Properties

Label 1305.2.c.f.784.2
Level $1305$
Weight $2$
Character 1305.784
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(784,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (of order \(2\), degree \(1\), 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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 784.2
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 1305.784
Dual form 1305.2.c.f.784.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.517638i q^{2} +1.73205 q^{4} +(1.73205 + 1.41421i) q^{5} -2.44949i q^{7} -1.93185i q^{8} +(0.732051 - 0.896575i) q^{10} +1.26795 q^{11} +1.79315i q^{13} -1.26795 q^{14} +2.46410 q^{16} -1.41421i q^{17} +3.26795 q^{19} +(3.00000 + 2.44949i) q^{20} -0.656339i q^{22} -6.31319i q^{23} +(1.00000 + 4.89898i) q^{25} +0.928203 q^{26} -4.24264i q^{28} +1.00000 q^{29} -8.73205 q^{31} -5.13922i q^{32} -0.732051 q^{34} +(3.46410 - 4.24264i) q^{35} +9.14162i q^{37} -1.69161i q^{38} +(2.73205 - 3.34607i) q^{40} +6.92820 q^{41} -9.14162i q^{43} +2.19615 q^{44} -3.26795 q^{46} -1.41421i q^{47} +1.00000 q^{49} +(2.53590 - 0.517638i) q^{50} +3.10583i q^{52} +5.93426i q^{53} +(2.19615 + 1.79315i) q^{55} -4.73205 q^{56} -0.517638i q^{58} -10.3923 q^{59} +2.92820 q^{61} +4.52004i q^{62} +2.26795 q^{64} +(-2.53590 + 3.10583i) q^{65} +4.24264i q^{67} -2.44949i q^{68} +(-2.19615 - 1.79315i) q^{70} -3.46410 q^{71} +7.34847i q^{73} +4.73205 q^{74} +5.66025 q^{76} -3.10583i q^{77} +4.19615 q^{79} +(4.26795 + 3.48477i) q^{80} -3.58630i q^{82} +10.1769i q^{83} +(2.00000 - 2.44949i) q^{85} -4.73205 q^{86} -2.44949i q^{88} +10.3923 q^{89} +4.39230 q^{91} -10.9348i q^{92} -0.732051 q^{94} +(5.66025 + 4.62158i) q^{95} -10.9348i q^{97} -0.517638i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{10} + 12 q^{11} - 12 q^{14} - 4 q^{16} + 20 q^{19} + 12 q^{20} + 4 q^{25} - 24 q^{26} + 4 q^{29} - 28 q^{31} + 4 q^{34} + 4 q^{40} - 12 q^{44} - 20 q^{46} + 4 q^{49} + 24 q^{50} - 12 q^{55} - 12 q^{56}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638i 0.366025i −0.983111 0.183013i \(-0.941415\pi\)
0.983111 0.183013i \(-0.0585849\pi\)
\(3\) 0 0
\(4\) 1.73205 0.866025
\(5\) 1.73205 + 1.41421i 0.774597 + 0.632456i
\(6\) 0 0
\(7\) 2.44949i 0.925820i −0.886405 0.462910i \(-0.846805\pi\)
0.886405 0.462910i \(-0.153195\pi\)
\(8\) 1.93185i 0.683013i
\(9\) 0 0
\(10\) 0.732051 0.896575i 0.231495 0.283522i
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 1.79315i 0.497331i 0.968589 + 0.248665i \(0.0799919\pi\)
−0.968589 + 0.248665i \(0.920008\pi\)
\(14\) −1.26795 −0.338874
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 1.41421i 0.342997i −0.985184 0.171499i \(-0.945139\pi\)
0.985184 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) 3.26795 0.749719 0.374859 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(20\) 3.00000 + 2.44949i 0.670820 + 0.547723i
\(21\) 0 0
\(22\) 0.656339i 0.139932i
