Properties

Label 441.4.p.c.80.2
Level $441$
Weight $4$
Character 441.80
Analytic conductor $26.020$
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] = [441,4,Mod(80,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(26.0198423125\)
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: \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.2
Root \(-3.91663 + 2.26127i\) of defining polynomial
Character \(\chi\) \(=\) 441.80
Dual form 441.4.p.c.215.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.91663 + 2.26127i) q^{2} +(6.22668 - 10.7849i) q^{4} +(0.632851 + 1.09613i) q^{5} +20.1405i q^{8} +O(q^{10})\) \(q+(-3.91663 + 2.26127i) q^{2} +(6.22668 - 10.7849i) q^{4} +(0.632851 + 1.09613i) q^{5} +20.1405i q^{8} +(-4.95730 - 2.86210i) q^{10} +(36.0248 + 20.7989i) q^{11} -85.7355i q^{13} +(4.27028 + 7.39634i) q^{16} +(-38.8929 + 67.3645i) q^{17} +(-42.1638 + 24.3433i) q^{19} +15.7623 q^{20} -188.128 q^{22} +(78.7639 - 45.4743i) q^{23} +(61.6990 - 106.866i) q^{25} +(193.871 + 335.795i) q^{26} +151.196i q^{29} +(-76.3661 - 44.0900i) q^{31} +(-172.988 - 99.8747i) q^{32} -351.790i q^{34} +(-45.2914 - 78.4470i) q^{37} +(110.093 - 190.687i) q^{38} +(-22.0767 + 12.7460i) q^{40} -383.530 q^{41} -227.894 q^{43} +(448.630 - 259.017i) q^{44} +(-205.660 + 356.213i) q^{46} +(-69.5529 - 120.469i) q^{47} +558.072i q^{50} +(-924.652 - 533.848i) q^{52} +(-289.749 - 167.287i) q^{53} +52.6505i q^{55} +(-341.895 - 592.179i) q^{58} +(440.050 - 762.189i) q^{59} +(-11.3944 + 6.57854i) q^{61} +398.797 q^{62} +835.050 q^{64} +(93.9774 - 54.2579i) q^{65} +(221.212 - 383.151i) q^{67} +(484.348 + 838.915i) q^{68} -341.552i q^{71} +(798.218 + 460.851i) q^{73} +(354.780 + 204.832i) q^{74} +606.311i q^{76} +(206.564 + 357.780i) q^{79} +(-5.40490 + 9.36157i) q^{80} +(1502.15 - 867.265i) q^{82} +954.307 q^{83} -98.4538 q^{85} +(892.579 - 515.331i) q^{86} +(-418.901 + 725.558i) q^{88} +(14.8490 + 25.7193i) q^{89} -1132.62i q^{92} +(544.826 + 314.556i) q^{94} +(-53.3668 - 30.8113i) q^{95} -1199.63i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 72 q^{10} - 188 q^{16} + 612 q^{19} + 528 q^{22} - 20 q^{25} - 1128 q^{31} - 1196 q^{37} + 3204 q^{40} + 328 q^{43} - 1392 q^{46} - 4452 q^{52} - 3372 q^{58} + 1632 q^{61} + 5432 q^{64} + 308 q^{67} - 4068 q^{73} - 2176 q^{79} + 10188 q^{82} - 4608 q^{85} + 708 q^{88} + 2916 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.91663 + 2.26127i −1.38474 + 0.799480i −0.992716 0.120476i \(-0.961558\pi\)
−0.392023 + 0.919955i \(0.628225\pi\)
\(3\) 0 0
\(4\) 6.22668 10.7849i 0.778336 1.34812i
\(5\) 0.632851 + 1.09613i 0.0566040 + 0.0980409i 0.892939 0.450178i \(-0.148639\pi\)
−0.836335 + 0.548219i \(0.815306\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 20.1405i 0.890094i
\(9\) 0 0
\(10\) −4.95730 2.86210i −0.156763 0.0905074i
\(11\) 36.0248 + 20.7989i 0.987443 + 0.570101i 0.904509 0.426454i \(-0.140237\pi\)
0.0829344 + 0.996555i \(0.473571\pi\)
\(12\) 0 0
\(13\) 85.7355i 1.82914i −0.404433 0.914568i \(-0.632531\pi\)
0.404433 0.914568i \(-0.367469\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.27028 + 7.39634i 0.0667231 + 0.115568i
\(17\) −38.8929 + 67.3645i −0.554878 + 0.961076i 0.443035 + 0.896504i \(0.353902\pi\)
−0.997913 + 0.0645722i \(0.979432\pi\)
\(18\) 0 0
\(19\) −42.1638 + 24.3433i −0.509107 + 0.293933i −0.732466 0.680803i \(-0.761631\pi\)
0.223360 + 0.974736i \(0.428298\pi\)
\(20\) 15.7623 0.176227
\(21\) 0 0
\(22\) −188.128 −1.82314
\(23\) 78.7639 45.4743i 0.714061 0.412263i −0.0985019 0.995137i \(-0.531405\pi\)
0.812563 + 0.582874i \(0.198072\pi\)
\(24\) 0 0
\(25\) 61.6990 106.866i 0.493592 0.854926i
\(26\) 193.871 + 335.795i 1.46236 + 2.53288i
\(27\) 0 0
\(28\) 0 0
\(29\) 151.196i 0.968151i 0.875026 + 0.484075i \(0.160844\pi\)
−0.875026 + 0.484075i \(0.839156\pi\)
\(30\) 0 0
\(31\) −76.3661 44.0900i −0.442444 0.255445i 0.262190 0.965016i \(-0.415555\pi\)
−0.704634 + 0.709571i \(0.748889\pi\)
\(32\) −172.988 99.8747i −0.955633 0.551735i
\(33\) 0 0
\(34\) 351.790i 1.77445i
\(35\) 0 0
\(36\) 0 0
\(37\) −45.2914 78.4470i −0.201239 0.348557i 0.747689 0.664050i \(-0.231164\pi\)
−0.948928 + 0.315493i \(0.897830\pi\)
\(38\) 110.093 190.687i 0.469987 0.814041i
\(39\) 0 0
\(40\) −22.0767 + 12.7460i −0.0872657 + 0.0503829i
\(41\) −383.530 −1.46091 −0.730455 0.682961i \(-0.760692\pi\)
−0.730455 + 0.682961i \(0.760692\pi\)
\(42\) 0 0
\(43\) −227.894 −0.808222 −0.404111 0.914710i \(-0.632419\pi\)
−0.404111 + 0.914710i \(0.632419\pi\)
\(44\) 448.630 259.017i 1.53712 0.887459i
\(45\) 0 0
\(46\) −205.660 + 356.213i −0.659192 + 1.14175i
\(47\) −69.5529 120.469i −0.215858 0.373877i 0.737680 0.675151i \(-0.235922\pi\)
−0.953538 + 0.301274i \(0.902588\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 558.072i 1.57847i
\(51\) 0 0
\(52\) −924.652 533.848i −2.46589 1.42368i
\(53\) −289.749 167.287i −0.750945 0.433558i 0.0750904 0.997177i \(-0.476075\pi\)
−0.826035 + 0.563619i \(0.809409\pi\)
\(54\) 0 0
\(55\) 52.6505i 0.129080i
\(56\) 0 0
\(57\) 0 0
\(58\) −341.895 592.179i −0.774017 1.34064i
\(59\) 440.050 762.189i 0.971010 1.68184i 0.278493 0.960438i \(-0.410165\pi\)
0.692518 0.721401i \(-0.256501\pi\)
\(60\) 0 0
\(61\) −11.3944 + 6.57854i −0.0239164 + 0.0138081i −0.511911 0.859039i \(-0.671062\pi\)
0.487994 + 0.872847i \(0.337729\pi\)
\(62\) 398.797 0.816892
\(63\) 0 0
\(64\) 835.050 1.63096
\(65\) 93.9774 54.2579i 0.179330 0.103536i
\(66\) 0 0
\(67\) 221.212 383.151i 0.403364 0.698647i −0.590766 0.806843i \(-0.701174\pi\)
0.994130 + 0.108197i \(0.0345076\pi\)
\(68\) 484.348 + 838.915i 0.863762 + 1.49608i
