\) 0 0
\(759\) 248.607 + 143.533i 0.327545 + 0.189108i
\(760\) 0 0
\(761\) 59.4109 34.3009i 0.0780694 0.0450734i −0.460457 0.887682i \(-0.652314\pi\)
0.538527 + 0.842609i \(0.318981\pi\)
\(762\) 0 0
\(763\) 955.212 + 293.436i 1.25192 + 0.384582i
\(764\) 0 0
\(765\) −78.9194 136.692i −0.103163 0.178683i
\(766\) 0 0
\(767\) 17.1113 29.6376i 0.0223094 0.0386409i
\(768\) 0 0
\(769\) 540.953i 0.703450i 0.936103 + 0.351725i \(0.114405\pi\)
−0.936103 + 0.351725i \(0.885595\pi\)
\(770\) 0 0
\(771\) −87.3977 −0.113356
\(772\) 0 0
\(773\) 223.462 + 129.016i 0.289085 + 0.166903i 0.637529 0.770426i \(-0.279957\pi\)
−0.348444 + 0.937329i \(0.613290\pi\)
\(774\) 0 0
\(775\) −264.651 + 152.797i −0.341486 + 0.197157i
\(776\) 0 0
\(777\) −15.2116 + 3.49044i −0.0195773 + 0.00449220i
\(778\) 0 0
\(779\) −660.134 1143.39i −0.847412 1.46776i
\(780\) 0 0
\(781\) 302.897 524.633i 0.387832 0.671745i
\(782\) 0 0
\(783\) 54.8686i 0.0700749i
\(784\) 0 0
\(785\) −303.880 −0.387109
\(786\) 0 0
\(787\) −156.307 90.2438i −0.198611 0.114668i 0.397396 0.917647i \(-0.369914\pi\)
−0.596007 + 0.802979i \(0.703247\pi\)
\(788\) 0 0
\(789\) 188.268 108.697i 0.238616 0.137765i
\(790\) 0 0
\(791\) 169.889 + 740.391i 0.214778 + 0.936018i
\(792\) 0 0
\(793\) 104.097 + 180.302i 0.131270 + 0.227367i
\(794\) 0 0
\(795\) 23.6387 40.9434i 0.0297342 0.0515011i
\(796\) 0 0
\(797\) 1387.83i 1.74132i 0.491886 + 0.870660i \(0.336308\pi\)
−0.491886 + 0.870660i \(0.663692\pi\)
\(798\) 0 0
\(799\) 909.543 1.13835
\(800\) 0 0
\(801\) 329.610 + 190.300i 0.411498 + 0.237578i
\(802\) 0 0
\(803\) 829.194 478.736i 1.03262 0.596184i
\(804\) 0 0
\(805\) −66.0643 + 215.057i −0.0820674 + 0.267151i
\(806\) 0 0
\(807\) −68.3767 118.432i −0.0847295 0.146756i
\(808\) 0 0
\(809\) 354.447 613.920i 0.438130 0.758863i −0.559415 0.828887i \(-0.688974\pi\)
0.997545 + 0.0700243i \(0.0223077\pi\)
\(810\) 0 0
\(811\) 281.652i 0.347289i −0.984808 0.173645i \(-0.944446\pi\)
0.984808 0.173645i \(-0.0555544\pi\)
\(812\) 0 0
\(813\) 52.8852 0.0650495
\(814\) 0 0
\(815\) 66.5533 + 38.4246i 0.0816605 + 0.0471467i
\(816\) 0 0
\(817\) 1302.87 752.212i 1.59470 0.920700i
\(818\) 0 0
\(819\) −105.702 113.656i −0.129062 0.138774i
\(820\) 0 0
\(821\) 610.722 + 1057.80i 0.743876 + 1.28843i 0.950718 + 0.310057i \(0.100348\pi\)
−0.206842 + 0.978374i \(0.566319\pi\)
\(822\) 0 0
\(823\) 41.7002 72.2268i 0.0506685 0.0877604i −0.839579 0.543238i \(-0.817198\pi\)
0.890247 + 0.455478i \(0.150531\pi\)
\(824\) 0 0
