Properties

Label 980.4.i.n.361.1
Level $980$
Weight $4$
Character 980.361
Analytic conductor $57.822$
Analytic rank $0$
Dimension $2$
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] = [980,4,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), 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: \(57.8218718056\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 980.361
Dual form 980.4.i.n.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{3} +(2.50000 + 4.33013i) q^{5} +(5.50000 + 9.52628i) q^{9} +O(q^{10})\) \(q+(2.00000 - 3.46410i) q^{3} +(2.50000 + 4.33013i) q^{5} +(5.50000 + 9.52628i) q^{9} +(30.0000 - 51.9615i) q^{11} -86.0000 q^{13} +20.0000 q^{15} +(9.00000 - 15.5885i) q^{17} +(22.0000 + 38.1051i) q^{19} +(-24.0000 - 41.5692i) q^{23} +(-12.5000 + 21.6506i) q^{25} +152.000 q^{27} -186.000 q^{29} +(88.0000 - 152.420i) q^{31} +(-120.000 - 207.846i) q^{33} +(-127.000 - 219.970i) q^{37} +(-172.000 + 297.913i) q^{39} -186.000 q^{41} -100.000 q^{43} +(-27.5000 + 47.6314i) q^{45} +(84.0000 + 145.492i) q^{47} +(-36.0000 - 62.3538i) q^{51} +(249.000 - 431.281i) q^{53} +300.000 q^{55} +176.000 q^{57} +(-126.000 + 218.238i) q^{59} +(-29.0000 - 50.2295i) q^{61} +(-215.000 - 372.391i) q^{65} +(518.000 - 897.202i) q^{67} -192.000 q^{69} +168.000 q^{71} +(253.000 - 438.209i) q^{73} +(50.0000 + 86.6025i) q^{75} +(-136.000 - 235.559i) q^{79} +(155.500 - 269.334i) q^{81} -948.000 q^{83} +90.0000 q^{85} +(-372.000 + 644.323i) q^{87} +(-507.000 - 878.150i) q^{89} +(-352.000 - 609.682i) q^{93} +(-110.000 + 190.526i) q^{95} +766.000 q^{97} +660.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 5 q^{5} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 5 q^{5} + 11 q^{9} + 60 q^{11} - 172 q^{13} + 40 q^{15} + 18 q^{17} + 44 q^{19} - 48 q^{23} - 25 q^{25} + 304 q^{27} - 372 q^{29} + 176 q^{31} - 240 q^{33} - 254 q^{37} - 344 q^{39} - 372 q^{41} - 200 q^{43} - 55 q^{45} + 168 q^{47} - 72 q^{51} + 498 q^{53} + 600 q^{55} + 352 q^{57} - 252 q^{59} - 58 q^{61} - 430 q^{65} + 1036 q^{67} - 384 q^{69} + 336 q^{71} + 506 q^{73} + 100 q^{75} - 272 q^{79} + 311 q^{81} - 1896 q^{83} + 180 q^{85} - 744 q^{87} - 1014 q^{89} - 704 q^{93} - 220 q^{95} + 1532 q^{97} + 1320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 2.00000 3.46410i 0.384900 0.666667i −0.606855 0.794812i \(-0.707569\pi\)
0.991755 + 0.128146i \(0.0409025\pi\)
\(4\) 0 0
\(5\) 2.50000 + 4.33013i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.50000 + 9.52628i 0.203704 + 0.352825i
\(10\) 0 0
\(11\) 30.0000 51.9615i 0.822304 1.42427i −0.0816590 0.996660i \(-0.526022\pi\)
0.903963 0.427611i \(-0.140645\pi\)
\(12\) 0 0
\(13\) −86.0000 −1.83478 −0.917389 0.397992i \(-0.869707\pi\)
−0.917389 + 0.397992i \(0.869707\pi\)
\(14\) 0 0
\(15\) 20.0000 0.344265
\(16\) 0 0
\(17\) 9.00000 15.5885i 0.128401 0.222397i −0.794656 0.607060i \(-0.792349\pi\)
0.923057 + 0.384662i \(0.125682\pi\)
\(18\) 0 0
\(19\) 22.0000 + 38.1051i 0.265639 + 0.460101i 0.967731 0.251986i \(-0.0810837\pi\)
−0.702092 + 0.712087i \(0.747750\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −24.0000 41.5692i −0.217580 0.376860i 0.736487 0.676451i \(-0.236483\pi\)
−0.954068 + 0.299591i \(0.903150\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 152.000 1.08342
\(28\) 0 0
\(29\) −186.000 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(30\) 0 0
\(31\) 88.0000 152.420i 0.509847 0.883081i −0.490088 0.871673i \(-0.663035\pi\)
0.999935 0.0114083i \(-0.00363144\pi\)
\(32\) 0 0
\(33\) −120.000 207.846i −0.633010 1.09640i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −127.000 219.970i −0.564288 0.977376i −0.997115 0.0758992i \(-0.975817\pi\)
0.432827 0.901477i \(-0.357516\pi\)
\(38\) 0 0
\(39\) −172.000 + 297.913i −0.706206 + 1.22319i
\(40\) 0 0
\(41\) −186.000 −0.708496 −0.354248 0.935152i \(-0.615263\pi\)
−0.354248 + 0.935152i \(0.615263\pi\)
\(42\) 0 0
\(43\) −100.000 −0.354648 −0.177324 0.984153i \(-0.556744\pi\)
−0.177324 + 0.984153i \(0.556744\pi\)
\(44\) 0 0
\(45\) −27.5000 + 47.6314i −0.0910991 + 0.157788i
\(46\) 0 0
\(47\) 84.0000 + 145.492i 0.260695 + 0.451537i 0.966427 0.256942i \(-0.0827150\pi\)
−0.705732 + 0.708479i \(0.749382\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −36.0000 62.3538i −0.0988433 0.171202i
\(52\) 0 0
\(53\) 249.000 431.281i 0.645335 1.11775i −0.338888 0.940827i \(-0.610051\pi\)
0.984224 0.176927i \(-0.0566157\pi\)
\(54\) 0 0
\(55\) 300.000 0.735491
\(56\) 0 0
\(57\) 176.000 0.408978
\(58\) 0 0
\(59\) −126.000 + 218.238i −0.278031 + 0.481563i −0.970895 0.239505i \(-0.923015\pi\)
0.692865 + 0.721068i \(0.256348\pi\)
\(60\) 0 0
\(61\) −29.0000 50.2295i −0.0608700 0.105430i 0.833985 0.551788i \(-0.186054\pi\)
−0.894855 + 0.446358i \(0.852721\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −215.000 372.391i −0.410269 0.710606i
\(66\) 0 0
\(67\) 518.000 897.202i 0.944534 1.63598i 0.187852 0.982197i \(-0.439847\pi\)
0.756682 0.653783i \(-0.226819\pi\)
\(68\) 0 0
\(69\) −192.000 −0.334987
\(70\) 0 0
\(71\) 168.000 0.280816 0.140408 0.990094i \(-0.455159\pi\)
0.140408 + 0.990094i \(0.455159\pi\)
\(72\) 0 0