\(23\) 6.31319i 1.31639i −0.752847 0.658196i \(-0.771320\pi\)
0.752847 0.658196i \(-0.228680\pi\)
\(24\) 0 0
\(25\) 1.00000 + 4.89898i 0.200000 + 0.979796i
\(26\) 0.928203 0.182036
\(27\) 0 0
\(28\) 4.24264i 0.801784i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.73205 −1.56832 −0.784161 0.620557i \(-0.786907\pi\)
−0.784161 + 0.620557i \(0.786907\pi\)
\(32\) 5.13922i 0.908494i
\(33\) 0 0
\(34\) −0.732051 −0.125546
\(35\) 3.46410 4.24264i 0.585540 0.717137i
\(36\) 0 0
\(37\) 9.14162i 1.50287i 0.659805 + 0.751437i \(0.270639\pi\)
−0.659805 + 0.751437i \(0.729361\pi\)
\(38\) 1.69161i 0.274416i
\(39\) 0 0
\(40\) 2.73205 3.34607i 0.431975 0.529059i
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 9.14162i 1.39408i −0.717030 0.697042i \(-0.754499\pi\)
0.717030 0.697042i \(-0.245501\pi\)
\(44\) 2.19615 0.331082
\(45\) 0 0
\(46\) −3.26795 −0.481833
\(47\) 1.41421i 0.206284i −0.994667 0.103142i \(-0.967110\pi\)
0.994667 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.53590 0.517638i 0.358630 0.0732051i
\(51\) 0 0
\(52\) 3.10583i 0.430701i
\(53\) 5.93426i 0.815133i 0.913176 + 0.407566i \(0.133622\pi\)
−0.913176 + 0.407566i \(0.866378\pi\)
\(54\) 0 0
\(55\) 2.19615 + 1.79315i 0.296129 + 0.241788i
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) 0.517638i 0.0679692i
\(59\) −10.3923 −1.35296 −0.676481 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 2.92820 0.374918 0.187459 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) 4.52004i 0.574046i
\(63\) 0 0
\(64\) 2.26795 0.283494
\(65\) −2.53590 + 3.10583i −0.314539 + 0.385231i
\(66\) 0 0
\(67\) 4.24264i 0.518321i 0.965834 + 0.259161i \(0.0834459\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) 2.44949i 0.297044i
\(69\) 0 0
\(70\) −2.19615 1.79315i −0.262490 0.214323i
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 7.34847i 0.860073i 0.902811 + 0.430037i \(0.141499\pi\)
−0.902811 + 0.430037i \(0.858501\pi\)
\(74\) 4.73205 0.550090
\(75\) 0 0
\(76\) 5.66025 0.649276
\(77\) 3.10583i 0.353942i
\(78\) 0 0
\(79\) 4.19615 0.472104 0.236052 0.971740i \(-0.424146\pi\)
0.236052 + 0.971740i \(0.424146\pi\)
\(80\) 4.26795 + 3.48477i 0.477171 + 0.389609i
\(81\) 0 0
\(82\) 3.58630i 0.396041i
\(83\) 10.1769i 1.11706i 0.829484 + 0.558530i \(0.188634\pi\)
−0.829484 + 0.558530i \(0.811366\pi\)
\(84\) 0 0
\(85\) 2.00000 2.44949i 0.216930 0.265684i
\(86\) −4.73205 −0.510270
\(87\) 0 0
\(88\) 2.44949i 0.261116i
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 4.39230 0.460439
\(92\) 10.9348i 1.14003i
\(93\) 0 0
\(94\) −0.732051 −0.0755053
\(95\) 5.66025 + 4.62158i 0.580730 + 0.474164i
\(96\) 0 0
\(97\) 10.9348i 1.11026i −0.831764 0.555129i \(-0.812669\pi\)
0.831764 0.555129i \(-0.187331\pi\)
\(98\) 0.517638i 0.0522893i
\(99\) 0 0
\(100\) 1.73205 + 8.48528i 0.173205 + 0.848528i
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 9.14162i 0.900751i −0.892839 0.450375i \(-0.851290\pi\)
