Properties

Label 1350.2.j.b.199.1
Level $1350$
Weight $2$
Character 1350.199
Analytic conductor $10.780$
Analytic rank $0$
Dimension $4$
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] = [1350,2,Mod(199,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.j (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: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.199
Dual form 1350.2.j.b.1099.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.73205 - 1.00000i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.73205 - 1.00000i) q^{7} +1.00000i q^{8} +(-3.46410 - 2.00000i) q^{13} +(-1.00000 + 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} +6.00000i q^{17} +7.00000 q^{19} +4.00000 q^{26} -2.00000i q^{28} +(3.00000 + 5.19615i) q^{29} +(5.00000 - 8.66025i) q^{31} +(0.866025 + 0.500000i) q^{32} +(-3.00000 - 5.19615i) q^{34} +2.00000i q^{37} +(-6.06218 + 3.50000i) q^{38} +(4.50000 - 7.79423i) q^{41} +(0.866025 - 0.500000i) q^{43} +(-5.19615 + 3.00000i) q^{47} +(-1.50000 + 2.59808i) q^{49} +(-3.46410 + 2.00000i) q^{52} -12.0000i q^{53} +(1.00000 + 1.73205i) q^{56} +(-5.19615 - 3.00000i) q^{58} +(4.50000 - 7.79423i) q^{59} +(2.00000 + 3.46410i) q^{61} +10.0000i q^{62} -1.00000 q^{64} +(11.2583 + 6.50000i) q^{67} +(5.19615 + 3.00000i) q^{68} -6.00000 q^{71} +1.00000i q^{73} +(-1.00000 - 1.73205i) q^{74} +(3.50000 - 6.06218i) q^{76} +(1.00000 + 1.73205i) q^{79} +9.00000i q^{82} +(7.79423 - 4.50000i) q^{83} +(-0.500000 + 0.866025i) q^{86} +15.0000 q^{89} -8.00000 q^{91} +(3.00000 - 5.19615i) q^{94} +(14.7224 - 8.50000i) q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{14} - 2 q^{16} + 28 q^{19} + 16 q^{26} + 12 q^{29} + 20 q^{31} - 12 q^{34} + 18 q^{41} - 6 q^{49} + 4 q^{56} + 18 q^{59} + 8 q^{61} - 4 q^{64} - 24 q^{71} - 4 q^{74} + 14 q^{76} + 4 q^{79} - 2 q^{86} + 60 q^{89} - 32 q^{91} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 0.866025i 0.250000 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.73205 1.00000i 0.654654 0.377964i −0.135583 0.990766i \(-0.543291\pi\)
0.790237 + 0.612801i \(0.209957\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −3.46410 2.00000i −0.960769 0.554700i −0.0643593 0.997927i \(-0.520500\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) −1.00000 + 1.73205i −0.267261 + 0.462910i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) 5.00000 8.66025i 0.898027 1.55543i 0.0680129 0.997684i \(-0.478334\pi\)
0.830014 0.557743i \(-0.188333\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −3.00000 5.19615i −0.514496 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) −6.06218 + 3.50000i −0.983415 + 0.567775i
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 7.79423i 0.702782 1.21725i −0.264704 0.964330i \(-0.585274\pi\)
0.967486 0.252924i \(-0.0813924\pi\)
\(42\) 0 0
\(43\) 0.866025 0.500000i 0.132068 0.0762493i −0.432511 0.901629i \(-0.642372\pi\)
0.564578 + 0.825380i \(0.309039\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.19615 + 3.00000i −0.757937 + 0.437595i −0.828554 0.559908i \(-0.810836\pi\)
0.0706177 + 0.997503i \(0.477503\pi\)
\(48\) 0 0
\(49\) −1.50000 + 2.59808i −0.214286 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.46410 + 2.00000i −0.480384 + 0.277350i
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 + 1.73205i 0.133631 + 0.231455i
\(57\) 0 0
\(58\) −5.19615 3.00000i −0.682288 0.393919i
\(59\) 4.50000 7.79423i 0.585850 1.01472i −0.408919 0.912571i \(-0.634094\pi\)
0.994769 0.102151i \(-0.0325726\pi\)
\(60\) 0 0
\(61\) 2.00000 + 3.46410i 0.256074 + 0.443533i 0.965187 0.261562i \(-0.0842377\pi\)
−0.709113 + 0.705095i \(0.750904\pi\)
\(62\) 10.0000i 1.27000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2583 + 6.50000i 1.37542 + 0.794101i 0.991605 0.129307i \(-0.0412752\pi\)
0.383819 + 0.923408i \(0.374609\pi\)
\(68\) 5.19615 + 3.00000i 0.630126 + 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) −1.00000 1.73205i −0.116248 0.201347i
\(75\) 0 0
\(76\) 3.50000 6.06218i 0.401478 0.695379i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 + 1.73205i 0.112509 + 0.194871i 0.916781 0.399390i \(-0.130778\pi\)
−0.804272 + 0.594261i \(0.797445\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.00000i 0.993884i
\(83\) 7.79423 4.50000i 0.855528 0.493939i −0.00698436 0.999976i \(-0.502223\pi\)
0.862512 + 0.506036i \(0.168890\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.500000 + 0.866025i −0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 3.00000 5.19615i 0.309426 0.535942i
