Properties

Label 112.4.p.g.47.2
Level $112$
Weight $4$
Character 112.47
Analytic conductor $6.608$
Analytic rank $0$
Dimension $6$
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] = [112,4,Mod(31,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.p (of order \(6\), degree \(2\), 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: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.12258833328.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 47.2
Root \(2.68858 + 4.65676i\) of defining polynomial
Character \(\chi\) \(=\) 112.47
Dual form 112.4.p.g.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.38417 + 4.12950i) q^{3} +(14.6315 + 8.44749i) q^{5} +(8.89981 - 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +O(q^{10})\) \(q+(2.38417 + 4.12950i) q^{3} +(14.6315 + 8.44749i) q^{5} +(8.89981 - 16.2417i) q^{7} +(2.13148 - 3.69182i) q^{9} +(-35.2840 + 20.3712i) q^{11} +56.7321i q^{13} +80.5609i q^{15} +(106.377 - 61.4170i) q^{17} +(-37.7473 + 65.3803i) q^{19} +(88.2889 - 1.97120i) q^{21} +(-91.9414 - 53.0824i) q^{23} +(80.2201 + 138.945i) q^{25} +149.072 q^{27} -146.703 q^{29} +(-21.2956 - 36.8850i) q^{31} +(-168.246 - 97.1369i) q^{33} +(267.419 - 162.459i) q^{35} +(-40.4571 + 70.0738i) q^{37} +(-234.275 + 135.259i) q^{39} -53.8052i q^{41} -341.166i q^{43} +(62.3733 - 36.0112i) q^{45} +(-2.06393 + 3.57482i) q^{47} +(-184.587 - 289.096i) q^{49} +(507.244 + 292.857i) q^{51} +(-139.849 - 242.225i) q^{53} -688.342 q^{55} -359.984 q^{57} +(-87.4155 - 151.408i) q^{59} +(-404.509 - 233.543i) q^{61} +(-40.9918 - 67.4754i) q^{63} +(-479.244 + 830.075i) q^{65} +(652.335 - 376.626i) q^{67} -506.229i q^{69} -669.583i q^{71} +(723.802 - 417.887i) q^{73} +(-382.516 + 662.538i) q^{75} +(16.8426 + 754.372i) q^{77} +(958.008 + 553.106i) q^{79} +(297.864 + 515.915i) q^{81} -552.905 q^{83} +2075.28 q^{85} +(-349.765 - 605.811i) q^{87} +(106.062 + 61.2347i) q^{89} +(921.427 + 504.906i) q^{91} +(101.545 - 175.880i) q^{93} +(-1104.60 + 637.740i) q^{95} -291.203i q^{97} +173.683i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} - 15 q^{21} + 15 q^{23} + 306 q^{25} - 362 q^{27} - 348 q^{29} - 205 q^{31} - 471 q^{33} + 1185 q^{35} - 249 q^{37} - 288 q^{39} + 2118 q^{45} - 75 q^{47} + 702 q^{49} + 2505 q^{51} - 645 q^{53} - 918 q^{55} - 6 q^{57} - 321 q^{59} - 1707 q^{61} + 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} + 555 q^{77} + 3447 q^{79} + 225 q^{81} - 24 q^{83} + 3786 q^{85} - 3642 q^{87} - 2607 q^{89} + 2448 q^{91} - 2991 q^{93} - 2085 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 2.38417 + 4.12950i 0.458834 + 0.794723i 0.998900 0.0468996i \(-0.0149341\pi\)
−0.540066 + 0.841623i \(0.681601\pi\)
\(4\) 0 0
\(5\) 14.6315 + 8.44749i 1.30868 + 0.755566i 0.981876 0.189526i \(-0.0606952\pi\)
0.326803 + 0.945092i \(0.394029\pi\)
\(6\) 0 0
\(7\) 8.89981 16.2417i 0.480545 0.876970i
\(8\) 0 0
\(9\) 2.13148 3.69182i 0.0789436 0.136734i
\(10\) 0 0
\(11\) −35.2840 + 20.3712i −0.967138 + 0.558378i −0.898363 0.439255i \(-0.855243\pi\)
−0.0687757 + 0.997632i \(0.521909\pi\)
\(12\) 0 0
\(13\) 56.7321i 1.21036i 0.796089 + 0.605179i \(0.206899\pi\)
−0.796089 + 0.605179i \(0.793101\pi\)
\(14\) 0 0
\(15\) 80.5609i 1.38672i
\(16\) 0 0
\(17\) 106.377 61.4170i 1.51767 0.876225i 0.517882 0.855452i \(-0.326721\pi\)
0.999784 0.0207724i \(-0.00661253\pi\)
\(18\) 0 0
\(19\) −37.7473 + 65.3803i −0.455780 + 0.789435i −0.998733 0.0503288i \(-0.983973\pi\)
0.542952 + 0.839763i \(0.317306\pi\)
\(20\) 0 0
\(21\) 88.2889 1.97120i 0.917438 0.0204833i
\(22\) 0 0
\(23\) −91.9414 53.0824i −0.833526 0.481236i 0.0215324 0.999768i \(-0.493146\pi\)
−0.855058 + 0.518532i \(0.826479\pi\)
\(24\) 0 0
\(25\) 80.2201 + 138.945i 0.641760 + 1.11156i
\(26\) 0 0
\(27\) 149.072 1.06255
\(28\) 0 0
\(29\) −146.703 −0.939382 −0.469691 0.882831i \(-0.655635\pi\)
−0.469691 + 0.882831i \(0.655635\pi\)
\(30\) 0 0
\(31\) −21.2956 36.8850i −0.123381 0.213702i 0.797718 0.603030i \(-0.206040\pi\)
−0.921099 + 0.389329i \(0.872707\pi\)
\(32\) 0 0
\(33\) −168.246 97.1369i −0.887511 0.512405i
\(34\) 0 0
\(35\) 267.419 162.459i 1.29149 0.784589i
\(36\) 0 0
\(37\) −40.4571 + 70.0738i −0.179760 + 0.311353i −0.941798 0.336179i \(-0.890865\pi\)
0.762039 + 0.647532i \(0.224199\pi\)
\(38\) 0 0
\(39\) −234.275 + 135.259i −0.961900 + 0.555353i
\(40\) 0 0
\(41\) 53.8052i 0.204950i −0.994736 0.102475i \(-0.967324\pi\)
0.994736 0.102475i \(-0.0326762\pi\)
\(42\) 0 0
\(43\) 341.166i 1.20994i −0.796249 0.604970i \(-0.793185\pi\)
0.796249 0.604970i \(-0.206815\pi\)
\(44\) 0 0
\(45\) 62.3733 36.0112i 0.206624 0.119294i
\(46\) 0 0
\(47\) −2.06393 + 3.57482i −0.00640542 + 0.0110945i −0.869210 0.494442i \(-0.835372\pi\)
0.862805 + 0.505537i \(0.168706\pi\)
\(48\) 0 0
\(49\) −184.587 289.096i −0.538153 0.842847i
\(50\) 0 0
\(51\) 507.244 + 292.857i 1.39271 + 0.804083i
\(52\) 0 0
\(53\) −139.849 242.225i −0.362447 0.627777i 0.625916 0.779891i \(-0.284725\pi\)
−0.988363 + 0.152114i \(0.951392\pi\)
\(54\) 0 0
\(55\) −688.342 −1.68756
\(56\) 0 0
\(57\) −359.984 −0.836509
\(58\) 0 0
\(59\) −87.4155 151.408i −0.192890 0.334096i 0.753317 0.657658i \(-0.228453\pi\)
−0.946207 + 0.323562i \(0.895119\pi\)
\(60\) 0 0
\(61\) −404.509 233.543i −0.849050 0.490199i 0.0112801 0.999936i \(-0.496409\pi\)
−0.860330 + 0.509737i \(0.829743\pi\)