\(73\) 253.000 438.209i 0.405636 0.702582i −0.588759 0.808308i \(-0.700383\pi\)
0.994395 + 0.105727i \(0.0337168\pi\)
\(74\) 0 0
\(75\) 50.0000 + 86.6025i 0.0769800 + 0.133333i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −136.000 235.559i −0.193686 0.335474i 0.752783 0.658269i \(-0.228711\pi\)
−0.946469 + 0.322795i \(0.895378\pi\)
\(80\) 0 0
\(81\) 155.500 269.334i 0.213306 0.369457i
\(82\) 0 0
\(83\) −948.000 −1.25369 −0.626846 0.779143i \(-0.715655\pi\)
−0.626846 + 0.779143i \(0.715655\pi\)
\(84\) 0 0
\(85\) 90.0000 0.114846
\(86\) 0 0
\(87\) −372.000 + 644.323i −0.458421 + 0.794008i
\(88\) 0 0
\(89\) −507.000 878.150i −0.603841 1.04588i −0.992233 0.124390i \(-0.960303\pi\)
0.388392 0.921494i \(-0.373031\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −352.000 609.682i −0.392481 0.679796i
\(94\) 0 0
\(95\) −110.000 + 190.526i −0.118797 + 0.205763i
\(96\) 0 0
\(97\) 766.000 0.801809 0.400905 0.916120i \(-0.368696\pi\)
0.400905 + 0.916120i \(0.368696\pi\)
\(98\) 0 0
\(99\) 660.000 0.670025
\(100\) 0 0
\(101\) −657.000 + 1137.96i −0.647267 + 1.12110i 0.336506 + 0.941681i \(0.390755\pi\)
−0.983773 + 0.179418i \(0.942579\pi\)
\(102\) 0 0
\(103\) −224.000 387.979i −0.214285 0.371153i 0.738766 0.673962i \(-0.235409\pi\)
−0.953051 + 0.302809i \(0.902076\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −774.000 1340.61i −0.699303 1.21123i −0.968708 0.248201i \(-0.920161\pi\)
0.269406 0.963027i \(-0.413173\pi\)
\(108\) 0 0
\(109\) −139.000 + 240.755i −0.122145 + 0.211561i −0.920613 0.390476i \(-0.872311\pi\)
0.798468 + 0.602037i \(0.205644\pi\)
\(110\) 0 0
\(111\) −1016.00 −0.868779
\(112\) 0 0
\(113\) −558.000 −0.464533 −0.232266 0.972652i \(-0.574614\pi\)
−0.232266 + 0.972652i \(0.574614\pi\)
\(114\) 0 0
\(115\) 120.000 207.846i 0.0973048 0.168537i
\(116\) 0 0
\(117\) −473.000 819.260i −0.373751 0.647356i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1134.50 1965.01i −0.852367 1.47634i
\(122\) 0 0
\(123\) −372.000 + 644.323i −0.272700 + 0.472330i
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 344.000 0.240355 0.120177 0.992752i \(-0.461654\pi\)
0.120177 + 0.992752i \(0.461654\pi\)
\(128\) 0 0
\(129\) −200.000 + 346.410i −0.136504 + 0.236432i
\(130\) 0 0
\(131\) 390.000 + 675.500i 0.260110 + 0.450524i 0.966271 0.257527i \(-0.0829077\pi\)
−0.706161 + 0.708052i \(0.749574\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 380.000 + 658.179i 0.242261 + 0.419608i
\(136\) 0 0
\(137\) −333.000 + 576.773i −0.207665 + 0.359686i −0.950979 0.309257i \(-0.899920\pi\)
0.743314 + 0.668943i \(0.233253\pi\)
\(138\) 0 0
\(139\) −884.000 −0.539424 −0.269712 0.962941i \(-0.586928\pi\)
−0.269712 + 0.962941i \(0.586928\pi\)
\(140\) 0 0
\(141\) 672.000 0.401366
\(142\) 0 0
\(143\) −2580.00 + 4468.69i −1.50874 + 2.61322i
\(144\) 0 0
\(145\) −465.000 805.404i −0.266318 0.461277i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 57.0000 + 98.7269i 0.0313397 + 0.0542820i 0.881270 0.472613i \(-0.156689\pi\)
−0.849930 + 0.526895i \(0.823356\pi\)
\(150\) 0 0
\(151\) 20.0000 34.6410i 0.0107787 0.0186692i −0.860586 0.509306i \(-0.829902\pi\)
0.871364 + 0.490636i \(0.163236\pi\)
\(152\) 0 0
\(153\) 198.000 0.104623
\(154\) 0 0
\(155\) 880.000 0.456021
\(156\) 0 0
\(157\) −77.0000 + 133.368i −0.0391418 + 0.0677957i −0.884933 0.465719i \(-0.845796\pi\)
0.845791 + 0.533515i \(0.179129\pi\)
\(158\) 0 0
\(159\) −996.000 1725.12i −0.496779 0.860447i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1090.00 1887.94i −0.523775 0.907206i −0.999617 0.0276746i \(-0.991190\pi\)
0.475842 0.879531i \(-0.342144\pi\)
\(164\) 0 0
\(165\) 600.000 1039.23i 0.283091 0.490327i
\(166\) 0 0
\(167\) −3696.00 −1.71261 −0.856303 0.516474i \(-0.827244\pi\)
−0.856303 + 0.516474i \(0.827244\pi\)
\(168\) 0 0
\(169\) 5199.00 2.36641
\(170\) 0 0
\(171\) −242.000 + 419.156i −0.108223 + 0.187448i
\(172\) 0 0
\(173\) 651.000 + 1127.57i 0.286096 + 0.495533i 0.972874 0.231334i \(-0.0743090\pi\)
−0.686778 + 0.726867i \(0.740976\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 504.000 + 872.954i 0.214028 + 0.370707i
\(178\) 0 0
\(179\) 2154.00 3730.84i 0.899427 1.55785i 0.0712000 0.997462i \(-0.477317\pi\)
0.828227 0.560392i \(-0.189350\pi\)
\(180\) 0 0
\(181\) −1550.00 −0.636523 −0.318261 0.948003i \(-0.603099\pi\)
−0.318261 + 0.948003i \(0.603099\pi\)
\(182\) 0 0
\(183\) −232.000 −0.0937155
\(184\) 0 0
\(185\) 635.000 1099.85i 0.252357 0.437096i
\(186\) 0 0
\(187\) −540.000 935.307i −0.211170 0.365756i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 41.5692i −0.00909204 0.0157479i 0.861444 0.507853i \(-0.169561\pi\)
−0.870536 + 0.492105i \(0.836227\pi\)
\(192\) 0 0
\(193\) −529.000 + 916.255i −0.197297 + 0.341728i −0.947651 0.319308i \(-0.896550\pi\)
0.750354 + 0.661036i \(0.229883\pi\)
\(194\) 0 0
\(195\) −1720.00 −0.631650
\(196\) 0 0
\(197\) −3714.00 −1.34321 −0.671603 0.740911i \(-0.734394\pi\)
−0.671603 + 0.740911i \(0.734394\pi\)
\(198\) 0 0
\(199\) −884.000 + 1531.13i −0.314900 + 0.545423i −0.979416 0.201851i \(-0.935304\pi\)
0.664516 + 0.747274i \(0.268638\pi\)
\(200\) 0 0
