Properties

Label 441.4.p.d.80.2
Level $441$
Weight $4$
Character 441.80
Analytic conductor $26.020$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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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: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.2
Character \(\chi\) \(=\) 441.80
Dual form 441.4.p.d.215.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.66169 + 2.69143i) q^{2} +(10.4876 - 18.1650i) q^{4} +(9.16288 + 15.8706i) q^{5} +69.8432i q^{8} +O(q^{10})\) \(q+(-4.66169 + 2.69143i) q^{2} +(10.4876 - 18.1650i) q^{4} +(9.16288 + 15.8706i) q^{5} +69.8432i q^{8} +(-85.4290 - 49.3225i) q^{10} +(-20.5857 - 11.8852i) q^{11} -11.7971i q^{13} +(-104.077 - 180.267i) q^{16} +(48.2099 - 83.5020i) q^{17} +(18.1933 - 10.5039i) q^{19} +384.385 q^{20} +127.952 q^{22} +(-132.203 + 76.3277i) q^{23} +(-105.417 + 182.587i) q^{25} +(31.7509 + 54.9943i) q^{26} -77.5438i q^{29} +(-172.974 - 99.8667i) q^{31} +(486.465 + 280.861i) q^{32} +519.014i q^{34} +(-82.1957 - 142.367i) q^{37} +(-56.5410 + 97.9319i) q^{38} +(-1108.45 + 639.965i) q^{40} -435.473 q^{41} -106.148 q^{43} +(-431.787 + 249.293i) q^{44} +(410.861 - 711.632i) q^{46} +(-5.52293 - 9.56599i) q^{47} -1134.89i q^{50} +(-214.294 - 123.722i) q^{52} +(96.0735 + 55.4681i) q^{53} -435.609i q^{55} +(208.703 + 361.485i) q^{58} +(-414.518 + 717.966i) q^{59} +(19.5452 - 11.2845i) q^{61} +1075.14 q^{62} -1358.43 q^{64} +(187.226 - 108.095i) q^{65} +(-199.265 + 345.137i) q^{67} +(-1011.21 - 1751.47i) q^{68} +228.153i q^{71} +(231.194 + 133.480i) q^{73} +(766.341 + 442.447i) q^{74} -440.641i q^{76} +(-460.333 - 797.321i) q^{79} +(1907.30 - 3303.54i) q^{80} +(2030.04 - 1172.04i) q^{82} -642.787 q^{83} +1766.97 q^{85} +(494.828 - 285.689i) q^{86} +(830.097 - 1437.77i) q^{88} +(-222.627 - 385.601i) q^{89} +3201.97i q^{92} +(51.4923 + 29.7291i) q^{94} +(333.406 + 192.492i) q^{95} -1617.57i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 96 q^{4} - 144 q^{16} + 1248 q^{22} - 312 q^{25} + 864 q^{37} + 2496 q^{43} + 3888 q^{46} + 7440 q^{58} - 6720 q^{64} + 2688 q^{67} - 480 q^{79} + 26496 q^{85} + 7248 q^{88}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.66169 + 2.69143i −1.64816 + 0.951563i −0.670351 + 0.742044i \(0.733857\pi\)
−0.977804 + 0.209519i \(0.932810\pi\)
\(3\) 0 0
\(4\) 10.4876 18.1650i 1.31095 2.27062i
\(5\) 9.16288 + 15.8706i 0.819553 + 1.41951i 0.906012 + 0.423252i \(0.139112\pi\)
−0.0864589 + 0.996255i \(0.527555\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 69.8432i 3.08666i
\(9\) 0 0
\(10\) −85.4290 49.3225i −2.70150 1.55971i
\(11\) −20.5857 11.8852i −0.564256 0.325773i 0.190596 0.981669i \(-0.438958\pi\)
−0.754852 + 0.655895i \(0.772291\pi\)
\(12\) 0 0
\(13\) 11.7971i 0.251686i −0.992050 0.125843i \(-0.959836\pi\)
0.992050 0.125843i \(-0.0401636\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −104.077 180.267i −1.62621 2.81668i
\(17\) 48.2099 83.5020i 0.687801 1.19131i −0.284746 0.958603i \(-0.591909\pi\)
0.972548 0.232704i \(-0.0747573\pi\)
\(18\) 0 0
\(19\) 18.1933 10.5039i 0.219675 0.126830i −0.386125 0.922447i \(-0.626187\pi\)
0.605800 + 0.795617i \(0.292853\pi\)
\(20\) 384.385 4.29756
\(21\) 0 0
\(22\) 127.952 1.23998
\(23\) −132.203 + 76.3277i −1.19854 + 0.691975i −0.960229 0.279215i \(-0.909926\pi\)
−0.238307 + 0.971190i \(0.576592\pi\)
\(24\) 0 0
\(25\) −105.417 + 182.587i −0.843334 + 1.46070i
\(26\) 31.7509 + 54.9943i 0.239495 + 0.414818i
\(27\) 0 0
\(28\) 0 0
\(29\) 77.5438i 0.496535i −0.968691 0.248268i \(-0.920139\pi\)
0.968691 0.248268i \(-0.0798612\pi\)
\(30\) 0 0
\(31\) −172.974 99.8667i −1.00216 0.578600i −0.0932764 0.995640i \(-0.529734\pi\)
−0.908888 + 0.417040i \(0.863067\pi\)
\(32\) 486.465 + 280.861i 2.68737 + 1.55155i
\(33\) 0 0
\(34\) 519.014i 2.61795i
\(35\) 0 0
\(36\) 0 0
\(37\) −82.1957 142.367i −0.365213 0.632568i 0.623597 0.781746i \(-0.285671\pi\)
−0.988810 + 0.149178i \(0.952337\pi\)
\(38\) −56.5410 + 97.9319i −0.241373 + 0.418070i
\(39\) 0 0
\(40\) −1108.45 + 639.965i −4.38154 + 2.52968i
\(41\) −435.473 −1.65877 −0.829384 0.558679i \(-0.811308\pi\)
−0.829384 + 0.558679i \(0.811308\pi\)
\(42\) 0 0
\(43\) −106.148 −0.376451 −0.188226 0.982126i \(-0.560274\pi\)
−0.188226 + 0.982126i \(0.560274\pi\)
\(44\) −431.787 + 249.293i −1.47942 + 0.854142i
\(45\) 0 0
\(46\) 410.861 711.632i 1.31692 2.28097i
\(47\) −5.52293 9.56599i −0.0171405 0.0296882i 0.857328 0.514771i \(-0.172123\pi\)
−0.874468 + 0.485082i \(0.838790\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1134.89i 3.20994i
\(51\) 0 0
\(52\) −214.294 123.722i −0.571484 0.329947i
\(53\) 96.0735 + 55.4681i 0.248995 + 0.143757i 0.619304 0.785151i \(-0.287415\pi\)
−0.370309 + 0.928909i \(0.620748\pi\)
\(54\) 0 0
\(55\) 435.609i 1.06795i
\(56\) 0 0
\(57\) 0 0
\(58\) 208.703 + 361.485i 0.472485 + 0.818367i
\(59\) −414.518 + 717.966i −0.914672 + 1.58426i −0.107291 + 0.994228i \(0.534218\pi\)
−0.807381 + 0.590031i \(0.799116\pi\)
\(60\) 0 0
\(61\) 19.5452 11.2845i 0.0410248 0.0236857i −0.479347 0.877625i \(-0.659127\pi\)
0.520372 + 0.853940i \(0.325793\pi\)
\(62\) 1075.14 2.20230
\(63\) 0 0
\(64\) −1358.43 −2.65318
\(65\) 187.226 108.095i 0.357270 0.206270i
\(66\) 0 0
\(67\) −199.265 + 345.137i −0.363345 + 0.629331i −0.988509 0.151162i \(-0.951699\pi\)
0.625164 + 0.780493i \(0.285032\pi\)
\(68\) −1011.21 1751.47i −1.80334 3.12348i
\(69\) 0 0
\(70\) 0 0
\(71\) 228.153i 0.381363i 0.981652 + 0.190682i \(0.0610698\pi\)
−0.981652 + 0.190682i \(0.938930\pi\)
