Properties

Label 448.4.p.f.383.2
Level $448$
Weight $4$
Character 448.383
Analytic conductor $26.433$
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] = [448,4,Mod(255,448)]
 
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([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("448.255");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.p (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: \(26.4328556826\)
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: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 383.2
Root \(2.68858 + 4.65676i\) of defining polynomial
Character \(\chi\) \(=\) 448.383
Dual form 448.4.p.f.255.2

$q$-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 \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 0.540066 0.841623i \(-0.318399\pi\)
−0.998900 + 0.0468996i \(0.985066\pi\)
\(4\) 0 0
\(5\) −14.6315 8.44749i −1.30868 0.755566i −0.326803 0.945092i \(-0.605971\pi\)
−0.981876 + 0.189526i \(0.939305\pi\)
\(6\) 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 0.0687757 0.997632i \(-0.478091\pi\)
0.898363 + 0.439255i \(0.144757\pi\)
\(12\) 0 0
\(13\) 56.7321i 1.21036i −0.796089 0.605179i \(-0.793101\pi\)
0.796089 0.605179i \(-0.206899\pi\)
\(14\) 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 −0.542952 0.839763i \(-0.682694\pi\)
0.998733 + 0.0503288i \(0.0160269\pi\)
\(20\) 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 \(-0.344365\pi\)
0.469691 + 0.882831i \(0.344365\pi\)
\(30\) 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 −0.762039 0.647532i \(-0.775801\pi\)
0.941798 + 0.336179i \(0.109135\pi\)
\(38\) 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 \(0.206815\pi\)
−0.796249 + 0.604970i \(0.793185\pi\)
\(44\) 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 0.988363 0.152114i \(-0.0486080\pi\)
−0.625916 + 0.779891i \(0.715275\pi\)
\(54\) 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 0.946207 0.323562i \(-0.104881\pi\)
−0.753317 + 0.657658i \(0.771547\pi\)
\(60\) 0 0
\(61\) 404.509 + 233.543i 0.849050 + 0.490199i 0.860330 0.509737i \(-0.170257\pi\)
−0.0112801 + 0.999936i \(0.503591\pi\)
\(62\) 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 −0.958189 0.286136i \(-0.907629\pi\)
−0.231294 + 0.972884i \(0.574296\pi\)
\(68\) 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 \(-0.380865\pi\)
0.365597 + 0.930773i \(0.380865\pi\)
\(84\) 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 0.201865 0.979413i \(-0.435300\pi\)
0.949129 + 0.314887i \(0.101966\pi\)
\(102\) 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 0.130916 0.991393i \(-0.458208\pi\)
−0.793114 + 0.609073i \(0.791541\pi\)
\(108\) 0 0
\(109\) 463.538 + 802.872i 0.407330 + 0.705516i 0.994590 0.103883i \(-0.0331267\pi\)
−0.587260 + 0.809398i \(0.699793\pi\)
\(110\) 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 −0.539045 0.842277i \(-0.681215\pi\)
0.998956 + 0.0456878i \(0.0145479\pi\)
\(132\) 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 \(-0.700404\pi\)
−0.588811 + 0.808271i \(0.700404\pi\)
\(140\) 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 \(-0.614674\pi\)
0.986690 0.162614i \(-0.0519924\pi\)
\(150\) 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 −0.516856 0.856072i \(-0.672898\pi\)
0.482952 + 0.875647i \(0.339565\pi\)
\(158\) 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 \(0.0504049\pi\)
0.630308 + 0.776345i \(0.282928\pi\)
\(164\) 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 0.933603 0.358308i \(-0.116646\pi\)
0.156498 + 0.987678i \(0.449980\pi\)
\(174\) 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 −0.229081 0.973407i \(-0.573572\pi\)
0.728455 + 0.685093i \(0.240239\pi\)
\(180\) 0 0
\(181\) 204.817i 0.0841103i 0.999115 + 0.0420551i \(0.0133905\pi\)
−0.999115 + 0.0420551i \(0.986609\pi\)
\(182\) 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 \(-0.478262\pi\)
0.0682395 + 0.997669i \(0.478262\pi\)
\(198\) 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 \(0.163414\pi\)
−0.871090 + 0.491123i \(0.836586\pi\)
\(212\) 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 0.951171 0.308665i \(-0.0998823\pi\)
−0.742897 + 0.669405i \(0.766549\pi\)
\(228\) 0 0