0.892839 0.450375i \(-0.148710\pi\)
\(104\) 3.46410 0.339683
\(105\) 0 0
\(106\) 3.07180 0.298359
\(107\) 18.6622i 1.80414i 0.431589 + 0.902070i \(0.357953\pi\)
−0.431589 + 0.902070i \(0.642047\pi\)
\(108\) 0 0
\(109\) −17.8564 −1.71033 −0.855167 0.518353i \(-0.826545\pi\)
−0.855167 + 0.518353i \(0.826545\pi\)
\(110\) 0.928203 1.13681i 0.0885007 0.108391i
\(111\) 0 0
\(112\) 6.03579i 0.570329i
\(113\) 13.0053i 1.22344i −0.791075 0.611719i \(-0.790478\pi\)
0.791075 0.611719i \(-0.209522\pi\)
\(114\) 0 0
\(115\) 8.92820 10.9348i 0.832559 1.01967i
\(116\) 1.73205 0.160817
\(117\) 0 0
\(118\) 5.37945i 0.495219i
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 1.51575i 0.137230i
\(123\) 0 0
\(124\) −15.1244 −1.35821
\(125\) −5.19615 + 9.89949i −0.464758 + 0.885438i
\(126\) 0 0
\(127\) 4.24264i 0.376473i 0.982124 + 0.188237i \(0.0602772\pi\)
−0.982124 + 0.188237i \(0.939723\pi\)
\(128\) 11.4524i 1.01226i
\(129\) 0 0
\(130\) 1.60770 + 1.31268i 0.141004 + 0.115129i
\(131\) 4.73205 0.413441 0.206721 0.978400i \(-0.433721\pi\)
0.206721 + 0.978400i \(0.433721\pi\)
\(132\) 0 0
\(133\) 8.00481i 0.694105i
\(134\) 2.19615 0.189719
\(135\) 0 0
\(136\) −2.73205 −0.234271
\(137\) 14.7985i 1.26432i −0.774838 0.632159i \(-0.782169\pi\)
0.774838 0.632159i \(-0.217831\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 6.00000 7.34847i 0.507093 0.621059i
\(141\) 0 0
\(142\) 1.79315i 0.150478i
\(143\) 2.27362i 0.190130i
\(144\) 0 0
\(145\) 1.73205 + 1.41421i 0.143839 + 0.117444i
\(146\) 3.80385 0.314809
\(147\) 0 0
\(148\) 15.8338i 1.30153i
\(149\) −0.928203 −0.0760414 −0.0380207 0.999277i \(-0.512105\pi\)
−0.0380207 + 0.999277i \(0.512105\pi\)
\(150\) 0 0
\(151\) −18.7846 −1.52867 −0.764335 0.644819i \(-0.776933\pi\)
−0.764335 + 0.644819i \(0.776933\pi\)
\(152\) 6.31319i 0.512068i
\(153\) 0 0
\(154\) −1.60770 −0.129552
\(155\) −15.1244 12.3490i −1.21482 0.991894i
\(156\) 0 0
\(157\) 10.9348i 0.872690i 0.899780 + 0.436345i \(0.143727\pi\)
−0.899780 + 0.436345i \(0.856273\pi\)
\(158\) 2.17209i 0.172802i
\(159\) 0 0
\(160\) 7.26795 8.90138i 0.574582 0.703716i
\(161\) −15.4641 −1.21874
\(162\) 0 0
\(163\) 14.0406i 1.09974i −0.835249 0.549872i \(-0.814676\pi\)
0.835249 0.549872i \(-0.185324\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 5.26795 0.408872
\(167\) 11.2122i 0.867624i −0.901003 0.433812i \(-0.857168\pi\)
0.901003 0.433812i \(-0.142832\pi\)
\(168\) 0 0
\(169\) 9.78461 0.752662
\(170\) −1.26795 1.03528i −0.0972473 0.0794021i
\(171\) 0 0
\(172\) 15.8338i 1.20731i
\(173\) 4.62158i 0.351372i 0.984446 + 0.175686i \(0.0562144\pi\)
−0.984446 + 0.175686i \(0.943786\pi\)
\(174\) 0 0
\(175\) 12.0000 2.44949i 0.907115 0.185164i
\(176\) 3.12436 0.235507
\(177\) 0 0
\(178\) 5.37945i 0.403207i
\(179\) 2.53590 0.189542 0.0947710 0.995499i \(-0.469788\pi\)
0.0947710 + 0.995499i \(0.469788\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 2.27362i 0.168532i