\(72\) 0 0
\(73\) 231.194 + 133.480i 0.370674 + 0.214009i 0.673753 0.738957i \(-0.264681\pi\)
−0.303079 + 0.952966i \(0.598014\pi\)
\(74\) 766.341 + 442.447i 1.20386 + 0.695047i
\(75\) 0 0
\(76\) 440.641i 0.665066i
\(77\) 0 0
\(78\) 0 0
\(79\) −460.333 797.321i −0.655589 1.13551i −0.981746 0.190198i \(-0.939087\pi\)
0.326156 0.945316i \(-0.394246\pi\)
\(80\) 1907.30 3303.54i 2.66553 4.61683i
\(81\) 0 0
\(82\) 2030.04 1172.04i 2.73391 1.57842i
\(83\) −642.787 −0.850061 −0.425031 0.905179i \(-0.639737\pi\)
−0.425031 + 0.905179i \(0.639737\pi\)
\(84\) 0 0
\(85\) 1766.97 2.25476
\(86\) 494.828 285.689i 0.620450 0.358217i
\(87\) 0 0
\(88\) 830.097 1437.77i 1.00555 1.74167i
\(89\) −222.627 385.601i −0.265150 0.459254i 0.702453 0.711731i \(-0.252088\pi\)
−0.967603 + 0.252476i \(0.918755\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3201.97i 3.62856i
\(93\) 0 0
\(94\) 51.4923 + 29.7291i 0.0565003 + 0.0326205i
\(95\) 333.406 + 192.492i 0.360071 + 0.207887i
\(96\) 0 0
\(97\) 1617.57i 1.69319i −0.532236 0.846596i \(-0.678648\pi\)
0.532236 0.846596i \(-0.321352\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2211.13 + 3829.79i 2.21113 + 3.82979i
\(101\) −37.5166 + 64.9806i −0.0369608 + 0.0640180i −0.883914 0.467649i \(-0.845101\pi\)
0.846953 + 0.531667i \(0.178434\pi\)
\(102\) 0 0
\(103\) 834.971 482.071i 0.798759 0.461164i −0.0442780 0.999019i \(-0.514099\pi\)
0.843037 + 0.537856i \(0.180765\pi\)
\(104\) 823.945 0.776870
\(105\) 0 0
\(106\) −597.153 −0.547176
\(107\) 1153.79 666.138i 1.04244 0.601851i 0.121914 0.992541i \(-0.461097\pi\)
0.920522 + 0.390690i \(0.127764\pi\)
\(108\) 0 0
\(109\) 445.080 770.901i 0.391110 0.677422i −0.601487 0.798883i \(-0.705425\pi\)
0.992596 + 0.121461i \(0.0387580\pi\)
\(110\) 1172.41 + 2030.67i 1.01623 + 1.76016i
\(111\) 0 0
\(112\) 0 0
\(113\) 1240.25i 1.03251i −0.856436 0.516253i \(-0.827326\pi\)
0.856436 0.516253i \(-0.172674\pi\)
\(114\) 0 0
\(115\) −2422.73 1398.76i −1.96453 1.13422i
\(116\) −1408.58 813.245i −1.12744 0.650930i
\(117\) 0 0
\(118\) 4462.58i 3.48147i
\(119\) 0 0
\(120\) 0 0
\(121\) −382.986 663.352i −0.287743 0.498386i
\(122\) −60.7426 + 105.209i −0.0450768 + 0.0780754i
\(123\) 0 0
\(124\) −3628.16 + 2094.72i −2.62756 + 1.51703i
\(125\) −1572.97 −1.12552
\(126\) 0 0
\(127\) 1338.38 0.935137 0.467569 0.883957i \(-0.345130\pi\)
0.467569 + 0.883957i \(0.345130\pi\)
\(128\) 2440.84 1409.22i 1.68548 0.973114i
\(129\) 0 0
\(130\) −581.860 + 1007.81i −0.392558 + 0.679930i
\(131\) −1156.91 2003.82i −0.771597 1.33645i −0.936687 0.350167i \(-0.886125\pi\)
0.165090 0.986279i \(-0.447209\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2145.23i 1.38298i
\(135\) 0 0
\(136\) 5832.05 + 3367.13i 3.67716 + 2.12301i
\(137\) −1915.10 1105.68i −1.19429 0.689524i −0.235014 0.971992i \(-0.575514\pi\)
−0.959277 + 0.282468i \(0.908847\pi\)
\(138\) 0 0
\(139\) 2718.66i 1.65895i 0.558545 + 0.829474i \(0.311360\pi\)
−0.558545 + 0.829474i \(0.688640\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −614.058 1063.58i −0.362891 0.628546i
\(143\) −140.210 + 242.851i −0.0819926 + 0.142015i
\(144\) 0 0
\(145\) 1230.66 710.524i 0.704835 0.406937i
\(146\) −1437.01 −0.814572
\(147\) 0 0
\(148\) −3448.13 −1.91510
\(149\) −1589.86 + 917.904i −0.874135 + 0.504682i −0.868720 0.495303i \(-0.835057\pi\)
−0.00541480 + 0.999985i \(0.501724\pi\)
\(150\) 0 0
\(151\) −1446.78 + 2505.90i −0.779718 + 1.35051i 0.152386 + 0.988321i \(0.451304\pi\)
−0.932104 + 0.362190i \(0.882029\pi\)
\(152\) 733.626 + 1270.68i 0.391480 + 0.678063i
\(153\) 0 0
\(154\) 0 0
\(155\) 3660.27i 1.89677i
\(156\) 0 0
\(157\) −1479.73 854.325i −0.752201 0.434284i 0.0742874 0.997237i \(-0.476332\pi\)
−0.826489 + 0.562953i \(0.809665\pi\)
\(158\) 4291.86 + 2477.91i 2.16103 + 1.24767i
\(159\) 0 0
\(160\) 10294.0i 5.08631i
\(161\) 0 0
\(162\) 0 0
\(163\) 1519.04 + 2631.05i 0.729941 + 1.26429i 0.956908 + 0.290392i \(0.0937858\pi\)
−0.226967 + 0.973903i \(0.572881\pi\)
\(164\) −4567.05 + 7910.36i −2.17455 + 3.76644i
\(165\) 0 0
\(166\) 2996.47 1730.02i 1.40103 0.808887i
\(167\) −702.416 −0.325477 −0.162738 0.986669i \(-0.552033\pi\)
−0.162738 + 0.986669i \(0.552033\pi\)
\(168\) 0 0
\(169\) 2057.83 0.936654
\(170\) −8237.05 + 4755.66i −3.71619 + 2.14555i
\(171\) 0 0
\(172\) −1113.23 + 1928.17i −0.493507 + 0.854779i
\(173\) −740.395 1282.40i −0.325383 0.563579i 0.656207 0.754581i \(-0.272160\pi\)
−0.981590 + 0.191002i \(0.938826\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4947.90i 2.11910i
\(177\) 0 0
\(178\) 2075.63 + 1198.37i 0.874019 + 0.504615i
\(179\) 829.096 + 478.679i 0.346199 + 0.199878i 0.663010 0.748611i \(-0.269279\pi\)
−0.316811 + 0.948489i \(0.602612\pi\)
\(180\) 0 0
\(181\) 505.013i 0.207389i 0.994609 + 0.103694i \(0.0330664\pi\)
−0.994609 + 0.103694i \(0.966934\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5330.97 9233.51i −2.13589 3.69948i
\(185\) 1506.30 2608.99i 0.598623 1.03685i
\(186\) 0 0
\(187\) −1984.87 + 1145.96i −0.776192 + 0.448135i
\(188\) −231.688 −0.0898808
\(189\) 0 0
\(190\) −2072.31 −0.791271
\(191\) 2019.91 1166.20i 0.765214 0.441796i −0.0659508 0.997823i \(-0.521008\pi\)
0.831165 + 0.556027i \(0.187675\pi\)
\(192\) 0 0
\(193\) 1962.92 3399.89i 0.732095 1.26803i −0.223891 0.974614i \(-0.571876\pi\)
0.955986 0.293412i \(-0.0947908\pi\)
\(194\) 4353.58 + 7540.62i 1.61118 + 2.79064i
\(195\) 0 0
\(196\) 0 0
\(197\) 1429.05i 0.516831i −0.966034 0.258416i \(-0.916800\pi\)
0.966034 0.258416i \(-0.0832004\pi\)
\(198\) 0 0