\(69\) 0 0
\(70\) 0 0
\(71\) 341.552i 0.570912i −0.958392 0.285456i \(-0.907855\pi\)
0.958392 0.285456i \(-0.0921450\pi\)
\(72\) 0 0
\(73\) 798.218 + 460.851i 1.27979 + 0.738885i 0.976809 0.214113i \(-0.0686862\pi\)
0.302977 + 0.952998i \(0.402019\pi\)
\(74\) 354.780 + 204.832i 0.557328 + 0.321774i
\(75\) 0 0
\(76\) 606.311i 0.915114i
\(77\) 0 0
\(78\) 0 0
\(79\) 206.564 + 357.780i 0.294181 + 0.509537i 0.974794 0.223107i \(-0.0716198\pi\)
−0.680613 + 0.732643i \(0.738286\pi\)
\(80\) −5.40490 + 9.36157i −0.00755358 + 0.0130832i
\(81\) 0 0
\(82\) 1502.15 867.265i 2.02298 1.16797i
\(83\) 954.307 1.26203 0.631017 0.775769i \(-0.282638\pi\)
0.631017 + 0.775769i \(0.282638\pi\)
\(84\) 0 0
\(85\) −98.4538 −0.125633
\(86\) 892.579 515.331i 1.11918 0.646157i
\(87\) 0 0
\(88\) −418.901 + 725.558i −0.507444 + 0.878918i
\(89\) 14.8490 + 25.7193i 0.0176853 + 0.0306319i 0.874733 0.484606i \(-0.161037\pi\)
−0.857047 + 0.515238i \(0.827704\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1132.62i 1.28352i
\(93\) 0 0
\(94\) 544.826 + 314.556i 0.597814 + 0.345148i
\(95\) −53.3668 30.8113i −0.0576349 0.0332755i
\(96\) 0 0
\(97\) 1199.63i 1.25572i −0.778328 0.627858i \(-0.783932\pi\)
0.778328 0.627858i \(-0.216068\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −768.360 1330.84i −0.768360 1.33084i
\(101\) −327.422 + 567.111i −0.322571 + 0.558710i −0.981018 0.193918i \(-0.937881\pi\)
0.658447 + 0.752628i \(0.271214\pi\)
\(102\) 0 0
\(103\) 1186.01 684.744i 1.13457 0.655047i 0.189493 0.981882i \(-0.439316\pi\)
0.945081 + 0.326836i \(0.105982\pi\)
\(104\) 1726.76 1.62810
\(105\) 0 0
\(106\) 1513.12 1.38648
\(107\) −371.311 + 214.377i −0.335477 + 0.193688i −0.658270 0.752782i \(-0.728711\pi\)
0.322793 + 0.946470i \(0.395378\pi\)
\(108\) 0 0
\(109\) 334.261 578.957i 0.293728 0.508752i −0.680960 0.732321i \(-0.738437\pi\)
0.974688 + 0.223568i \(0.0717706\pi\)
\(110\) −119.057 206.213i −0.103197 0.178742i
\(111\) 0 0
\(112\) 0 0
\(113\) 914.837i 0.761598i −0.924658 0.380799i \(-0.875649\pi\)
0.924658 0.380799i \(-0.124351\pi\)
\(114\) 0 0
\(115\) 99.6917 + 57.5570i 0.0808374 + 0.0466715i
\(116\) 1630.64 + 941.449i 1.30518 + 0.753546i
\(117\) 0 0
\(118\) 3980.29i 3.10521i
\(119\) 0 0
\(120\) 0 0
\(121\) 199.690 + 345.873i 0.150030 + 0.259859i
\(122\) 29.7517 51.5314i 0.0220786 0.0382413i
\(123\) 0 0
\(124\) −951.015 + 549.069i −0.688739 + 0.397644i
\(125\) 314.398 0.224965
\(126\) 0 0
\(127\) 1260.95 0.881034 0.440517 0.897744i \(-0.354795\pi\)
0.440517 + 0.897744i \(0.354795\pi\)
\(128\) −1886.68 + 1089.28i −1.30282 + 0.752182i
\(129\) 0 0
\(130\) −245.383 + 425.016i −0.165550 + 0.286742i
\(131\) −683.600 1184.03i −0.455926 0.789688i 0.542814 0.839853i \(-0.317359\pi\)
−0.998741 + 0.0501648i \(0.984025\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2000.88i 1.28992i
\(135\) 0 0
\(136\) −1356.76 783.325i −0.855449 0.493894i
\(137\) −953.631 550.579i −0.594702 0.343351i 0.172252 0.985053i \(-0.444896\pi\)
−0.766955 + 0.641701i \(0.778229\pi\)
\(138\) 0 0
\(139\) 2306.56i 1.40748i −0.710458 0.703739i \(-0.751512\pi\)
0.710458 0.703739i \(-0.248488\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 772.341 + 1337.73i 0.456432 + 0.790564i
\(143\) 1783.21 3088.60i 1.04279 1.80617i
\(144\) 0 0
\(145\) −165.730 + 95.6845i −0.0949184 + 0.0548012i
\(146\) −4168.44 −2.36289
\(147\) 0 0
\(148\) −1128.06 −0.626527
\(149\) 1520.57 877.901i 0.836040 0.482688i −0.0198764 0.999802i \(-0.506327\pi\)
0.855916 + 0.517115i \(0.172994\pi\)
\(150\) 0 0
\(151\) 262.491 454.647i 0.141465 0.245024i −0.786584 0.617484i \(-0.788152\pi\)
0.928048 + 0.372460i \(0.121485\pi\)
\(152\) −490.286 849.201i −0.261628 0.453153i
\(153\) 0 0
\(154\) 0 0
\(155\) 111.610i 0.0578368i
\(156\) 0 0
\(157\) 1141.44 + 659.009i 0.580233 + 0.334998i 0.761226 0.648487i \(-0.224598\pi\)
−0.180993 + 0.983484i \(0.557931\pi\)
\(158\) −1618.07 934.195i −0.814728 0.470384i
\(159\) 0 0
\(160\) 252.823i 0.124921i
\(161\) 0 0
\(162\) 0 0
\(163\) 223.916 + 387.834i 0.107598 + 0.186365i 0.914797 0.403915i \(-0.132351\pi\)
−0.807199 + 0.590280i \(0.799017\pi\)
\(164\) −2388.12 + 4136.35i −1.13708 + 1.96948i
\(165\) 0 0
\(166\) −3737.67 + 2157.95i −1.74759 + 1.00897i
\(167\) −811.124 −0.375848 −0.187924 0.982184i \(-0.560176\pi\)
−0.187924 + 0.982184i \(0.560176\pi\)
\(168\) 0 0
\(169\) −5153.58 −2.34574
\(170\) 385.608 222.631i 0.173969 0.100441i
\(171\) 0 0
\(172\) −1419.03 + 2457.82i −0.629068 + 1.08958i
\(173\) −1121.24 1942.04i −0.492751 0.853470i 0.507214 0.861820i \(-0.330675\pi\)
−0.999965 + 0.00834994i \(0.997342\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 355.269i 0.152156i
\(177\) 0 0
\(178\) −116.317 67.1554i −0.0489792 0.0282781i
\(179\) −2531.77 1461.72i −1.05717 0.610357i −0.132521 0.991180i \(-0.542307\pi\)
−0.924648 + 0.380824i \(0.875641\pi\)
\(180\) 0 0
\(181\) 282.859i 0.116159i 0.998312 + 0.0580794i \(0.0184977\pi\)
−0.998312 + 0.0580794i \(0.981502\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 915.878 + 1586.35i 0.366953 + 0.635582i
\(185\) 57.3255 99.2906i 0.0227819 0.0394594i
\(186\) 0 0
\(187\) −2802.22 + 1617.86i −1.09582 + 0.632672i
\(188\) −1732.34 −0.672040
\(189\) 0 0
\(190\) 278.691 0.106412
\(191\) −3998.63 + 2308.61i −1.51482 + 0.874582i −0.514971 + 0.857208i \(0.672197\pi\)
−0.999849 + 0.0173741i \(0.994469\pi\)
\(192\) 0 0
\(193\) 2077.73 3598.73i 0.774912 1.34219i −0.159933 0.987128i \(-0.551128\pi\)
0.934844 0.355058i \(-0.115539\pi\)
\(194\) 2712.70 + 4698.53i 1.00392 + 1.73884i
\(195\) 0 0
\(196\) 0 0
\(197\) 1626.36i 0.588190i −0.955776 0.294095i \(-0.904982\pi\)