\(95\) 0 0
\(96\) 0 0
\(97\) 14.7224 8.50000i 1.49484 0.863044i 0.494854 0.868976i \(-0.335222\pi\)
0.999982 + 0.00593185i \(0.00188818\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 + 10.3923i 0.597022 + 1.03407i 0.993258 + 0.115924i \(0.0369830\pi\)
−0.396236 + 0.918149i \(0.629684\pi\)
\(102\) 0 0
\(103\) 1.73205 + 1.00000i 0.170664 + 0.0985329i 0.582899 0.812545i \(-0.301918\pi\)
−0.412235 + 0.911078i \(0.635252\pi\)
\(104\) 2.00000 3.46410i 0.196116 0.339683i
\(105\) 0 0
\(106\) 6.00000 + 10.3923i 0.582772 + 1.00939i
\(107\) 9.00000i 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.73205 1.00000i −0.163663 0.0944911i
\(113\) −2.59808 1.50000i −0.244406 0.141108i 0.372794 0.927914i \(-0.378400\pi\)
−0.617200 + 0.786806i \(0.711733\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 9.00000i 0.828517i
\(119\) 6.00000 + 10.3923i 0.550019 + 0.952661i
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) −3.46410 2.00000i −0.313625 0.181071i
\(123\) 0 0
\(124\) −5.00000 8.66025i −0.449013 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000i 1.77471i 0.461084 + 0.887357i \(0.347461\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0.866025 0.500000i 0.0765466 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 + 10.3923i −0.524222 + 0.907980i 0.475380 + 0.879781i \(0.342311\pi\)
−0.999602 + 0.0281993i \(0.991023\pi\)
\(132\) 0 0
\(133\) 12.1244 7.00000i 1.05131 0.606977i
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 2.59808 1.50000i 0.221969 0.128154i −0.384893 0.922961i \(-0.625762\pi\)
0.606861 + 0.794808i \(0.292428\pi\)
\(138\) 0 0
\(139\) −2.00000 + 3.46410i −0.169638 + 0.293821i −0.938293 0.345843i \(-0.887593\pi\)
0.768655 + 0.639664i \(0.220926\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.19615 3.00000i 0.436051 0.251754i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −0.500000 0.866025i −0.0413803 0.0716728i
\(147\) 0 0
\(148\) 1.73205 + 1.00000i 0.142374 + 0.0821995i
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) 2.00000 + 3.46410i 0.162758 + 0.281905i 0.935857 0.352381i \(-0.114628\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 7.00000i 0.567775i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.1244 7.00000i −0.967629 0.558661i −0.0691164 0.997609i \(-0.522018\pi\)
−0.898513 + 0.438948i \(0.855351\pi\)
\(158\) −1.73205 1.00000i −0.137795 0.0795557i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.00000i 0.548282i 0.961689 + 0.274141i \(0.0883936\pi\)
−0.961689 + 0.274141i \(0.911606\pi\)
\(164\) −4.50000 7.79423i −0.351391 0.608627i
\(165\) 0 0
\(166\) −4.50000 + 7.79423i −0.349268 + 0.604949i
\(167\) 10.3923 + 6.00000i 0.804181 + 0.464294i 0.844931 0.534875i \(-0.179641\pi\)
−0.0407502 + 0.999169i \(0.512975\pi\)
\(168\) 0 0
\(169\) 1.50000 + 2.59808i 0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000i 0.0762493i
\(173\) −10.3923 + 6.00000i −0.790112 + 0.456172i −0.840002 0.542583i \(-0.817446\pi\)
0.0498898 + 0.998755i \(0.484113\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −12.9904 + 7.50000i −0.973670 + 0.562149i
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 6.92820 4.00000i 0.513553 0.296500i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) 1.73205 + 1.00000i 0.124676 + 0.0719816i 0.561041 0.827788i \(-0.310401\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(194\) −8.50000 + 14.7224i −0.610264 + 1.05701i
\(195\) 0 0
\(196\) 1.50000 + 2.59808i 0.107143 + 0.185577i
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.3923 6.00000i −0.731200 0.422159i
\(203\) 10.3923 + 6.00000i 0.729397 + 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.50000 + 4.33013i −0.172107 + 0.298098i −0.939156 0.343490i \(-0.888391\pi\)
0.767049 + 0.641588i \(0.221724\pi\)
\(212\) −10.3923 6.00000i −0.713746 0.412082i
\(213\) 0 0
\(214\) 4.50000 + 7.79423i 0.307614 + 0.532803i
\(215\) 0 0
\(216\) 0 0
\(217\) 20.0000i 1.35769i
\(218\) 1.73205 1.00000i 0.117309 0.0677285i
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 20.7846i 0.807207 1.39812i
\(222\) 0 0
\(223\) 13.8564 8.00000i 0.927894 0.535720i 0.0417488 0.999128i \(-0.486707\pi\)
0.886145 + 0.463409i \(0.153374\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) 12.9904 7.50000i 0.862202 0.497792i −0.00254715 0.999997i \(-0.500811\pi\)
0.864749 + 0.502204i \(0.167477\pi\)
\(228\) 0 0
\(229\) −5.00000 + 8.66025i −0.330409 + 0.572286i −0.982592 0.185776i \(-0.940520\pi\)