\(199\) −4276.35 2468.95i −1.52333 0.879494i −0.999619 0.0275964i \(-0.991215\pi\)
−0.523709 0.851897i \(-0.675452\pi\)
\(200\) −12752.5 7362.65i −4.50868 2.60309i
\(201\) 0 0
\(202\) 403.893i 0.140682i
\(203\) 0 0
\(204\) 0 0
\(205\) −3990.19 6911.21i −1.35945 2.35463i
\(206\) −2594.92 + 4494.53i −0.877653 + 1.52014i
\(207\) 0 0
\(208\) −2126.63 + 1227.81i −0.708918 + 0.409294i
\(209\) −499.362 −0.165271
\(210\) 0 0
\(211\) 1803.90 0.588557 0.294279 0.955720i \(-0.404921\pi\)
0.294279 + 0.955720i \(0.404921\pi\)
\(212\) 2015.15 1163.45i 0.652836 0.376915i
\(213\) 0 0
\(214\) −3585.73 + 6210.66i −1.14540 + 1.98389i
\(215\) −972.620 1684.63i −0.308522 0.534375i
\(216\) 0 0
\(217\) 0 0
\(218\) 4791.60i 1.48866i
\(219\) 0 0
\(220\) −7912.83 4568.48i −2.42492 1.40003i
\(221\) −985.079 568.736i −0.299835 0.173110i
\(222\) 0 0
\(223\) 2540.18i 0.762795i −0.924411 0.381397i \(-0.875443\pi\)
0.924411 0.381397i \(-0.124557\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3338.05 + 5781.67i 0.982495 + 1.70173i
\(227\) −1989.54 + 3445.98i −0.581719 + 1.00757i 0.413557 + 0.910478i \(0.364286\pi\)
−0.995276 + 0.0970885i \(0.969047\pi\)
\(228\) 0 0
\(229\) −1917.78 + 1107.23i −0.553407 + 0.319510i −0.750495 0.660876i \(-0.770185\pi\)
0.197088 + 0.980386i \(0.436852\pi\)
\(230\) 15058.7 4.31713
\(231\) 0 0
\(232\) 5415.91 1.53264
\(233\) 4060.10 2344.10i 1.14157 0.659087i 0.194752 0.980852i \(-0.437610\pi\)
0.946819 + 0.321766i \(0.104276\pi\)
\(234\) 0 0
\(235\) 101.212 175.304i 0.0280950 0.0486620i
\(236\) 8694.57 + 15059.4i 2.39817 + 4.15375i
\(237\) 0 0
\(238\) 0 0
\(239\) 6575.90i 1.77975i 0.456207 + 0.889874i \(0.349208\pi\)
−0.456207 + 0.889874i \(0.650792\pi\)
\(240\) 0 0
\(241\) 3358.45 + 1939.00i 0.897662 + 0.518265i 0.876441 0.481510i \(-0.159911\pi\)
0.0212209 + 0.999775i \(0.493245\pi\)
\(242\) 3570.73 + 2061.56i 0.948492 + 0.547612i
\(243\) 0 0
\(244\) 473.386i 0.124202i
\(245\) 0 0
\(246\) 0 0
\(247\) −123.915 214.628i −0.0319212 0.0552892i
\(248\) 6975.01 12081.1i 1.78594 3.09334i
\(249\) 0 0
\(250\) 7332.68 4233.53i 1.85504 1.07101i
\(251\) −2789.25 −0.701418 −0.350709 0.936484i \(-0.614059\pi\)
−0.350709 + 0.936484i \(0.614059\pi\)
\(252\) 0 0
\(253\) 3628.66 0.901708
\(254\) −6239.13 + 3602.16i −1.54125 + 0.889842i
\(255\) 0 0
\(256\) −2151.92 + 3727.24i −0.525371 + 0.909970i
\(257\) −1502.32 2602.09i −0.364638 0.631572i 0.624080 0.781361i \(-0.285474\pi\)
−0.988718 + 0.149789i \(0.952141\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4534.62i 1.08163i
\(261\) 0 0
\(262\) 10786.3 + 6227.45i 2.54343 + 1.46845i
\(263\) 3452.66 + 1993.39i 0.809506 + 0.467368i 0.846784 0.531937i \(-0.178536\pi\)
−0.0372784 + 0.999305i \(0.511869\pi\)
\(264\) 0 0
\(265\) 2032.99i 0.471266i
\(266\) 0 0
\(267\) 0 0
\(268\) 4179.61 + 7239.29i 0.952650 + 1.65004i
\(269\) 2608.45 4517.97i 0.591227 1.02403i −0.402841 0.915270i \(-0.631977\pi\)
0.994068 0.108765i \(-0.0346895\pi\)
\(270\) 0 0
\(271\) 232.269 134.101i 0.0520640 0.0300591i −0.473742 0.880664i \(-0.657097\pi\)
0.525806 + 0.850604i \(0.323764\pi\)
\(272\) −20070.3 −4.47404
\(273\) 0 0
\(274\) 11903.5 2.62450
\(275\) 4340.16 2505.79i 0.951713 0.549472i
\(276\) 0 0
\(277\) −1859.88 + 3221.41i −0.403427 + 0.698756i −0.994137 0.108128i \(-0.965514\pi\)
0.590710 + 0.806884i \(0.298848\pi\)
\(278\) −7317.08 12673.6i −1.57859 2.73421i
\(279\) 0 0
\(280\) 0 0
\(281\) 4360.92i 0.925802i 0.886410 + 0.462901i \(0.153191\pi\)
−0.886410 + 0.462901i \(0.846809\pi\)
\(282\) 0 0
\(283\) 2540.70 + 1466.87i 0.533671 + 0.308115i 0.742510 0.669835i \(-0.233635\pi\)
−0.208839 + 0.977950i \(0.566968\pi\)
\(284\) 4144.40 + 2392.77i 0.865933 + 0.499946i
\(285\) 0 0
\(286\) 1509.46i 0.312085i
\(287\) 0 0
\(288\) 0 0
\(289\) −2191.89 3796.47i −0.446141 0.772739i
\(290\) −3824.65 + 6624.49i −0.774452 + 1.34139i
\(291\) 0 0
\(292\) 4849.33 2799.76i 0.971868 0.561108i
\(293\) −1081.19 −0.215577 −0.107789 0.994174i \(-0.534377\pi\)
−0.107789 + 0.994174i \(0.534377\pi\)
\(294\) 0 0
\(295\) −15192.7 −2.99849
\(296\) 9943.37 5740.81i 1.95252 1.12729i
\(297\) 0 0
\(298\) 4940.94 8557.97i 0.960474 1.66359i
\(299\) 900.443 + 1559.61i 0.174160 + 0.301655i
\(300\) 0 0
\(301\) 0 0
\(302\) 15575.6i 2.96780i
\(303\) 0 0
\(304\) −3787.02 2186.44i −0.714476 0.412503i
\(305\) 358.182 + 206.796i 0.0672440 + 0.0388233i
\(306\) 0 0
\(307\) 4327.83i 0.804567i 0.915515 + 0.402284i \(0.131783\pi\)
−0.915515 + 0.402284i \(0.868217\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 9851.35 + 17063.0i 1.80490 + 3.12618i
\(311\) −762.084 + 1319.97i −0.138951 + 0.240671i −0.927100 0.374814i \(-0.877706\pi\)
0.788149 + 0.615485i \(0.211040\pi\)
\(312\) 0 0
\(313\) −5520.33 + 3187.16i −0.996893 + 0.575556i −0.907327 0.420425i \(-0.861881\pi\)
−0.0895652 + 0.995981i \(0.528548\pi\)
\(314\) 9197.41 1.65299
\(315\) 0 0
\(316\) −19311.1 −3.43777
\(317\) −809.520 + 467.377i −0.143430 + 0.0828091i −0.569998 0.821646i \(-0.693056\pi\)
0.426568 + 0.904456i \(0.359723\pi\)
\(318\) 0 0
\(319\) −921.620 + 1596.29i −0.161758 + 0.280173i
\(320\) −12447.1 21559.0i −2.17442 3.76620i
\(321\) 0 0
\(322\) 0 0
\(323\) 2025.57i 0.348934i
\(324\) 0 0
\(325\) 2153.99 + 1243.61i 0.367637 + 0.212255i
\(326\) −14162.6 8176.77i −2.40611 1.38917i
\(327\) 0 0
\(328\) 30414.8i 5.12006i
\(329\) 0 0
\(330\) 0 0
\(331\) 2488.04 + 4309.41i 0.413157 + 0.715609i 0.995233 0.0975255i \(-0.0310928\pi\)
−0.582076 + 0.813134i \(0.697759\pi\)
\(332\) −6741.27 + 11676.2i −1.11438 + 1.93017i
\(333\) 0 0