\(229\) −206.746 119.365i −0.0596602 0.0344448i 0.469873 0.882734i \(-0.344300\pi\)
−0.529533 + 0.848289i \(0.677633\pi\)
\(230\) 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 \(-0.416188\pi\)
0.260271 + 0.965536i \(0.416188\pi\)
\(252\) 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 −0.724636 0.689132i \(-0.757992\pi\)
0.234488 + 0.972119i \(0.424659\pi\)
\(270\) 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 0.745483 0.666524i \(-0.232219\pi\)
−0.949969 + 0.312345i \(0.898885\pi\)
\(278\) 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 0.599306 0.800520i \(-0.295443\pi\)
−0.992924 + 0.118754i \(0.962110\pi\)
\(284\) 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 \(-0.776167\pi\)
0.762782 0.646655i \(-0.223833\pi\)
\(294\) 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 \(-0.809276\pi\)
−0.825800 + 0.563963i \(0.809276\pi\)
\(308\) 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 −0.980327 0.197382i \(-0.936756\pi\)
0.661101 + 0.750297i \(0.270089\pi\)
\(318\) 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 0.718397 0.695633i \(-0.244876\pi\)
−0.243238 + 0.969967i \(0.578210\pi\)
\(332\) 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 −0.515684 0.856779i \(-0.672462\pi\)
0.484150 + 0.874985i \(0.339129\pi\)
\(348\) 0 0
\(349\) 2984.02i 0.457681i 0.973464 + 0.228841i \(0.0734935\pi\)
−0.973464 + 0.228841i \(0.926507\pi\)
\(350\) 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 \(-0.594721\pi\)
0.974565 0.224105i \(-0.0719459\pi\)
\(374\) 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 \(-0.913102\pi\)
0.962967 0.269621i \(-0.0868984\pi\)
\(380\) 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 \(0.143429\pi\)
−0.0729372 + 0.997337i \(0.523237\pi\)
\(390\) 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 0.382254 0.924057i \(-0.375148\pi\)
−0.609130 + 0.793070i \(0.708481\pi\)
\(398\) 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 \(-0.723209\pi\)
−0.645158 + 0.764049i \(0.723209\pi\)
\(420\) 0 0
\(421\) −15288.1 −1.76982 −0.884910 0.465761i \(-0.845781\pi\)
−0.884910 + 0.465761i \(0.845781\pi\)
\(422\) 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 −0.211856 0.977301i \(-0.567951\pi\)
−0.952295 + 0.305177i \(0.901284\pi\)
\(444\) 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 \(-0.894803\pi\)
0.945885 0.324501i \(-0.105197\pi\)
\(462\) 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 −0.835844 0.548967i \(-0.815021\pi\)
0.893341 + 0.449379i \(0.148355\pi\)
\(468\) 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 \(0.000468851\pi\)
−0.999999 + 0.00147294i \(0.999531\pi\)
\(492\) 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 0.178493 0.983941i \(-0.442878\pi\)
−0.762871 + 0.646550i \(0.776211\pi\)
\(500\) 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 \(0.0270100\pi\)
0.571599 + 0.820533i \(0.306323\pi\)
\(510\) 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 \(0.412427\pi\)
−0.969287 + 0.245931i \(0.920906\pi\)
\(524\) 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 −0.799014 0.601313i \(-0.794645\pi\)
0.920259 + 0.391310i \(0.127978\pi\)
\(542\) 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 \(-0.844301\pi\)
0.882736 0.469870i \(-0.155699\pi\)
\(548\) 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 0.998377 0.0569444i \(-0.0181358\pi\)
−0.548504 + 0.836148i \(0.684802\pi\)
\(558\) 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 \(0.00447558\pi\)
−0.487774 + 0.872970i \(0.662191\pi\)
\(564\) 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 0.0569155 0.998379i \(-0.481873\pi\)
0.893079 + 0.449899i \(0.148540\pi\)
\(572\) 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 \(-0.415223\pi\)
0.263199 + 0.964742i \(0.415223\pi\)
\(588\) 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 0.955615 0.294617i \(-0.0951920\pi\)
−0.732953 + 0.680279i \(0.761859\pi\)
\(614\) 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 \(0.0674206\pi\)
−0.306764 + 0.951786i \(0.599246\pi\)
\(620\) 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 \(-0.357240\pi\)
0.433610 + 0.901101i \(0.357240\pi\)
\(644\) 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 −0.764427 0.644710i \(-0.776978\pi\)