0.652183 + 0.758062i \(0.273853\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.19615 + 3.00000i −0.341144 + 0.196960i
\(233\) 3.00000i 0.196537i 0.995160 + 0.0982683i \(0.0313303\pi\)
−0.995160 + 0.0982683i \(0.968670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.50000 7.79423i −0.292925 0.507361i
\(237\) 0 0
\(238\) −10.3923 6.00000i −0.673633 0.388922i
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 9.50000 + 16.4545i 0.611949 + 1.05993i 0.990912 + 0.134515i \(0.0429475\pi\)
−0.378963 + 0.925412i \(0.623719\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) −24.2487 14.0000i −1.54291 0.890799i
\(248\) 8.66025 + 5.00000i 0.549927 + 0.317500i
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −10.0000 17.3205i −0.627456 1.08679i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) 2.59808 + 1.50000i 0.162064 + 0.0935674i 0.578838 0.815442i \(-0.303506\pi\)
−0.416775 + 0.909010i \(0.636840\pi\)
\(258\) 0 0
\(259\) 2.00000 + 3.46410i 0.124274 + 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) −5.19615 + 3.00000i −0.320408 + 0.184988i −0.651575 0.758585i \(-0.725891\pi\)
0.331166 + 0.943572i \(0.392558\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −7.00000 + 12.1244i −0.429198 + 0.743392i
\(267\) 0 0
\(268\) 11.2583 6.50000i 0.687712 0.397051i
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 5.19615 3.00000i 0.315063 0.181902i
\(273\) 0 0
\(274\) −1.50000 + 2.59808i −0.0906183 + 0.156956i
\(275\) 0 0
\(276\) 0 0
\(277\) 1.73205 1.00000i 0.104069 0.0600842i −0.447062 0.894503i \(-0.647530\pi\)
0.551131 + 0.834419i \(0.314196\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −9.00000 15.5885i −0.536895 0.929929i −0.999069 0.0431402i \(-0.986264\pi\)
0.462174 0.886789i \(-0.347070\pi\)
\(282\) 0 0
\(283\) −16.4545 9.50000i −0.978117 0.564716i −0.0764162 0.997076i \(-0.524348\pi\)
−0.901701 + 0.432360i \(0.857681\pi\)
\(284\) −3.00000 + 5.19615i −0.178017 + 0.308335i
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0000i 1.06251i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0.866025 + 0.500000i 0.0506803 + 0.0292603i
\(293\) −10.3923 6.00000i −0.607125 0.350524i 0.164714 0.986341i \(-0.447330\pi\)
−0.771839 + 0.635818i \(0.780663\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 6.00000i 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 1.73205i 0.0576390 0.0998337i
\(302\) −3.46410 2.00000i −0.199337 0.115087i
\(303\) 0 0
\(304\) −3.50000 6.06218i −0.200739 0.347690i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.0000 + 25.9808i −0.850572 + 1.47323i 0.0301210 + 0.999546i \(0.490411\pi\)
−0.880693 + 0.473688i \(0.842923\pi\)
\(312\) 0 0
\(313\) 16.4545 9.50000i 0.930062 0.536972i 0.0432311 0.999065i \(-0.486235\pi\)
0.886831 + 0.462093i \(0.152902\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −25.9808 + 15.0000i −1.45922 + 0.842484i −0.998973 0.0453045i \(-0.985574\pi\)
−0.460252 + 0.887788i \(0.652241\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 42.0000i 2.33694i
\(324\) 0 0
\(325\) 0 0
\(326\) −3.50000 6.06218i −0.193847 0.335753i
\(327\) 0 0
\(328\) 7.79423 + 4.50000i 0.430364 + 0.248471i
\(329\) −6.00000 + 10.3923i −0.330791 + 0.572946i
\(330\) 0 0
\(331\) −5.50000 9.52628i −0.302307 0.523612i 0.674351 0.738411i \(-0.264424\pi\)
−0.976658 + 0.214799i \(0.931090\pi\)
\(332\) 9.00000i 0.493939i
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −12.1244 7.00000i −0.660456 0.381314i 0.131995 0.991250i \(-0.457862\pi\)
−0.792451 + 0.609936i \(0.791195\pi\)
\(338\) −2.59808 1.50000i −0.141317 0.0815892i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0.500000 + 0.866025i 0.0269582 + 0.0466930i
\(345\) 0 0
\(346\) 6.00000 10.3923i 0.322562 0.558694i
\(347\) −10.3923 6.00000i −0.557888 0.322097i 0.194409 0.980921i \(-0.437721\pi\)
−0.752297 + 0.658824i \(0.771054\pi\)
\(348\) 0 0
\(349\) −5.00000 8.66025i −0.267644 0.463573i 0.700609 0.713545i \(-0.252912\pi\)
−0.968253 + 0.249973i \(0.919578\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.9904 7.50000i 0.691408 0.399185i −0.112731 0.993626i \(-0.535960\pi\)
0.804139 + 0.594441i \(0.202627\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.50000 12.9904i 0.397499 0.688489i
\(357\) 0 0
\(358\) 7.79423 4.50000i 0.411938 0.237832i
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 13.8564 8.00000i 0.728277 0.420471i
\(363\) 0 0
\(364\) −4.00000 + 6.92820i −0.209657 + 0.363137i
\(365\) 0 0