\(334\) 3274.45 1890.50i 0.536436 0.309712i
\(335\) −7303.37 −1.19112
\(336\) 0 0
\(337\) −5127.06 −0.828750 −0.414375 0.910106i \(-0.636000\pi\)
−0.414375 + 0.910106i \(0.636000\pi\)
\(338\) −9592.96 + 5538.50i −1.54375 + 0.891286i
\(339\) 0 0
\(340\) 18531.2 32096.9i 2.95587 5.11971i
\(341\) 2373.86 + 4111.65i 0.376985 + 0.652957i
\(342\) 0 0
\(343\) 0 0
\(344\) 7413.71i 1.16198i
\(345\) 0 0
\(346\) 6902.98 + 3985.44i 1.07256 + 0.619244i
\(347\) −4466.14 2578.52i −0.690936 0.398912i 0.113027 0.993592i \(-0.463945\pi\)
−0.803963 + 0.594680i \(0.797279\pi\)
\(348\) 0 0
\(349\) 2961.44i 0.454218i −0.973869 0.227109i \(-0.927073\pi\)
0.973869 0.227109i \(-0.0729274\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6676.15 11563.4i −1.01091 1.75094i
\(353\) 5649.11 9784.54i 0.851761 1.47529i −0.0278569 0.999612i \(-0.508868\pi\)
0.879618 0.475681i \(-0.157798\pi\)
\(354\) 0 0
\(355\) −3620.92 + 2090.54i −0.541348 + 0.312547i
\(356\) −9339.25 −1.39039
\(357\) 0 0
\(358\) −5153.32 −0.760786
\(359\) 2275.44 1313.72i 0.334521 0.193136i −0.323326 0.946288i \(-0.604801\pi\)
0.657847 + 0.753152i \(0.271468\pi\)
\(360\) 0 0
\(361\) −3208.84 + 5557.87i −0.467829 + 0.810303i
\(362\) −1359.21 2354.21i −0.197343 0.341809i
\(363\) 0 0
\(364\) 0 0
\(365\) 4892.25i 0.701567i
\(366\) 0 0
\(367\) 2320.20 + 1339.57i 0.330009 + 0.190531i 0.655845 0.754895i \(-0.272312\pi\)
−0.325836 + 0.945426i \(0.605646\pi\)
\(368\) 27518.8 + 15888.0i 3.89814 + 2.25059i
\(369\) 0 0
\(370\) 16216.4i 2.27851i
\(371\) 0 0
\(372\) 0 0
\(373\) 2593.86 + 4492.70i 0.360068 + 0.623655i 0.987972 0.154636i \(-0.0494203\pi\)
−0.627904 + 0.778291i \(0.716087\pi\)
\(374\) 6168.56 10684.3i 0.852857 1.47719i
\(375\) 0 0
\(376\) 668.119 385.739i 0.0916373 0.0529068i
\(377\) −914.789 −0.124971
\(378\) 0 0
\(379\) −2469.97 −0.334760 −0.167380 0.985892i \(-0.553531\pi\)
−0.167380 + 0.985892i \(0.553531\pi\)
\(380\) 6993.23 4037.54i 0.944067 0.545057i
\(381\) 0 0
\(382\) −6277.47 + 10872.9i −0.840794 + 1.45630i
\(383\) −5632.62 9755.99i −0.751472 1.30159i −0.947109 0.320911i \(-0.896011\pi\)
0.195638 0.980676i \(-0.437322\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 21132.3i 2.78654i
\(387\) 0 0
\(388\) −29383.2 16964.4i −3.84460 2.21968i
\(389\) −6658.54 3844.31i −0.867870 0.501065i −0.00122995 0.999999i \(-0.500392\pi\)
−0.866640 + 0.498934i \(0.833725\pi\)
\(390\) 0 0
\(391\) 14719.0i 1.90376i
\(392\) 0 0
\(393\) 0 0
\(394\) 3846.19 + 6661.80i 0.491798 + 0.851819i
\(395\) 8435.96 14611.5i 1.07458 1.86123i
\(396\) 0 0
\(397\) −700.905 + 404.668i −0.0886081 + 0.0511579i −0.543649 0.839312i \(-0.682958\pi\)
0.455041 + 0.890470i \(0.349625\pi\)
\(398\) 26580.0 3.34758
\(399\) 0 0
\(400\) 43886.0 5.48575
\(401\) −3813.91 + 2201.96i −0.474956 + 0.274216i −0.718312 0.695721i \(-0.755085\pi\)
0.243356 + 0.969937i \(0.421752\pi\)
\(402\) 0 0
\(403\) −1178.13 + 2040.59i −0.145625 + 0.252231i
\(404\) 786.915 + 1362.98i 0.0969072 + 0.167848i
\(405\) 0 0
\(406\) 0 0
\(407\) 3907.63i 0.475907i
\(408\) 0 0
\(409\) −50.4345 29.1184i −0.00609737 0.00352032i 0.496948 0.867780i \(-0.334454\pi\)
−0.503046 + 0.864260i \(0.667787\pi\)
\(410\) 37202.0 + 21478.6i 4.48116 + 2.58720i
\(411\) 0 0
\(412\) 20223.0i 2.41824i
\(413\) 0 0
\(414\) 0 0
\(415\) −5889.78 10201.4i −0.696670 1.20667i
\(416\) 3313.33 5738.86i 0.390504 0.676372i
\(417\) 0 0
\(418\) 2327.87 1344.00i 0.272392 0.157266i
\(419\) 6309.98 0.735711 0.367855 0.929883i \(-0.380092\pi\)
0.367855 + 0.929883i \(0.380092\pi\)
\(420\) 0 0
\(421\) 939.246 0.108732 0.0543658 0.998521i \(-0.482686\pi\)
0.0543658 + 0.998521i \(0.482686\pi\)
\(422\) −8409.22 + 4855.07i −0.970034 + 0.560050i
\(423\) 0 0
\(424\) −3874.07 + 6710.08i −0.443730 + 0.768562i
\(425\) 10164.3 + 17605.0i 1.16009 + 2.00934i
\(426\) 0 0
\(427\) 0 0
\(428\) 27944.7i 3.15597i
\(429\) 0 0
\(430\) 9068.11 + 5235.47i 1.01698 + 0.587156i
\(431\) −8233.84 4753.81i −0.920209 0.531283i −0.0365070 0.999333i \(-0.511623\pi\)
−0.883702 + 0.468051i \(0.844956\pi\)
\(432\) 0 0
\(433\) 9603.12i 1.06581i 0.846175 + 0.532906i \(0.178900\pi\)
−0.846175 + 0.532906i \(0.821100\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9335.61 16169.8i −1.02545 1.77613i
\(437\) −1603.48 + 2777.30i −0.175526 + 0.304019i
\(438\) 0 0
\(439\) 756.045 436.503i 0.0821960 0.0474559i −0.458339 0.888778i \(-0.651555\pi\)
0.540535 + 0.841322i \(0.318222\pi\)
\(440\) 30424.3 3.29642
\(441\) 0 0
\(442\) 6122.84 0.658900
\(443\) 696.335 402.029i 0.0746815 0.0431174i −0.462194 0.886779i \(-0.652938\pi\)
0.536876 + 0.843661i \(0.319604\pi\)
\(444\) 0 0
\(445\) 4079.81 7066.43i 0.434610 0.752766i
\(446\) 6836.72 + 11841.5i 0.725847 + 1.25720i
\(447\) 0 0
\(448\) 0 0
\(449\) 12649.5i 1.32955i 0.747046 + 0.664773i \(0.231472\pi\)
−0.747046 + 0.664773i \(0.768528\pi\)
\(450\) 0 0
\(451\) 8964.51 + 5175.66i 0.935970 + 0.540383i
\(452\) −22529.2 13007.2i −2.34443 1.35356i
\(453\) 0 0
\(454\) 21418.8i 2.21417i
\(455\) 0 0
\(456\) 0 0
\(457\) 3813.36 + 6604.93i 0.390331 + 0.676073i 0.992493 0.122301i \(-0.0390272\pi\)
−0.602162 + 0.798374i \(0.705694\pi\)
\(458\) 5960.05 10323.1i 0.608067 1.05320i
\(459\) 0 0
\(460\) −50817.0 + 29339.2i −5.15078 + 2.97380i
\(461\) 15305.4 1.54630 0.773148 0.634226i \(-0.218681\pi\)
0.773148 + 0.634226i \(0.218681\pi\)
\(462\) 0 0
\(463\) −9675.77 −0.971212 −0.485606 0.874178i \(-0.661401\pi\)
−0.485606 + 0.874178i \(0.661401\pi\)
\(464\) −13978.6 + 8070.56i −1.39858 + 0.807470i
\(465\) 0 0
\(466\) −12618.0 + 21854.9i −1.25433 + 2.17256i