0.940549 + 0.339659i \(0.110312\pi\)
\(654\) 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 \(-0.894578\pi\)
0.945655 0.325172i \(-0.105422\pi\)
\(660\) 0 0
\(661\) −3196.72 + 1845.63i −0.188106 + 0.108603i −0.591095 0.806602i \(-0.701304\pi\)
0.402990 + 0.915205i \(0.367971\pi\)
\(662\) 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 0.0584939 0.998288i \(-0.481370\pi\)
−0.835296 + 0.549801i \(0.814704\pi\)
\(678\) 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 0.123619 0.992330i \(-0.460550\pi\)
0.921192 + 0.389108i \(0.127217\pi\)
\(684\) 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 −0.916241 0.400627i \(-0.868792\pi\)
0.805074 + 0.593175i \(0.202126\pi\)
\(692\) 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 \(-0.603190\pi\)
−0.318533 + 0.947912i \(0.603190\pi\)
\(702\) 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 \(-0.588206\pi\)
0.969775 0.244003i \(-0.0784606\pi\)
\(710\) 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 0.453030 0.891495i \(-0.350343\pi\)
−0.545543 + 0.838083i \(0.683676\pi\)
\(734\) 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 −0.711987 0.702193i \(-0.752204\pi\)
0.252124 + 0.967695i \(0.418871\pi\)
\(740\) 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 \(-0.577641\pi\)
−0.241506 + 0.970399i \(0.577641\pi\)
\(758\) 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 −0.332126 0.943235i \(-0.607766\pi\)
0.650803 + 0.759247i \(0.274432\pi\)
\(774\) 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 0.699552 0.714582i \(-0.253383\pi\)
−0.968622 + 0.248539i \(0.920050\pi\)
\(788\) 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 \(-0.927193\pi\)
0.973955 0.226740i \(-0.0728068\pi\)
\(798\) 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 \(-0.462462\pi\)
0.117657 + 0.993054i \(0.462462\pi\)
\(812\) 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 −0.887802 0.460226i \(-0.847768\pi\)
0.842468 + 0.538746i \(0.181102\pi\)
\(822\) 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 \(0.193840\pi\)
−0.820239 + 0.572021i \(0.806160\pi\)
\(828\) 0 0
\(829\) −14900.9 + 8603.02i −0.624281 + 0.360429i −0.778534 0.627603i \(-0.784036\pi\)
0.154253 + 0.988031i \(0.450703\pi\)
\(830\) 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 \(-0.945771\pi\)
0.985523 0.169543i \(-0.0542293\pi\)
\(854\) 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 −0.782173 0.623061i \(-0.785889\pi\)
0.930673 + 0.365852i \(0.119222\pi\)
\(860\) 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 −0.723226 0.690611i \(-0.757342\pi\)
0.959700 + 0.281027i \(0.0906750\pi\)
\(878\) 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 \(0.0577884\pi\)
−0.983565 + 0.180552i \(0.942212\pi\)
\(884\) 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 −0.873140 0.487469i \(-0.837920\pi\)
−0.0144093 + 0.999896i \(0.504587\pi\)
\(908\) 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 −0.871857 0.489761i \(-0.837084\pi\)
−0.0117829 + 0.999931i \(0.503751\pi\)
\(942\) 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 0.778504 0.627639i \(-0.215979\pi\)
−0.154299 + 0.988024i \(0.549312\pi\)
\(948\) 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 \(0.533938\pi\)
−0.807900 + 0.589320i \(0.799396\pi\)
\(972\) 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 −0.974637 0.223791i \(-0.928157\pi\)
−0.293510 + 0.955956i \(0.594823\pi\)
\(998\) 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 448.4.p.f.383.2 6
4.3 odd 2 448.4.p.g.383.2 6
7.3 odd 6 448.4.p.g.255.2 6
8.3 odd 2 112.4.p.f.47.2 yes 6
8.5 even 2 112.4.p.g.47.2 yes 6
28.3 even 6 inner 448.4.p.f.255.2 6
56.3 even 6 112.4.p.g.31.2 yes 6
56.5 odd 6 784.4.f.h.783.4 6
56.19 even 6 784.4.f.g.783.4 6
56.37 even 6 784.4.f.g.783.3 6
56.45 odd 6 112.4.p.f.31.2 6
56.51 odd 6 784.4.f.h.783.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.p.f.31.2 6 56.45 odd 6
112.4.p.f.47.2 yes 6 8.3 odd 2
112.4.p.g.31.2 yes 6 56.3 even 6
112.4.p.g.47.2 yes 6 8.5 even 2
448.4.p.f.255.2 6 28.3 even 6 inner
448.4.p.f.383.2 6 1.1 even 1 trivial
448.4.p.g.255.2 6 7.3 odd 6
448.4.p.g.383.2 6 4.3 odd 2
784.4.f.g.783.3 6 56.37 even 6
784.4.f.g.783.4 6 56.19 even 6
784.4.f.h.783.3 6 56.51 odd 6
784.4.f.h.783.4 6 56.5 odd 6