\(366\) 0 0
\(367\) −8.66025 + 5.00000i −0.452062 + 0.260998i −0.708700 0.705509i \(-0.750718\pi\)
0.256639 + 0.966507i \(0.417385\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 20.7846i −0.623009 1.07908i
\(372\) 0 0
\(373\) −3.46410 2.00000i −0.179364 0.103556i 0.407630 0.913147i \(-0.366355\pi\)
−0.586994 + 0.809591i \(0.699689\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.00000 5.19615i −0.154713 0.267971i
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.19615 3.00000i −0.265858 0.153493i
\(383\) −20.7846 12.0000i −1.06204 0.613171i −0.136047 0.990702i \(-0.543440\pi\)
−0.925997 + 0.377531i \(0.876773\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 17.0000i 0.863044i
\(389\) −3.00000 5.19615i −0.152106 0.263455i 0.779895 0.625910i \(-0.215272\pi\)
−0.932002 + 0.362454i \(0.881939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.59808 1.50000i −0.131223 0.0757614i
\(393\) 0 0
\(394\) 9.00000 + 15.5885i 0.453413 + 0.785335i
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000i 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) −13.8564 + 8.00000i −0.694559 + 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 15.5885i 0.449439 0.778450i −0.548911 0.835881i \(-0.684957\pi\)
0.998350 + 0.0574304i \(0.0182907\pi\)
\(402\) 0 0
\(403\) −34.6410 + 20.0000i −1.72559 + 0.996271i
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −15.5000 + 26.8468i −0.766426 + 1.32749i 0.173064 + 0.984911i \(0.444633\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.73205 1.00000i 0.0853320 0.0492665i
\(413\) 18.0000i 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 3.46410i −0.0980581 0.169842i
\(417\) 0 0
\(418\) 0 0
\(419\) −1.50000 + 2.59808i −0.0732798 + 0.126924i −0.900337 0.435194i \(-0.856680\pi\)
0.827057 + 0.562118i \(0.190013\pi\)
\(420\) 0 0
\(421\) 5.00000 + 8.66025i 0.243685 + 0.422075i 0.961761 0.273890i \(-0.0883103\pi\)
−0.718076 + 0.695965i \(0.754977\pi\)
\(422\) 5.00000i 0.243396i
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 6.92820 + 4.00000i 0.335279 + 0.193574i
\(428\) −7.79423 4.50000i −0.376748 0.217516i
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 10.0000 + 17.3205i 0.480015 + 0.831411i
\(435\) 0 0
\(436\) −1.00000 + 1.73205i −0.0478913 + 0.0829502i
\(437\) 0 0
\(438\) 0 0
\(439\) −2.00000 3.46410i −0.0954548 0.165333i 0.814344 0.580383i \(-0.197097\pi\)
−0.909798 + 0.415051i \(0.863764\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.0000i 1.14156i
\(443\) 31.1769 18.0000i 1.48126 0.855206i 0.481486 0.876454i \(-0.340097\pi\)
0.999774 + 0.0212481i \(0.00676401\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 + 13.8564i −0.378811 + 0.656120i
\(447\) 0 0
\(448\) −1.73205 + 1.00000i −0.0818317 + 0.0472456i
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.59808 + 1.50000i −0.122203 + 0.0705541i
\(453\) 0 0
\(454\) −7.50000 + 12.9904i −0.351992 + 0.609669i
\(455\) 0 0
\(456\) 0 0
\(457\) −16.4545 + 9.50000i −0.769708 + 0.444391i −0.832771 0.553618i \(-0.813247\pi\)
0.0630623 + 0.998010i \(0.479913\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 + 5.19615i 0.139724 + 0.242009i 0.927392 0.374091i \(-0.122045\pi\)
−0.787668 + 0.616100i \(0.788712\pi\)
\(462\) 0 0
\(463\) 22.5167 + 13.0000i 1.04644 + 0.604161i 0.921650 0.388022i \(-0.126842\pi\)
0.124788 + 0.992183i \(0.460175\pi\)
\(464\) 3.00000 5.19615i 0.139272 0.241225i
\(465\) 0 0
\(466\) −1.50000 2.59808i −0.0694862 0.120354i
\(467\) 15.0000i 0.694117i 0.937843 + 0.347059i \(0.112820\pi\)
−0.937843 + 0.347059i \(0.887180\pi\)
\(468\) 0 0
\(469\) 26.0000 1.20057
\(470\) 0 0
\(471\) 0 0
\(472\) 7.79423 + 4.50000i 0.358758 + 0.207129i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) 0 0
\(479\) −18.0000 31.1769i −0.822441 1.42451i −0.903859 0.427830i \(-0.859278\pi\)
0.0814184 0.996680i \(-0.474055\pi\)
\(480\) 0 0
\(481\) 4.00000 6.92820i 0.182384 0.315899i
\(482\) −16.4545 9.50000i −0.749481 0.432713i
\(483\) 0 0
\(484\) −5.50000 9.52628i −0.250000 0.433013i
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) −3.46410 + 2.00000i −0.156813 + 0.0905357i
\(489\) 0 0
\(490\) 0 0
\(491\) −1.50000 + 2.59808i −0.0676941 + 0.117250i −0.897886 0.440228i \(-0.854898\pi\)
0.830192 + 0.557478i \(0.188231\pi\)
\(492\) 0 0
\(493\) −31.1769 + 18.0000i −1.40414 + 0.810679i
\(494\) 28.0000 1.25978
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −10.3923 + 6.00000i −0.466159 + 0.269137i
\(498\) 0 0