\(467\) 1743.83 + 3020.40i 0.172794 + 0.299288i 0.939396 0.342835i \(-0.111387\pi\)
−0.766602 + 0.642123i \(0.778054\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1089.62i 0.106937i
\(471\) 0 0
\(472\) −50145.1 28951.3i −4.89007 2.82328i
\(473\) 2185.13 + 1261.58i 0.212415 + 0.122638i
\(474\) 0 0
\(475\) 4429.15i 0.427839i
\(476\) 0 0
\(477\) 0 0
\(478\) −17698.6 30654.8i −1.69354 2.93330i
\(479\) −9087.22 + 15739.5i −0.866818 + 1.50137i −0.00158646 + 0.999999i \(0.500505\pi\)
−0.865231 + 0.501373i \(0.832828\pi\)
\(480\) 0 0
\(481\) −1679.51 + 969.668i −0.159208 + 0.0919190i
\(482\) −20874.7 −1.97265
\(483\) 0 0
\(484\) −16066.4 −1.50886
\(485\) 25671.8 14821.6i 2.40350 1.38766i
\(486\) 0 0
\(487\) 7405.34 12826.4i 0.689052 1.19347i −0.283094 0.959092i \(-0.591361\pi\)
0.972145 0.234380i \(-0.0753060\pi\)
\(488\) 788.142 + 1365.10i 0.0731097 + 0.126630i
\(489\) 0 0
\(490\) 0 0
\(491\) 5187.59i 0.476808i −0.971166 0.238404i \(-0.923376\pi\)
0.971166 0.238404i \(-0.0766242\pi\)
\(492\) 0 0
\(493\) −6475.06 3738.38i −0.591526 0.341517i
\(494\) 1155.31 + 667.018i 0.105222 + 0.0607501i
\(495\) 0 0
\(496\) 41575.5i 3.76370i
\(497\) 0 0
\(498\) 0 0
\(499\) 2292.61 + 3970.91i 0.205674 + 0.356238i 0.950347 0.311192i \(-0.100728\pi\)
−0.744673 + 0.667429i \(0.767395\pi\)
\(500\) −16496.6 + 28572.9i −1.47550 + 2.55564i
\(501\) 0 0
\(502\) 13002.6 7507.06i 1.15605 0.667444i
\(503\) −2600.54 −0.230521 −0.115261 0.993335i \(-0.536770\pi\)
−0.115261 + 0.993335i \(0.536770\pi\)
\(504\) 0 0
\(505\) −1375.04 −0.121165
\(506\) −16915.7 + 9766.29i −1.48616 + 0.858032i
\(507\) 0 0
\(508\) 14036.4 24311.7i 1.22591 2.12334i
\(509\) −5759.80 9976.27i −0.501569 0.868743i −0.999998 0.00181295i \(-0.999423\pi\)
0.498429 0.866930i \(-0.333910\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 619.438i 0.0534678i
\(513\) 0 0
\(514\) 14006.7 + 8086.76i 1.20196 + 0.693953i
\(515\) 15301.5 + 8834.32i 1.30925 + 0.755896i
\(516\) 0 0
\(517\) 262.563i 0.0223356i
\(518\) 0 0
\(519\) 0 0
\(520\) 7549.71 + 13076.5i 0.636686 + 1.10277i
\(521\) −3361.14 + 5821.67i −0.282638 + 0.489543i −0.972034 0.234842i \(-0.924543\pi\)
0.689396 + 0.724385i \(0.257876\pi\)
\(522\) 0 0
\(523\) −19252.8 + 11115.6i −1.60969 + 0.929352i −0.620246 + 0.784407i \(0.712967\pi\)
−0.989440 + 0.144945i \(0.953700\pi\)
\(524\) −48532.5 −4.04609
\(525\) 0 0
\(526\) −21460.3 −1.77892
\(527\) −16678.1 + 9629.13i −1.37858 + 0.795923i
\(528\) 0 0
\(529\) 5568.33 9644.63i 0.457659 0.792688i
\(530\) −5471.64 9477.16i −0.448440 0.776720i
\(531\) 0 0
\(532\) 0 0
\(533\) 5137.31i 0.417489i
\(534\) 0 0
\(535\) 21144.0 + 12207.5i 1.70866 + 0.986497i
\(536\) −24105.5 13917.3i −1.94253 1.12152i
\(537\) 0 0
\(538\) 28081.8i 2.25036i
\(539\) 0 0
\(540\) 0 0
\(541\) 2578.35 + 4465.84i 0.204902 + 0.354901i 0.950102 0.311941i \(-0.100979\pi\)
−0.745199 + 0.666842i \(0.767646\pi\)
\(542\) −721.844 + 1250.27i −0.0572064 + 0.0990843i
\(543\) 0 0
\(544\) 46904.9 27080.5i 3.69675 2.13432i
\(545\) 16312.9 1.28214
\(546\) 0 0
\(547\) 22545.5 1.76230 0.881149 0.472839i \(-0.156771\pi\)
0.881149 + 0.472839i \(0.156771\pi\)
\(548\) −40169.4 + 23191.8i −3.13130 + 1.80786i
\(549\) 0 0
\(550\) −13488.3 + 23362.4i −1.04571 + 1.81123i
\(551\) −814.512 1410.78i −0.0629753 0.109076i
\(552\) 0 0
\(553\) 0 0
\(554\) 20022.9i 1.53555i
\(555\) 0 0
\(556\) 49384.5 + 28512.1i 3.76685 + 2.17479i
\(557\) 8246.05 + 4760.86i 0.627282 + 0.362162i 0.779699 0.626155i \(-0.215372\pi\)
−0.152417 + 0.988316i \(0.548706\pi\)
\(558\) 0 0
\(559\) 1252.23i 0.0947475i
\(560\) 0 0
\(561\) 0 0
\(562\) −11737.1 20329.2i −0.880959 1.52587i
\(563\) −2761.09 + 4782.34i −0.206689 + 0.357996i −0.950670 0.310205i \(-0.899602\pi\)
0.743981 + 0.668201i \(0.232935\pi\)
\(564\) 0 0
\(565\) 19683.5 11364.3i 1.46565 0.846194i
\(566\) −15791.9 −1.17276
\(567\) 0 0
\(568\) −15934.9 −1.17714
\(569\) 2537.61 1465.09i 0.186963 0.107943i −0.403597 0.914937i \(-0.632240\pi\)
0.590560 + 0.806994i \(0.298907\pi\)
\(570\) 0 0
\(571\) −4359.17 + 7550.30i −0.319484 + 0.553363i −0.980381 0.197114i \(-0.936843\pi\)
0.660896 + 0.750477i \(0.270176\pi\)
\(572\) 2940.92 + 5093.82i 0.214976 + 0.372349i
\(573\) 0 0
\(574\) 0 0
\(575\) 32184.9i 2.33427i
\(576\) 0 0
\(577\) −14243.4 8223.43i −1.02766 0.593320i −0.111347 0.993782i \(-0.535517\pi\)
−0.916314 + 0.400461i \(0.868850\pi\)
\(578\) 20435.8 + 11798.6i 1.47062 + 0.849063i
\(579\) 0 0
\(580\) 29806.7i 2.13389i
\(581\) 0 0
\(582\) 0 0
\(583\) −1318.49 2283.70i −0.0936645 0.162232i
\(584\) −9322.67 + 16147.3i −0.660574 + 1.14415i
\(585\) 0 0
\(586\) 5040.19 2909.96i 0.355305 0.205135i
\(587\) −2346.72 −0.165008 −0.0825040 0.996591i \(-0.526292\pi\)
−0.0825040 + 0.996591i \(0.526292\pi\)
\(588\) 0 0
\(589\) −4195.96 −0.293534
\(590\) 70823.7 40890.1i 4.94198 2.85325i
\(591\) 0 0
\(592\) −17109.4 + 29634.4i −1.18783 + 2.05738i
\(593\) −7971.40 13806.9i −0.552017 0.956122i −0.998129 0.0611443i \(-0.980525\pi\)
0.446112 0.894977i \(-0.352808\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 38506.3i 2.64644i
\(597\) 0 0
\(598\) −8395.17 4846.95i −0.574087 0.331449i
\(599\) −17836.6 10298.0i −1.21667 0.702445i −0.252466 0.967606i \(-0.581242\pi\)
−0.964204 + 0.265161i \(0.914575\pi\)
\(600\) 0 0
\(601\) 662.921i 0.0449935i 0.999747 + 0.0224968i \(0.00716154\pi\)
−0.999747 + 0.0224968i \(0.992838\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 30346.4 + 52561.5i 2.04434 + 3.54089i
\(605\) 7018.52 12156.4i 0.471642 0.816908i
\(606\) 0 0