0.955776 0.294095i \(-0.0950183\pi\)
\(198\) 0 0
\(199\) −150.861 87.0995i −0.0537399 0.0310267i 0.472889 0.881122i \(-0.343211\pi\)
−0.526629 + 0.850095i \(0.676544\pi\)
\(200\) 2152.33 + 1242.65i 0.760965 + 0.439344i
\(201\) 0 0
\(202\) 2961.56i 1.03156i
\(203\) 0 0
\(204\) 0 0
\(205\) −242.718 420.399i −0.0826933 0.143229i
\(206\) −3096.78 + 5363.78i −1.04739 + 1.81414i
\(207\) 0 0
\(208\) 634.129 366.115i 0.211389 0.122046i
\(209\) −2025.25 −0.670286
\(210\) 0 0
\(211\) −2942.35 −0.959999 −0.479999 0.877269i \(-0.659363\pi\)
−0.479999 + 0.877269i \(0.659363\pi\)
\(212\) −3608.35 + 2083.28i −1.16897 + 0.674907i
\(213\) 0 0
\(214\) 969.527 1679.27i 0.309699 0.536414i
\(215\) −144.223 249.802i −0.0457486 0.0792389i
\(216\) 0 0
\(217\) 0 0
\(218\) 3023.42i 0.939319i
\(219\) 0 0
\(220\) 567.832 + 327.838i 0.174015 + 0.100467i
\(221\) 5775.53 + 3334.51i 1.75794 + 1.01495i
\(222\) 0 0
\(223\) 3374.75i 1.01341i −0.862120 0.506704i \(-0.830864\pi\)
0.862120 0.506704i \(-0.169136\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2068.69 + 3583.08i 0.608882 + 1.05461i
\(227\) −1515.43 + 2624.79i −0.443094 + 0.767461i −0.997917 0.0645069i \(-0.979453\pi\)
0.554823 + 0.831968i \(0.312786\pi\)
\(228\) 0 0
\(229\) −960.030 + 554.274i −0.277033 + 0.159945i −0.632080 0.774904i \(-0.717798\pi\)
0.355046 + 0.934849i \(0.384465\pi\)
\(230\) −520.608 −0.149252
\(231\) 0 0
\(232\) −3045.17 −0.861746
\(233\) 2684.57 1549.94i 0.754815 0.435793i −0.0726160 0.997360i \(-0.523135\pi\)
0.827431 + 0.561567i \(0.189801\pi\)
\(234\) 0 0
\(235\) 88.0333 152.478i 0.0244368 0.0423259i
\(236\) −5480.10 9491.82i −1.51154 2.61807i
\(237\) 0 0
\(238\) 0 0
\(239\) 1735.25i 0.469640i 0.972039 + 0.234820i \(0.0754501\pi\)
−0.972039 + 0.234820i \(0.924550\pi\)
\(240\) 0 0
\(241\) 1039.26 + 600.019i 0.277779 + 0.160376i 0.632418 0.774628i \(-0.282063\pi\)
−0.354638 + 0.935004i \(0.615396\pi\)
\(242\) −1564.22 903.104i −0.415504 0.239892i
\(243\) 0 0
\(244\) 163.850i 0.0429894i
\(245\) 0 0
\(246\) 0 0
\(247\) 2087.08 + 3614.93i 0.537643 + 0.931225i
\(248\) 887.996 1538.05i 0.227370 0.393817i
\(249\) 0 0
\(250\) −1231.38 + 710.939i −0.311518 + 0.179855i
\(251\) 3712.56 0.933603 0.466802 0.884362i \(-0.345406\pi\)
0.466802 + 0.884362i \(0.345406\pi\)
\(252\) 0 0
\(253\) 3783.27 0.940126
\(254\) −4938.69 + 2851.35i −1.22000 + 0.704369i
\(255\) 0 0
\(256\) 1586.09 2747.20i 0.387230 0.670702i
\(257\) 389.574 + 674.762i 0.0945563 + 0.163776i 0.909423 0.415872i \(-0.136523\pi\)
−0.814867 + 0.579648i \(0.803190\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1351.39i 0.322344i
\(261\) 0 0
\(262\) 5354.82 + 3091.61i 1.26268 + 0.729008i
\(263\) 1702.07 + 982.690i 0.399065 + 0.230400i 0.686080 0.727526i \(-0.259330\pi\)
−0.287015 + 0.957926i \(0.592663\pi\)
\(264\) 0 0
\(265\) 423.470i 0.0981644i
\(266\) 0 0
\(267\) 0 0
\(268\) −2754.84 4771.52i −0.627905 1.08756i
\(269\) 1236.41 2141.53i 0.280243 0.485395i −0.691201 0.722662i \(-0.742918\pi\)
0.971444 + 0.237267i \(0.0762516\pi\)
\(270\) 0 0
\(271\) −4095.79 + 2364.71i −0.918088 + 0.530058i −0.883025 0.469327i \(-0.844497\pi\)
−0.0350633 + 0.999385i \(0.511163\pi\)
\(272\) −664.334 −0.148093
\(273\) 0 0
\(274\) 4980.03 1.09801
\(275\) 4445.39 2566.54i 0.974788 0.562794i
\(276\) 0 0
\(277\) −586.579 + 1015.98i −0.127235 + 0.220378i −0.922604 0.385747i \(-0.873944\pi\)
0.795369 + 0.606125i \(0.207277\pi\)
\(278\) 5215.74 + 9033.93i 1.12525 + 1.94899i
\(279\) 0 0
\(280\) 0 0
\(281\) 8195.18i 1.73980i −0.493229 0.869899i \(-0.664184\pi\)
0.493229 0.869899i \(-0.335816\pi\)
\(282\) 0 0
\(283\) −2242.44 1294.67i −0.471021 0.271944i 0.245646 0.969360i \(-0.421000\pi\)
−0.716667 + 0.697415i \(0.754333\pi\)
\(284\) −3683.61 2126.74i −0.769656 0.444361i
\(285\) 0 0
\(286\) 16129.2i 3.33476i
\(287\) 0 0
\(288\) 0 0
\(289\) −568.820 985.225i −0.115779 0.200534i
\(290\) 432.737 749.523i 0.0876248 0.151771i
\(291\) 0 0
\(292\) 9940.50 5739.15i 1.99221 1.15020i
\(293\) 8871.16 1.76880 0.884400 0.466729i \(-0.154568\pi\)
0.884400 + 0.466729i \(0.154568\pi\)
\(294\) 0 0
\(295\) 1113.94 0.219852
\(296\) 1579.96 912.193i 0.310249 0.179122i
\(297\) 0 0
\(298\) −3970.34 + 6876.84i −0.771798 + 1.33679i
\(299\) −3898.77 6752.86i −0.754085 1.30611i
\(300\) 0 0
\(301\) 0 0
\(302\) 2374.25i 0.452393i
\(303\) 0 0
\(304\) −360.102 207.905i −0.0679383 0.0392242i
\(305\) −14.4219 8.32647i −0.00270752 0.00156319i
\(306\) 0 0
\(307\) 2707.52i 0.503344i 0.967813 + 0.251672i \(0.0809804\pi\)
−0.967813 + 0.251672i \(0.919020\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 252.379 + 437.134i 0.0462393 + 0.0800889i
\(311\) 1080.55 1871.56i 0.197017 0.341243i −0.750543 0.660822i \(-0.770208\pi\)
0.947560 + 0.319579i \(0.103541\pi\)
\(312\) 0 0
\(313\) 7300.25 4214.80i 1.31832 0.761133i 0.334863 0.942267i \(-0.391310\pi\)
0.983459 + 0.181133i \(0.0579766\pi\)
\(314\) −5960.79 −1.07130
\(315\) 0 0
\(316\) 5144.84 0.915886
\(317\) −8310.07 + 4797.82i −1.47237 + 0.850071i −0.999517 0.0310734i \(-0.990107\pi\)
−0.472848 + 0.881144i \(0.656774\pi\)
\(318\) 0 0
\(319\) −3144.71 + 5446.80i −0.551943 + 0.955994i
\(320\) 528.463 + 915.324i 0.0923186 + 0.159901i
\(321\) 0 0
\(322\) 0 0
\(323\) 3787.12i 0.652387i
\(324\) 0 0
\(325\) −9162.20 5289.80i −1.56378 0.902847i
\(326\) −1753.99 1012.67i −0.297990 0.172045i
\(327\) 0 0
\(328\) 7724.50i 1.30035i
\(329\) 0 0
\(330\) 0 0
\(331\) −4271.96 7399.25i −0.709390 1.22870i −0.965084 0.261941i \(-0.915637\pi\)
0.255694 0.966758i \(-0.417696\pi\)
\(332\) 5942.17 10292.1i 0.982286 1.70137i