\(499\) 5.50000 9.52628i 0.246214 0.426455i −0.716258 0.697835i \(-0.754147\pi\)
0.962472 + 0.271380i \(0.0874801\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.9904 7.50000i 0.579789 0.334741i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 17.3205 + 10.0000i 0.768473 + 0.443678i
\(509\) 18.0000 31.1769i 0.797836 1.38189i −0.123187 0.992384i \(-0.539311\pi\)
0.921023 0.389509i \(-0.127355\pi\)
\(510\) 0 0
\(511\) 1.00000 + 1.73205i 0.0442374 + 0.0766214i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −3.46410 2.00000i −0.152204 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 1.00000i 0.0437269i 0.999761 + 0.0218635i \(0.00695991\pi\)
−0.999761 + 0.0218635i \(0.993040\pi\)
\(524\) 6.00000 + 10.3923i 0.262111 + 0.453990i
\(525\) 0 0
\(526\) 3.00000 5.19615i 0.130806 0.226563i
\(527\) 51.9615 + 30.0000i 2.26348 + 1.30682i
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 14.0000i 0.606977i
\(533\) −31.1769 + 18.0000i −1.35042 + 0.779667i
\(534\) 0 0
\(535\) 0 0
\(536\) −6.50000 + 11.2583i −0.280757 + 0.486286i
\(537\) 0 0
\(538\) 25.9808 15.0000i 1.12011 0.646696i
\(539\) 0 0
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 24.2487 14.0000i 1.04157 0.601351i
\(543\) 0 0
\(544\) −3.00000 + 5.19615i −0.128624 + 0.222783i
\(545\) 0 0
\(546\) 0 0
\(547\) 4.33013 2.50000i 0.185143 0.106892i −0.404564 0.914510i \(-0.632577\pi\)
0.589707 + 0.807617i \(0.299243\pi\)
\(548\) 3.00000i 0.128154i
\(549\) 0 0
\(550\) 0 0
\(551\) 21.0000 + 36.3731i 0.894630 + 1.54954i
\(552\) 0 0
\(553\) 3.46410 + 2.00000i 0.147309 + 0.0850487i
\(554\) −1.00000 + 1.73205i −0.0424859 + 0.0735878i
\(555\) 0 0
\(556\) 2.00000 + 3.46410i 0.0848189 + 0.146911i
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 15.5885 + 9.00000i 0.657559 + 0.379642i
\(563\) −7.79423 4.50000i −0.328488 0.189652i 0.326682 0.945134i \(-0.394069\pi\)
−0.655169 + 0.755482i \(0.727403\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 19.0000 0.798630
\(567\) 0 0
\(568\) 6.00000i 0.251754i
\(569\) 21.0000 + 36.3731i 0.880366 + 1.52484i 0.850935 + 0.525271i \(0.176036\pi\)
0.0294311 + 0.999567i \(0.490630\pi\)
\(570\) 0 0
\(571\) −17.5000 + 30.3109i −0.732352 + 1.26847i 0.223523 + 0.974699i \(0.428244\pi\)
−0.955875 + 0.293773i \(0.905089\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.00000 + 15.5885i 0.375653 + 0.650650i
\(575\) 0 0
\(576\) 0 0
\(577\) 35.0000i 1.45707i 0.685009 + 0.728535i \(0.259798\pi\)
−0.685009 + 0.728535i \(0.740202\pi\)
\(578\) 16.4545 9.50000i 0.684416 0.395148i
\(579\) 0 0
\(580\) 0 0
\(581\) 9.00000 15.5885i 0.373383 0.646718i
\(582\) 0 0
\(583\) 0 0
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) −31.1769 + 18.0000i −1.28681 + 0.742940i −0.978084 0.208212i \(-0.933236\pi\)
−0.308725 + 0.951151i \(0.599902\pi\)
\(588\) 0 0
\(589\) 35.0000 60.6218i 1.44215 2.49788i
\(590\) 0 0
\(591\) 0 0
\(592\) 1.73205 1.00000i 0.0711868 0.0410997i
\(593\) 9.00000i 0.369586i −0.982777 0.184793i \(-0.940839\pi\)
0.982777 0.184793i \(-0.0591614\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.00000 5.19615i −0.122885 0.212843i
\(597\) 0 0
\(598\) 0 0
\(599\) −15.0000 + 25.9808i −0.612883 + 1.06155i 0.377869 + 0.925859i \(0.376657\pi\)
−0.990752 + 0.135686i \(0.956676\pi\)
\(600\) 0 0
\(601\) 5.00000 + 8.66025i 0.203954 + 0.353259i 0.949799 0.312861i \(-0.101287\pi\)
−0.745845 + 0.666120i \(0.767954\pi\)
\(602\) 2.00000i 0.0815139i
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) 34.6410 + 20.0000i 1.40604 + 0.811775i 0.995003 0.0998457i \(-0.0318349\pi\)
0.411033 + 0.911621i \(0.365168\pi\)
\(608\) 6.06218 + 3.50000i 0.245854 + 0.141944i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 44.0000i 1.77714i −0.458738 0.888572i \(-0.651698\pi\)
0.458738 0.888572i \(-0.348302\pi\)
\(614\) −4.00000 6.92820i −0.161427 0.279600i
\(615\) 0 0
\(616\) 0 0
\(617\) −7.79423 4.50000i −0.313784 0.181163i 0.334835 0.942277i \(-0.391320\pi\)
−0.648618 + 0.761114i \(0.724653\pi\)
\(618\) 0 0
\(619\) 11.5000 + 19.9186i 0.462224 + 0.800595i 0.999071 0.0430838i \(-0.0137183\pi\)
−0.536847 + 0.843679i \(0.680385\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.0000i 1.20289i
\(623\) 25.9808 15.0000i 1.04090 0.600962i
\(624\) 0 0
\(625\) 0 0
\(626\) −9.50000 + 16.4545i −0.379696 + 0.657653i
\(627\) 0 0
\(628\) −12.1244 + 7.00000i −0.483814 + 0.279330i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −1.73205 + 1.00000i −0.0688973 + 0.0397779i
\(633\) 0 0