\(201\) −2072.00 3588.81i −0.727103 1.25938i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −465.000 805.404i −0.158424 0.274399i
\(206\) 0 0
\(207\) 264.000 457.261i 0.0886438 0.153536i
\(208\) 0 0
\(209\) 2640.00 0.873745
\(210\) 0 0
\(211\) −4036.00 −1.31682 −0.658412 0.752658i \(-0.728771\pi\)
−0.658412 + 0.752658i \(0.728771\pi\)
\(212\) 0 0
\(213\) 336.000 581.969i 0.108086 0.187211i
\(214\) 0 0
\(215\) −250.000 433.013i −0.0793017 0.137355i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1012.00 1752.84i −0.312259 0.540848i
\(220\) 0 0
\(221\) −774.000 + 1340.61i −0.235588 + 0.408050i
\(222\) 0 0
\(223\) −680.000 −0.204198 −0.102099 0.994774i \(-0.532556\pi\)
−0.102099 + 0.994774i \(0.532556\pi\)
\(224\) 0 0
\(225\) −275.000 −0.0814815
\(226\) 0 0
\(227\) 1194.00 2068.07i 0.349113 0.604681i −0.636979 0.770881i \(-0.719816\pi\)
0.986092 + 0.166200i \(0.0531497\pi\)
\(228\) 0 0
\(229\) −1937.00 3354.98i −0.558954 0.968137i −0.997584 0.0694695i \(-0.977869\pi\)
0.438630 0.898668i \(-0.355464\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1581.00 2738.37i −0.444527 0.769943i 0.553492 0.832854i \(-0.313295\pi\)
−0.998019 + 0.0629112i \(0.979962\pi\)
\(234\) 0 0
\(235\) −420.000 + 727.461i −0.116586 + 0.201933i
\(236\) 0 0
\(237\) −1088.00 −0.298199
\(238\) 0 0
\(239\) 5424.00 1.46799 0.733995 0.679155i \(-0.237654\pi\)
0.733995 + 0.679155i \(0.237654\pi\)
\(240\) 0 0
\(241\) −1943.00 + 3365.37i −0.519335 + 0.899514i 0.480413 + 0.877042i \(0.340487\pi\)
−0.999747 + 0.0224714i \(0.992847\pi\)
\(242\) 0 0
\(243\) 1430.00 + 2476.83i 0.377508 + 0.653864i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1892.00 3277.04i −0.487389 0.844182i
\(248\) 0 0
\(249\) −1896.00 + 3283.97i −0.482547 + 0.835795i
\(250\) 0 0
\(251\) 5100.00 1.28251 0.641253 0.767329i \(-0.278415\pi\)
0.641253 + 0.767329i \(0.278415\pi\)
\(252\) 0 0
\(253\) −2880.00 −0.715668
\(254\) 0 0
\(255\) 180.000 311.769i 0.0442041 0.0765637i
\(256\) 0 0
\(257\) 1089.00 + 1886.20i 0.264319 + 0.457814i 0.967385 0.253311i \(-0.0815195\pi\)
−0.703066 + 0.711125i \(0.748186\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1023.00 1771.89i −0.242613 0.420219i
\(262\) 0 0
\(263\) 3072.00 5320.86i 0.720257 1.24752i −0.240639 0.970615i \(-0.577357\pi\)
0.960897 0.276907i \(-0.0893095\pi\)
\(264\) 0 0
\(265\) 2490.00 0.577206
\(266\) 0 0
\(267\) −4056.00 −0.929675
\(268\) 0 0
\(269\) 411.000 711.873i 0.0931566 0.161352i −0.815681 0.578502i \(-0.803638\pi\)
0.908838 + 0.417150i \(0.136971\pi\)
\(270\) 0 0
\(271\) 4240.00 + 7343.90i 0.950412 + 1.64616i 0.744534 + 0.667584i \(0.232672\pi\)
0.205878 + 0.978578i \(0.433995\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 750.000 + 1299.04i 0.164461 + 0.284854i
\(276\) 0 0
\(277\) 569.000 985.537i 0.123422 0.213773i −0.797693 0.603064i \(-0.793946\pi\)
0.921115 + 0.389291i \(0.127280\pi\)
\(278\) 0 0
\(279\) 1936.00 0.415431
\(280\) 0 0
\(281\) 5706.00 1.21136 0.605679 0.795709i \(-0.292902\pi\)
0.605679 + 0.795709i \(0.292902\pi\)
\(282\) 0 0
\(283\) −1514.00 + 2622.32i −0.318014 + 0.550816i −0.980074 0.198635i \(-0.936349\pi\)
0.662060 + 0.749451i \(0.269683\pi\)
\(284\) 0 0
\(285\) 440.000 + 762.102i 0.0914504 + 0.158397i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2294.50 + 3974.19i 0.467026 + 0.808913i
\(290\) 0 0
\(291\) 1532.00 2653.50i 0.308617 0.534540i
\(292\) 0 0
\(293\) −3390.00 −0.675925 −0.337962 0.941160i \(-0.609738\pi\)
−0.337962 + 0.941160i \(0.609738\pi\)
\(294\) 0 0
\(295\) −1260.00 −0.248678
\(296\) 0 0
\(297\) 4560.00 7898.15i 0.890902 1.54309i
\(298\) 0 0
\(299\) 2064.00 + 3574.95i 0.399211 + 0.691454i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2628.00 + 4551.83i 0.498266 + 0.863022i
\(304\) 0 0
\(305\) 145.000 251.147i 0.0272219 0.0471497i
\(306\) 0 0
\(307\) 4156.00 0.772624 0.386312 0.922368i \(-0.373749\pi\)
0.386312 + 0.922368i \(0.373749\pi\)
\(308\) 0 0
\(309\) −1792.00 −0.329914
\(310\) 0 0
\(311\) 3276.00 5674.20i 0.597315 1.03458i −0.395901 0.918293i \(-0.629568\pi\)
0.993216 0.116286i \(-0.0370990\pi\)
\(312\) 0 0
\(313\) −683.000 1182.99i −0.123340 0.213631i 0.797743 0.602998i \(-0.206027\pi\)
−0.921083 + 0.389367i \(0.872694\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1299.00 2249.93i −0.230155 0.398640i 0.727699 0.685897i \(-0.240590\pi\)
−0.957854 + 0.287257i \(0.907257\pi\)
\(318\) 0 0
\(319\) −5580.00 + 9664.84i −0.979373 + 1.69632i
\(320\) 0 0
\(321\) −6192.00 −1.07665
\(322\) 0 0
\(323\) 792.000 0.136434
\(324\) 0 0
\(325\) 1075.00 1861.95i 0.183478 0.317793i
\(326\) 0 0
\(327\) 556.000 + 963.020i 0.0940271 + 0.162860i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1646.00 + 2850.96i 0.273330 + 0.473422i 0.969713 0.244249i \(-0.0785415\pi\)
−0.696382 + 0.717671i \(0.745208\pi\)
\(332\) 0 0
\(333\) 1397.00 2419.67i 0.229895 0.398190i
\(334\) 0 0
\(335\) 5180.00 0.844817
\(336\) 0 0
\(337\) 6194.00 1.00121 0.500606 0.865675i \(-0.333110\pi\)
0.500606 + 0.865675i \(0.333110\pi\)
\(338\) 0 0
\(339\) −1116.00 + 1932.97i −0.178799 + 0.309689i
\(340\) 0 0