\(333\) 0 0
\(334\) 3176.88 1834.17i 0.520452 0.300483i
\(335\) 559.978 0.0913280
\(336\) 0 0
\(337\) 598.875 0.0968036 0.0484018 0.998828i \(-0.484587\pi\)
0.0484018 + 0.998828i \(0.484587\pi\)
\(338\) 20184.7 11653.6i 3.24823 1.87537i
\(339\) 0 0
\(340\) −613.041 + 1061.82i −0.0977847 + 0.169368i
\(341\) −1834.05 3176.66i −0.291259 0.504475i
\(342\) 0 0
\(343\) 0 0
\(344\) 4589.91i 0.719394i
\(345\) 0 0
\(346\) 8782.94 + 5070.83i 1.36466 + 0.787889i
\(347\) −6149.62 3550.49i −0.951381 0.549280i −0.0578712 0.998324i \(-0.518431\pi\)
−0.893510 + 0.449044i \(0.851765\pi\)
\(348\) 0 0
\(349\) 3620.71i 0.555336i 0.960677 + 0.277668i \(0.0895616\pi\)
−0.960677 + 0.277668i \(0.910438\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4154.57 7195.92i −0.629089 1.08961i
\(353\) −1164.89 + 2017.64i −0.175639 + 0.304216i −0.940382 0.340119i \(-0.889533\pi\)
0.764743 + 0.644335i \(0.222866\pi\)
\(354\) 0 0
\(355\) 374.386 216.152i 0.0559727 0.0323159i
\(356\) 369.841 0.0550605
\(357\) 0 0
\(358\) 13221.3 1.95187
\(359\) −1522.43 + 878.975i −0.223818 + 0.129222i −0.607717 0.794154i \(-0.707914\pi\)
0.383899 + 0.923375i \(0.374581\pi\)
\(360\) 0 0
\(361\) −2244.31 + 3887.26i −0.327207 + 0.566739i
\(362\) −639.621 1107.86i −0.0928667 0.160850i
\(363\) 0 0
\(364\) 0 0
\(365\) 1166.60i 0.167295i
\(366\) 0 0
\(367\) −1458.89 842.290i −0.207503 0.119802i 0.392648 0.919689i \(-0.371559\pi\)
−0.600150 + 0.799887i \(0.704893\pi\)
\(368\) 672.687 + 388.376i 0.0952887 + 0.0550150i
\(369\) 0 0
\(370\) 518.513i 0.0728547i
\(371\) 0 0
\(372\) 0 0
\(373\) 148.646 + 257.462i 0.0206343 + 0.0357397i 0.876158 0.482024i \(-0.160098\pi\)
−0.855524 + 0.517763i \(0.826765\pi\)
\(374\) 7316.84 12673.1i 1.01162 1.75217i
\(375\) 0 0
\(376\) 2426.31 1400.83i 0.332786 0.192134i
\(377\) 12962.9 1.77088
\(378\) 0 0
\(379\) −7402.78 −1.00331 −0.501656 0.865067i \(-0.667276\pi\)
−0.501656 + 0.865067i \(0.667276\pi\)
\(380\) −664.596 + 383.705i −0.0897186 + 0.0517991i
\(381\) 0 0
\(382\) 10440.8 18084.0i 1.39842 2.42214i
\(383\) 6263.77 + 10849.2i 0.835676 + 1.44743i 0.893479 + 0.449105i \(0.148257\pi\)
−0.0578031 + 0.998328i \(0.518410\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18793.2i 2.47810i
\(387\) 0 0
\(388\) −12938.0 7469.75i −1.69285 0.977368i
\(389\) 6904.14 + 3986.11i 0.899881 + 0.519547i 0.877162 0.480195i \(-0.159434\pi\)
0.0227196 + 0.999742i \(0.492767\pi\)
\(390\) 0 0
\(391\) 7074.52i 0.915023i
\(392\) 0 0
\(393\) 0 0
\(394\) 3677.64 + 6369.87i 0.470246 + 0.814491i
\(395\) −261.449 + 452.843i −0.0333036 + 0.0576836i
\(396\) 0 0
\(397\) −10832.5 + 6254.16i −1.36944 + 0.790648i −0.990857 0.134918i \(-0.956923\pi\)
−0.378586 + 0.925566i \(0.623590\pi\)
\(398\) 787.822 0.0992210
\(399\) 0 0
\(400\) 1053.89 0.131736
\(401\) −6943.55 + 4008.86i −0.864699 + 0.499234i −0.865583 0.500765i \(-0.833052\pi\)
0.000883860 1.00000i \(0.499719\pi\)
\(402\) 0 0
\(403\) −3780.08 + 6547.29i −0.467243 + 0.809289i
\(404\) 4077.51 + 7062.45i 0.502137 + 0.869728i
\(405\) 0 0
\(406\) 0 0
\(407\) 3768.05i 0.458907i
\(408\) 0 0
\(409\) 7566.04 + 4368.26i 0.914711 + 0.528109i 0.881944 0.471354i \(-0.156235\pi\)
0.0327670 + 0.999463i \(0.489568\pi\)
\(410\) 1901.27 + 1097.70i 0.229017 + 0.132223i
\(411\) 0 0
\(412\) 17054.7i 2.03938i
\(413\) 0 0
\(414\) 0 0
\(415\) 603.935 + 1046.05i 0.0714361 + 0.123731i
\(416\) −8562.81 + 14831.2i −1.00920 + 1.74798i
\(417\) 0 0
\(418\) 7932.18 4579.64i 0.928171 0.535880i
\(419\) −3926.67 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(420\) 0 0
\(421\) 1443.44 0.167100 0.0835499 0.996504i \(-0.473374\pi\)
0.0835499 + 0.996504i \(0.473374\pi\)
\(422\) 11524.1 6653.45i 1.32935 0.767500i
\(423\) 0 0
\(424\) 3369.24 5835.70i 0.385908 0.668412i
\(425\) 4799.31 + 8312.65i 0.547766 + 0.948759i
\(426\) 0 0
\(427\) 0 0
\(428\) 5339.42i 0.603016i
\(429\) 0 0
\(430\) 1129.74 + 652.255i 0.126700 + 0.0731501i
\(431\) 12820.5 + 7401.92i 1.43281 + 0.827234i 0.997334 0.0729655i \(-0.0232463\pi\)
0.435477 + 0.900200i \(0.356580\pi\)
\(432\) 0 0
\(433\) 15872.1i 1.76158i −0.473508 0.880790i \(-0.657012\pi\)
0.473508 0.880790i \(-0.342988\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4162.67 7209.96i −0.457238 0.791960i
\(437\) −2213.99 + 3834.74i −0.242356 + 0.419772i
\(438\) 0 0
\(439\) 2626.58 1516.45i 0.285557 0.164867i −0.350379 0.936608i \(-0.613947\pi\)
0.635937 + 0.771741i \(0.280614\pi\)
\(440\) −1060.41 −0.114893
\(441\) 0 0
\(442\) −30160.9 −3.24572
\(443\) 11126.8 6424.08i 1.19334 0.688978i 0.234281 0.972169i \(-0.424726\pi\)
0.959064 + 0.283191i \(0.0913930\pi\)
\(444\) 0 0
\(445\) −18.7945 + 32.5530i −0.00200212 + 0.00346777i
\(446\) 7631.22 + 13217.7i 0.810199 + 1.40331i
\(447\) 0 0
\(448\) 0 0
\(449\) 107.668i 0.0113166i −0.999984 0.00565831i \(-0.998199\pi\)
0.999984 0.00565831i \(-0.00180111\pi\)
\(450\) 0 0
\(451\) −13816.6 7977.01i −1.44257 0.832866i
\(452\) −9866.45 5696.40i −1.02672 0.592779i
\(453\) 0 0
\(454\) 13707.1i 1.41698i
\(455\) 0 0
\(456\) 0 0
\(457\) 4888.53 + 8467.18i 0.500385 + 0.866691i 1.00000 0.000444115i \(0.000141366\pi\)
−0.499615 + 0.866247i \(0.666525\pi\)
\(458\) 2506.72 4341.77i 0.255746 0.442965i
\(459\) 0 0
\(460\) 1241.50 716.779i 0.125837 0.0726521i
\(461\) 638.874 0.0645452 0.0322726 0.999479i \(-0.489726\pi\)
0.0322726 + 0.999479i \(0.489726\pi\)
\(462\) 0 0
\(463\) −5602.26 −0.562331 −0.281165 0.959659i \(-0.590721\pi\)
−0.281165 + 0.959659i \(0.590721\pi\)
\(464\) −1118.30 + 645.648i −0.111887 + 0.0645980i
\(465\) 0 0
\(466\) −7009.65 + 12141.1i −0.696815 + 1.20692i