\(634\) 15.0000 25.9808i 0.595726 1.03183i
\(635\) 0 0
\(636\) 0 0
\(637\) 10.3923 6.00000i 0.411758 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5000 + 28.5788i 0.651711 + 1.12880i 0.982708 + 0.185164i \(0.0592817\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(642\) 0 0
\(643\) −11.2583 6.50000i −0.443985 0.256335i 0.261301 0.965257i \(-0.415848\pi\)
−0.705287 + 0.708922i \(0.749182\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 36.3731i −0.826234 1.43108i
\(647\) 48.0000i 1.88707i 0.331266 + 0.943537i \(0.392524\pi\)
−0.331266 + 0.943537i \(0.607476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 6.06218 + 3.50000i 0.237413 + 0.137071i
\(653\) 31.1769 + 18.0000i 1.22005 + 0.704394i 0.964928 0.262515i \(-0.0845520\pi\)
0.255119 + 0.966910i \(0.417885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) −16.5000 28.5788i −0.642749 1.11327i −0.984817 0.173598i \(-0.944461\pi\)
0.342068 0.939675i \(-0.388873\pi\)
\(660\) 0 0
\(661\) −1.00000 + 1.73205i −0.0388955 + 0.0673690i −0.884818 0.465937i \(-0.845717\pi\)
0.845922 + 0.533306i \(0.179051\pi\)
\(662\) 9.52628 + 5.50000i 0.370249 + 0.213764i
\(663\) 0 0
\(664\) 4.50000 + 7.79423i 0.174634 + 0.302475i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 10.3923 6.00000i 0.402090 0.232147i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.73205 + 1.00000i −0.0667657 + 0.0385472i −0.533011 0.846108i \(-0.678940\pi\)
0.466246 + 0.884655i \(0.345606\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 10.3923 6.00000i 0.399409 0.230599i −0.286820 0.957984i \(-0.592598\pi\)
0.686229 + 0.727386i \(0.259265\pi\)
\(678\) 0 0
\(679\) 17.0000 29.4449i 0.652400 1.12999i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.00000i 0.114792i −0.998351 0.0573959i \(-0.981720\pi\)
0.998351 0.0573959i \(-0.0182797\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.0000 17.3205i −0.381802 0.661300i
\(687\) 0 0
\(688\) −0.866025 0.500000i −0.0330169 0.0190623i
\(689\) −24.0000 + 41.5692i −0.914327 + 1.58366i
\(690\) 0 0
\(691\) −17.5000 30.3109i −0.665731 1.15308i −0.979086 0.203445i \(-0.934786\pi\)
0.313355 0.949636i \(-0.398547\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 46.7654 + 27.0000i 1.77136 + 1.02270i
\(698\) 8.66025 + 5.00000i 0.327795 + 0.189253i
\(699\) 0 0
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 14.0000i 0.528020i
\(704\) 0 0
\(705\) 0 0
\(706\) −7.50000 + 12.9904i −0.282266 + 0.488899i
\(707\) 20.7846 + 12.0000i 0.781686 + 0.451306i
\(708\) 0 0
\(709\) −14.0000 24.2487i −0.525781 0.910679i −0.999549 0.0300298i \(-0.990440\pi\)
0.473768 0.880650i \(-0.342894\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.0000i 0.562149i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −4.50000 + 7.79423i −0.168173 + 0.291284i
\(717\) 0 0
\(718\) −10.3923 + 6.00000i −0.387837 + 0.223918i
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −25.9808 + 15.0000i −0.966904 + 0.558242i
\(723\) 0 0
\(724\) −8.00000 + 13.8564i −0.297318 + 0.514969i
\(725\) 0 0
\(726\) 0 0
\(727\) −34.6410 + 20.0000i −1.28476 + 0.741759i −0.977715 0.209935i \(-0.932675\pi\)
−0.307049 + 0.951694i \(0.599341\pi\)
\(728\) 8.00000i 0.296500i
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) 0 0
\(733\) −19.0526 11.0000i −0.703722 0.406294i 0.105010 0.994471i \(-0.466513\pi\)
−0.808732 + 0.588177i \(0.799846\pi\)
\(734\) 5.00000 8.66025i 0.184553 0.319656i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 20.7846 + 12.0000i 0.763027 + 0.440534i
\(743\) −5.19615 3.00000i −0.190628 0.110059i 0.401648 0.915794i \(-0.368437\pi\)
−0.592277 + 0.805735i \(0.701771\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 0 0
\(749\) −9.00000 15.5885i −0.328853 0.569590i
\(750\) 0 0
\(751\) 2.00000 3.46410i 0.0729810 0.126407i −0.827225 0.561870i \(-0.810082\pi\)
0.900207 + 0.435463i \(0.143415\pi\)
\(752\) 5.19615 + 3.00000i 0.189484 + 0.109399i
\(753\) 0 0
\(754\) 12.0000 + 20.7846i 0.437014 + 0.756931i
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000i 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 17.3205 10.0000i 0.629109 0.363216i
\(759\) 0 0
\(760\) 0 0
\(761\) 10.5000 18.1865i 0.380625 0.659261i −0.610527 0.791995i \(-0.709042\pi\)
0.991152 + 0.132734i \(0.0423756\pi\)
\(762\) 0 0
\(763\) −3.46410 + 2.00000i −0.125409 + 0.0724049i
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −31.1769 + 18.0000i −1.12573 + 0.649942i
\(768\) 0 0