\(825\) 288.978i 0.350277i
\(826\) 0 0
\(827\) 804.602 0.972916 0.486458 0.873704i \(-0.338289\pi\)
0.486458 + 0.873704i \(0.338289\pi\)
\(828\) 0 0
\(829\) −891.302 514.593i −1.07515 0.620740i −0.145568 0.989348i \(-0.546501\pi\)
−0.929585 + 0.368608i \(0.879834\pi\)
\(830\) 0 0
\(831\) −187.691 + 108.363i −0.225862 + 0.130401i
\(832\) 0 0
\(833\) −662.249 48.0902i −0.795017 0.0577314i
\(834\) 0 0
\(835\) 174.301 + 301.899i 0.208744 + 0.361555i
\(836\) 0 0
\(837\) 96.6963 167.483i 0.115527 0.200099i
\(838\) 0 0
\(839\) 864.563i 1.03047i −0.857049 0.515234i \(-0.827705\pi\)
0.857049 0.515234i \(-0.172295\pi\)
\(840\) 0 0
\(841\) −826.822 −0.983141
\(842\) 0 0
\(843\) 168.050 + 97.0237i 0.199348 + 0.115093i
\(844\) 0 0
\(845\) 196.942 113.705i 0.233068 0.134562i
\(846\) 0 0
\(847\) 516.425 480.284i 0.609711 0.567041i
\(848\) 0 0
\(849\) 172.562 + 298.886i 0.203253 + 0.352044i
\(850\) 0 0
\(851\) −30.2550 + 52.4032i −0.0355523 + 0.0615784i
\(852\) 0 0
\(853\) 285.568i 0.334781i 0.985891 + 0.167391i \(0.0535341\pi\)
−0.985891 + 0.167391i \(0.946466\pi\)
\(854\) 0 0
\(855\) −339.693 −0.397302
\(856\) 0 0
\(857\) −565.510 326.497i −0.659872 0.380977i 0.132356 0.991202i \(-0.457746\pi\)
−0.792228 + 0.610225i \(0.791079\pi\)
\(858\) 0 0
\(859\) −75.6507 + 43.6770i −0.0880683 + 0.0508463i −0.543387 0.839482i \(-0.682859\pi\)
0.455319 + 0.890328i \(0.349525\pi\)
\(860\) 0 0
\(861\) 255.305 + 78.4283i 0.296522 + 0.0910898i
\(862\) 0 0
\(863\) −213.639 370.034i −0.247554 0.428776i 0.715293 0.698825i \(-0.246293\pi\)
−0.962847 + 0.270049i \(0.912960\pi\)
\(864\) 0 0
\(865\) −182.433 + 315.983i −0.210905 + 0.365299i
\(866\) 0 0
\(867\) 88.8083i 0.102432i
\(868\) 0 0
\(869\) −1199.05 −1.37981
\(870\) 0 0
\(871\) 106.184 + 61.3056i 0.121911 + 0.0703853i
\(872\) 0 0
\(873\) −963.923 + 556.521i −1.10415 + 0.637482i
\(874\) 0 0
\(875\) −460.401 + 105.643i −0.526173 + 0.120735i
\(876\) 0 0
\(877\) −290.651 503.422i −0.331415 0.574028i 0.651374 0.758756i \(-0.274193\pi\)
−0.982790 + 0.184729i \(0.940859\pi\)
\(878\) 0 0
\(879\) 184.617 319.765i 0.210030 0.363783i
\(880\) 0 0
\(881\) 617.976i 0.701448i −0.936479 0.350724i \(-0.885936\pi\)
0.936479 0.350724i \(-0.114064\pi\)
\(882\) 0 0
\(883\) −150.782 −0.170761 −0.0853807 0.996348i \(-0.527211\pi\)
−0.0853807 + 0.996348i \(0.527211\pi\)
\(884\) 0 0
\(885\) 13.1215 + 7.57573i 0.0148266 + 0.00856015i
\(886\) 0 0
\(887\) 1158.00 668.569i 1.30552 0.753742i 0.324174 0.945997i \(-0.394914\pi\)