\(183\) 0 0
\(184\) −12.1962 −0.899112
\(185\) −12.9282 + 15.8338i −0.950500 + 1.16412i
\(186\) 0 0
\(187\) 1.79315i 0.131128i
\(188\) 2.44949i 0.178647i
\(189\) 0 0
\(190\) 2.39230 2.92996i 0.173556 0.212562i
\(191\) −15.1244 −1.09436 −0.547180 0.837015i \(-0.684299\pi\)
−0.547180 + 0.837015i \(0.684299\pi\)
\(192\) 0 0
\(193\) 0.656339i 0.0472443i 0.999721 + 0.0236222i \(0.00751986\pi\)
−0.999721 + 0.0236222i \(0.992480\pi\)
\(194\) −5.66025 −0.406383
\(195\) 0 0
\(196\) 1.73205 0.123718
\(197\) 22.4243i 1.59767i 0.601551 + 0.798834i \(0.294550\pi\)
−0.601551 + 0.798834i \(0.705450\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 9.46410 1.93185i 0.669213 0.136603i
\(201\) 0 0
\(202\) 3.10583i 0.218525i
\(203\) 2.44949i 0.171920i
\(204\) 0 0
\(205\) 12.0000 + 9.79796i 0.838116 + 0.684319i
\(206\) −4.73205 −0.329698
\(207\) 0 0
\(208\) 4.41851i 0.306368i
\(209\) 4.14359 0.286618
\(210\) 0 0
\(211\) −14.7321 −1.01420 −0.507098 0.861888i \(-0.669282\pi\)
−0.507098 + 0.861888i \(0.669282\pi\)
\(212\) 10.2784i 0.705926i
\(213\) 0 0
\(214\) 9.66025 0.660361
\(215\) 12.9282 15.8338i 0.881696 1.07985i
\(216\) 0 0
\(217\) 21.3891i 1.45198i
\(218\) 9.24316i 0.626026i
\(219\) 0 0
\(220\) 3.80385 + 3.10583i 0.256455 + 0.209395i
\(221\) 2.53590 0.170583
\(222\) 0 0
\(223\) 15.3533i 1.02813i 0.857751 + 0.514066i \(0.171861\pi\)
−0.857751 + 0.514066i \(0.828139\pi\)
\(224\) −12.5885 −0.841102
\(225\) 0 0
\(226\) −6.73205 −0.447809
\(227\) 20.4553i 1.35767i 0.734292 + 0.678834i \(0.237514\pi\)
−0.734292 + 0.678834i \(0.762486\pi\)
\(228\) 0 0
\(229\) −18.7846 −1.24132 −0.620661 0.784079i \(-0.713136\pi\)
−0.620661 + 0.784079i \(0.713136\pi\)
\(230\) −5.66025 4.62158i −0.373226 0.304738i
\(231\) 0 0
\(232\) 1.93185i 0.126832i
\(233\) 3.38323i 0.221643i −0.993840 0.110821i \(-0.964652\pi\)
0.993840 0.110821i \(-0.0353481\pi\)
\(234\) 0 0
\(235\) 2.00000 2.44949i 0.130466 0.159787i
\(236\) −18.0000 −1.17170
\(237\) 0 0
\(238\) 1.79315i 0.116233i
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 11.4641 0.738468 0.369234 0.929337i \(-0.379620\pi\)
0.369234 + 0.929337i \(0.379620\pi\)
\(242\) 4.86181i 0.312529i
\(243\) 0 0
\(244\) 5.07180 0.324689
\(245\) 1.73205 + 1.41421i 0.110657 + 0.0903508i
\(246\) 0 0
\(247\) 5.85993i 0.372858i
\(248\) 16.8690i 1.07118i
\(249\) 0 0
\(250\) 5.12436 + 2.68973i 0.324093 + 0.170113i
\(251\) −15.1244 −0.954641 −0.477320 0.878729i \(-0.658392\pi\)
−0.477320 + 0.878729i \(0.658392\pi\)
\(252\) 0 0
\(253\) 8.00481i 0.503258i
\(254\) 2.19615 0.137799
\(255\) 0 0
\(256\) −1.39230 −0.0870191
\(257\) 9.52056i 0.593876i 0.954897 + 0.296938i \(0.0959655\pi\)
−0.954897 + 0.296938i \(0.904035\pi\)
\(258\) 0 0
\(259\) 22.3923 1.39139
\(260\) −4.39230 + 5.37945i −0.272399 + 0.333620i
\(261\) 0 0
\(262\) 2.44949i 0.151330i
\(263\) 4.79744i 0.295823i 0.989001 + 0.147912i \(0.0472551\pi\)