\(62\) 0 0
\(63\) −40.9918 67.4754i −0.0819759 0.134938i
\(64\) 0 0
\(65\) −479.244 + 830.075i −0.914506 + 1.58397i
\(66\) 0 0
\(67\) 652.335 376.626i 1.18948 0.686748i 0.231294 0.972884i \(-0.425704\pi\)
0.958189 + 0.286136i \(0.0923708\pi\)
\(68\) 0 0
\(69\) 506.229i 0.883230i
\(70\) 0 0
\(71\) 669.583i 1.11922i −0.828755 0.559611i \(-0.810950\pi\)
0.828755 0.559611i \(-0.189050\pi\)
\(72\) 0 0
\(73\) 723.802 417.887i 1.16047 0.670000i 0.209057 0.977904i \(-0.432961\pi\)
0.951418 + 0.307904i \(0.0996274\pi\)
\(74\) 0 0
\(75\) −382.516 + 662.538i −0.588922 + 1.02004i
\(76\) 0 0
\(77\) 16.8426 + 754.372i 0.0249272 + 1.11648i
\(78\) 0 0
\(79\) 958.008 + 553.106i 1.36436 + 0.787713i 0.990201 0.139652i \(-0.0445983\pi\)
0.374159 + 0.927365i \(0.377932\pi\)
\(80\) 0 0
\(81\) 297.864 + 515.915i 0.408592 + 0.707703i
\(82\) 0 0
\(83\) −552.905 −0.731195 −0.365597 0.930773i \(-0.619135\pi\)
−0.365597 + 0.930773i \(0.619135\pi\)
\(84\) 0 0
\(85\) 2075.28 2.64818
\(86\) 0 0
\(87\) −349.765 605.811i −0.431020 0.746548i
\(88\) 0 0
\(89\) 106.062 + 61.2347i 0.126320 + 0.0729310i 0.561829 0.827254i \(-0.310098\pi\)
−0.435508 + 0.900185i \(0.643431\pi\)
\(90\) 0 0
\(91\) 921.427 + 504.906i 1.06145 + 0.581632i
\(92\) 0 0
\(93\) 101.545 175.880i 0.113222 0.196107i
\(94\) 0 0
\(95\) −1104.60 + 637.740i −1.19294 + 0.688744i
\(96\) 0 0
\(97\) 291.203i 0.304816i −0.988318 0.152408i \(-0.951297\pi\)
0.988318 0.152408i \(-0.0487028\pi\)
\(98\) 0 0
\(99\) 173.683i 0.176321i
\(100\) 0 0
\(101\) −1168.30 + 674.520i −1.15099 + 0.664527i −0.949129 0.314887i \(-0.898034\pi\)
−0.201865 + 0.979413i \(0.564700\pi\)
\(102\) 0 0
\(103\) −989.035 + 1713.06i −0.946141 + 1.63876i −0.192690 + 0.981260i \(0.561721\pi\)
−0.753451 + 0.657504i \(0.771612\pi\)
\(104\) 0 0
\(105\) 1308.45 + 716.977i 1.21611 + 0.666379i
\(106\) 0 0
\(107\) 732.931 + 423.158i 0.662198 + 0.382320i 0.793114 0.609073i \(-0.208459\pi\)
−0.130916 + 0.991393i \(0.541792\pi\)
\(108\) 0 0
\(109\) −463.538 802.872i −0.407330 0.705516i 0.587260 0.809398i \(-0.300207\pi\)
−0.994590 + 0.103883i \(0.966873\pi\)
\(110\) 0 0
\(111\) −385.826 −0.329919
\(112\) 0 0
\(113\) −599.053 −0.498709 −0.249355 0.968412i \(-0.580218\pi\)
−0.249355 + 0.968412i \(0.580218\pi\)
\(114\) 0 0
\(115\) −896.825 1553.35i −0.727212 1.25957i
\(116\) 0 0
\(117\) 209.445 + 120.923i 0.165497 + 0.0955500i
\(118\) 0 0
\(119\) −50.7787 2274.35i −0.0391166 1.75201i
\(120\) 0 0
\(121\) 164.473 284.876i 0.123571 0.214031i
\(122\) 0 0
\(123\) 222.189 128.281i 0.162879 0.0940380i
\(124\) 0 0
\(125\) 598.760i 0.428438i
\(126\) 0 0
\(127\) 1833.90i 1.28136i 0.767810 + 0.640678i \(0.221347\pi\)
−0.767810 + 0.640678i \(0.778653\pi\)
\(128\) 0 0
\(129\) 1408.85 813.398i 0.961567 0.555161i
\(130\) 0 0
\(131\) −689.574 + 1194.38i −0.459911 + 0.796589i −0.998956 0.0456878i \(-0.985452\pi\)
0.539045 + 0.842277i \(0.318785\pi\)
\(132\) 0 0
\(133\) 725.943 + 1194.95i 0.473288 + 0.779064i
\(134\) 0 0
\(135\) 2181.15 + 1259.29i 1.39054 + 0.802831i
\(136\) 0 0
\(137\) −736.714 1276.03i −0.459429 0.795754i 0.539502 0.841984i \(-0.318612\pi\)
−0.998931 + 0.0462305i \(0.985279\pi\)
\(138\) 0 0
\(139\) 1929.87 1.17762 0.588811 0.808271i \(-0.299596\pi\)
0.588811 + 0.808271i \(0.299596\pi\)
\(140\) 0 0
\(141\) −19.6830 −0.0117561
\(142\) 0 0
\(143\) −1155.70 2001.74i −0.675837 1.17058i
\(144\) 0 0
\(145\) −2146.48 1239.27i −1.22935 0.709765i
\(146\) 0 0
\(147\) 753.739 1451.51i 0.422907 0.814409i
\(148\) 0 0
\(149\) −1153.42 + 1997.78i −0.634172 + 1.09842i 0.352517 + 0.935805i \(0.385326\pi\)
−0.986690 + 0.162614i \(0.948008\pi\)
\(150\) 0 0
\(151\) 887.906 512.633i 0.478522 0.276275i −0.241278 0.970456i \(-0.577567\pi\)
0.719800 + 0.694181i \(0.244233\pi\)
\(152\) 0 0
\(153\) 523.636i 0.276689i
\(154\) 0 0
\(155\) 719.577i 0.372889i
\(156\) 0 0
\(157\) 66.6966 38.5073i 0.0339043 0.0195746i −0.482952 0.875647i \(-0.660435\pi\)
0.516856 + 0.856072i \(0.327102\pi\)
\(158\) 0 0
\(159\) 666.846 1155.01i 0.332606 0.576090i
\(160\) 0 0
\(161\) −1680.41 + 1020.86i −0.822577 + 0.499722i
\(162\) 0 0
\(163\) −3366.71 1943.77i −1.61780 0.934036i −0.987489 0.157691i \(-0.949595\pi\)
−0.630308 0.776345i \(-0.717072\pi\)
\(164\) 0 0
\(165\) −1641.12 2842.51i −0.774311 1.34115i
\(166\) 0 0
\(167\) −117.863 −0.0546141 −0.0273070 0.999627i \(-0.508693\pi\)
−0.0273070 + 0.999627i \(0.508693\pi\)
\(168\) 0 0
\(169\) −1021.54 −0.464969
\(170\) 0 0
\(171\) 160.915 + 278.713i 0.0719618 + 0.124642i
\(172\) 0 0
\(173\) −2480.48 1432.11i −1.09010 0.629370i −0.156498 0.987678i \(-0.550020\pi\)
−0.933603 + 0.358308i \(0.883354\pi\)
\(174\) 0 0
\(175\) 2970.65 66.3248i 1.28320 0.0286496i
\(176\) 0 0
\(177\) 416.826 721.965i 0.177009 0.306589i
\(178\) 0 0
\(179\) −1195.93 + 690.471i −0.499375 + 0.288314i −0.728455 0.685093i \(-0.759761\pi\)
0.229081 + 0.973407i \(0.426428\pi\)
\(180\) 0 0
\(181\) 204.817i 0.0841103i −0.999115 0.0420551i \(-0.986609\pi\)
0.999115 0.0420551i \(-0.0133905\pi\)
\(182\) 0 0
\(183\) 2227.23i 0.899680i
\(184\) 0 0
\(185\) −1183.89 + 683.522i −0.470495 + 0.271641i
\(186\) 0 0
\(187\) −2502.28 + 4334.08i −0.978528 + 1.69486i
\(188\) 0 0
\(189\) 1326.72 2421.19i 0.510605 0.931829i
\(190\) 0 0
\(191\) 3609.38 + 2083.88i 1.36736 + 0.789445i 0.990590 0.136862i \(-0.0437018\pi\)