0.981345 + 0.192256i \(0.0615804\pi\)
\(888\) 0 0
\(889\) −218.844 953.737i −0.246169 1.07282i
\(890\) 0 0
\(891\) 464.058 + 803.773i 0.520829 + 0.902102i
\(892\) 0 0
\(893\) 978.738 1695.22i 1.09601 1.89835i
\(894\) 0 0
\(895\) 194.152i 0.216930i
\(896\) 0 0
\(897\) 51.5630 0.0574839
\(898\) 0 0
\(899\) 43.2789 + 24.9871i 0.0481412 + 0.0277943i
\(900\) 0 0
\(901\) 468.514 270.497i 0.519993 0.300218i
\(902\) 0 0
\(903\) −89.3678 + 290.916i −0.0989677 + 0.322166i
\(904\) 0 0
\(905\) −80.4398 139.326i −0.0888838 0.153951i
\(906\) 0 0
\(907\) 322.677 558.894i 0.355763 0.616200i −0.631485 0.775388i \(-0.717554\pi\)
0.987248 + 0.159188i \(0.0508876\pi\)
\(908\) 0 0
\(909\) 1635.01i 1.79869i
\(910\) 0 0
\(911\) −865.195 −0.949720 −0.474860 0.880061i \(-0.657501\pi\)
−0.474860 + 0.880061i \(0.657501\pi\)
\(912\) 0 0
\(913\) 1571.58 + 907.350i 1.72133 + 0.993812i
\(914\) 0 0
\(915\) −79.8257 + 46.0874i −0.0872412 + 0.0503687i
\(916\) 0 0
\(917\) −452.998 487.086i −0.494000 0.531173i
\(918\) 0 0
\(919\) 31.6093 + 54.7489i 0.0343953 + 0.0595745i 0.882711 0.469917i \(-0.155716\pi\)
−0.848315 + 0.529491i \(0.822383\pi\)
\(920\) 0 0
\(921\) 57.5268 99.6393i 0.0624612 0.108186i
\(922\) 0 0
\(923\) 108.813i 0.117891i
\(924\) 0 0
\(925\) −60.9130 −0.0658519
\(926\) 0 0
\(927\) 1006.13 + 580.889i 1.08536 + 0.626633i
\(928\) 0 0
\(929\) 1109.30 640.454i 1.19408 0.689402i 0.234850 0.972032i \(-0.424540\pi\)
0.959229 + 0.282630i \(0.0912069\pi\)
\(930\) 0 0
\(931\) −802.262 + 1182.56i −0.861721 + 1.27021i
\(932\) 0 0
\(933\) 244.799 + 424.004i 0.262378 + 0.454452i
\(934\) 0 0
\(935\) 141.767 245.548i 0.151623 0.262618i
\(936\) 0 0
\(937\) 1176.84i 1.25597i 0.778227 + 0.627983i \(0.216119\pi\)
−0.778227 + 0.627983i \(0.783881\pi\)
\(938\) 0 0
\(939\) −157.769 −0.168018
\(940\) 0 0
\(941\) −1184.64 683.950i −1.25891 0.726833i −0.286048 0.958215i \(-0.592342\pi\)
−0.972863 + 0.231382i \(0.925675\pi\)
\(942\) 0 0
\(943\) 896.772 517.752i 0.950978 0.549047i
\(944\) 0 0
\(945\) 104.950 97.6054i 0.111058 0.103286i
\(946\) 0 0
\(947\) 87.6049 + 151.736i 0.0925078 + 0.160228i 0.908566 0.417742i \(-0.137178\pi\)
−0.816058 + 0.577970i \(0.803845\pi\)
\(948\) 0 0
\(949\) 85.9908 148.940i 0.0906120 0.156945i
\(950\) 0 0
\(951\) 139.809i 0.147013i
\(952\) 0 0
\(953\) 89.5669 0.0939842 0.0469921 0.998895i \(-0.485036\pi\)
0.0469921 + 0.998895i \(0.485036\pi\)
\(954\) 0 0
\(955\) 185.484 + 107.089i 0.194224 + 0.112135i
\(956\) 0 0
\(957\) 40.9259 23.6286i 0.0427648 0.0246903i