\(467\) −2759.44 4779.49i −0.273430 0.473594i 0.696308 0.717743i \(-0.254825\pi\)
−0.969738 + 0.244149i \(0.921491\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 796.268i 0.0781470i
\(471\) 0 0
\(472\) 15350.9 + 8862.84i 1.49700 + 0.864291i
\(473\) −8209.84 4739.95i −0.798074 0.460768i
\(474\) 0 0
\(475\) 6007.82i 0.580332i
\(476\) 0 0
\(477\) 0 0
\(478\) −3923.87 6796.34i −0.375468 0.650329i
\(479\) −2734.40 + 4736.11i −0.260830 + 0.451771i −0.966463 0.256807i \(-0.917330\pi\)
0.705632 + 0.708578i \(0.250663\pi\)
\(480\) 0 0
\(481\) −6725.70 + 3883.08i −0.637558 + 0.368094i
\(482\) −5427.22 −0.512869
\(483\) 0 0
\(484\) 4973.62 0.467094
\(485\) 1314.96 759.190i 0.123112 0.0710785i
\(486\) 0 0
\(487\) 5866.72 10161.5i 0.545886 0.945502i −0.452665 0.891681i \(-0.649527\pi\)
0.998551 0.0538213i \(-0.0171401\pi\)
\(488\) −132.495 229.489i −0.0122905 0.0212878i
\(489\) 0 0
\(490\) 0 0
\(491\) 3514.92i 0.323068i −0.986867 0.161534i \(-0.948356\pi\)
0.986867 0.161534i \(-0.0516441\pi\)
\(492\) 0 0
\(493\) −10185.2 5880.45i −0.930467 0.537205i
\(494\) −16348.7 9438.91i −1.48899 0.859670i
\(495\) 0 0
\(496\) 753.106i 0.0681763i
\(497\) 0 0
\(498\) 0 0
\(499\) −4944.49 8564.11i −0.443579 0.768301i 0.554373 0.832268i \(-0.312958\pi\)
−0.997952 + 0.0639672i \(0.979625\pi\)
\(500\) 1957.66 3390.76i 0.175098 0.303279i
\(501\) 0 0
\(502\) −14540.7 + 8395.09i −1.29280 + 0.746397i
\(503\) −10172.2 −0.901698 −0.450849 0.892600i \(-0.648879\pi\)
−0.450849 + 0.892600i \(0.648879\pi\)
\(504\) 0 0
\(505\) −828.838 −0.0730353
\(506\) −14817.7 + 8554.99i −1.30183 + 0.751612i
\(507\) 0 0
\(508\) 7851.55 13599.3i 0.685740 1.18774i
\(509\) −2149.56 3723.15i −0.187186 0.324216i 0.757125 0.653270i \(-0.226603\pi\)
−0.944311 + 0.329054i \(0.893270\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3082.06i 0.266033i
\(513\) 0 0
\(514\) −3051.64 1761.86i −0.261872 0.151192i
\(515\) 1501.14 + 866.682i 0.128443 + 0.0741565i
\(516\) 0 0
\(517\) 5786.50i 0.492243i
\(518\) 0 0
\(519\) 0 0
\(520\) 1092.78 + 1892.76i 0.0921571 + 0.159621i
\(521\) −5496.48 + 9520.18i −0.462198 + 0.800550i −0.999070 0.0431133i \(-0.986272\pi\)
0.536872 + 0.843664i \(0.319606\pi\)
\(522\) 0 0
\(523\) −7386.80 + 4264.77i −0.617595 + 0.356569i −0.775932 0.630817i \(-0.782720\pi\)
0.158337 + 0.987385i \(0.449387\pi\)
\(524\) −17026.2 −1.41946
\(525\) 0 0
\(526\) −8888.51 −0.736801
\(527\) 5940.20 3429.58i 0.491004 0.283481i
\(528\) 0 0
\(529\) −1947.67 + 3373.46i −0.160078 + 0.277263i
\(530\) 957.581 + 1658.58i 0.0784805 + 0.135932i
\(531\) 0 0
\(532\) 0 0
\(533\) 32882.2i 2.67220i
\(534\) 0 0
\(535\) −469.970 271.337i −0.0379786 0.0219270i
\(536\) 7716.86 + 4455.33i 0.621862 + 0.359032i
\(537\) 0 0
\(538\) 11183.4i 0.896195i
\(539\) 0 0
\(540\) 0 0
\(541\) 4352.93 + 7539.49i 0.345928 + 0.599165i 0.985522 0.169548i \(-0.0542308\pi\)
−0.639594 + 0.768713i \(0.720897\pi\)
\(542\) 10694.5 18523.4i 0.847542 1.46799i
\(543\) 0 0
\(544\) 13456.0 7768.84i 1.06052 0.612291i
\(545\) 846.150 0.0665047
\(546\) 0 0
\(547\) 17183.8 1.34319 0.671596 0.740917i \(-0.265609\pi\)
0.671596 + 0.740917i \(0.265609\pi\)
\(548\) −11875.9 + 6856.56i −0.925756 + 0.534485i
\(549\) 0 0
\(550\) −11607.3 + 20104.4i −0.899885 + 1.55865i
\(551\) −3680.60 6374.99i −0.284571 0.492892i
\(552\) 0 0
\(553\) 0 0
\(554\) 5305.66i 0.406888i
\(555\) 0 0
\(556\) −24876.0 14362.2i −1.89744 1.09549i
\(557\) 8989.79 + 5190.26i 0.683859 + 0.394826i 0.801307 0.598253i \(-0.204138\pi\)
−0.117448 + 0.993079i \(0.537471\pi\)
\(558\) 0 0
\(559\) 19538.6i 1.47835i
\(560\) 0 0
\(561\) 0 0
\(562\) 18531.5 + 32097.5i 1.39093 + 2.40917i
\(563\) 9248.22 16018.4i 0.692302 1.19910i −0.278780 0.960355i \(-0.589930\pi\)
0.971082 0.238747i \(-0.0767366\pi\)
\(564\) 0 0
\(565\) 1002.78 578.956i 0.0746678 0.0431095i
\(566\) 11710.4 0.869656
\(567\) 0 0
\(568\) 6879.04 0.508166
\(569\) 3493.45 2016.94i 0.257386 0.148602i −0.365755 0.930711i \(-0.619189\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(570\) 0 0
\(571\) −6430.01 + 11137.1i −0.471257 + 0.816241i −0.999459 0.0328777i \(-0.989533\pi\)
0.528203 + 0.849118i \(0.322866\pi\)
\(572\) −22206.9 38463.5i −1.62328 2.81161i
\(573\) 0 0
\(574\) 0 0
\(575\) 11222.9i 0.813959i
\(576\) 0 0
\(577\) 17669.2 + 10201.3i 1.27483 + 0.736026i 0.975894 0.218246i \(-0.0700335\pi\)
0.298940 + 0.954272i \(0.403367\pi\)
\(578\) 4455.72 + 2572.51i 0.320646 + 0.185125i
\(579\) 0 0
\(580\) 2383.19i 0.170615i
\(581\) 0 0
\(582\) 0 0
\(583\) −6958.76 12052.9i −0.494344 0.856228i
\(584\) −9281.80 + 16076.5i −0.657677 + 1.13913i
\(585\) 0 0
\(586\) −34745.1 + 20060.1i −2.44933 + 1.41412i
\(587\) −16279.7 −1.14470 −0.572348 0.820011i \(-0.693967\pi\)
−0.572348 + 0.820011i \(0.693967\pi\)
\(588\) 0 0
\(589\) 4293.17 0.300335
\(590\) −4362.91 + 2518.93i −0.304438 + 0.175767i
\(591\) 0 0
\(592\) 386.814 669.981i 0.0268546 0.0465136i
\(593\) 1342.16 + 2324.68i 0.0929440 + 0.160984i 0.908749 0.417344i \(-0.137039\pi\)
−0.815805 + 0.578328i \(0.803706\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21865.7i 1.50277i
\(597\) 0 0
\(598\) 30540.1 + 17632.3i 2.08842 + 1.20575i
\(599\) −12224.6 7057.90i −0.833865 0.481432i 0.0213091 0.999773i \(-0.493217\pi\)
−0.855174 + 0.518341i \(0.826550\pi\)
\(600\) 0 0
\(601\) 11096.1i 0.753109i 0.926394 + 0.376555i \(0.122891\pi\)
−0.926394 + 0.376555i \(0.877109\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3268.89 5661.89i −0.220214 0.381422i
\(605\) −252.748 + 437.772i −0.0169846 + 0.0294181i
\(606\) 0 0
\(607\) 9592.70 5538.35i 0.641442 0.370337i −0.143728 0.989617i \(-0.545909\pi\)