0.376769 + 0.926307i \(0.377035\pi\)
\(192\) 0 0
\(193\) 2392.97 + 4144.74i 0.892485 + 1.54583i 0.836887 + 0.547376i \(0.184373\pi\)
0.0555978 + 0.998453i \(0.482294\pi\)
\(194\) 0 0
\(195\) −4570.39 −1.67842
\(196\) 0 0
\(197\) −377.368 −0.136479 −0.0682395 0.997669i \(-0.521738\pi\)
−0.0682395 + 0.997669i \(0.521738\pi\)
\(198\) 0 0
\(199\) 809.764 + 1402.55i 0.288455 + 0.499620i 0.973441 0.228937i \(-0.0735249\pi\)
−0.684986 + 0.728556i \(0.740192\pi\)
\(200\) 0 0
\(201\) 3110.55 + 1795.88i 1.09155 + 0.630206i
\(202\) 0 0
\(203\) −1305.63 + 2382.71i −0.451415 + 0.823810i
\(204\) 0 0
\(205\) 454.519 787.249i 0.154853 0.268214i
\(206\) 0 0
\(207\) −391.942 + 226.288i −0.131603 + 0.0759810i
\(208\) 0 0
\(209\) 3075.83i 1.01799i
\(210\) 0 0
\(211\) 3010.54i 0.982247i −0.871090 0.491123i \(-0.836586\pi\)
0.871090 0.491123i \(-0.163414\pi\)
\(212\) 0 0
\(213\) 2765.04 1596.40i 0.889472 0.513537i
\(214\) 0 0
\(215\) 2882.00 4991.77i 0.914189 1.58342i
\(216\) 0 0
\(217\) −788.603 + 17.6069i −0.246700 + 0.00550798i
\(218\) 0 0
\(219\) 3451.33 + 1992.63i 1.06493 + 0.614837i
\(220\) 0 0
\(221\) 3484.32 + 6035.02i 1.06055 + 1.83692i
\(222\) 0 0
\(223\) 879.692 0.264164 0.132082 0.991239i \(-0.457834\pi\)
0.132082 + 0.991239i \(0.457834\pi\)
\(224\) 0 0
\(225\) 683.948 0.202651
\(226\) 0 0
\(227\) −712.315 1233.77i −0.208273 0.360740i 0.742897 0.669405i \(-0.233451\pi\)
−0.951171 + 0.308665i \(0.900118\pi\)
\(228\) 0 0
\(229\) 206.746 + 119.365i 0.0596602 + 0.0344448i 0.529533 0.848289i \(-0.322367\pi\)
−0.469873 + 0.882734i \(0.655700\pi\)
\(230\) 0 0
\(231\) −3075.03 + 1868.10i −0.875852 + 0.532087i
\(232\) 0 0
\(233\) −2920.86 + 5059.08i −0.821253 + 1.42245i 0.0834970 + 0.996508i \(0.473391\pi\)
−0.904750 + 0.425943i \(0.859942\pi\)
\(234\) 0 0
\(235\) −60.3966 + 34.8700i −0.0167653 + 0.00967943i
\(236\) 0 0
\(237\) 5274.80i 1.44572i
\(238\) 0 0
\(239\) 2101.91i 0.568876i 0.958694 + 0.284438i \(0.0918070\pi\)
−0.958694 + 0.284438i \(0.908193\pi\)
\(240\) 0 0
\(241\) 1883.76 1087.59i 0.503501 0.290697i −0.226657 0.973975i \(-0.572780\pi\)
0.730158 + 0.683278i \(0.239446\pi\)
\(242\) 0 0
\(243\) 592.161 1025.65i 0.156326 0.270764i
\(244\) 0 0
\(245\) −258.636 5789.20i −0.0674435 1.50963i
\(246\) 0 0
\(247\) −3709.16 2141.49i −0.955499 0.551658i
\(248\) 0 0
\(249\) −1318.22 2283.22i −0.335497 0.581097i
\(250\) 0 0
\(251\) −2069.98 −0.520542 −0.260271 0.965536i \(-0.583812\pi\)
−0.260271 + 0.965536i \(0.583812\pi\)
\(252\) 0 0
\(253\) 4325.41 1.07485
\(254\) 0 0
\(255\) 4947.81 + 8569.87i 1.21508 + 2.10457i
\(256\) 0 0
\(257\) 4498.76 + 2597.36i 1.09192 + 0.630423i 0.934088 0.357042i \(-0.116215\pi\)
0.157836 + 0.987465i \(0.449548\pi\)
\(258\) 0 0
\(259\) 778.057 + 1280.74i 0.186665 + 0.307263i
\(260\) 0 0
\(261\) −312.694 + 541.602i −0.0741581 + 0.128446i
\(262\) 0 0
\(263\) −1580.24 + 912.352i −0.370501 + 0.213909i −0.673677 0.739026i \(-0.735286\pi\)
0.303176 + 0.952934i \(0.401953\pi\)
\(264\) 0 0
\(265\) 4725.48i 1.09541i
\(266\) 0 0
\(267\) 583.975i 0.133853i
\(268\) 0 0
\(269\) 2162.50 1248.52i 0.490148 0.282987i −0.234488 0.972119i \(-0.575341\pi\)
0.724636 + 0.689132i \(0.242008\pi\)
\(270\) 0 0
\(271\) −961.292 + 1665.01i −0.215477 + 0.373217i −0.953420 0.301646i \(-0.902464\pi\)
0.737943 + 0.674863i \(0.235797\pi\)
\(272\) 0 0
\(273\) 111.830 + 5008.82i 0.0247922 + 1.11043i
\(274\) 0 0
\(275\) −5660.97 3268.36i −1.24134 0.716689i
\(276\) 0 0
\(277\) 942.717 + 1632.83i 0.204485 + 0.354179i 0.949969 0.312345i \(-0.101115\pi\)
−0.745483 + 0.666524i \(0.767781\pi\)
\(278\) 0 0
\(279\) −181.564 −0.0389604
\(280\) 0 0
\(281\) −2666.13 −0.566006 −0.283003 0.959119i \(-0.591331\pi\)
−0.283003 + 0.959119i \(0.591331\pi\)
\(282\) 0 0
\(283\) 1873.94 + 3245.75i 0.393618 + 0.681766i 0.992924 0.118754i \(-0.0378899\pi\)
−0.599306 + 0.800520i \(0.704557\pi\)
\(284\) 0 0
\(285\) −5267.09 3040.96i −1.09472 0.632038i
\(286\) 0 0
\(287\) −873.889 478.856i −0.179735 0.0984878i
\(288\) 0 0
\(289\) 5087.61 8811.99i 1.03554 1.79361i
\(290\) 0 0
\(291\) 1202.52 694.276i 0.242244 0.139860i
\(292\) 0 0
\(293\) 6486.41i 1.29331i 0.762782 + 0.646655i \(0.223833\pi\)
−0.762782 + 0.646655i \(0.776167\pi\)
\(294\) 0 0
\(295\) 2953.76i 0.582965i
\(296\) 0 0
\(297\) −5259.87 + 3036.78i −1.02764 + 0.593307i
\(298\) 0 0
\(299\) 3011.48 5216.03i 0.582469 1.00887i
\(300\) 0 0
\(301\) −5541.13 3036.32i −1.06108 0.581430i
\(302\) 0 0
\(303\) −5570.86 3216.34i −1.05623 0.609814i
\(304\) 0 0
\(305\) −3945.71 6834.17i −0.740756 1.28303i
\(306\) 0 0
\(307\) 8884.08 1.65160 0.825800 0.563963i \(-0.190724\pi\)
0.825800 + 0.563963i \(0.190724\pi\)
\(308\) 0 0
\(309\) −9432.11 −1.73648
\(310\) 0 0
\(311\) −3500.88 6063.71i −0.638318 1.10560i −0.985802 0.167913i \(-0.946297\pi\)
0.347484 0.937686i \(-0.387036\pi\)
\(312\) 0 0
\(313\) 4140.54 + 2390.54i 0.747722 + 0.431697i 0.824870 0.565322i \(-0.191248\pi\)
−0.0771483 + 0.997020i \(0.524581\pi\)
\(314\) 0 0
\(315\) −29.7735 1333.54i −0.00532555 0.238529i
\(316\) 0 0
\(317\) 1801.72 3120.66i 0.319226 0.552915i −0.661101 0.750297i \(-0.729911\pi\)
0.980327 + 0.197382i \(0.0632439\pi\)
\(318\) 0 0
\(319\) 5176.27 2988.52i 0.908512 0.524530i
\(320\) 0 0
\(321\) 4035.52i 0.701685i
\(322\) 0 0
\(323\) 9273.31i 1.59746i
\(324\) 0 0
\(325\) −7882.66 + 4551.06i −1.34539 + 0.776760i