\(341\) −5280.00 9145.23i −0.838499 1.45232i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −480.000 831.384i −0.0749053 0.129740i
\(346\) 0 0
\(347\) 5010.00 8677.57i 0.775075 1.34247i −0.159678 0.987169i \(-0.551046\pi\)
0.934753 0.355299i \(-0.115621\pi\)
\(348\) 0 0
\(349\) 3130.00 0.480072 0.240036 0.970764i \(-0.422841\pi\)
0.240036 + 0.970764i \(0.422841\pi\)
\(350\) 0 0
\(351\) −13072.0 −1.98784
\(352\) 0 0
\(353\) 2097.00 3632.11i 0.316181 0.547642i −0.663506 0.748171i \(-0.730932\pi\)
0.979688 + 0.200528i \(0.0642658\pi\)
\(354\) 0 0
\(355\) 420.000 + 727.461i 0.0627924 + 0.108760i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2052.00 + 3554.17i 0.301672 + 0.522512i 0.976515 0.215450i \(-0.0691218\pi\)
−0.674842 + 0.737962i \(0.735788\pi\)
\(360\) 0 0
\(361\) 2461.50 4263.44i 0.358872 0.621584i
\(362\) 0 0
\(363\) −9076.00 −1.31230
\(364\) 0 0
\(365\) 2530.00 0.362812
\(366\) 0 0
\(367\) 3748.00 6491.73i 0.533090 0.923339i −0.466163 0.884699i \(-0.654364\pi\)
0.999253 0.0386401i \(-0.0123026\pi\)
\(368\) 0 0
\(369\) −1023.00 1771.89i −0.144323 0.249975i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2921.00 + 5059.32i 0.405479 + 0.702310i 0.994377 0.105897i \(-0.0337714\pi\)
−0.588898 + 0.808207i \(0.700438\pi\)
\(374\) 0 0
\(375\) −250.000 + 433.013i −0.0344265 + 0.0596285i
\(376\) 0 0
\(377\) 15996.0 2.18524
\(378\) 0 0
\(379\) −412.000 −0.0558391 −0.0279195 0.999610i \(-0.508888\pi\)
−0.0279195 + 0.999610i \(0.508888\pi\)
\(380\) 0 0
\(381\) 688.000 1191.65i 0.0925126 0.160237i
\(382\) 0 0
\(383\) 1284.00 + 2223.95i 0.171304 + 0.296707i 0.938876 0.344256i \(-0.111869\pi\)
−0.767572 + 0.640963i \(0.778535\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −550.000 952.628i −0.0722431 0.125129i
\(388\) 0 0
\(389\) −6543.00 + 11332.8i −0.852810 + 1.47711i 0.0258510 + 0.999666i \(0.491770\pi\)
−0.878662 + 0.477445i \(0.841563\pi\)
\(390\) 0 0
\(391\) −864.000 −0.111750
\(392\) 0 0
\(393\) 3120.00 0.400466
\(394\) 0 0
\(395\) 680.000 1177.79i 0.0866190 0.150029i
\(396\) 0 0
\(397\) 5227.00 + 9053.43i 0.660795 + 1.14453i 0.980407 + 0.196982i \(0.0631141\pi\)
−0.319612 + 0.947548i \(0.603553\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5415.00 + 9379.06i 0.674345 + 1.16800i 0.976660 + 0.214791i \(0.0689071\pi\)
−0.302315 + 0.953208i \(0.597760\pi\)
\(402\) 0 0
\(403\) −7568.00 + 13108.2i −0.935456 + 1.62026i
\(404\) 0 0
\(405\) 1555.00 0.190787
\(406\) 0 0
\(407\) −15240.0 −1.85607
\(408\) 0 0
\(409\) −4283.00 + 7418.37i −0.517801 + 0.896858i 0.481985 + 0.876180i \(0.339916\pi\)
−0.999786 + 0.0206786i \(0.993417\pi\)
\(410\) 0 0
\(411\) 1332.00 + 2307.09i 0.159861 + 0.276887i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2370.00 4104.96i −0.280334 0.485553i
\(416\) 0 0
\(417\) −1768.00 + 3062.27i −0.207624 + 0.359616i
\(418\) 0 0
\(419\) −13884.0 −1.61880 −0.809401 0.587257i \(-0.800208\pi\)
−0.809401 + 0.587257i \(0.800208\pi\)
\(420\) 0 0
\(421\) 4286.00 0.496168 0.248084 0.968738i \(-0.420199\pi\)
0.248084 + 0.968738i \(0.420199\pi\)
\(422\) 0 0
\(423\) −924.000 + 1600.41i −0.106209 + 0.183959i
\(424\) 0 0
\(425\) 225.000 + 389.711i 0.0256802 + 0.0444795i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 10320.0 + 17874.8i 1.16143 + 2.01166i
\(430\) 0 0
\(431\) −3168.00 + 5487.14i −0.354054 + 0.613239i −0.986956 0.160993i \(-0.948530\pi\)
0.632902 + 0.774232i \(0.281864\pi\)
\(432\) 0 0
\(433\) 8974.00 0.995988 0.497994 0.867180i \(-0.334070\pi\)
0.497994 + 0.867180i \(0.334070\pi\)
\(434\) 0 0
\(435\) −3720.00 −0.410024
\(436\) 0 0
\(437\) 1056.00 1829.05i 0.115596 0.200218i
\(438\) 0 0
\(439\) −1484.00 2570.36i −0.161338 0.279446i 0.774011 0.633173i \(-0.218248\pi\)
−0.935349 + 0.353727i \(0.884914\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6186.00 + 10714.5i 0.663444 + 1.14912i 0.979705 + 0.200446i \(0.0642393\pi\)
−0.316261 + 0.948672i \(0.602427\pi\)
\(444\) 0 0
\(445\) 2535.00 4390.75i 0.270046 0.467734i
\(446\) 0 0
\(447\) 456.000 0.0482507
\(448\) 0 0
\(449\) 11394.0 1.19759 0.598793 0.800904i \(-0.295647\pi\)
0.598793 + 0.800904i \(0.295647\pi\)
\(450\) 0 0
\(451\) −5580.00 + 9664.84i −0.582599 + 1.00909i
\(452\) 0 0
\(453\) −80.0000 138.564i −0.00829741 0.0143715i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 179.000 + 310.037i 0.0183222 + 0.0317351i 0.875041 0.484049i \(-0.160834\pi\)
−0.856719 + 0.515784i \(0.827501\pi\)
\(458\) 0 0
\(459\) 1368.00 2369.45i 0.139113 0.240950i
\(460\) 0 0
\(461\) 7530.00 0.760753 0.380376 0.924832i \(-0.375794\pi\)
0.380376 + 0.924832i \(0.375794\pi\)
\(462\) 0 0
\(463\) −13768.0 −1.38197 −0.690986 0.722868i \(-0.742823\pi\)
−0.690986 + 0.722868i \(0.742823\pi\)
\(464\) 0 0
\(465\) 1760.00 3048.41i 0.175523 0.304014i
\(466\) 0 0
\(467\) 6690.00 + 11587.4i 0.662904 + 1.14818i 0.979849 + 0.199740i \(0.0640097\pi\)
−0.316945 + 0.948444i \(0.602657\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 308.000 + 533.472i 0.0301314 + 0.0521891i
\(472\) 0 0
\(473\) −3000.00 + 5196.15i −0.291628 + 0.505115i
\(474\) 0 0
\(475\) −1100.00 −0.106256
\(476\) 0 0
\(477\) 5478.00 0.525829
\(478\) 0 0
\(479\) −3168.00 + 5487.14i −0.302191 + 0.523411i −0.976632 0.214918i \(-0.931051\pi\)