0.785170 + 0.619280i \(0.212576\pi\)
\(608\) 9725.10 0.648692
\(609\) 0 0
\(610\) 75.3136 0.00499895
\(611\) −10328.5 + 5963.15i −0.683872 + 0.394834i
\(612\) 0 0
\(613\) 3801.34 6584.11i 0.250464 0.433817i −0.713189 0.700971i \(-0.752750\pi\)
0.963654 + 0.267154i \(0.0860834\pi\)
\(614\) −6122.44 10604.4i −0.402413 0.697000i
\(615\) 0 0
\(616\) 0 0
\(617\) 11325.9i 0.738998i −0.929231 0.369499i \(-0.879529\pi\)
0.929231 0.369499i \(-0.120471\pi\)
\(618\) 0 0
\(619\) 16595.2 + 9581.22i 1.07757 + 0.622136i 0.930240 0.366952i \(-0.119599\pi\)
0.147330 + 0.989087i \(0.452932\pi\)
\(620\) −1203.70 694.958i −0.0779707 0.0450164i
\(621\) 0 0
\(622\) 9773.64i 0.630044i
\(623\) 0 0
\(624\) 0 0
\(625\) −7513.41 13013.6i −0.480858 0.832871i
\(626\) −19061.6 + 33015.7i −1.21702 + 2.10794i
\(627\) 0 0
\(628\) 14214.7 8206.88i 0.903232 0.521481i
\(629\) 7046.06 0.446653
\(630\) 0 0
\(631\) 10140.7 0.639768 0.319884 0.947457i \(-0.396356\pi\)
0.319884 + 0.947457i \(0.396356\pi\)
\(632\) −7205.88 + 4160.32i −0.453536 + 0.261849i
\(633\) 0 0
\(634\) 21698.3 37582.6i 1.35923 2.35425i
\(635\) 797.995 + 1382.17i 0.0498700 + 0.0863774i
\(636\) 0 0
\(637\) 0 0
\(638\) 28444.2i 1.76507i
\(639\) 0 0
\(640\) −2387.98 1378.70i −0.147489 0.0851530i
\(641\) 2950.66 + 1703.56i 0.181816 + 0.104972i 0.588146 0.808755i \(-0.299858\pi\)
−0.406330 + 0.913727i \(0.633192\pi\)
\(642\) 0 0
\(643\) 659.110i 0.0404242i 0.999796 + 0.0202121i \(0.00643415\pi\)
−0.999796 + 0.0202121i \(0.993566\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8563.71 + 14832.8i 0.521570 + 0.903386i
\(647\) 3303.47 5721.78i 0.200731 0.347676i −0.748033 0.663661i \(-0.769002\pi\)
0.948764 + 0.315985i \(0.102335\pi\)
\(648\) 0 0
\(649\) 31705.4 18305.1i 1.91764 1.10715i
\(650\) 47846.6 2.88723
\(651\) 0 0
\(652\) 5577.01 0.334989
\(653\) −22417.1 + 12942.5i −1.34342 + 0.775622i −0.987307 0.158823i \(-0.949230\pi\)
−0.356109 + 0.934444i \(0.615897\pi\)
\(654\) 0 0
\(655\) 865.234 1498.63i 0.0516145 0.0893989i
\(656\) −1637.78 2836.72i −0.0974764 0.168834i
\(657\) 0 0
\(658\) 0 0
\(659\) 7468.86i 0.441495i 0.975331 + 0.220748i \(0.0708497\pi\)
−0.975331 + 0.220748i \(0.929150\pi\)
\(660\) 0 0
\(661\) 5501.96 + 3176.56i 0.323754 + 0.186919i 0.653065 0.757302i \(-0.273483\pi\)
−0.329311 + 0.944222i \(0.606816\pi\)
\(662\) 33463.4 + 19320.1i 1.96464 + 1.13429i
\(663\) 0 0
\(664\) 19220.3i 1.12333i
\(665\) 0 0
\(666\) 0 0
\(667\) 6875.53 + 11908.8i 0.399133 + 0.691319i
\(668\) −5050.61 + 8747.92i −0.292536 + 0.506687i
\(669\) 0 0
\(670\) −2193.23 + 1266.26i −0.126465 + 0.0730148i
\(671\) −547.306 −0.0314881
\(672\) 0 0
\(673\) −20238.2 −1.15918 −0.579589 0.814909i \(-0.696787\pi\)
−0.579589 + 0.814909i \(0.696787\pi\)
\(674\) −2345.58 + 1354.22i −0.134048 + 0.0773925i
\(675\) 0 0
\(676\) −32089.7 + 55581.1i −1.82577 + 3.16233i
\(677\) 5658.37 + 9800.59i 0.321224 + 0.556377i 0.980741 0.195313i \(-0.0625723\pi\)
−0.659516 + 0.751690i \(0.729239\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1982.91i 0.111825i
\(681\) 0 0
\(682\) 14366.6 + 8294.55i 0.806635 + 0.465711i
\(683\) −14869.4 8584.87i −0.833035 0.480953i 0.0218557 0.999761i \(-0.493043\pi\)
−0.854891 + 0.518808i \(0.826376\pi\)
\(684\) 0 0
\(685\) 1393.74i 0.0777402i
\(686\) 0 0
\(687\) 0 0
\(688\) −973.172 1685.58i −0.0539271 0.0934044i
\(689\) −14342.4 + 24841.8i −0.793037 + 1.37358i
\(690\) 0 0
\(691\) 7238.59 4179.20i 0.398508 0.230079i −0.287332 0.957831i \(-0.592768\pi\)
0.685840 + 0.727752i \(0.259435\pi\)
\(692\) −27926.3 −1.53410
\(693\) 0 0
\(694\) 32114.4 1.75655
\(695\) 2528.29 1459.71i 0.137990 0.0796688i
\(696\) 0 0
\(697\) 14916.6 25836.3i 0.810627 1.40405i
\(698\) −8187.41 14181.0i −0.443980 0.768996i
\(699\) 0 0
\(700\) 0 0
\(701\) 19235.8i 1.03641i −0.855256 0.518206i \(-0.826600\pi\)
0.855256 0.518206i \(-0.173400\pi\)
\(702\) 0 0
\(703\) 3819.31 + 2205.08i 0.204905 + 0.118302i
\(704\) 30082.5 + 17368.1i 1.61048 + 0.929810i
\(705\) 0 0
\(706\) 10536.5i 0.561680i
\(707\) 0 0
\(708\) 0 0
\(709\) 5160.17 + 8937.67i 0.273334 + 0.473429i 0.969714 0.244245i \(-0.0785401\pi\)
−0.696379 + 0.717674i \(0.745207\pi\)
\(710\) −977.554 + 1693.17i −0.0516718 + 0.0894981i
\(711\) 0 0
\(712\) −518.001 + 299.068i −0.0272653 + 0.0157416i
\(713\) −8019.85 −0.421242
\(714\) 0 0
\(715\) 4514.02 0.236105
\(716\) −31529.0 + 18203.3i −1.64566 + 0.950124i
\(717\) 0 0
\(718\) 3975.20 6885.25i 0.206620 0.357876i
\(719\) −11679.8 20229.9i −0.605815 1.04930i −0.991922 0.126849i \(-0.959514\pi\)
0.386107 0.922454i \(-0.373820\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 20300.0i 1.04638i
\(723\) 0 0
\(724\) 3050.62 + 1761.28i 0.156596 + 0.0904106i
\(725\) 16157.7 + 9328.63i 0.827698 + 0.477871i
\(726\) 0 0
\(727\) 22260.4i 1.13561i −0.823162 0.567807i \(-0.807792\pi\)
0.823162 0.567807i \(-0.192208\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2638.00 4569.15i −0.133749 0.231660i
\(731\) 8863.48 15352.0i 0.448464 0.776763i
\(732\) 0 0
\(733\) −9047.84 + 5223.77i −0.455920 + 0.263226i −0.710327 0.703872i \(-0.751453\pi\)
0.254407 + 0.967097i \(0.418120\pi\)
\(734\) 7618.58 0.383116
\(735\) 0 0
\(736\) −18166.9 −0.909840
\(737\) 15938.2 9201.95i 0.796598 0.459916i
\(738\) 0 0
\(739\) −6595.44 + 11423.6i −0.328304 + 0.568640i −0.982176 0.187966i \(-0.939811\pi\)
0.653871 + 0.756606i \(0.273144\pi\)
\(740\) −713.895 1236.50i −0.0354639 0.0614253i
\(741\) 0 0
\(742\) 0 0
\(743\) 35382.2i 1.74703i 0.486795 + 0.873517i \(0.338166\pi\)
−0.486795 + 0.873517i \(0.661834\pi\)
\(744\) 0 0
\(745\) 1924.59 + 1111.16i 0.0946463 + 0.0546441i