\(326\) 0 0
\(327\) 2210.31 3828.37i 0.373793 0.647428i
\(328\) 0 0
\(329\) 39.6927 + 65.3370i 0.00665146 + 0.0109488i
\(330\) 0 0
\(331\) −2861.42 1652.04i −0.475159 0.274333i 0.243238 0.969967i \(-0.421790\pi\)
−0.718397 + 0.695633i \(0.755124\pi\)
\(332\) 0 0
\(333\) 172.467 + 298.721i 0.0283817 + 0.0491586i
\(334\) 0 0
\(335\) 12726.2 2.07554
\(336\) 0 0
\(337\) −6650.35 −1.07498 −0.537489 0.843271i \(-0.680627\pi\)
−0.537489 + 0.843271i \(0.680627\pi\)
\(338\) 0 0
\(339\) −1428.24 2473.79i −0.228825 0.396336i
\(340\) 0 0
\(341\) 1502.79 + 867.634i 0.238652 + 0.137786i
\(342\) 0 0
\(343\) −6338.21 + 425.098i −0.997758 + 0.0669188i
\(344\) 0 0
\(345\) 4276.37 7406.88i 0.667339 1.15586i
\(346\) 0 0
\(347\) 203.832 117.683i 0.0315339 0.0182061i −0.484150 0.874985i \(-0.660871\pi\)
0.515684 + 0.856779i \(0.327538\pi\)
\(348\) 0 0
\(349\) 2984.02i 0.457681i −0.973464 0.228841i \(-0.926507\pi\)
0.973464 0.228841i \(-0.0734935\pi\)
\(350\) 0 0
\(351\) 8457.19i 1.28607i
\(352\) 0 0
\(353\) −2118.54 + 1223.14i −0.319429 + 0.184422i −0.651138 0.758959i \(-0.725708\pi\)
0.331709 + 0.943382i \(0.392375\pi\)
\(354\) 0 0
\(355\) 5656.29 9796.98i 0.845647 1.46470i
\(356\) 0 0
\(357\) 9270.88 5632.13i 1.37442 0.834969i
\(358\) 0 0
\(359\) −4931.22 2847.04i −0.724957 0.418554i 0.0916173 0.995794i \(-0.470796\pi\)
−0.816574 + 0.577240i \(0.804130\pi\)
\(360\) 0 0
\(361\) 579.782 + 1004.21i 0.0845286 + 0.146408i
\(362\) 0 0
\(363\) 1568.53 0.226794
\(364\) 0 0
\(365\) 14120.4 2.02492
\(366\) 0 0
\(367\) 168.161 + 291.263i 0.0239180 + 0.0414272i 0.877737 0.479143i \(-0.159053\pi\)
−0.853819 + 0.520571i \(0.825719\pi\)
\(368\) 0 0
\(369\) −198.639 114.684i −0.0280237 0.0161795i
\(370\) 0 0
\(371\) −5178.78 + 115.625i −0.724714 + 0.0161804i
\(372\) 0 0
\(373\) −4908.42 + 8501.64i −0.681363 + 1.18016i 0.293202 + 0.956051i \(0.405279\pi\)
−0.974565 + 0.224105i \(0.928054\pi\)
\(374\) 0 0
\(375\) −2472.58 + 1427.54i −0.340489 + 0.196582i
\(376\) 0 0
\(377\) 8322.78i 1.13699i
\(378\) 0 0
\(379\) 3978.71i 0.539242i 0.962967 + 0.269621i \(0.0868984\pi\)
−0.962967 + 0.269621i \(0.913102\pi\)
\(380\) 0 0
\(381\) −7573.09 + 4372.33i −1.01832 + 0.587929i
\(382\) 0 0
\(383\) −1862.99 + 3226.80i −0.248550 + 0.430501i −0.963124 0.269059i \(-0.913287\pi\)
0.714574 + 0.699560i \(0.246621\pi\)
\(384\) 0 0
\(385\) −6126.12 + 11179.9i −0.810950 + 1.47994i
\(386\) 0 0
\(387\) −1259.53 727.188i −0.165440 0.0955169i
\(388\) 0 0
\(389\) −6346.89 10993.1i −0.827250 1.43284i −0.900187 0.435503i \(-0.856571\pi\)
0.0729372 0.997337i \(-0.476763\pi\)
\(390\) 0 0
\(391\) −13040.6 −1.68669
\(392\) 0 0
\(393\) −6576.24 −0.844091
\(394\) 0 0
\(395\) 9344.72 + 16185.5i 1.19034 + 2.06173i
\(396\) 0 0
\(397\) 1794.63 + 1036.13i 0.226877 + 0.130987i 0.609130 0.793070i \(-0.291519\pi\)
−0.382254 + 0.924057i \(0.624852\pi\)
\(398\) 0 0
\(399\) −3203.79 + 5846.76i −0.401980 + 0.733594i
\(400\) 0 0
\(401\) 4928.69 8536.75i 0.613784 1.06310i −0.376813 0.926289i \(-0.622980\pi\)
0.990597 0.136815i \(-0.0436866\pi\)
\(402\) 0 0
\(403\) 2092.57 1208.14i 0.258656 0.149335i
\(404\) 0 0
\(405\) 10064.8i 1.23487i
\(406\) 0 0
\(407\) 3296.64i 0.401495i
\(408\) 0 0
\(409\) 1190.33 687.236i 0.143907 0.0830847i −0.426318 0.904573i \(-0.640189\pi\)
0.570225 + 0.821489i \(0.306856\pi\)
\(410\) 0 0
\(411\) 3512.90 6084.52i 0.421603 0.730237i
\(412\) 0 0
\(413\) −3237.11 + 72.2738i −0.385684 + 0.00861104i
\(414\) 0 0
\(415\) −8089.81 4670.66i −0.956899 0.552466i
\(416\) 0 0
\(417\) 4601.14 + 7969.40i 0.540332 + 0.935883i
\(418\) 0 0
\(419\) 11066.7 1.29032 0.645158 0.764049i \(-0.276791\pi\)
0.645158 + 0.764049i \(0.276791\pi\)
\(420\) 0 0
\(421\) 15288.1 1.76982 0.884910 0.465761i \(-0.154219\pi\)
0.884910 + 0.465761i \(0.154219\pi\)
\(422\) 0 0
\(423\) 8.79842 + 15.2393i 0.00101133 + 0.00175168i
\(424\) 0 0
\(425\) 17067.2 + 9853.76i 1.94796 + 1.12465i
\(426\) 0 0
\(427\) −7393.20 + 4491.43i −0.837897 + 0.509029i
\(428\) 0 0
\(429\) 5510.78 9544.95i 0.620194 1.07421i
\(430\) 0 0
\(431\) −2265.08 + 1307.74i −0.253143 + 0.146152i −0.621203 0.783650i \(-0.713356\pi\)
0.368059 + 0.929802i \(0.380022\pi\)
\(432\) 0 0
\(433\) 3152.43i 0.349875i 0.984580 + 0.174938i \(0.0559723\pi\)
−0.984580 + 0.174938i \(0.944028\pi\)
\(434\) 0 0
\(435\) 11818.5i 1.30266i
\(436\) 0 0
\(437\) 6941.08 4007.43i 0.759810 0.438676i
\(438\) 0 0
\(439\) −3470.23 + 6010.62i −0.377278 + 0.653465i −0.990665 0.136317i \(-0.956473\pi\)
0.613387 + 0.789783i \(0.289807\pi\)
\(440\) 0 0
\(441\) −1460.74 + 65.2592i −0.157730 + 0.00704667i
\(442\) 0 0
\(443\) 10854.6 + 6266.93i 1.16415 + 0.672123i 0.952295 0.305177i \(-0.0987158\pi\)
0.211856 + 0.977301i \(0.432049\pi\)
\(444\) 0 0
\(445\) 1034.56 + 1791.91i 0.110208 + 0.190887i
\(446\) 0 0
\(447\) −10999.8 −1.16392
\(448\) 0 0
\(449\) 8637.25 0.907833 0.453917 0.891044i \(-0.350026\pi\)
0.453917 + 0.891044i \(0.350026\pi\)
\(450\) 0 0
\(451\) 1096.08 + 1898.46i 0.114440 + 0.198215i
\(452\) 0 0
\(453\) 4233.84 + 2444.41i 0.439124 + 0.253528i
\(454\) 0 0
\(455\) 9216.66 + 15171.3i 0.949634 + 1.56316i
\(456\) 0 0
\(457\) −8248.34 + 14286.5i −0.844291 + 1.46236i 0.0419444 + 0.999120i \(0.486645\pi\)
−0.886235 + 0.463235i \(0.846689\pi\)
\(458\) 0 0
\(459\) 15857.9 9155.58i 1.61260 0.931037i
\(460\) 0 0