0.674441 + 0.738329i \(0.264385\pi\)
\(480\) 0 0
\(481\) 10922.0 + 18917.5i 1.03534 + 1.79327i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1915.00 + 3316.88i 0.179290 + 0.310539i
\(486\) 0 0
\(487\) 2504.00 4337.06i 0.232992 0.403554i −0.725695 0.688016i \(-0.758482\pi\)
0.958687 + 0.284462i \(0.0918151\pi\)
\(488\) 0 0
\(489\) −8720.00 −0.806405
\(490\) 0 0
\(491\) 12900.0 1.18568 0.592840 0.805320i \(-0.298007\pi\)
0.592840 + 0.805320i \(0.298007\pi\)
\(492\) 0 0
\(493\) −1674.00 + 2899.45i −0.152927 + 0.264878i
\(494\) 0 0
\(495\) 1650.00 + 2857.88i 0.149822 + 0.259500i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4058.00 + 7028.66i 0.364050 + 0.630553i 0.988623 0.150413i \(-0.0480603\pi\)
−0.624573 + 0.780966i \(0.714727\pi\)
\(500\) 0 0
\(501\) −7392.00 + 12803.3i −0.659182 + 1.14174i
\(502\) 0 0
\(503\) 4944.00 0.438255 0.219127 0.975696i \(-0.429679\pi\)
0.219127 + 0.975696i \(0.429679\pi\)
\(504\) 0 0
\(505\) −6570.00 −0.578933
\(506\) 0 0
\(507\) 10398.0 18009.9i 0.910831 1.57761i
\(508\) 0 0
\(509\) −2733.00 4733.69i −0.237992 0.412215i 0.722146 0.691741i \(-0.243156\pi\)
−0.960138 + 0.279526i \(0.909823\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3344.00 + 5791.98i 0.287800 + 0.498484i
\(514\) 0 0
\(515\) 1120.00 1939.90i 0.0958313 0.165985i
\(516\) 0 0
\(517\) 10080.0 0.857481
\(518\) 0 0
\(519\) 5208.00 0.440474
\(520\) 0 0
\(521\) 5037.00 8724.34i 0.423560 0.733628i −0.572724 0.819748i \(-0.694113\pi\)
0.996285 + 0.0861198i \(0.0274468\pi\)
\(522\) 0 0
\(523\) −6914.00 11975.4i −0.578065 1.00124i −0.995701 0.0926239i \(-0.970475\pi\)
0.417636 0.908614i \(-0.362859\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1584.00 2743.57i −0.130930 0.226777i
\(528\) 0 0
\(529\) 4931.50 8541.61i 0.405318 0.702031i
\(530\) 0 0
\(531\) −2772.00 −0.226543
\(532\) 0 0
\(533\) 15996.0 1.29993
\(534\) 0 0
\(535\) 3870.00 6703.04i 0.312738 0.541678i
\(536\) 0 0
\(537\) −8616.00 14923.3i −0.692380 1.19924i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7613.00 + 13186.1i 0.605006 + 1.04790i 0.992050 + 0.125841i \(0.0401628\pi\)
−0.387044 + 0.922061i \(0.626504\pi\)
\(542\) 0 0
\(543\) −3100.00 + 5369.36i −0.244998 + 0.424348i
\(544\) 0 0
\(545\) −1390.00 −0.109250
\(546\) 0 0
\(547\) −13228.0 −1.03398 −0.516991 0.855991i \(-0.672948\pi\)
−0.516991 + 0.855991i \(0.672948\pi\)
\(548\) 0 0
\(549\) 319.000 552.524i 0.0247989 0.0429529i
\(550\) 0 0
\(551\) −4092.00 7087.55i −0.316379 0.547985i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2540.00 4399.41i −0.194265 0.336477i
\(556\) 0 0
\(557\) 4245.00 7352.56i 0.322920 0.559314i −0.658169 0.752870i \(-0.728669\pi\)
0.981089 + 0.193556i \(0.0620022\pi\)
\(558\) 0 0
\(559\) 8600.00 0.650700
\(560\) 0 0
\(561\) −4320.00 −0.325117
\(562\) 0 0
\(563\) −5142.00 + 8906.21i −0.384919 + 0.666699i −0.991758 0.128125i \(-0.959104\pi\)
0.606839 + 0.794825i \(0.292437\pi\)
\(564\) 0 0
\(565\) −1395.00 2416.21i −0.103873 0.179913i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −885.000 1532.86i −0.0652041 0.112937i 0.831580 0.555404i \(-0.187437\pi\)
−0.896785 + 0.442468i \(0.854103\pi\)
\(570\) 0 0
\(571\) −3034.00 + 5255.04i −0.222362 + 0.385143i −0.955525 0.294911i \(-0.904710\pi\)
0.733162 + 0.680054i \(0.238043\pi\)
\(572\) 0 0
\(573\) −192.000 −0.0139981
\(574\) 0 0
\(575\) 1200.00 0.0870321
\(576\) 0 0
\(577\) 10753.0 18624.7i 0.775829 1.34377i −0.158499 0.987359i \(-0.550665\pi\)
0.934327 0.356416i \(-0.116001\pi\)
\(578\) 0 0
\(579\) 2116.00 + 3665.02i 0.151879 + 0.263062i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −14940.0 25876.8i −1.06132 1.83827i
\(584\) 0 0
\(585\) 2365.00 4096.30i 0.167147 0.289506i
\(586\) 0 0
\(587\) −12108.0 −0.851364 −0.425682 0.904873i \(-0.639966\pi\)
−0.425682 + 0.904873i \(0.639966\pi\)
\(588\) 0 0
\(589\) 7744.00 0.541742
\(590\) 0 0
\(591\) −7428.00 + 12865.7i −0.517000 + 0.895471i
\(592\) 0 0
\(593\) 7737.00 + 13400.9i 0.535785 + 0.928007i 0.999125 + 0.0418262i \(0.0133176\pi\)
−0.463340 + 0.886181i \(0.653349\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3536.00 + 6124.53i 0.242410 + 0.419867i
\(598\) 0 0
\(599\) 1260.00 2182.38i 0.0859469 0.148864i −0.819847 0.572582i \(-0.805942\pi\)
0.905794 + 0.423718i \(0.139275\pi\)
\(600\) 0 0
\(601\) 12790.0 0.868078 0.434039 0.900894i \(-0.357088\pi\)
0.434039 + 0.900894i \(0.357088\pi\)
\(602\) 0 0
\(603\) 11396.0 0.769620
\(604\) 0 0
\(605\) 5672.50 9825.06i 0.381190 0.660240i
\(606\) 0 0
\(607\) 5788.00 + 10025.1i 0.387031 + 0.670357i 0.992049 0.125855i \(-0.0401674\pi\)
−0.605018 + 0.796212i \(0.706834\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7224.00 12512.3i −0.478317 0.828470i
\(612\) 0 0
\(613\) −10063.0 + 17429.6i −0.663035 + 1.14841i 0.316778 + 0.948500i \(0.397399\pi\)
−0.979814 + 0.199912i \(0.935935\pi\)
\(614\) 0 0
\(615\) −3720.00 −0.243910
\(616\) 0 0
\(617\) −27942.0 −1.82318 −0.911590 0.411100i \(-0.865145\pi\)
−0.911590 + 0.411100i \(0.865145\pi\)
\(618\) 0 0
\(619\) −11270.0 + 19520.2i −0.731792 + 1.26750i 0.224324 + 0.974515i \(0.427983\pi\)