\(607\) 11688.2 6748.21i 0.781567 0.451238i −0.0554186 0.998463i \(-0.517649\pi\)
0.836985 + 0.547226i \(0.184316\pi\)
\(608\) 11800.5 0.787130
\(609\) 0 0
\(610\) −2226.31 −0.147771
\(611\) −112.851 + 65.1543i −0.00747209 + 0.00431401i
\(612\) 0 0
\(613\) −10037.0 + 17384.6i −0.661322 + 1.14544i 0.318946 + 0.947773i \(0.396671\pi\)
−0.980268 + 0.197671i \(0.936662\pi\)
\(614\) −11648.0 20175.0i −0.765597 1.32605i
\(615\) 0 0
\(616\) 0 0
\(617\) 21664.4i 1.41358i 0.707425 + 0.706788i \(0.249857\pi\)
−0.707425 + 0.706788i \(0.750143\pi\)
\(618\) 0 0
\(619\) 1881.89 + 1086.51i 0.122196 + 0.0705501i 0.559852 0.828592i \(-0.310858\pi\)
−0.437656 + 0.899143i \(0.644191\pi\)
\(620\) −66488.7 38387.3i −4.30686 2.48657i
\(621\) 0 0
\(622\) 8204.38i 0.528884i
\(623\) 0 0
\(624\) 0 0
\(625\) −1235.81 2140.48i −0.0790916 0.136991i
\(626\) 17156.0 29715.1i 1.09536 1.89721i
\(627\) 0 0
\(628\) −31037.6 + 17919.6i −1.97219 + 1.13864i
\(629\) −15850.6 −1.00478
\(630\) 0 0
\(631\) −7823.69 −0.493591 −0.246796 0.969068i \(-0.579378\pi\)
−0.246796 + 0.969068i \(0.579378\pi\)
\(632\) 55687.5 32151.2i 3.50495 2.02358i
\(633\) 0 0
\(634\) 2515.82 4357.53i 0.157596 0.272965i
\(635\) 12263.5 + 21240.9i 0.766395 + 1.32743i
\(636\) 0 0
\(637\) 0 0
\(638\) 9921.89i 0.615692i
\(639\) 0 0
\(640\) 44730.3 + 25825.0i 2.76269 + 1.59504i
\(641\) −268.735 155.154i −0.0165591 0.00956040i 0.491698 0.870766i \(-0.336377\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(642\) 0 0
\(643\) 28064.5i 1.72124i −0.509250 0.860618i \(-0.670077\pi\)
0.509250 0.860618i \(-0.329923\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5451.67 + 9442.57i 0.332033 + 0.575098i
\(647\) 4219.67 7308.68i 0.256402 0.444101i −0.708873 0.705336i \(-0.750796\pi\)
0.965275 + 0.261234i \(0.0841295\pi\)
\(648\) 0 0
\(649\) 17066.3 9853.22i 1.03222 0.595952i
\(650\) −13388.3 −0.807898
\(651\) 0 0
\(652\) 63724.1 3.82765
\(653\) −22966.4 + 13259.6i −1.37633 + 0.794624i −0.991716 0.128453i \(-0.958999\pi\)
−0.384614 + 0.923077i \(0.625665\pi\)
\(654\) 0 0
\(655\) 21201.2 36721.5i 1.26473 2.19058i
\(656\) 45322.9 + 78501.6i 2.69750 + 4.67221i
\(657\) 0 0
\(658\) 0 0
\(659\) 29625.1i 1.75118i −0.483054 0.875591i \(-0.660473\pi\)
0.483054 0.875591i \(-0.339527\pi\)
\(660\) 0 0
\(661\) −18868.3 10893.6i −1.11027 0.641018i −0.171374 0.985206i \(-0.554821\pi\)
−0.938901 + 0.344188i \(0.888154\pi\)
\(662\) −23196.9 13392.7i −1.36189 0.786290i
\(663\) 0 0
\(664\) 44894.3i 2.62385i
\(665\) 0 0
\(666\) 0 0
\(667\) 5918.74 + 10251.6i 0.343590 + 0.595115i
\(668\) −7366.63 + 12759.4i −0.426682 + 0.739035i
\(669\) 0 0
\(670\) 34046.0 19656.5i 1.96315 1.13343i
\(671\) −536.470 −0.0308647
\(672\) 0 0
\(673\) 14551.5 0.833462 0.416731 0.909030i \(-0.363176\pi\)
0.416731 + 0.909030i \(0.363176\pi\)
\(674\) 23900.8 13799.1i 1.36591 0.788608i
\(675\) 0 0
\(676\) 21581.6 37380.4i 1.22790 2.12679i
\(677\) 6158.05 + 10666.0i 0.349591 + 0.605509i 0.986177 0.165697i \(-0.0529873\pi\)
−0.636586 + 0.771206i \(0.719654\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 123411.i 6.95968i
\(681\) 0 0
\(682\) −22132.4 12778.2i −1.24266 0.717450i
\(683\) 9488.46 + 5478.16i 0.531575 + 0.306905i 0.741658 0.670779i \(-0.234040\pi\)
−0.210083 + 0.977684i \(0.567373\pi\)
\(684\) 0 0
\(685\) 40524.9i 2.26041i
\(686\) 0 0
\(687\) 0 0
\(688\) 11047.6 + 19135.0i 0.612188 + 1.06034i
\(689\) 654.361 1133.39i 0.0361816 0.0626684i
\(690\) 0 0
\(691\) −20240.0 + 11685.6i −1.11428 + 0.643328i −0.939934 0.341357i \(-0.889113\pi\)
−0.174343 + 0.984685i \(0.555780\pi\)
\(692\) −31059.7 −1.70623
\(693\) 0 0
\(694\) 27759.6 1.51836
\(695\) −43146.7 + 24910.8i −2.35489 + 1.35960i
\(696\) 0 0
\(697\) −20994.1 + 36362.9i −1.14090 + 1.97610i
\(698\) 7970.49 + 13805.3i 0.432217 + 0.748622i
\(699\) 0 0
\(700\) 0 0
\(701\) 13644.5i 0.735160i 0.929992 + 0.367580i \(0.119814\pi\)
−0.929992 + 0.367580i \(0.880186\pi\)
\(702\) 0 0
\(703\) −2990.82 1726.75i −0.160457 0.0926396i
\(704\) 27964.1 + 16145.1i 1.49707 + 0.864335i
\(705\) 0 0
\(706\) 60816.6i 3.24202i
\(707\) 0 0
\(708\) 0 0
\(709\) 11964.6 + 20723.3i 0.633767 + 1.09772i 0.986775 + 0.162096i \(0.0518256\pi\)
−0.353008 + 0.935620i \(0.614841\pi\)
\(710\) 11253.1 19490.9i 0.594817 1.03025i
\(711\) 0 0
\(712\) 26931.6 15549.0i 1.41756 0.818430i
\(713\) 30490.4 1.60151
\(714\) 0 0
\(715\) −5138.91 −0.268789
\(716\) 17390.4 10040.3i 0.907695 0.524058i
\(717\) 0 0
\(718\) −7071.59 + 12248.4i −0.367562 + 0.636636i
\(719\) −14781.6 25602.4i −0.766703 1.32797i −0.939341 0.342984i \(-0.888562\pi\)
0.172638 0.984985i \(-0.444771\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 34545.4i 1.78067i
\(723\) 0 0
\(724\) 9173.56 + 5296.36i 0.470902 + 0.271875i
\(725\) 14158.5 + 8174.42i 0.725288 + 0.418745i
\(726\) 0 0
\(727\) 3335.51i 0.170161i 0.996374 + 0.0850805i \(0.0271148\pi\)
−0.996374 + 0.0850805i \(0.972885\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −13167.1 22806.1i −0.667585 1.15629i
\(731\) −5117.38 + 8863.56i −0.258924 + 0.448469i
\(732\) 0 0
\(733\) 10582.6 6109.89i 0.533258 0.307877i −0.209084 0.977898i \(-0.567048\pi\)
0.742342 + 0.670021i \(0.233715\pi\)
\(734\) −14421.4 −0.725209
\(735\) 0 0
\(736\) −85749.8 −4.29454
\(737\) 8204.01 4736.59i 0.410039 0.236736i
\(738\) 0 0
\(739\) −8879.68 + 15380.1i −0.442009 + 0.765581i −0.997838 0.0657148i \(-0.979067\pi\)
0.555830 + 0.831296i \(0.312401\pi\)
\(740\) −31594.8 54723.8i −1.56952 2.71850i
\(741\) 0 0
\(742\) 0 0
\(743\) 5371.15i 0.265207i −0.991169 0.132603i \(-0.957666\pi\)