\(461\) 6423.89i 0.649003i 0.945885 + 0.324501i \(0.105197\pi\)
−0.945885 + 0.324501i \(0.894803\pi\)
\(462\) 0 0
\(463\) 3390.22i 0.340295i 0.985419 + 0.170148i \(0.0544245\pi\)
−0.985419 + 0.170148i \(0.945576\pi\)
\(464\) 0 0
\(465\) 2971.49 1715.59i 0.296343 0.171094i
\(466\) 0 0
\(467\) −580.262 + 1005.04i −0.0574975 + 0.0995886i −0.893341 0.449379i \(-0.851645\pi\)
0.835844 + 0.548967i \(0.184979\pi\)
\(468\) 0 0
\(469\) −311.388 13946.9i −0.0306580 1.37315i
\(470\) 0 0
\(471\) 318.032 + 183.616i 0.0311128 + 0.0179630i
\(472\) 0 0
\(473\) 6949.97 + 12037.7i 0.675603 + 1.17018i
\(474\) 0 0
\(475\) −12112.4 −1.17001
\(476\) 0 0
\(477\) −1192.34 −0.114451
\(478\) 0 0
\(479\) −2042.46 3537.65i −0.194828 0.337452i 0.752016 0.659145i \(-0.229081\pi\)
−0.946844 + 0.321693i \(0.895748\pi\)
\(480\) 0 0
\(481\) −3975.43 2295.22i −0.376849 0.217574i
\(482\) 0 0
\(483\) −8222.03 4505.35i −0.774566 0.424431i
\(484\) 0 0
\(485\) 2459.93 4260.72i 0.230309 0.398906i
\(486\) 0 0
\(487\) 6340.08 3660.44i 0.589931 0.340597i −0.175139 0.984544i \(-0.556038\pi\)
0.765070 + 0.643947i \(0.222704\pi\)
\(488\) 0 0
\(489\) 18537.1i 1.71427i
\(490\) 0 0
\(491\) 32.0507i 0.00294588i −0.999999 0.00147294i \(-0.999531\pi\)
0.999999 0.00147294i \(-0.000468851\pi\)
\(492\) 0 0
\(493\) −15605.9 + 9010.07i −1.42567 + 0.823110i
\(494\) 0 0
\(495\) −1467.19 + 2541.24i −0.133222 + 0.230748i
\(496\) 0 0
\(497\) −10875.2 5959.16i −0.981525 0.537837i
\(498\) 0 0
\(499\) 6513.95 + 3760.83i 0.584378 + 0.337391i 0.762871 0.646550i \(-0.223789\pi\)
−0.178493 + 0.983941i \(0.557122\pi\)
\(500\) 0 0
\(501\) −281.006 486.717i −0.0250588 0.0434031i
\(502\) 0 0
\(503\) 4599.64 0.407729 0.203865 0.978999i \(-0.434650\pi\)
0.203865 + 0.978999i \(0.434650\pi\)
\(504\) 0 0
\(505\) −22792.0 −2.00838
\(506\) 0 0
\(507\) −2435.51 4218.43i −0.213343 0.369521i
\(508\) 0 0
\(509\) −18006.2 10395.9i −1.56800 0.905286i −0.996402 0.0847525i \(-0.972990\pi\)
−0.571599 0.820533i \(-0.693677\pi\)
\(510\) 0 0
\(511\) −345.503 15474.9i −0.0299103 1.33967i
\(512\) 0 0
\(513\) −5627.08 + 9746.39i −0.484292 + 0.838818i
\(514\) 0 0
\(515\) −28942.1 + 16709.7i −2.47639 + 1.42974i
\(516\) 0 0
\(517\) 168.179i 0.0143066i
\(518\) 0 0
\(519\) 13657.5i 1.15511i
\(520\) 0 0
\(521\) 16707.9 9646.31i 1.40496 0.811156i 0.410067 0.912055i \(-0.365505\pi\)
0.994897 + 0.100899i \(0.0321718\pi\)
\(522\) 0 0
\(523\) 8344.02 14452.3i 0.697626 1.20832i −0.271661 0.962393i \(-0.587573\pi\)
0.969287 0.245931i \(-0.0790936\pi\)
\(524\) 0 0
\(525\) 7356.43 + 12109.2i 0.611544 + 1.00664i
\(526\) 0 0
\(527\) −4530.74 2615.82i −0.374501 0.216218i
\(528\) 0 0
\(529\) −448.024 776.000i −0.0368229 0.0637791i
\(530\) 0 0
\(531\) −745.296 −0.0609098
\(532\) 0 0
\(533\) 3052.48 0.248063
\(534\) 0 0
\(535\) 7149.24 + 12382.9i 0.577736 + 1.00067i
\(536\) 0 0
\(537\) −5702.60 3292.40i −0.458260 0.264576i
\(538\) 0 0
\(539\) 12402.2 + 6440.22i 0.991096 + 0.514657i
\(540\) 0 0
\(541\) −1525.67 + 2642.54i −0.121245 + 0.210003i −0.920259 0.391310i \(-0.872022\pi\)
0.799014 + 0.601313i \(0.205355\pi\)
\(542\) 0 0
\(543\) 845.794 488.320i 0.0668444 0.0385926i
\(544\) 0 0
\(545\) 15662.9i 1.23106i
\(546\) 0 0
\(547\) 12022.3i 0.939740i 0.882736 + 0.469870i \(0.155699\pi\)
−0.882736 + 0.469870i \(0.844301\pi\)
\(548\) 0 0
\(549\) −1724.40 + 995.584i −0.134054 + 0.0773962i
\(550\) 0 0
\(551\) 5537.65 9591.48i 0.428152 0.741581i
\(552\) 0 0
\(553\) 17509.5 10637.2i 1.34644 0.817971i
\(554\) 0 0
\(555\) −5645.21 3259.26i −0.431758 0.249276i
\(556\) 0 0
\(557\) −5913.89 10243.2i −0.449873 0.779204i 0.548504 0.836148i \(-0.315198\pi\)
−0.998377 + 0.0569444i \(0.981864\pi\)
\(558\) 0 0
\(559\) 19355.1 1.46446
\(560\) 0 0
\(561\) −23863.4 −1.79593
\(562\) 0 0
\(563\) −6841.32 11849.5i −0.512127 0.887030i −0.999901 0.0140600i \(-0.995524\pi\)
0.487774 0.872970i \(-0.337809\pi\)
\(564\) 0 0
\(565\) −8765.03 5060.49i −0.652650 0.376808i
\(566\) 0 0
\(567\) 11030.3 246.269i 0.816981 0.0182405i
\(568\) 0 0
\(569\) −816.737 + 1414.63i −0.0601747 + 0.104226i −0.894543 0.446981i \(-0.852499\pi\)
0.834369 + 0.551207i \(0.185832\pi\)
\(570\) 0 0
\(571\) −12962.1 + 7483.67i −0.949995 + 0.548480i −0.893079 0.449899i \(-0.851460\pi\)
−0.0569155 + 0.998379i \(0.518127\pi\)
\(572\) 0 0
\(573\) 19873.3i 1.44890i
\(574\) 0 0
\(575\) 17033.1i 1.23535i
\(576\) 0 0
\(577\) 11197.4 6464.81i 0.807891 0.466436i −0.0383317 0.999265i \(-0.512204\pi\)
0.846223 + 0.532829i \(0.178871\pi\)
\(578\) 0 0
\(579\) −11410.5 + 19763.5i −0.819004 + 1.41856i
\(580\) 0 0
\(581\) −4920.75 + 8980.12i −0.351372 + 0.641236i
\(582\) 0 0
\(583\) 9868.84 + 5697.78i 0.701073 + 0.404765i
\(584\) 0 0
\(585\) 2042.99 + 3538.57i 0.144389 + 0.250089i
\(586\) 0 0
\(587\) −7486.36 −0.526397 −0.263199 0.964742i \(-0.584777\pi\)
−0.263199 + 0.964742i \(0.584777\pi\)
\(588\) 0 0
\(589\) 3215.40 0.224938
\(590\) 0 0
\(591\) −899.710 1558.34i −0.0626212 0.108463i
\(592\) 0 0
\(593\) 2294.33 + 1324.63i 0.158882 + 0.0917304i 0.577333 0.816509i \(-0.304094\pi\)
−0.418451 + 0.908239i \(0.637427\pi\)
\(594\) 0 0
\(595\) 18469.6 33706.1i 1.27257 2.32238i
\(596\) 0 0
\(597\) −3861.23 + 6687.84i −0.264706 + 0.458484i
\(598\) 0 0
\(599\) 15164.6 8755.28i 1.03440 0.597213i 0.116161 0.993230i \(-0.462941\pi\)
0.918243 + 0.396017i \(0.129608\pi\)