\(746\) −1164.38 672.258i −0.0571463 0.0329934i
\(747\) 0 0
\(748\) 40295.6i 1.96973i
\(749\) 0 0
\(750\) 0 0
\(751\) 14692.3 + 25447.8i 0.713888 + 1.23649i 0.963387 + 0.268116i \(0.0864010\pi\)
−0.249498 + 0.968375i \(0.580266\pi\)
\(752\) 594.020 1028.87i 0.0288054 0.0498925i
\(753\) 0 0
\(754\) −50770.8 + 29312.5i −2.45221 + 1.41578i
\(755\) 664.470 0.0320299
\(756\) 0 0
\(757\) 11329.1 0.543939 0.271969 0.962306i \(-0.412325\pi\)
0.271969 + 0.962306i \(0.412325\pi\)
\(758\) 28994.0 16739.7i 1.38932 0.802127i
\(759\) 0 0
\(760\) 620.557 1074.84i 0.0296184 0.0513005i
\(761\) 12696.5 + 21990.9i 0.604792 + 1.04753i 0.992084 + 0.125574i \(0.0400772\pi\)
−0.387292 + 0.921957i \(0.626589\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 57499.9i 2.72287i
\(765\) 0 0
\(766\) −49065.8 28328.2i −2.31439 1.33621i
\(767\) −65346.7 37727.9i −3.07631 1.77611i
\(768\) 0 0
\(769\) 18120.8i 0.849744i −0.905253 0.424872i \(-0.860319\pi\)
0.905253 0.424872i \(-0.139681\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25874.7 44816.3i −1.20628 2.08934i
\(773\) 8673.05 15022.2i 0.403555 0.698978i −0.590597 0.806967i \(-0.701108\pi\)
0.994152 + 0.107989i \(0.0344410\pi\)
\(774\) 0 0
\(775\) −9423.42 + 5440.61i −0.436773 + 0.252171i
\(776\) 24161.3 1.11771
\(777\) 0 0
\(778\) −36054.7 −1.66147
\(779\) 16171.1 9336.37i 0.743759 0.429410i
\(780\) 0 0
\(781\) 7103.91 12304.3i 0.325477 0.563743i
\(782\) −15997.4 27708.3i −0.731542 1.26707i
\(783\) 0 0
\(784\) 0 0
\(785\) 1668.22i 0.0758488i
\(786\) 0 0
\(787\) −1801.52 1040.11i −0.0815977 0.0471104i 0.458646 0.888619i \(-0.348335\pi\)
−0.540244 + 0.841509i \(0.681668\pi\)
\(788\) −17540.2 10126.8i −0.792949 0.457810i
\(789\) 0 0
\(790\) 2364.83i 0.106502i
\(791\) 0 0
\(792\) 0 0
\(793\) 564.014 + 976.901i 0.0252569 + 0.0437463i
\(794\) 28284.7 48990.5i 1.26421 2.18968i
\(795\) 0 0
\(796\) −1878.73 + 1084.68i −0.0836554 + 0.0482984i
\(797\) −39737.3 −1.76608 −0.883041 0.469297i \(-0.844507\pi\)
−0.883041 + 0.469297i \(0.844507\pi\)
\(798\) 0 0
\(799\) 10820.5 0.479099
\(800\) −21346.4 + 12324.3i −0.943385 + 0.544664i
\(801\) 0 0
\(802\) 18130.2 31402.5i 0.798255 1.38262i
\(803\) 19170.4 + 33204.1i 0.842477 + 1.45921i
\(804\) 0 0
\(805\) 0 0
\(806\) 34191.1i 1.49421i
\(807\) 0 0
\(808\) −11421.9 6594.45i −0.497305 0.287119i
\(809\) 9713.37 + 5608.02i 0.422131 + 0.243717i 0.695989 0.718053i \(-0.254966\pi\)
−0.273858 + 0.961770i \(0.588300\pi\)
\(810\) 0 0
\(811\) 14792.0i 0.640465i 0.947339 + 0.320232i \(0.103761\pi\)
−0.947339 + 0.320232i \(0.896239\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8520.57 + 14758.1i 0.366887 + 0.635467i
\(815\) −283.411 + 490.882i −0.0121809 + 0.0210980i
\(816\) 0 0
\(817\) 9608.88 5547.69i 0.411471 0.237563i
\(818\) −39511.2 −1.68885
\(819\) 0 0
\(820\) −6045.30 −0.257453
\(821\) 28632.7 16531.1i 1.21716 0.702726i 0.252849 0.967506i \(-0.418632\pi\)
0.964309 + 0.264779i \(0.0852992\pi\)
\(822\) 0 0
\(823\) −10045.4 + 17399.1i −0.425467 + 0.736930i −0.996464 0.0840220i \(-0.973223\pi\)
0.570997 + 0.820952i \(0.306557\pi\)
\(824\) 13791.1 + 23886.9i 0.583053 + 1.00988i
\(825\) 0 0
\(826\) 0 0
\(827\) 36882.5i 1.55082i 0.631458 + 0.775410i \(0.282457\pi\)
−0.631458 + 0.775410i \(0.717543\pi\)
\(828\) 0 0
\(829\) 32697.0 + 18877.6i 1.36986 + 0.790888i 0.990910 0.134530i \(-0.0429524\pi\)
0.378949 + 0.925418i \(0.376286\pi\)
\(830\) −4730.78 2731.32i −0.197841 0.114223i
\(831\) 0 0
\(832\) 71593.5i 2.98324i
\(833\) 0 0
\(834\) 0 0
\(835\) −513.321 889.098i −0.0212745 0.0368485i
\(836\) −12610.6 + 21842.2i −0.521707 + 0.903623i
\(837\) 0 0
\(838\) 15379.3 8879.25i 0.633973 0.366025i
\(839\) −44250.4 −1.82085 −0.910426 0.413672i \(-0.864246\pi\)
−0.910426 + 0.413672i \(0.864246\pi\)
\(840\) 0 0
\(841\) 1528.81 0.0626844
\(842\) −5653.43 + 3264.01i −0.231390 + 0.133593i
\(843\) 0 0
\(844\) −18321.1 + 31733.1i −0.747201 + 1.29419i
\(845\) −3261.45 5649.00i −0.132778 0.229978i
\(846\) 0 0
\(847\) 0 0
\(848\) 2857.44i 0.115713i
\(849\) 0 0
\(850\) −37594.3 21705.1i −1.51703 0.875856i
\(851\) −7134.65 4119.19i −0.287394 0.165927i
\(852\) 0 0
\(853\) 14952.2i 0.600179i −0.953911 0.300089i \(-0.902983\pi\)
0.953911 0.300089i \(-0.0970165\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4317.66 7478.41i −0.172400 0.298606i
\(857\) −1224.76 + 2121.35i −0.0488181 + 0.0845555i −0.889402 0.457126i \(-0.848879\pi\)
0.840584 + 0.541682i \(0.182212\pi\)
\(858\) 0 0
\(859\) 5751.97 3320.90i 0.228469 0.131906i −0.381397 0.924411i \(-0.624557\pi\)
0.609865 + 0.792505i \(0.291223\pi\)
\(860\) −3592.13 −0.142431
\(861\) 0 0
\(862\) −66951.0 −2.64543
\(863\) −16339.2 + 9433.46i −0.644489 + 0.372096i −0.786342 0.617792i \(-0.788027\pi\)
0.141853 + 0.989888i \(0.454694\pi\)
\(864\) 0 0
\(865\) 1419.15 2458.04i 0.0557833 0.0966196i
\(866\) 35891.1 + 62165.2i 1.40835 + 2.43933i
\(867\) 0 0
\(868\) 0 0
\(869\) 17185.3i 0.670851i
\(870\) 0 0
\(871\) −32849.6 18965.8i −1.27792 0.737807i
\(872\) 11660.5 + 6732.19i 0.452837 + 0.261446i
\(873\) 0 0
\(874\) 20025.7i 0.775033i
\(875\) 0 0
\(876\) 0 0
\(877\) −5989.84 10374.7i −0.230630 0.399463i 0.727364 0.686252i \(-0.240745\pi\)
−0.957994 + 0.286789i \(0.907412\pi\)
\(878\) −6858.23 + 11878.8i −0.263615 + 0.456594i
\(879\) 0 0
\(880\) −389.421 + 224.832i −0.0149175 + 0.00861261i
\(881\) 34504.0 1.31949 0.659743 0.751491i \(-0.270665\pi\)
0.659743 + 0.751491i \(0.270665\pi\)
\(882\) 0 0
\(883\) −8148.85 −0.310567 −0.155283 0.987870i \(-0.549629\pi\)
−0.155283 + 0.987870i \(0.549629\pi\)
\(884\) 71924.9 41525.8i 2.73653 1.57994i