\(769\) 2.50000 4.33013i 0.0901523 0.156148i −0.817423 0.576038i \(-0.804598\pi\)
0.907575 + 0.419890i \(0.137931\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.73205 1.00000i 0.0623379 0.0359908i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.50000 + 14.7224i 0.305132 + 0.528505i
\(777\) 0 0
\(778\) 5.19615 + 3.00000i 0.186291 + 0.107555i
\(779\) 31.5000 54.5596i 1.12860 1.95480i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) −17.3205 10.0000i −0.617409 0.356462i 0.158450 0.987367i \(-0.449350\pi\)
−0.775860 + 0.630905i \(0.782684\pi\)
\(788\) −15.5885 9.00000i −0.555316 0.320612i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 16.0000i 0.568177i
\(794\) 11.0000 + 19.0526i 0.390375 + 0.676150i
\(795\) 0 0
\(796\) 8.00000 13.8564i 0.283552 0.491127i
\(797\) −31.1769 18.0000i −1.10434 0.637593i −0.166985 0.985959i \(-0.553403\pi\)
−0.937358 + 0.348367i \(0.886736\pi\)
\(798\) 0 0
\(799\) −18.0000 31.1769i −0.636794 1.10296i
\(800\) 0 0
\(801\) 0 0
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 20.0000 34.6410i 0.704470 1.22018i
\(807\) 0 0
\(808\) −10.3923 + 6.00000i −0.365600 + 0.211079i
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) 10.3923 6.00000i 0.364698 0.210559i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.06218 3.50000i 0.212089 0.122449i
\(818\) 31.0000i 1.08389i
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 20.7846i −0.418803 0.725388i 0.577016 0.816733i \(-0.304217\pi\)
−0.995819 + 0.0913446i \(0.970884\pi\)
\(822\) 0 0
\(823\) 12.1244 + 7.00000i 0.422628 + 0.244005i 0.696201 0.717847i \(-0.254872\pi\)
−0.273573 + 0.961851i \(0.588205\pi\)
\(824\) −1.00000 + 1.73205i −0.0348367 + 0.0603388i
\(825\) 0 0
\(826\) 9.00000 + 15.5885i 0.313150 + 0.542392i
\(827\) 27.0000i 0.938882i 0.882964 + 0.469441i \(0.155545\pi\)
−0.882964 + 0.469441i \(0.844455\pi\)
\(828\) 0 0
\(829\) −32.0000 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.46410 + 2.00000i 0.120096 + 0.0693375i
\(833\) −15.5885 9.00000i −0.540108 0.311832i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 3.00000i 0.103633i
\(839\) 21.0000 + 36.3731i 0.725001 + 1.25574i 0.958974 + 0.283495i \(0.0914938\pi\)
−0.233973 + 0.972243i \(0.575173\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) −8.66025 5.00000i −0.298452 0.172311i
\(843\) 0 0
\(844\) 2.50000 + 4.33013i 0.0860535 + 0.149049i
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) −10.3923 + 6.00000i −0.356873 + 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.66025 5.00000i 0.296521 0.171197i −0.344358 0.938839i \(-0.611903\pi\)
0.640879 + 0.767642i \(0.278570\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) 33.7750 19.5000i 1.15373 0.666107i 0.203938 0.978984i \(-0.434626\pi\)
0.949794 + 0.312877i \(0.101293\pi\)
\(858\) 0 0
\(859\) −15.5000 + 26.8468i −0.528853 + 0.916001i 0.470581 + 0.882357i \(0.344044\pi\)
−0.999434 + 0.0336436i \(0.989289\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 5.19615 3.00000i 0.176982 0.102180i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.00000 + 1.73205i 0.0339814 + 0.0588575i
\(867\) 0 0
\(868\) −17.3205 10.0000i −0.587896 0.339422i
\(869\) 0 0
\(870\) 0 0
\(871\) −26.0000 45.0333i −0.880976 1.52590i
\(872\) 2.00000i 0.0677285i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.66025 + 5.00000i 0.292436 + 0.168838i 0.639040 0.769174i \(-0.279332\pi\)
−0.346604 + 0.938012i \(0.612665\pi\)
\(878\) 3.46410 + 2.00000i 0.116908 + 0.0674967i
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 20.0000i 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) −12.0000 20.7846i −0.403604 0.699062i
\(885\) 0 0
\(886\) −18.0000 + 31.1769i −0.604722 + 1.04741i
\(887\) 5.19615 + 3.00000i 0.174470 + 0.100730i 0.584692 0.811256i \(-0.301215\pi\)
−0.410222 + 0.911986i \(0.634549\pi\)
\(888\) 0 0
\(889\) 20.0000 + 34.6410i 0.670778 + 1.16182i
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000i 0.535720i
\(893\) −36.3731 + 21.0000i −1.21718 + 0.702738i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 1.73205i 0.0334077 0.0578638i
\(897\) 0 0
\(898\) 12.9904 7.50000i 0.433495 0.250278i
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) 0 0
\(904\) 1.50000 2.59808i 0.0498893 0.0864107i
\(905\) 0 0
\(906\) 0 0
\(907\) −6.06218 + 3.50000i −0.201291 + 0.116216i −0.597258 0.802049i \(-0.703743\pi\)
0.395966 + 0.918265i \(0.370410\pi\)
\(908\) 15.0000i 0.497792i
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00000 + 10.3923i 0.198789 + 0.344312i 0.948136 0.317865i \(-0.102966\pi\)