−0.956116 + 0.292987i \(0.905351\pi\)
\(620\) 0 0
\(621\) −3648.00 6318.52i −0.235731 0.408299i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 5280.00 9145.23i 0.336304 0.582496i
\(628\) 0 0
\(629\) −4572.00 −0.289821
\(630\) 0 0
\(631\) −5128.00 −0.323522 −0.161761 0.986830i \(-0.551717\pi\)
−0.161761 + 0.986830i \(0.551717\pi\)
\(632\) 0 0
\(633\) −8072.00 + 13981.1i −0.506845 + 0.877882i
\(634\) 0 0
\(635\) 860.000 + 1489.56i 0.0537450 + 0.0930890i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 924.000 + 1600.41i 0.0572032 + 0.0990789i
\(640\) 0 0
\(641\) 6399.00 11083.4i 0.394298 0.682945i −0.598713 0.800964i \(-0.704321\pi\)
0.993011 + 0.118019i \(0.0376543\pi\)
\(642\) 0 0
\(643\) 21148.0 1.29704 0.648519 0.761198i \(-0.275389\pi\)
0.648519 + 0.761198i \(0.275389\pi\)
\(644\) 0 0
\(645\) −2000.00 −0.122093
\(646\) 0 0
\(647\) 8232.00 14258.2i 0.500206 0.866382i −0.499794 0.866144i \(-0.666591\pi\)
1.00000 0.000237943i \(-7.57396e-5\pi\)
\(648\) 0 0
\(649\) 7560.00 + 13094.3i 0.457251 + 0.791982i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12117.0 + 20987.3i 0.726148 + 1.25773i 0.958500 + 0.285094i \(0.0920248\pi\)
−0.232351 + 0.972632i \(0.574642\pi\)
\(654\) 0 0
\(655\) −1950.00 + 3377.50i −0.116325 + 0.201481i
\(656\) 0 0
\(657\) 5566.00 0.330518
\(658\) 0 0
\(659\) −22836.0 −1.34987 −0.674935 0.737877i \(-0.735828\pi\)
−0.674935 + 0.737877i \(0.735828\pi\)
\(660\) 0 0
\(661\) 13159.0 22792.1i 0.774320 1.34116i −0.160855 0.986978i \(-0.551425\pi\)
0.935176 0.354184i \(-0.115241\pi\)
\(662\) 0 0
\(663\) 3096.00 + 5362.43i 0.181355 + 0.314117i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4464.00 + 7731.87i 0.259141 + 0.448845i
\(668\) 0 0
\(669\) −1360.00 + 2355.59i −0.0785959 + 0.136132i
\(670\) 0 0
\(671\) −3480.00 −0.200214
\(672\) 0 0
\(673\) 28802.0 1.64968 0.824841 0.565365i \(-0.191265\pi\)
0.824841 + 0.565365i \(0.191265\pi\)
\(674\) 0 0
\(675\) −1900.00 + 3290.90i −0.108342 + 0.187654i
\(676\) 0 0
\(677\) 1263.00 + 2187.58i 0.0717002 + 0.124188i 0.899647 0.436619i \(-0.143824\pi\)
−0.827946 + 0.560807i \(0.810491\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −4776.00 8272.27i −0.268747 0.465483i
\(682\) 0 0
\(683\) 11538.0 19984.4i 0.646397 1.11959i −0.337580 0.941297i \(-0.609608\pi\)
0.983977 0.178296i \(-0.0570584\pi\)
\(684\) 0 0
\(685\) −3330.00 −0.185741
\(686\) 0 0
\(687\) −15496.0 −0.860567
\(688\) 0 0
\(689\) −21414.0 + 37090.1i −1.18405 + 2.05083i
\(690\) 0 0
\(691\) 3934.00 + 6813.89i 0.216579 + 0.375127i 0.953760 0.300569i \(-0.0971766\pi\)
−0.737181 + 0.675696i \(0.763843\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2210.00 3827.83i −0.120619 0.208918i
\(696\) 0 0
\(697\) −1674.00 + 2899.45i −0.0909717 + 0.157568i
\(698\) 0 0
\(699\) −12648.0 −0.684394
\(700\) 0 0
\(701\) 21510.0 1.15895 0.579473 0.814991i \(-0.303258\pi\)
0.579473 + 0.814991i \(0.303258\pi\)
\(702\) 0 0
\(703\) 5588.00 9678.70i 0.299794 0.519259i
\(704\) 0 0
\(705\) 1680.00 + 2909.85i 0.0897482 + 0.155448i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15007.0 25992.9i −0.794922 1.37685i −0.922889 0.385067i \(-0.874178\pi\)
0.127967 0.991778i \(-0.459155\pi\)
\(710\) 0 0
\(711\) 1496.00 2591.15i 0.0789091 0.136675i
\(712\) 0 0
\(713\) −8448.00 −0.443731
\(714\) 0 0
\(715\) −25800.0 −1.34946
\(716\) 0 0
\(717\) 10848.0 18789.3i 0.565029 0.978659i
\(718\) 0 0
\(719\) −408.000 706.677i −0.0211625 0.0366545i 0.855250 0.518215i \(-0.173403\pi\)
−0.876413 + 0.481561i \(0.840070\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 7772.00 + 13461.5i 0.399784 + 0.692446i
\(724\) 0 0
\(725\) 2325.00 4027.02i 0.119101 0.206289i
\(726\) 0 0
\(727\) 9952.00 0.507702 0.253851 0.967243i \(-0.418303\pi\)
0.253851 + 0.967243i \(0.418303\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) −900.000 + 1558.85i −0.0455372 + 0.0788728i
\(732\) 0 0
\(733\) −16973.0 29398.1i −0.855269 1.48137i −0.876395 0.481592i \(-0.840059\pi\)
0.0211266 0.999777i \(-0.493275\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31080.0 53832.1i −1.55339 2.69055i
\(738\) 0 0
\(739\) −11710.0 + 20282.3i −0.582895 + 1.00960i 0.412239 + 0.911076i \(0.364747\pi\)
−0.995134 + 0.0985280i \(0.968587\pi\)
\(740\) 0 0
\(741\) −15136.0 −0.750384
\(742\) 0 0
\(743\) −14592.0 −0.720496 −0.360248 0.932857i \(-0.617308\pi\)
−0.360248 + 0.932857i \(0.617308\pi\)
\(744\) 0 0
\(745\) −285.000 + 493.634i −0.0140156 + 0.0242757i
\(746\) 0 0
\(747\) −5214.00 9030.91i −0.255382 0.442334i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4528.00 7842.73i −0.220012 0.381072i 0.734799 0.678285i \(-0.237276\pi\)
−0.954811 + 0.297213i \(0.903943\pi\)
\(752\) 0 0
\(753\) 10200.0 17666.9i 0.493637 0.855004i
\(754\) 0 0
\(755\) 200.000 0.00964072
\(756\) 0 0
\(757\) −17554.0 −0.842815 −0.421408 0.906871i \(-0.638464\pi\)
−0.421408 + 0.906871i \(0.638464\pi\)
\(758\) 0 0
\(759\) −5760.00 + 9976.61i −0.275461 + 0.477112i
\(760\) 0 0
\(761\) −18219.0 31556.2i −0.867856 1.50317i −0.864183 0.503177i \(-0.832164\pi\)
−0.00367239 0.999993i \(-0.501169\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 495.000 + 857.365i 0.0233945 + 0.0405204i