0.991169 0.132603i \(-0.0423336\pi\)
\(744\) 0 0
\(745\) −29135.3 16821.3i −1.43280 0.827227i
\(746\) −24183.6 13962.4i −1.18689 0.685254i
\(747\) 0 0
\(748\) 48073.5i 2.34992i
\(749\) 0 0
\(750\) 0 0
\(751\) −11636.4 20154.8i −0.565403 0.979306i −0.997012 0.0772457i \(-0.975387\pi\)
0.431609 0.902061i \(-0.357946\pi\)
\(752\) −1149.62 + 1991.21i −0.0557480 + 0.0965583i
\(753\) 0 0
\(754\) 4264.46 2462.09i 0.205972 0.118918i
\(755\) −53026.8 −2.55608
\(756\) 0 0
\(757\) 16172.1 0.776465 0.388233 0.921561i \(-0.373086\pi\)
0.388233 + 0.921561i \(0.373086\pi\)
\(758\) 11514.2 6647.76i 0.551737 0.318545i
\(759\) 0 0
\(760\) −13444.3 + 23286.1i −0.641677 + 1.11142i
\(761\) 9460.31 + 16385.7i 0.450639 + 0.780529i 0.998426 0.0560887i \(-0.0178630\pi\)
−0.547787 + 0.836618i \(0.684530\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 48922.3i 2.31668i
\(765\) 0 0
\(766\) 52515.1 + 30319.6i 2.47709 + 1.43015i
\(767\) 8469.90 + 4890.10i 0.398736 + 0.230210i
\(768\) 0 0
\(769\) 33747.4i 1.58253i −0.611475 0.791264i \(-0.709423\pi\)
0.611475 0.791264i \(-0.290577\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −41172.6 71313.0i −1.91947 3.32463i
\(773\) −10806.4 + 18717.2i −0.502817 + 0.870905i 0.497178 + 0.867649i \(0.334370\pi\)
−0.999995 + 0.00325584i \(0.998964\pi\)
\(774\) 0 0
\(775\) 36468.8 21055.3i 1.69032 0.975906i
\(776\) 112976. 5.22631
\(777\) 0 0
\(778\) 41386.7 1.90718
\(779\) −7922.69 + 4574.17i −0.364390 + 0.210381i
\(780\) 0 0
\(781\) 2711.63 4696.69i 0.124238 0.215187i
\(782\) −39615.1 68615.4i −1.81155 3.13770i
\(783\) 0 0
\(784\) 0 0
\(785\) 31312.3i 1.42367i
\(786\) 0 0
\(787\) 21260.3 + 12274.6i 0.962957 + 0.555963i 0.897082 0.441865i \(-0.145683\pi\)
0.0658748 + 0.997828i \(0.479016\pi\)
\(788\) −25958.7 14987.3i −1.17353 0.677538i
\(789\) 0 0
\(790\) 90819.1i 4.09013i
\(791\) 0 0
\(792\) 0 0
\(793\) −133.123 230.577i −0.00596135 0.0103254i
\(794\) 2178.27 3772.87i 0.0973600 0.168632i
\(795\) 0 0
\(796\) −89696.9 + 51786.5i −3.99400 + 2.30594i
\(797\) 12323.1 0.547687 0.273844 0.961774i \(-0.411705\pi\)
0.273844 + 0.961774i \(0.411705\pi\)
\(798\) 0 0
\(799\) −1065.04 −0.0471569
\(800\) −102563. + 59214.9i −4.53270 + 2.61695i
\(801\) 0 0
\(802\) 11852.8 20529.7i 0.521868 0.903901i
\(803\) −3172.86 5495.56i −0.139437 0.241512i
\(804\) 0 0
\(805\) 0 0
\(806\) 12683.5i 0.554287i
\(807\) 0 0
\(808\) −4538.46 2620.28i −0.197602 0.114086i
\(809\) 3177.11 + 1834.31i 0.138073 + 0.0797167i 0.567445 0.823411i \(-0.307932\pi\)
−0.429372 + 0.903128i \(0.641265\pi\)
\(810\) 0 0
\(811\) 24967.7i 1.08105i −0.841327 0.540527i \(-0.818225\pi\)
0.841327 0.540527i \(-0.181775\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −10517.1 18216.2i −0.452856 0.784369i
\(815\) −27837.6 + 48216.1i −1.19645 + 2.07231i
\(816\) 0 0
\(817\) −1931.18 + 1114.97i −0.0826970 + 0.0477451i
\(818\) 313.480 0.0133992
\(819\) 0 0
\(820\) −167389. −7.12865
\(821\) −5425.51 + 3132.42i −0.230635 + 0.133157i −0.610865 0.791735i \(-0.709178\pi\)
0.380230 + 0.924892i \(0.375845\pi\)
\(822\) 0 0
\(823\) −1666.86 + 2887.08i −0.0705990 + 0.122281i −0.899164 0.437612i \(-0.855824\pi\)
0.828565 + 0.559893i \(0.189158\pi\)
\(824\) 33669.4 + 58317.1i 1.42346 + 2.46550i
\(825\) 0 0
\(826\) 0 0
\(827\) 27785.4i 1.16831i −0.811643 0.584154i \(-0.801426\pi\)
0.811643 0.584154i \(-0.198574\pi\)
\(828\) 0 0
\(829\) −7667.54 4426.86i −0.321236 0.185466i 0.330707 0.943733i \(-0.392713\pi\)
−0.651943 + 0.758268i \(0.726046\pi\)
\(830\) 54912.7 + 31703.8i 2.29644 + 1.32585i
\(831\) 0 0
\(832\) 16025.4i 0.667767i
\(833\) 0 0
\(834\) 0 0
\(835\) −6436.16 11147.8i −0.266745 0.462017i
\(836\) −5237.09 + 9070.91i −0.216661 + 0.375268i
\(837\) 0 0
\(838\) −29415.2 + 16982.9i −1.21257 + 0.700075i
\(839\) 92.8332 0.00381997 0.00190999 0.999998i \(-0.499392\pi\)
0.00190999 + 0.999998i \(0.499392\pi\)
\(840\) 0 0
\(841\) 18376.0 0.753453
\(842\) −4378.47 + 2527.91i −0.179207 + 0.103465i
\(843\) 0 0
\(844\) 18918.5 32767.8i 0.771566 1.33639i
\(845\) 18855.6 + 32658.9i 0.767638 + 1.32959i
\(846\) 0 0
\(847\) 0 0
\(848\) 23091.9i 0.935116i
\(849\) 0 0
\(850\) −94765.3 54712.8i −3.82403 2.20780i
\(851\) 21733.1 + 12547.6i 0.875442 + 0.505437i
\(852\) 0 0
\(853\) 8509.71i 0.341579i 0.985308 + 0.170790i \(0.0546318\pi\)
−0.985308 + 0.170790i \(0.945368\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 46525.2 + 80584.1i 1.85771 + 3.21765i
\(857\) −19517.2 + 33804.9i −0.777942 + 1.34744i 0.155184 + 0.987886i \(0.450403\pi\)
−0.933126 + 0.359550i \(0.882930\pi\)
\(858\) 0 0
\(859\) −30440.2 + 17574.7i −1.20909 + 0.698067i −0.962560 0.271069i \(-0.912623\pi\)
−0.246527 + 0.969136i \(0.579290\pi\)
\(860\) −40801.7 −1.61782
\(861\) 0 0
\(862\) 51178.1 2.02220
\(863\) 13889.2 8018.96i 0.547851 0.316302i −0.200404 0.979713i \(-0.564225\pi\)
0.748255 + 0.663411i \(0.230892\pi\)
\(864\) 0 0
\(865\) 13568.3 23501.0i 0.533336 0.923766i
\(866\) −25846.1 44766.7i −1.01419 1.75662i
\(867\) 0 0
\(868\) 0 0
\(869\) 21884.5i 0.854295i
\(870\) 0 0
\(871\) 4071.60 + 2350.74i 0.158394 + 0.0914488i
\(872\) 53842.2 + 31085.8i 2.09097 + 1.20722i
\(873\) 0 0
\(874\) 17262.6i 0.668095i
\(875\) 0 0
\(876\) 0 0
\(877\) −11223.0 19438.8i −0.432126 0.748464i 0.564930 0.825139i \(-0.308903\pi\)
−0.997056 + 0.0766749i \(0.975570\pi\)
\(878\) −2349.63 + 4069.68i −0.0903146 + 0.156429i
\(879\) 0 0
\(880\) −78526.1 + 45337.1i −3.00808 + 1.73672i
\(881\) 4544.20 0.173778 0.0868888 0.996218i \(-0.472308\pi\)