\(600\) 0 0
\(601\) 12124.7i 0.822922i 0.911427 + 0.411461i \(0.134981\pi\)
−0.911427 + 0.411461i \(0.865019\pi\)
\(602\) 0 0
\(603\) 3211.07i 0.216857i
\(604\) 0 0
\(605\) 4812.96 2778.77i 0.323429 0.186732i
\(606\) 0 0
\(607\) −5706.27 + 9883.55i −0.381566 + 0.660891i −0.991286 0.131725i \(-0.957948\pi\)
0.609721 + 0.792616i \(0.291282\pi\)
\(608\) 0 0
\(609\) −12952.2 + 289.180i −0.861825 + 0.0192417i
\(610\) 0 0
\(611\) −202.807 117.091i −0.0134283 0.00775285i
\(612\) 0 0
\(613\) −3379.38 5853.26i −0.222662 0.385662i 0.732953 0.680279i \(-0.238141\pi\)
−0.955615 + 0.294617i \(0.904808\pi\)
\(614\) 0 0
\(615\) 4334.60 0.284208
\(616\) 0 0
\(617\) −9991.14 −0.651909 −0.325955 0.945385i \(-0.605686\pi\)
−0.325955 + 0.945385i \(0.605686\pi\)
\(618\) 0 0
\(619\) −10332.1 17895.6i −0.670889 1.16201i −0.977652 0.210228i \(-0.932579\pi\)
0.306764 0.951786i \(-0.400754\pi\)
\(620\) 0 0
\(621\) −13705.9 7913.11i −0.885667 0.511340i
\(622\) 0 0
\(623\) 1938.48 1177.64i 0.124661 0.0757325i
\(624\) 0 0
\(625\) 4969.49 8607.41i 0.318047 0.550874i
\(626\) 0 0
\(627\) 12701.7 7333.31i 0.809020 0.467088i
\(628\) 0 0
\(629\) 9939.02i 0.630039i
\(630\) 0 0
\(631\) 28295.8i 1.78516i 0.450887 + 0.892581i \(0.351108\pi\)
−0.450887 + 0.892581i \(0.648892\pi\)
\(632\) 0 0
\(633\) 12432.0 7177.63i 0.780614 0.450688i
\(634\) 0 0
\(635\) −15491.8 + 26832.6i −0.968149 + 1.67688i
\(636\) 0 0
\(637\) 16401.1 10472.0i 1.02015 0.651359i
\(638\) 0 0
\(639\) −2471.98 1427.20i −0.153036 0.0883554i
\(640\) 0 0
\(641\) 8207.85 + 14216.4i 0.505758 + 0.875998i 0.999978 + 0.00666106i \(0.00212030\pi\)
−0.494220 + 0.869337i \(0.664546\pi\)
\(642\) 0 0
\(643\) −14139.9 −0.867220 −0.433610 0.901101i \(-0.642760\pi\)
−0.433610 + 0.901101i \(0.642760\pi\)
\(644\) 0 0
\(645\) 27484.7 1.67784
\(646\) 0 0
\(647\) −10974.5 19008.5i −0.666853 1.15502i −0.978779 0.204917i \(-0.934308\pi\)
0.311926 0.950106i \(-0.399026\pi\)
\(648\) 0 0
\(649\) 6168.73 + 3561.52i 0.373103 + 0.215411i
\(650\) 0 0
\(651\) −1952.87 3214.56i −0.117571 0.193531i
\(652\) 0 0
\(653\) −2938.88 + 5090.29i −0.176121 + 0.305051i −0.940549 0.339659i \(-0.889688\pi\)
0.764427 + 0.644710i \(0.223022\pi\)
\(654\) 0 0
\(655\) −20179.0 + 11650.3i −1.20375 + 0.694987i
\(656\) 0 0
\(657\) 3562.87i 0.211569i
\(658\) 0 0
\(659\) 11002.0i 0.650344i 0.945655 + 0.325172i \(0.105422\pi\)
−0.945655 + 0.325172i \(0.894578\pi\)
\(660\) 0 0
\(661\) 3196.72 1845.63i 0.188106 0.108603i −0.402990 0.915205i \(-0.632029\pi\)
0.591095 + 0.806602i \(0.298696\pi\)
\(662\) 0 0
\(663\) −16614.4 + 28777.0i −0.973229 + 1.68568i
\(664\) 0 0
\(665\) 527.274 + 23616.3i 0.0307471 + 1.37715i
\(666\) 0 0
\(667\) 13488.1 + 7787.35i 0.782999 + 0.452065i
\(668\) 0 0
\(669\) 2097.33 + 3632.69i 0.121207 + 0.209937i
\(670\) 0 0
\(671\) 19030.2 1.09487
\(672\) 0 0
\(673\) 6819.19 0.390580 0.195290 0.980746i \(-0.437435\pi\)
0.195290 + 0.980746i \(0.437435\pi\)
\(674\) 0 0
\(675\) 11958.6 + 20712.9i 0.681906 + 1.18110i
\(676\) 0 0
\(677\) 13683.4 + 7900.10i 0.776802 + 0.448487i 0.835296 0.549801i \(-0.185296\pi\)
−0.0584939 + 0.998288i \(0.518630\pi\)
\(678\) 0 0
\(679\) −4729.63 2591.65i −0.267314 0.146478i
\(680\) 0 0
\(681\) 3396.56 5883.02i 0.191126 0.331039i
\(682\) 0 0
\(683\) −18649.6 + 10767.3i −1.04481 + 0.603222i −0.921192 0.389108i \(-0.872783\pi\)
−0.123619 + 0.992330i \(0.539450\pi\)
\(684\) 0 0
\(685\) 24893.5i 1.38851i
\(686\) 0 0
\(687\) 1138.35i 0.0632177i
\(688\) 0 0
\(689\) 13742.0 7933.92i 0.759835 0.438691i
\(690\) 0 0
\(691\) 2019.27 3497.48i 0.111167 0.192548i −0.805074 0.593175i \(-0.797874\pi\)
0.916241 + 0.400627i \(0.131208\pi\)
\(692\) 0 0
\(693\) 2820.91 + 1545.75i 0.154628 + 0.0847302i
\(694\) 0 0
\(695\) 28236.8 + 16302.6i 1.54113 + 0.889771i
\(696\) 0 0
\(697\) −3304.56 5723.66i −0.179582 0.311046i
\(698\) 0 0
\(699\) −27855.3 −1.50727
\(700\) 0 0
\(701\) 11823.9 0.637066 0.318533 0.947912i \(-0.396810\pi\)
0.318533 + 0.947912i \(0.396810\pi\)
\(702\) 0 0
\(703\) −3054.29 5290.19i −0.163862 0.283817i
\(704\) 0 0
\(705\) −287.991 166.272i −0.0153849 0.00888249i
\(706\) 0 0
\(707\) 557.683 + 24978.3i 0.0296659 + 1.32872i
\(708\) 0 0
\(709\) −13143.3 + 22764.8i −0.696200 + 1.20585i 0.273575 + 0.961851i \(0.411794\pi\)
−0.969775 + 0.244003i \(0.921539\pi\)
\(710\) 0 0
\(711\) 4083.94 2357.87i 0.215415 0.124370i
\(712\) 0 0
\(713\) 4521.68i 0.237501i
\(714\) 0 0
\(715\) 39051.1i 2.04256i
\(716\) 0 0
\(717\) −8679.85 + 5011.32i −0.452099 + 0.261020i
\(718\) 0 0
\(719\) 9614.97 16653.6i 0.498717 0.863804i −0.501282 0.865284i \(-0.667138\pi\)
0.999999 + 0.00148052i \(0.000471263\pi\)
\(720\) 0 0
\(721\) 19020.8 + 31309.5i 0.982484 + 1.61724i
\(722\) 0 0
\(723\) 8982.42 + 5186.00i 0.462047 + 0.266763i
\(724\) 0 0
\(725\) −11768.5 20383.7i −0.602858 1.04418i
\(726\) 0 0
\(727\) −33771.2 −1.72284 −0.861419 0.507895i \(-0.830424\pi\)
−0.861419 + 0.507895i \(0.830424\pi\)
\(728\) 0 0
\(729\) 21731.9 1.10409
\(730\) 0 0
\(731\) −20953.4 36292.4i −1.06018 1.83628i
\(732\) 0 0
\(733\) 1835.93 + 1059.97i 0.0925125 + 0.0534121i 0.545543 0.838083i \(-0.316324\pi\)
−0.453030 + 0.891495i \(0.649657\pi\)
\(734\) 0 0
\(735\) 23289.9 14870.5i 1.16879 0.746266i
\(736\) 0 0
\(737\) −15344.6 + 26577.7i −0.766930 + 1.32836i
\(738\) 0 0