−0.749347 + 0.662177i \(0.769633\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 9.50000 16.4545i 0.314232 0.544266i
\(915\) 0 0
\(916\) 5.00000 + 8.66025i 0.165205 + 0.286143i
\(917\) 24.0000i 0.792550i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −5.19615 3.00000i −0.171126 0.0987997i
\(923\) 20.7846 + 12.0000i 0.684134 + 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 0 0
\(928\) 6.00000i 0.196960i
\(929\) 27.0000 + 46.7654i 0.885841 + 1.53432i 0.844746 + 0.535167i \(0.179751\pi\)
0.0410949 + 0.999155i \(0.486915\pi\)
\(930\) 0 0
\(931\) −10.5000 + 18.1865i −0.344124 + 0.596040i
\(932\) 2.59808 + 1.50000i 0.0851028 + 0.0491341i
\(933\) 0 0
\(934\) −7.50000 12.9904i −0.245407 0.425058i
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0000i 0.751377i 0.926746 + 0.375689i \(0.122594\pi\)
−0.926746 + 0.375689i \(0.877406\pi\)
\(938\) −22.5167 + 13.0000i −0.735195 + 0.424465i
\(939\) 0 0
\(940\) 0 0
\(941\) −27.0000 + 46.7654i −0.880175 + 1.52451i −0.0290288 + 0.999579i \(0.509241\pi\)
−0.851146 + 0.524929i \(0.824092\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 0 0
\(947\) 18.1865 10.5000i 0.590983 0.341204i −0.174503 0.984657i \(-0.555832\pi\)
0.765486 + 0.643452i \(0.222499\pi\)
\(948\) 0 0
\(949\) 2.00000 3.46410i 0.0649227 0.112449i
\(950\) 0 0
\(951\) 0 0
\(952\) −10.3923 + 6.00000i −0.336817 + 0.194461i
\(953\) 30.0000i 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 31.1769 + 18.0000i 1.00728 + 0.581554i
\(959\) 3.00000 5.19615i 0.0968751 0.167793i
\(960\) 0 0
\(961\) −34.5000 59.7558i −1.11290 1.92760i
\(962\) 8.00000i 0.257930i
\(963\) 0 0
\(964\) 19.0000 0.611949
\(965\) 0 0
\(966\) 0 0
\(967\) −17.3205 10.0000i −0.556990 0.321578i 0.194946 0.980814i \(-0.437547\pi\)
−0.751936 + 0.659236i \(0.770880\pi\)
\(968\) 9.52628 + 5.50000i 0.306186 + 0.176777i
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) −1.00000 1.73205i −0.0320421 0.0554985i
\(975\) 0 0
\(976\) 2.00000 3.46410i 0.0640184 0.110883i
\(977\) −12.9904 7.50000i −0.415599 0.239946i 0.277594 0.960699i \(-0.410463\pi\)
−0.693193 + 0.720752i \(0.743796\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 3.00000i 0.0957338i
\(983\) −25.9808 + 15.0000i −0.828658 + 0.478426i −0.853393 0.521268i \(-0.825459\pi\)
0.0247352 + 0.999694i \(0.492126\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 31.1769i 0.573237 0.992875i
\(987\) 0 0
\(988\) −24.2487 + 14.0000i −0.771454 + 0.445399i
\(989\) 0 0
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 8.66025 5.00000i 0.274963 0.158750i
\(993\) 0 0
\(994\) 6.00000 10.3923i 0.190308 0.329624i
\(995\) 0 0
\(996\) 0 0
\(997\) −24.2487 + 14.0000i −0.767964 + 0.443384i −0.832148 0.554554i \(-0.812889\pi\)
0.0641836 + 0.997938i \(0.479556\pi\)
\(998\) 11.0000i 0.348199i
\(999\) 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 1350.2.j.b.199.1 4
3.2 odd 2 450.2.j.b.49.2 4
5.2 odd 4 1350.2.e.d.901.1 2
5.3 odd 4 1350.2.e.h.901.1 2
5.4 even 2 inner 1350.2.j.b.199.2 4
9.2 odd 6 450.2.j.b.349.1 4
9.4 even 3 4050.2.c.m.649.1 2
9.5 odd 6 4050.2.c.h.649.2 2
9.7 even 3 inner 1350.2.j.b.1099.2 4
15.2 even 4 450.2.e.f.301.1 yes 2
15.8 even 4 450.2.e.c.301.1 yes 2
15.14 odd 2 450.2.j.b.49.1 4
45.2 even 12 450.2.e.f.151.1 yes 2
45.4 even 6 4050.2.c.m.649.2 2
45.7 odd 12 1350.2.e.d.451.1 2
45.13 odd 12 4050.2.a.o.1.1 1
45.14 odd 6 4050.2.c.h.649.1 2
45.22 odd 12 4050.2.a.u.1.1 1
45.23 even 12 4050.2.a.bg.1.1 1
45.29 odd 6 450.2.j.b.349.2 4
45.32 even 12 4050.2.a.d.1.1 1
45.34 even 6 inner 1350.2.j.b.1099.1 4
45.38 even 12 450.2.e.c.151.1 2
45.43 odd 12 1350.2.e.h.451.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.2.e.c.151.1 2 45.38 even 12
450.2.e.c.301.1 yes 2 15.8 even 4
450.2.e.f.151.1 yes 2 45.2 even 12
450.2.e.f.301.1 yes 2 15.2 even 4
450.2.j.b.49.1 4 15.14 odd 2
450.2.j.b.49.2 4 3.2 odd 2
450.2.j.b.349.1 4 9.2 odd 6
450.2.j.b.349.2 4 45.29 odd 6
1350.2.e.d.451.1 2 45.7 odd 12
1350.2.e.d.901.1 2 5.2 odd 4
1350.2.e.h.451.1 2 45.43 odd 12
1350.2.e.h.901.1 2 5.3 odd 4
1350.2.j.b.199.1 4 1.1 even 1 trivial
1350.2.j.b.199.2 4 5.4 even 2 inner
1350.2.j.b.1099.1 4 45.34 even 6 inner
1350.2.j.b.1099.2 4 9.7 even 3 inner
4050.2.a.d.1.1 1 45.32 even 12
4050.2.a.o.1.1 1 45.13 odd 12
4050.2.a.u.1.1 1 45.22 odd 12
4050.2.a.bg.1.1 1 45.23 even 12
4050.2.c.h.649.1 2 45.14 odd 6
4050.2.c.h.649.2 2 9.5 odd 6
4050.2.c.m.649.1 2 9.4 even 3
4050.2.c.m.649.2 2 45.4 even 6