\(766\) 0 0
\(767\) 10836.0 18768.5i 0.510124 0.883561i
\(768\) 0 0
\(769\) 9022.00 0.423071 0.211536 0.977370i \(-0.432154\pi\)
0.211536 + 0.977370i \(0.432154\pi\)
\(770\) 0 0
\(771\) 8712.00 0.406946
\(772\) 0 0
\(773\) 735.000 1273.06i 0.0341994 0.0592350i −0.848419 0.529325i \(-0.822445\pi\)
0.882618 + 0.470090i \(0.155779\pi\)
\(774\) 0 0
\(775\) 2200.00 + 3810.51i 0.101969 + 0.176616i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4092.00 7087.55i −0.188204 0.325979i
\(780\) 0 0
\(781\) 5040.00 8729.54i 0.230916 0.399958i
\(782\) 0 0
\(783\) −28272.0 −1.29037
\(784\) 0 0
\(785\) −770.000 −0.0350095
\(786\) 0 0
\(787\) 2626.00 4548.37i 0.118941 0.206012i −0.800407 0.599457i \(-0.795383\pi\)
0.919348 + 0.393444i \(0.128717\pi\)
\(788\) 0 0
\(789\) −12288.0 21283.4i −0.554454 0.960343i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2494.00 + 4319.73i 0.111683 + 0.193440i
\(794\) 0 0
\(795\) 4980.00 8625.61i 0.222167 0.384804i
\(796\) 0 0
\(797\) −12294.0 −0.546394 −0.273197 0.961958i \(-0.588081\pi\)
−0.273197 + 0.961958i \(0.588081\pi\)
\(798\) 0 0
\(799\) 3024.00 0.133894
\(800\) 0 0
\(801\) 5577.00 9659.65i 0.246009 0.426101i
\(802\) 0 0
\(803\) −15180.0 26292.5i −0.667112 1.15547i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1644.00 2847.49i −0.0717119 0.124209i
\(808\) 0 0
\(809\) −7773.00 + 13463.2i −0.337805 + 0.585095i −0.984020 0.178061i \(-0.943018\pi\)
0.646215 + 0.763156i \(0.276351\pi\)
\(810\) 0 0
\(811\) −19364.0 −0.838424 −0.419212 0.907888i \(-0.637694\pi\)
−0.419212 + 0.907888i \(0.637694\pi\)
\(812\) 0 0
\(813\) 33920.0 1.46326
\(814\) 0 0
\(815\) 5450.00 9439.68i 0.234239 0.405715i
\(816\) 0 0
\(817\) −2200.00 3810.51i −0.0942084 0.163174i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3657.00 + 6334.11i 0.155457 + 0.269259i 0.933225 0.359292i \(-0.116982\pi\)
−0.777768 + 0.628551i \(0.783648\pi\)
\(822\) 0 0
\(823\) −5992.00 + 10378.4i −0.253789 + 0.439575i −0.964566 0.263843i \(-0.915010\pi\)
0.710777 + 0.703417i \(0.248343\pi\)
\(824\) 0 0
\(825\) 6000.00 0.253204
\(826\) 0 0
\(827\) 13500.0 0.567643 0.283822 0.958877i \(-0.408398\pi\)
0.283822 + 0.958877i \(0.408398\pi\)
\(828\) 0 0
\(829\) −22301.0 + 38626.5i −0.934313 + 1.61828i −0.158459 + 0.987365i \(0.550653\pi\)
−0.775854 + 0.630913i \(0.782681\pi\)
\(830\) 0 0
\(831\) −2276.00 3942.15i −0.0950103 0.164563i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9240.00 16004.1i −0.382950 0.663289i
\(836\) 0 0
\(837\) 13376.0 23167.9i 0.552380 0.956751i
\(838\) 0 0
\(839\) −35448.0 −1.45864 −0.729321 0.684172i \(-0.760164\pi\)
−0.729321 + 0.684172i \(0.760164\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) 0 0
\(843\) 11412.0 19766.2i 0.466252 0.807572i
\(844\) 0 0
\(845\) 12997.5 + 22512.3i 0.529145 + 0.916506i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 6056.00 + 10489.3i 0.244807 + 0.424019i
\(850\) 0 0
\(851\) −6096.00 + 10558.6i −0.245556 + 0.425316i
\(852\) 0 0
\(853\) −12590.0 −0.505362 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(854\) 0 0
\(855\) −2420.00 −0.0967980
\(856\) 0 0
\(857\) 12453.0 21569.2i 0.496367 0.859733i −0.503624 0.863923i \(-0.668000\pi\)
0.999991 + 0.00419015i \(0.00133377\pi\)
\(858\) 0 0
\(859\) 11602.0 + 20095.3i 0.460833 + 0.798185i 0.999003 0.0446509i \(-0.0142175\pi\)
−0.538170 + 0.842836i \(0.680884\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9924.00 17188.9i −0.391445 0.678002i 0.601196 0.799102i \(-0.294691\pi\)
−0.992640 + 0.121100i \(0.961358\pi\)
\(864\) 0 0
\(865\) −3255.00 + 5637.83i −0.127946 + 0.221609i
\(866\) 0 0
\(867\) 18356.0 0.719034
\(868\) 0 0
\(869\) −16320.0 −0.637075
\(870\) 0 0
\(871\) −44548.0 + 77159.4i −1.73301 + 3.00166i
\(872\) 0 0
\(873\) 4213.00 + 7297.13i 0.163332 + 0.282899i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13771.0 23852.1i −0.530232 0.918389i −0.999378 0.0352683i \(-0.988771\pi\)
0.469146 0.883121i \(-0.344562\pi\)
\(878\) 0 0
\(879\) −6780.00 + 11743.3i −0.260164 + 0.450616i
\(880\) 0 0
\(881\) 20718.0 0.792290 0.396145 0.918188i \(-0.370348\pi\)
0.396145 + 0.918188i \(0.370348\pi\)
\(882\) 0 0
\(883\) 25172.0 0.959349 0.479675 0.877446i \(-0.340755\pi\)
0.479675 + 0.877446i \(0.340755\pi\)
\(884\) 0 0
\(885\) −2520.00 + 4364.77i −0.0957162 + 0.165785i
\(886\) 0 0
\(887\) −6432.00 11140.6i −0.243478 0.421717i 0.718224 0.695812i \(-0.244955\pi\)
−0.961703 + 0.274095i \(0.911622\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9330.00 16160.0i −0.350804 0.607611i
\(892\) 0 0
\(893\) −3696.00 + 6401.66i −0.138502 + 0.239892i
\(894\) 0 0
\(895\) 21540.0 0.804472
\(896\) 0 0
\(897\) 16512.0 0.614626
\(898\) 0 0
\(899\) −16368.0 + 28350.2i −0.607234 + 1.05176i
\(900\) 0 0
\(901\) −4482.00 7763.05i −0.165724 0.287042i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3875.00 6711.70i −0.142331 0.246524i
\(906\) 0 0
\(907\) 11546.0 19998.3i 0.422689 0.732118i −0.573513 0.819197i \(-0.694420\pi\)
0.996201 + 0.0870783i \(0.0277530\pi\)
\(908\) 0 0
\(909\) −14454.0 −0.527403
\(910\) 0 0
\(911\) −14208.0 −0.516720 −0.258360 0.966049i \(-0.583182\pi\)