\(739\) 9238.36 5333.77i 0.459863 0.265502i −0.252124 0.967695i \(-0.581129\pi\)
0.711987 + 0.702193i \(0.247796\pi\)
\(740\) 0 0
\(741\) 20422.7i 1.01248i
\(742\) 0 0
\(743\) 26684.2i 1.31756i −0.752335 0.658781i \(-0.771072\pi\)
0.752335 0.658781i \(-0.228928\pi\)
\(744\) 0 0
\(745\) −33752.4 + 19487.0i −1.65986 + 0.958318i
\(746\) 0 0
\(747\) −1178.50 + 2041.23i −0.0577231 + 0.0999794i
\(748\) 0 0
\(749\) 13395.8 8138.03i 0.653499 0.397006i
\(750\) 0 0
\(751\) 30415.3 + 17560.3i 1.47785 + 0.853240i 0.999687 0.0250284i \(-0.00796761\pi\)
0.478168 + 0.878268i \(0.341301\pi\)
\(752\) 0 0
\(753\) −4935.19 8547.99i −0.238842 0.413687i
\(754\) 0 0
\(755\) 17321.8 0.834975
\(756\) 0 0
\(757\) 10060.1 0.483011 0.241506 0.970399i \(-0.422359\pi\)
0.241506 + 0.970399i \(0.422359\pi\)
\(758\) 0 0
\(759\) 10312.5 + 17861.8i 0.493176 + 0.854205i
\(760\) 0 0
\(761\) −31480.5 18175.3i −1.49956 0.865774i −0.499565 0.866276i \(-0.666507\pi\)
−1.00000 0.000502093i \(0.999840\pi\)
\(762\) 0 0
\(763\) −17165.4 + 383.247i −0.814456 + 0.0181841i
\(764\) 0 0
\(765\) 4423.41 7661.56i 0.209057 0.362097i
\(766\) 0 0
\(767\) 8589.70 4959.27i 0.404376 0.233466i
\(768\) 0 0
\(769\) 17664.9i 0.828366i −0.910194 0.414183i \(-0.864067\pi\)
0.910194 0.414183i \(-0.135933\pi\)
\(770\) 0 0
\(771\) 24770.2i 1.15704i
\(772\) 0 0
\(773\) −6848.89 + 3954.21i −0.318677 + 0.183988i −0.650803 0.759247i \(-0.725568\pi\)
0.332126 + 0.943235i \(0.392234\pi\)
\(774\) 0 0
\(775\) 3416.67 5917.84i 0.158362 0.274290i
\(776\) 0 0
\(777\) −3433.78 + 6266.48i −0.158541 + 0.289329i
\(778\) 0 0
\(779\) 3517.80 + 2031.00i 0.161795 + 0.0934123i
\(780\) 0 0
\(781\) 13640.2 + 23625.5i 0.624949 + 1.08244i
\(782\) 0 0
\(783\) −21869.4 −0.998145
\(784\) 0 0
\(785\) 1301.16 0.0591598
\(786\) 0 0
\(787\) 5940.56 + 10289.4i 0.269070 + 0.466043i 0.968622 0.248539i \(-0.0799503\pi\)
−0.699552 + 0.714582i \(0.746617\pi\)
\(788\) 0 0
\(789\) −7535.12 4350.40i −0.339997 0.196297i
\(790\) 0 0
\(791\) −5331.46 + 9729.65i −0.239652 + 0.437353i
\(792\) 0 0
\(793\) 13249.4 22948.7i 0.593317 1.02766i
\(794\) 0 0
\(795\) 19513.9 11266.3i 0.870549 0.502611i
\(796\) 0 0
\(797\) 10203.4i 0.453480i 0.973955 + 0.226740i \(0.0728068\pi\)
−0.973955 + 0.226740i \(0.927193\pi\)
\(798\) 0 0
\(799\) 507.041i 0.0224503i
\(800\) 0 0
\(801\) 452.135 261.040i 0.0199443 0.0115149i
\(802\) 0 0
\(803\) −17025.7 + 29489.5i −0.748226 + 1.29597i
\(804\) 0 0
\(805\) −33210.6 + 741.482i −1.45406 + 0.0324643i
\(806\) 0 0
\(807\) 10311.5 + 5953.36i 0.449793 + 0.259688i
\(808\) 0 0
\(809\) −10471.1 18136.4i −0.455059 0.788185i 0.543633 0.839323i \(-0.317048\pi\)
−0.998692 + 0.0511383i \(0.983715\pi\)
\(810\) 0 0
\(811\) −5434.74 −0.235314 −0.117657 0.993054i \(-0.537538\pi\)
−0.117657 + 0.993054i \(0.537538\pi\)
\(812\) 0 0
\(813\) −9167.53 −0.395473
\(814\) 0 0
\(815\) −32839.9 56880.4i −1.41145 2.44471i
\(816\) 0 0
\(817\) 22305.5 + 12878.1i 0.955168 + 0.551467i
\(818\) 0 0
\(819\) 3828.02 2325.55i 0.163323 0.0992203i
\(820\) 0 0
\(821\) 1066.43 1847.11i 0.0453334 0.0785197i −0.842468 0.538746i \(-0.818898\pi\)
0.887802 + 0.460226i \(0.152232\pi\)
\(822\) 0 0
\(823\) 16591.2 9578.95i 0.702714 0.405712i −0.105643 0.994404i \(-0.533690\pi\)
0.808358 + 0.588692i \(0.200357\pi\)
\(824\) 0 0
\(825\) 31169.3i 1.31536i
\(826\) 0 0
\(827\) 27208.2i 1.14404i −0.820239 0.572021i \(-0.806160\pi\)
0.820239 0.572021i \(-0.193840\pi\)
\(828\) 0 0
\(829\) 14900.9 8603.02i 0.624281 0.360429i −0.154253 0.988031i \(-0.549297\pi\)
0.778534 + 0.627603i \(0.215964\pi\)
\(830\) 0 0
\(831\) −4495.19 + 7785.91i −0.187649 + 0.325018i
\(832\) 0 0
\(833\) −37391.3 19416.6i −1.55526 0.807617i
\(834\) 0 0
\(835\) −1724.52 995.650i −0.0714723 0.0412645i
\(836\) 0 0
\(837\) −3174.58 5498.54i −0.131099 0.227070i
\(838\) 0 0
\(839\) −9526.10 −0.391988 −0.195994 0.980605i \(-0.562793\pi\)
−0.195994 + 0.980605i \(0.562793\pi\)
\(840\) 0 0
\(841\) −2867.21 −0.117562
\(842\) 0 0
\(843\) −6356.49 11009.8i −0.259703 0.449818i
\(844\) 0 0
\(845\) −14946.6 8629.41i −0.608495 0.351315i
\(846\) 0 0
\(847\) −3163.09 5206.66i −0.128318 0.211220i
\(848\) 0 0
\(849\) −8935.56 + 15476.8i −0.361210 + 0.625635i
\(850\) 0 0
\(851\) 7439.36 4295.12i 0.299669 0.173014i
\(852\) 0 0
\(853\) 8447.62i 0.339087i 0.985523 + 0.169543i \(0.0542293\pi\)
−0.985523 + 0.169543i \(0.945771\pi\)
\(854\) 0 0
\(855\) 5437.31i 0.217488i
\(856\) 0 0
\(857\) −22211.6 + 12823.9i −0.885337 + 0.511150i −0.872414 0.488767i \(-0.837447\pi\)
−0.0129226 + 0.999916i \(0.504114\pi\)
\(858\) 0 0
\(859\) −3738.66 + 6475.54i −0.148500 + 0.257209i −0.930673 0.365852i \(-0.880778\pi\)
0.782173 + 0.623061i \(0.214111\pi\)
\(860\) 0 0
\(861\) −106.061 4750.40i −0.00419807 0.188029i
\(862\) 0 0
\(863\) −20094.6 11601.6i −0.792616 0.457617i 0.0482670 0.998834i \(-0.484630\pi\)
−0.840883 + 0.541218i \(0.817964\pi\)
\(864\) 0 0
\(865\) −24195.4 41907.7i −0.951062 1.64729i
\(866\) 0 0
\(867\) 48518.9 1.90056
\(868\) 0 0
\(869\) −45069.8 −1.75937
\(870\) 0 0
\(871\) 21366.8 + 37008.3i 0.831212 + 1.43970i
\(872\) 0 0
\(873\) −1075.07 620.691i −0.0416788 0.0240633i
\(874\) 0 0
\(875\) 9724.89 + 5328.85i 0.375727 + 0.205883i
\(876\) 0 0
\(877\) −6141.61 + 10637.6i −0.236474 + 0.409584i −0.959700 0.281027i \(-0.909325\pi\)
0.723226 + 0.690611i \(0.242658\pi\)
\(878\) 0 0