\(885\) 0 0
\(886\) −29053.2 + 50321.6i −1.10165 + 1.90811i
\(887\) −17477.6 30272.1i −0.661602 1.14593i −0.980195 0.198036i \(-0.936544\pi\)
0.318593 0.947892i \(-0.396790\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 169.998i 0.00640262i
\(891\) 0 0
\(892\) −36396.4 21013.5i −1.36619 0.788771i
\(893\) 5865.22 + 3386.29i 0.219790 + 0.126896i
\(894\) 0 0
\(895\) 3700.20i 0.138194i
\(896\) 0 0
\(897\) 0 0
\(898\) 243.466 + 421.696i 0.00904741 + 0.0156706i
\(899\) 6666.22 11546.2i 0.247309 0.428352i
\(900\) 0 0
\(901\) 22538.4 13012.5i 0.833365 0.481144i
\(902\) 72152.7 2.66344
\(903\) 0 0
\(904\) 18425.3 0.677894
\(905\) −310.051 + 179.008i −0.0113883 + 0.00657505i
\(906\) 0 0
\(907\) 12494.5 21641.1i 0.457412 0.792261i −0.541411 0.840758i \(-0.682110\pi\)
0.998823 + 0.0484970i \(0.0154431\pi\)
\(908\) 18872.2 + 32687.5i 0.689752 + 1.19468i
\(909\) 0 0
\(910\) 0 0
\(911\) 41609.1i 1.51325i 0.653849 + 0.756625i \(0.273153\pi\)
−0.653849 + 0.756625i \(0.726847\pi\)
\(912\) 0 0
\(913\) 34378.7 + 19848.5i 1.24619 + 0.719486i
\(914\) −38293.2 22108.6i −1.38580 0.800095i
\(915\) 0 0
\(916\) 13805.1i 0.497964i
\(917\) 0 0
\(918\) 0 0
\(919\) −25496.0 44160.4i −0.915164 1.58511i −0.806661 0.591014i \(-0.798728\pi\)
−0.108502 0.994096i \(-0.534605\pi\)
\(920\) −1159.23 + 2007.84i −0.0415420 + 0.0719529i
\(921\) 0 0
\(922\) −2502.24 + 1444.67i −0.0893782 + 0.0516025i
\(923\) −29283.1 −1.04428
\(924\) 0 0
\(925\) −11177.7 −0.397321
\(926\) 21942.0 12668.2i 0.778682 0.449572i
\(927\) 0 0
\(928\) 15100.6 26155.1i 0.534162 0.925196i
\(929\) 23746.5 + 41130.2i 0.838642 + 1.45257i 0.891031 + 0.453943i \(0.149983\pi\)
−0.0523888 + 0.998627i \(0.516684\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 38603.9i 1.35677i
\(933\) 0 0
\(934\) 21615.4 + 12479.7i 0.757258 + 0.437203i
\(935\) −3546.78 2047.73i −0.124056 0.0716235i
\(936\) 0 0
\(937\) 6811.98i 0.237500i 0.992924 + 0.118750i \(0.0378887\pi\)
−0.992924 + 0.118750i \(0.962111\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1096.31 1898.87i −0.0380401 0.0658874i
\(941\) −16271.0 + 28182.2i −0.563677 + 0.976318i 0.433494 + 0.901156i \(0.357280\pi\)
−0.997171 + 0.0751612i \(0.976053\pi\)
\(942\) 0 0
\(943\) −30208.3 + 17440.8i −1.04318 + 0.602280i
\(944\) 7516.54 0.259155
\(945\) 0 0
\(946\) 42873.3 1.47350
\(947\) −10119.9 + 5842.70i −0.347256 + 0.200488i −0.663476 0.748198i \(-0.730920\pi\)
0.316220 + 0.948686i \(0.397586\pi\)
\(948\) 0 0
\(949\) 39511.3 68435.7i 1.35152 2.34090i
\(950\) −13585.3 23530.4i −0.463963 0.803608i
\(951\) 0 0
\(952\) 0 0
\(953\) 46457.5i 1.57912i −0.613671 0.789562i \(-0.710308\pi\)
0.613671 0.789562i \(-0.289692\pi\)
\(954\) 0 0
\(955\) −5061.08 2922.01i −0.171490 0.0990096i
\(956\) 18714.6 + 10804.9i 0.633130 + 0.365538i
\(957\) 0 0
\(958\) 24732.8i 0.834114i
\(959\) 0 0
\(960\) 0 0
\(961\) −11007.6 19065.8i −0.369496 0.639986i
\(962\) 17561.4 30417.2i 0.588568 1.01943i
\(963\) 0 0
\(964\) 12942.3 7472.25i 0.432411 0.249653i
\(965\) 5259.57 0.175452
\(966\) 0 0
\(967\) −27949.1 −0.929455 −0.464728 0.885454i \(-0.653848\pi\)
−0.464728 + 0.885454i \(0.653848\pi\)
\(968\) −6966.06 + 4021.86i −0.231299 + 0.133541i
\(969\) 0 0
\(970\) −3433.47 + 5946.94i −0.113652 + 0.196850i
\(971\) 1609.85 + 2788.33i 0.0532053 + 0.0921544i 0.891401 0.453215i \(-0.149723\pi\)
−0.838196 + 0.545369i \(0.816390\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 53064.9i 1.74570i
\(975\) 0 0
\(976\) −97.3141 56.1843i −0.00319155 0.00184264i
\(977\) 33258.6 + 19201.9i 1.08909 + 0.628785i 0.933333 0.359011i \(-0.116886\pi\)
0.155754 + 0.987796i \(0.450219\pi\)
\(978\) 0 0
\(979\) 1235.38i 0.0403297i
\(980\) 0 0
\(981\) 0 0
\(982\) 7948.19 + 13766.7i 0.258286 + 0.447364i
\(983\) 13709.1 23744.8i 0.444814 0.770440i −0.553226 0.833031i \(-0.686603\pi\)
0.998039 + 0.0625917i \(0.0199366\pi\)
\(984\) 0 0
\(985\) 1782.71 1029.25i 0.0576667 0.0332939i
\(986\) 53189.1 1.71794
\(987\) 0 0
\(988\) 51982.4 1.67387
\(989\) −17949.8 + 10363.3i −0.577120 + 0.333200i
\(990\) 0 0
\(991\) −21857.9 + 37859.1i −0.700646 + 1.21355i 0.267594 + 0.963532i \(0.413771\pi\)
−0.968240 + 0.250023i \(0.919562\pi\)
\(992\) 8806.94 + 15254.1i 0.281876 + 0.488223i
\(993\) 0 0
\(994\) 0 0
\(995\) 220.484i 0.00702495i
\(996\) 0 0
\(997\) 21930.8 + 12661.7i 0.696644 + 0.402208i 0.806096 0.591784i \(-0.201576\pi\)
−0.109452 + 0.993992i \(0.534910\pi\)
\(998\) 38731.5 + 22361.7i 1.22848 + 0.709264i
\(999\) 0 0
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 441.4.p.c.80.2 16
3.2 odd 2 inner 441.4.p.c.80.7 16
7.2 even 3 63.4.p.a.26.7 yes 16
7.3 odd 6 441.4.c.a.440.4 16
7.4 even 3 441.4.c.a.440.3 16
7.5 odd 6 inner 441.4.p.c.215.7 16
7.6 odd 2 63.4.p.a.17.2 16
21.2 odd 6 63.4.p.a.26.2 yes 16
21.5 even 6 inner 441.4.p.c.215.2 16
21.11 odd 6 441.4.c.a.440.14 16
21.17 even 6 441.4.c.a.440.13 16
21.20 even 2 63.4.p.a.17.7 yes 16
28.23 odd 6 1008.4.bt.a.593.5 16
28.27 even 2 1008.4.bt.a.17.4 16
84.23 even 6 1008.4.bt.a.593.4 16
84.83 odd 2 1008.4.bt.a.17.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.2 16 7.6 odd 2
63.4.p.a.17.7 yes 16 21.20 even 2
63.4.p.a.26.2 yes 16 21.2 odd 6
63.4.p.a.26.7 yes 16 7.2 even 3
441.4.c.a.440.3 16 7.4 even 3
441.4.c.a.440.4 16 7.3 odd 6
441.4.c.a.440.13 16 21.17 even 6
441.4.c.a.440.14 16 21.11 odd 6
441.4.p.c.80.2 16 1.1 even 1 trivial
441.4.p.c.80.7 16 3.2 odd 2 inner
441.4.p.c.215.2 16 21.5 even 6 inner
441.4.p.c.215.7 16 7.5 odd 6 inner
1008.4.bt.a.17.4 16 28.27 even 2
1008.4.bt.a.17.5 16 84.83 odd 2
1008.4.bt.a.593.4 16 84.23 even 6
1008.4.bt.a.593.5 16 28.23 odd 6