−0.258360 + 0.966049i \(0.583182\pi\)
\(912\) 0 0
\(913\) −28440.0 + 49259.5i −1.03092 + 1.78560i
\(914\) 0 0
\(915\) −580.000 1004.59i −0.0209554 0.0362959i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 13292.0 + 23022.4i 0.477108 + 0.826376i 0.999656 0.0262342i \(-0.00835157\pi\)
−0.522547 + 0.852610i \(0.675018\pi\)
\(920\) 0 0
\(921\) 8312.00 14396.8i 0.297383 0.515082i
\(922\) 0 0
\(923\) −14448.0 −0.515235
\(924\) 0 0
\(925\) 6350.00 0.225715
\(926\) 0 0
\(927\) 2464.00 4267.77i 0.0873014 0.151210i
\(928\) 0 0
\(929\) 81.0000 + 140.296i 0.00286063 + 0.00495475i 0.867452 0.497521i \(-0.165756\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −13104.0 22696.8i −0.459813 0.796420i
\(934\) 0 0
\(935\) 2700.00 4676.54i 0.0944379 0.163571i
\(936\) 0 0
\(937\) 29734.0 1.03668 0.518339 0.855175i \(-0.326551\pi\)
0.518339 + 0.855175i \(0.326551\pi\)
\(938\) 0 0
\(939\) −5464.00 −0.189894
\(940\) 0 0
\(941\) 8571.00 14845.4i 0.296925 0.514290i −0.678506 0.734595i \(-0.737372\pi\)
0.975431 + 0.220306i \(0.0707055\pi\)
\(942\) 0 0
\(943\) 4464.00 + 7731.87i 0.154155 + 0.267004i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13218.0 22894.2i −0.453566 0.785600i 0.545038 0.838411i \(-0.316515\pi\)
−0.998605 + 0.0528113i \(0.983182\pi\)
\(948\) 0 0
\(949\) −21758.0 + 37686.0i −0.744251 + 1.28908i
\(950\) 0 0
\(951\) −10392.0 −0.354347
\(952\) 0 0
\(953\) 27882.0 0.947730 0.473865 0.880598i \(-0.342858\pi\)
0.473865 + 0.880598i \(0.342858\pi\)
\(954\) 0 0
\(955\) 120.000 207.846i 0.00406608 0.00704266i
\(956\) 0 0
\(957\) 22320.0 + 38659.4i 0.753922 + 1.30583i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −592.500 1026.24i −0.0198886 0.0344480i
\(962\) 0 0
\(963\) 8514.00 14746.7i 0.284901 0.493463i
\(964\) 0 0
\(965\) −5290.00 −0.176467
\(966\) 0 0
\(967\) 12656.0 0.420879 0.210439 0.977607i \(-0.432511\pi\)
0.210439 + 0.977607i \(0.432511\pi\)
\(968\) 0 0
\(969\) 1584.00 2743.57i 0.0525133 0.0909557i
\(970\) 0 0
\(971\) 1458.00 + 2525.33i 0.0481869 + 0.0834621i 0.889113 0.457688i \(-0.151322\pi\)
−0.840926 + 0.541150i \(0.817989\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4300.00 7447.82i −0.141241 0.244637i
\(976\) 0 0
\(977\) 3447.00 5970.38i 0.112875 0.195506i −0.804053 0.594558i \(-0.797327\pi\)
0.916928 + 0.399052i \(0.130661\pi\)
\(978\) 0 0
\(979\) −60840.0 −1.98616
\(980\) 0 0
\(981\) −3058.00 −0.0995254
\(982\) 0 0
\(983\) −22632.0 + 39199.8i −0.734332 + 1.27190i 0.220683 + 0.975346i \(0.429171\pi\)
−0.955016 + 0.296555i \(0.904162\pi\)
\(984\) 0 0
\(985\) −9285.00 16082.1i −0.300350 0.520221i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2400.00 + 4156.92i 0.0771644 + 0.133653i
\(990\) 0 0
\(991\) −26008.0 + 45047.2i −0.833674 + 1.44397i 0.0614307 + 0.998111i \(0.480434\pi\)
−0.895105 + 0.445855i \(0.852900\pi\)
\(992\) 0 0
\(993\) 13168.0 0.420820
\(994\) 0 0
\(995\) −8840.00 −0.281655
\(996\) 0 0
\(997\) −6929.00 + 12001.4i −0.220104 + 0.381231i −0.954839 0.297123i \(-0.903973\pi\)
0.734735 + 0.678354i \(0.237306\pi\)
\(998\) 0 0
\(999\) −19304.0 33435.5i −0.611363 1.05891i
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 980.4.i.n.361.1 2
7.2 even 3 inner 980.4.i.n.961.1 2
7.3 odd 6 20.4.a.a.1.1 1
7.4 even 3 980.4.a.c.1.1 1
7.5 odd 6 980.4.i.e.961.1 2
7.6 odd 2 980.4.i.e.361.1 2
21.17 even 6 180.4.a.a.1.1 1
28.3 even 6 80.4.a.c.1.1 1
35.3 even 12 100.4.c.a.49.2 2
35.17 even 12 100.4.c.a.49.1 2
35.24 odd 6 100.4.a.a.1.1 1
56.3 even 6 320.4.a.k.1.1 1
56.45 odd 6 320.4.a.d.1.1 1
63.31 odd 6 1620.4.i.d.541.1 2
63.38 even 6 1620.4.i.j.1081.1 2
63.52 odd 6 1620.4.i.d.1081.1 2
63.59 even 6 1620.4.i.j.541.1 2
77.10 even 6 2420.4.a.d.1.1 1
84.59 odd 6 720.4.a.k.1.1 1
105.17 odd 12 900.4.d.k.649.1 2
105.38 odd 12 900.4.d.k.649.2 2
105.59 even 6 900.4.a.m.1.1 1
112.3 even 12 1280.4.d.c.641.1 2
112.45 odd 12 1280.4.d.n.641.2 2
112.59 even 12 1280.4.d.c.641.2 2
112.101 odd 12 1280.4.d.n.641.1 2
140.3 odd 12 400.4.c.j.49.1 2
140.59 even 6 400.4.a.o.1.1 1
140.87 odd 12 400.4.c.j.49.2 2
280.59 even 6 1600.4.a.p.1.1 1
280.269 odd 6 1600.4.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.4.a.a.1.1 1 7.3 odd 6
80.4.a.c.1.1 1 28.3 even 6
100.4.a.a.1.1 1 35.24 odd 6
100.4.c.a.49.1 2 35.17 even 12
100.4.c.a.49.2 2 35.3 even 12
180.4.a.a.1.1 1 21.17 even 6
320.4.a.d.1.1 1 56.45 odd 6
320.4.a.k.1.1 1 56.3 even 6
400.4.a.o.1.1 1 140.59 even 6
400.4.c.j.49.1 2 140.3 odd 12
400.4.c.j.49.2 2 140.87 odd 12
720.4.a.k.1.1 1 84.59 odd 6
900.4.a.m.1.1 1 105.59 even 6
900.4.d.k.649.1 2 105.17 odd 12
900.4.d.k.649.2 2 105.38 odd 12
980.4.a.c.1.1 1 7.4 even 3
980.4.i.e.361.1 2 7.6 odd 2
980.4.i.e.961.1 2 7.5 odd 6
980.4.i.n.361.1 2 1.1 even 1 trivial
980.4.i.n.961.1 2 7.2 even 3 inner
1280.4.d.c.641.1 2 112.3 even 12
1280.4.d.c.641.2 2 112.59 even 12
1280.4.d.n.641.1 2 112.101 odd 12
1280.4.d.n.641.2 2 112.45 odd 12
1600.4.a.p.1.1 1 280.59 even 6
1600.4.a.bl.1.1 1 280.269 odd 6
1620.4.i.d.541.1 2 63.31 odd 6
1620.4.i.d.1081.1 2 63.52 odd 6
1620.4.i.j.541.1 2 63.59 even 6
1620.4.i.j.1081.1 2 63.38 even 6
2420.4.a.d.1.1 1 77.10 even 6