\(958\) 0 0
\(959\) −1027.59 315.670i −1.07152 0.329166i
\(960\) 0 0
\(961\) −392.429 679.707i −0.408355 0.707292i
\(962\) 0 0
\(963\) −439.436 + 761.126i −0.456320 + 0.790370i
\(964\) 0 0
\(965\) 340.472i 0.352820i
\(966\) 0 0
\(967\) −640.201 −0.662049 −0.331024 0.943622i \(-0.607394\pi\)
−0.331024 + 0.943622i \(0.607394\pi\)
\(968\) 0 0
\(969\) 288.443 + 166.533i 0.297671 + 0.171860i
\(970\) 0 0
\(971\) −990.220 + 571.704i −1.01979 + 0.588779i −0.914044 0.405614i \(-0.867058\pi\)
−0.105750 + 0.994393i \(0.533724\pi\)
\(972\) 0 0
\(973\) 1484.14 340.549i 1.52532 0.349999i
\(974\) 0 0
\(975\) 25.9532 + 44.9523i 0.0266187 + 0.0461049i
\(976\) 0 0
\(977\) −801.534 + 1388.30i −0.820403 + 1.42098i 0.0849795 + 0.996383i \(0.472918\pi\)
−0.905382 + 0.424597i \(0.860416\pi\)
\(978\) 0 0
\(979\) 683.694i 0.698359i
\(980\) 0 0
\(981\) −1183.37 −1.20629
\(982\) 0 0
\(983\) −850.237 490.885i −0.864941 0.499374i 0.000722851 1.00000i \(-0.499770\pi\)
−0.865664 + 0.500626i \(0.833103\pi\)
\(984\) 0 0
\(985\) 240.350 138.766i 0.244010 0.140879i
\(986\) 0 0
\(987\) 88.5602 + 385.952i 0.0897267 + 0.391035i
\(988\) 0 0
\(989\) 589.970 + 1021.86i 0.596531 + 1.03322i
\(990\) 0 0
\(991\) 413.854 716.817i 0.417613 0.723327i −0.578086 0.815976i \(-0.696200\pi\)
0.995699 + 0.0926491i \(0.0295335\pi\)
\(992\) 0 0
\(993\) 352.455i 0.354939i
\(994\) 0 0
\(995\) −230.671 −0.231830
\(996\) 0 0
\(997\) 156.575 + 90.3985i 0.157046 + 0.0906705i 0.576463 0.817123i \(-0.304432\pi\)
−0.419418 + 0.907793i \(0.637766\pi\)
\(998\) 0 0
\(999\) 33.3839 19.2742i 0.0334173 0.0192935i
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 448.3.s.h.129.5 16
4.3 odd 2 inner 448.3.s.h.129.4 16
7.5 odd 6 inner 448.3.s.h.257.5 16
8.3 odd 2 224.3.s.b.129.5 yes 16
8.5 even 2 224.3.s.b.129.4 yes 16
28.19 even 6 inner 448.3.s.h.257.4 16
56.3 even 6 1568.3.c.g.97.10 16
56.5 odd 6 224.3.s.b.33.4 16
56.11 odd 6 1568.3.c.g.97.7 16
56.19 even 6 224.3.s.b.33.5 yes 16
56.45 odd 6 1568.3.c.g.97.8 16
56.53 even 6 1568.3.c.g.97.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.s.b.33.4 16 56.5 odd 6
224.3.s.b.33.5 yes 16 56.19 even 6
224.3.s.b.129.4 yes 16 8.5 even 2
224.3.s.b.129.5 yes 16 8.3 odd 2
448.3.s.h.129.4 16 4.3 odd 2 inner
448.3.s.h.129.5 16 1.1 even 1 trivial
448.3.s.h.257.4 16 28.19 even 6 inner
448.3.s.h.257.5 16 7.5 odd 6 inner
1568.3.c.g.97.7 16 56.11 odd 6
1568.3.c.g.97.8 16 56.45 odd 6
1568.3.c.g.97.9 16 56.53 even 6
1568.3.c.g.97.10 16 56.3 even 6