\(879\) −26785.6 + 15464.7i −1.02782 + 0.593414i
\(880\) 0 0
\(881\) 44697.8i 1.70931i 0.519193 + 0.854657i \(0.326233\pi\)
−0.519193 + 0.854657i \(0.673767\pi\)
\(882\) 0 0
\(883\) 9474.88i 0.361104i −0.983565 0.180552i \(-0.942212\pi\)
0.983565 0.180552i \(-0.0577884\pi\)
\(884\) 0 0
\(885\) 12197.6 7042.27i 0.463296 0.267484i
\(886\) 0 0
\(887\) 7502.29 12994.3i 0.283993 0.491891i −0.688371 0.725359i \(-0.741674\pi\)
0.972365 + 0.233468i \(0.0750073\pi\)
\(888\) 0 0
\(889\) 29785.7 + 16321.4i 1.12371 + 0.615749i
\(890\) 0 0
\(891\) −21019.6 12135.7i −0.790330 0.456298i
\(892\) 0 0
\(893\) −155.815 269.880i −0.00583892 0.0101133i
\(894\) 0 0
\(895\) −23331.0 −0.871361
\(896\) 0 0
\(897\) 28719.5 1.06902
\(898\) 0 0
\(899\) 3124.13 + 5411.15i 0.115902 + 0.200747i
\(900\) 0 0
\(901\) −29753.5 17178.2i −1.10015 0.635170i
\(902\) 0 0
\(903\) −672.506 30121.2i −0.0247836 1.11004i
\(904\) 0 0
\(905\) 1730.19 2996.78i 0.0635509 0.110073i
\(906\) 0 0
\(907\) 24244.0 13997.3i 0.887549 0.512427i 0.0144093 0.999896i \(-0.495413\pi\)
0.873140 + 0.487469i \(0.162080\pi\)
\(908\) 0 0
\(909\) 5750.89i 0.209840i
\(910\) 0 0
\(911\) 27683.1i 1.00679i 0.864057 + 0.503393i \(0.167915\pi\)
−0.864057 + 0.503393i \(0.832085\pi\)
\(912\) 0 0
\(913\) 19508.7 11263.3i 0.707167 0.408283i
\(914\) 0 0
\(915\) 18814.5 32587.6i 0.679767 1.17739i
\(916\) 0 0
\(917\) 13261.7 + 21829.6i 0.477577 + 0.786125i
\(918\) 0 0
\(919\) 15.0992 + 8.71755i 0.000541978 + 0.000312911i 0.500271 0.865869i \(-0.333234\pi\)
−0.499729 + 0.866182i \(0.666567\pi\)
\(920\) 0 0
\(921\) 21181.2 + 36686.8i 0.757810 + 1.31256i
\(922\) 0 0
\(923\) 37986.9 1.35466
\(924\) 0 0
\(925\) −12981.9 −0.461451
\(926\) 0 0
\(927\) 4216.21 + 7302.69i 0.149383 + 0.258740i
\(928\) 0 0
\(929\) 36888.5 + 21297.6i 1.30277 + 0.752155i 0.980878 0.194621i \(-0.0623478\pi\)
0.321892 + 0.946776i \(0.395681\pi\)
\(930\) 0 0
\(931\) 25868.8 1155.71i 0.910652 0.0406839i
\(932\) 0 0
\(933\) 16693.4 28913.8i 0.585764 1.01457i
\(934\) 0 0
\(935\) −73224.1 + 42275.9i −2.56116 + 1.47869i
\(936\) 0 0
\(937\) 26316.4i 0.917522i −0.888560 0.458761i \(-0.848293\pi\)
0.888560 0.458761i \(-0.151707\pi\)
\(938\) 0 0
\(939\) 22797.8i 0.792309i
\(940\) 0 0
\(941\) 25507.0 14726.5i 0.883640 0.510170i 0.0117829 0.999931i \(-0.496249\pi\)
0.871857 + 0.489761i \(0.162916\pi\)
\(942\) 0 0
\(943\) −2856.11 + 4946.92i −0.0986295 + 0.170831i
\(944\) 0 0
\(945\) 39864.8 24218.2i 1.37228 0.833669i
\(946\) 0 0
\(947\) −18190.8 10502.5i −0.624205 0.360385i 0.154299 0.988024i \(-0.450688\pi\)
−0.778504 + 0.627639i \(0.784021\pi\)
\(948\) 0 0
\(949\) 23707.6 + 41062.8i 0.810940 + 1.40459i
\(950\) 0 0
\(951\) 17182.4 0.585885
\(952\) 0 0
\(953\) 10707.4 0.363952 0.181976 0.983303i \(-0.441751\pi\)
0.181976 + 0.983303i \(0.441751\pi\)
\(954\) 0 0
\(955\) 35207.0 + 60980.4i 1.19296 + 2.06626i
\(956\) 0 0
\(957\) 24682.2 + 14250.3i 0.833712 + 0.481344i
\(958\) 0 0
\(959\) −27281.5 + 609.104i −0.918628 + 0.0205099i
\(960\) 0 0
\(961\) 13988.5 24228.8i 0.469554 0.813292i
\(962\) 0 0
\(963\) 3124.45 1803.90i 0.104552 0.0603634i
\(964\) 0 0
\(965\) 80858.2i 2.69733i
\(966\) 0 0
\(967\) 20353.4i 0.676859i 0.940992 + 0.338429i \(0.109896\pi\)
−0.940992 + 0.338429i \(0.890104\pi\)
\(968\) 0 0
\(969\) −38294.2 + 22109.1i −1.26954 + 0.732970i
\(970\) 0 0
\(971\) 27664.6 47916.6i 0.914316 1.58364i 0.106416 0.994322i \(-0.466062\pi\)
0.807900 0.589320i \(-0.200604\pi\)
\(972\) 0 0
\(973\) 17175.5 31344.4i 0.565900 1.03274i
\(974\) 0 0
\(975\) −37587.2 21701.0i −1.23462 0.712808i
\(976\) 0 0
\(977\) −14962.5 25915.8i −0.489963 0.848640i 0.509971 0.860192i \(-0.329656\pi\)
−0.999933 + 0.0115516i \(0.996323\pi\)
\(978\) 0 0
\(979\) −4989.70 −0.162892
\(980\) 0 0
\(981\) −3952.08 −0.128624
\(982\) 0 0
\(983\) −2281.10 3950.97i −0.0740139 0.128196i 0.826643 0.562727i \(-0.190248\pi\)
−0.900657 + 0.434531i \(0.856914\pi\)
\(984\) 0 0
\(985\) −5521.45 3187.81i −0.178607 0.103119i
\(986\) 0 0
\(987\) −175.175 + 319.686i −0.00564932 + 0.0103097i
\(988\) 0 0
\(989\) −18109.9 + 31367.3i −0.582267 + 1.00852i
\(990\) 0 0
\(991\) 1060.34 612.186i 0.0339886 0.0196233i −0.482909 0.875670i \(-0.660420\pi\)
0.516898 + 0.856047i \(0.327087\pi\)
\(992\) 0 0
\(993\) 15755.0i 0.503493i
\(994\) 0 0
\(995\) 27361.9i 0.871789i
\(996\) 0 0
\(997\) 39922.0 23049.0i 1.26815 0.732165i 0.293510 0.955956i \(-0.405177\pi\)
0.974637 + 0.223791i \(0.0718433\pi\)
\(998\) 0 0
\(999\) −6031.03 + 10446.1i −0.191004 + 0.330829i
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 112.4.p.g.47.2 yes 6
4.3 odd 2 112.4.p.f.47.2 yes 6
7.2 even 3 784.4.f.g.783.3 6
7.3 odd 6 112.4.p.f.31.2 6
7.5 odd 6 784.4.f.h.783.4 6
8.3 odd 2 448.4.p.g.383.2 6
8.5 even 2 448.4.p.f.383.2 6
28.3 even 6 inner 112.4.p.g.31.2 yes 6
28.19 even 6 784.4.f.g.783.4 6
28.23 odd 6 784.4.f.h.783.3 6
56.3 even 6 448.4.p.f.255.2 6
56.45 odd 6 448.4.p.g.255.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.p.f.31.2 6 7.3 odd 6
112.4.p.f.47.2 yes 6 4.3 odd 2
112.4.p.g.31.2 yes 6 28.3 even 6 inner
112.4.p.g.47.2 yes 6 1.1 even 1 trivial
448.4.p.f.255.2 6 56.3 even 6
448.4.p.f.383.2 6 8.5 even 2
448.4.p.g.255.2 6 56.45 odd 6
448.4.p.g.383.2 6 8.3 odd 2
784.4.f.g.783.3 6 7.2 even 3
784.4.f.g.783.4 6 28.19 even 6
784.4.f.h.783.3 6 28.23 odd 6
784.4.f.h.783.4 6 7.5 odd 6