0.0868888 + 0.996218i \(0.472308\pi\)
\(882\) 0 0
\(883\) 18749.1 0.714561 0.357280 0.933997i \(-0.383704\pi\)
0.357280 + 0.933997i \(0.383704\pi\)
\(884\) −20662.2 + 11929.3i −0.786135 + 0.453875i
\(885\) 0 0
\(886\) −2164.07 + 3748.27i −0.0820578 + 0.142128i
\(887\) −12553.5 21743.4i −0.475205 0.823079i 0.524392 0.851477i \(-0.324293\pi\)
−0.999597 + 0.0283981i \(0.990959\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 43922.0i 1.65423i
\(891\) 0 0
\(892\) −46142.4 26640.3i −1.73202 0.999982i
\(893\) −200.960 116.025i −0.00753067 0.00434783i
\(894\) 0 0
\(895\) 17544.3i 0.655242i
\(896\) 0 0
\(897\) 0 0
\(898\) −34045.1 58967.9i −1.26515 2.19130i
\(899\) −7744.04 + 13413.1i −0.287295 + 0.497610i
\(900\) 0 0
\(901\) 9263.39 5348.22i 0.342518 0.197753i
\(902\) −55719.7 −2.05683
\(903\) 0 0
\(904\) 86623.3 3.18700
\(905\) −8014.85 + 4627.38i −0.294390 + 0.169966i
\(906\) 0 0
\(907\) 5098.56 8830.96i 0.186654 0.323293i −0.757479 0.652860i \(-0.773569\pi\)
0.944132 + 0.329566i \(0.106902\pi\)
\(908\) 41730.8 + 72279.8i 1.52520 + 2.64173i
\(909\) 0 0
\(910\) 0 0
\(911\) 27309.8i 0.993208i 0.867977 + 0.496604i \(0.165420\pi\)
−0.867977 + 0.496604i \(0.834580\pi\)
\(912\) 0 0
\(913\) 13232.2 + 7639.62i 0.479652 + 0.276927i
\(914\) −35553.4 20526.7i −1.28665 0.742849i
\(915\) 0 0
\(916\) 46448.5i 1.67544i
\(917\) 0 0
\(918\) 0 0
\(919\) 117.243 + 203.071i 0.00420837 + 0.00728911i 0.868122 0.496351i \(-0.165327\pi\)
−0.863914 + 0.503640i \(0.831994\pi\)
\(920\) 97694.1 169211.i 3.50096 6.06383i
\(921\) 0 0
\(922\) −71348.9 + 41193.3i −2.54854 + 1.47140i
\(923\) 2691.54 0.0959838
\(924\) 0 0
\(925\) 34659.2 1.23199
\(926\) 45105.4 26041.6i 1.60071 0.924169i
\(927\) 0 0
\(928\) 21779.0 37722.3i 0.770400 1.33437i
\(929\) 12740.7 + 22067.6i 0.449956 + 0.779347i 0.998383 0.0568516i \(-0.0181062\pi\)
−0.548426 + 0.836199i \(0.684773\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 98335.6i 3.45611i
\(933\) 0 0
\(934\) −16258.4 9386.76i −0.569582 0.328848i
\(935\) −36374.2 21000.7i −1.27226 0.734541i
\(936\) 0 0
\(937\) 36561.0i 1.27470i −0.770574 0.637350i \(-0.780031\pi\)
0.770574 0.637350i \(-0.219969\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2122.93 3677.02i −0.0736621 0.127587i
\(941\) 17521.1 30347.4i 0.606984 1.05133i −0.384751 0.923020i \(-0.625713\pi\)
0.991735 0.128306i \(-0.0409540\pi\)
\(942\) 0 0
\(943\) 57571.0 33238.7i 1.98809 1.14783i
\(944\) 172568. 5.94979
\(945\) 0 0
\(946\) −13581.8 −0.466790
\(947\) −30868.9 + 17822.1i −1.05924 + 0.611554i −0.925223 0.379424i \(-0.876122\pi\)
−0.134020 + 0.990979i \(0.542789\pi\)
\(948\) 0 0
\(949\) 1574.67 2727.41i 0.0538631 0.0932936i
\(950\) −11920.7 20647.3i −0.407116 0.705145i
\(951\) 0 0
\(952\) 0 0
\(953\) 35376.7i 1.20248i 0.799068 + 0.601240i \(0.205327\pi\)
−0.799068 + 0.601240i \(0.794673\pi\)
\(954\) 0 0
\(955\) 37016.5 + 21371.5i 1.25427 + 0.724151i
\(956\) 119451. + 68965.2i 4.04114 + 2.33315i
\(957\) 0 0
\(958\) 97830.3i 3.29933i
\(959\) 0 0
\(960\) 0 0
\(961\) 5051.23 + 8748.98i 0.169556 + 0.293679i
\(962\) 5219.58 9040.58i 0.174934 0.302994i
\(963\) 0 0
\(964\) 70443.8 40670.8i 2.35357 1.35883i
\(965\) 71944.2 2.39996
\(966\) 0 0
\(967\) −27697.2 −0.921078 −0.460539 0.887640i \(-0.652344\pi\)
−0.460539 + 0.887640i \(0.652344\pi\)
\(968\) 46330.6 26749.0i 1.53835 0.888166i
\(969\) 0 0
\(970\) −79782.7 + 138188.i −2.64089 + 4.57416i
\(971\) −15880.0 27505.0i −0.524834 0.909040i −0.999582 0.0289176i \(-0.990794\pi\)
0.474747 0.880122i \(-0.342539\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 79723.8i 2.62270i
\(975\) 0 0
\(976\) −4068.44 2348.91i −0.133430 0.0770357i
\(977\) −989.925 571.533i −0.0324161 0.0187154i 0.483704 0.875231i \(-0.339291\pi\)
−0.516120 + 0.856516i \(0.672624\pi\)
\(978\) 0 0
\(979\) 10583.8i 0.345516i
\(980\) 0 0
\(981\) 0 0
\(982\) 13962.0 + 24182.9i 0.453713 + 0.785853i
\(983\) −6695.23 + 11596.5i −0.217238 + 0.376267i −0.953962 0.299926i \(-0.903038\pi\)
0.736725 + 0.676193i \(0.236371\pi\)
\(984\) 0 0
\(985\) 22679.9 13094.2i 0.733646 0.423571i
\(986\) 40246.3 1.29990
\(987\) 0 0
\(988\) −5198.28 −0.167388
\(989\) 14033.1 8102.02i 0.451190 0.260495i
\(990\) 0 0
\(991\) −8007.84 + 13870.0i −0.256688 + 0.444596i −0.965353 0.260949i \(-0.915964\pi\)
0.708665 + 0.705545i \(0.249298\pi\)
\(992\) −56097.3 97163.4i −1.79545 3.10982i
\(993\) 0 0
\(994\) 0 0
\(995\) 90490.8i 2.88317i
\(996\) 0 0
\(997\) −25588.5 14773.6i −0.812836 0.469291i 0.0351038 0.999384i \(-0.488824\pi\)
−0.847940 + 0.530093i \(0.822157\pi\)
\(998\) −21374.9 12340.8i −0.677965 0.391423i
\(999\) 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.d.80.2 48
3.2 odd 2 inner 441.4.p.d.80.23 48
7.2 even 3 inner 441.4.p.d.215.24 48
7.3 odd 6 441.4.c.b.440.1 24
7.4 even 3 441.4.c.b.440.23 yes 24
7.5 odd 6 inner 441.4.p.d.215.23 48
7.6 odd 2 inner 441.4.p.d.80.1 48
21.2 odd 6 inner 441.4.p.d.215.1 48
21.5 even 6 inner 441.4.p.d.215.2 48
21.11 odd 6 441.4.c.b.440.2 yes 24
21.17 even 6 441.4.c.b.440.24 yes 24
21.20 even 2 inner 441.4.p.d.80.24 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.c.b.440.1 24 7.3 odd 6
441.4.c.b.440.2 yes 24 21.11 odd 6
441.4.c.b.440.23 yes 24 7.4 even 3
441.4.c.b.440.24 yes 24 21.17 even 6
441.4.p.d.80.1 48 7.6 odd 2 inner
441.4.p.d.80.2 48 1.1 even 1 trivial
441.4.p.d.80.23 48 3.2 odd 2 inner
441.4.p.d.80.24 48 21.20 even 2 inner
441.4.p.d.215.1 48 21.2 odd 6 inner
441.4.p.d.215.2 48 21.5 even 6 inner
441.4.p.d.215.23 48 7.5 odd 6 inner
441.4.p.d.215.24 48 7.2 even 3 inner