−0.989001 + 0.147912i \(0.952745\pi\)
\(264\) 0 0
\(265\) −8.39230 + 10.2784i −0.515535 + 0.631399i
\(266\) −4.14359 −0.254060
\(267\) 0 0
\(268\) 7.34847i 0.448879i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −26.7321 −1.62386 −0.811928 0.583757i \(-0.801582\pi\)
−0.811928 + 0.583757i \(0.801582\pi\)
\(272\) 3.48477i 0.211295i
\(273\) 0 0
\(274\) −7.66025 −0.462773
\(275\) 1.26795 + 6.21166i 0.0764602 + 0.374577i
\(276\) 0 0
\(277\) 18.7637i 1.12740i 0.825979 + 0.563701i \(0.190623\pi\)
−0.825979 + 0.563701i \(0.809377\pi\)
\(278\) 4.14110i 0.248367i
\(279\) 0 0
\(280\) −8.19615 6.69213i −0.489814 0.399931i
\(281\) −32.7846 −1.95577 −0.977883 0.209153i \(-0.932929\pi\)
−0.977883 + 0.209153i \(0.932929\pi\)
\(282\) 0 0
\(283\) 27.4249i 1.63024i 0.579293 + 0.815119i \(0.303329\pi\)
−0.579293 + 0.815119i \(0.696671\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 1.17691 0.0695924
\(287\) 16.9706i 1.00174i
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0.732051 0.896575i 0.0429875 0.0526487i
\(291\) 0 0
\(292\) 12.7279i 0.744845i
\(293\) 30.7338i 1.79549i 0.440520 + 0.897743i \(0.354794\pi\)
−0.440520 + 0.897743i \(0.645206\pi\)
\(294\) 0 0
\(295\) −18.0000 14.6969i −1.04800 0.855689i
\(296\) 17.6603 1.02648
\(297\) 0 0
\(298\) 0.480473i 0.0278331i
\(299\) 11.3205 0.654682
\(300\) 0 0
\(301\) −22.3923 −1.29067
\(302\) 9.72363i 0.559532i
\(303\) 0 0
\(304\) 8.05256 0.461846
\(305\) 5.07180 + 4.14110i 0.290410 + 0.237119i
\(306\) 0 0
\(307\) 10.4543i 0.596658i 0.954463 + 0.298329i \(0.0964293\pi\)
−0.954463 + 0.298329i \(0.903571\pi\)
\(308\) 5.37945i 0.306523i
\(309\) 0 0
\(310\) −6.39230 + 7.82894i −0.363059 + 0.444654i
\(311\) 6.58846 0.373597 0.186799 0.982398i \(-0.440189\pi\)
0.186799 + 0.982398i \(0.440189\pi\)
\(312\) 0 0
\(313\) 23.6627i 1.33749i 0.743490 + 0.668747i \(0.233169\pi\)
−0.743490 + 0.668747i \(0.766831\pi\)
\(314\) 5.66025 0.319427
\(315\) 0 0
\(316\) 7.26795 0.408854
\(317\) 5.75839i 0.323423i 0.986838 + 0.161712i \(0.0517015\pi\)
−0.986838 + 0.161712i \(0.948299\pi\)
\(318\) 0 0
\(319\) 1.26795 0.0709915
\(320\) 3.92820 + 3.20736i 0.219593 + 0.179297i
\(321\) 0 0
\(322\) 8.00481i 0.446091i
\(323\) 4.62158i 0.257151i
\(324\) 0 0
\(325\) −8.78461 + 1.79315i −0.487282 + 0.0994661i
\(326\) −7.26795 −0.402534
\(327\) 0 0
\(328\) 13.3843i 0.739022i
\(329\) −3.46410 −0.190982
\(330\) 0 0
\(331\) −18.1962 −1.00015 −0.500075 0.865982i \(-0.666694\pi\)
−0.500075 + 0.865982i \(0.666694\pi\)
\(332\) 17.6269i 0.967402i
\(333\) 0 0
\(334\) −5.80385 −0.317572
\(335\) −6.00000 + 7.34847i −0.327815 + 0.401490i
\(336\) 0 0
\(337\) 27.9053i 1.52010i −0.649864 0.760050i \(-0.725174\pi\)
0.649864 0.760050i \(-0.274826\pi\)
\(338\) 5.06489i 0.275494i
\(339\) 0 0
\(340\) 3.46410 4.24264i 0.187867 0.230089i
\(341\) −11.0718 −0.599571
\(342\) 0 0
\(343\) 19.5959i 1.05808i
\(344\) −17.6603 −0.952177
\(345\) 0 0
\(346\) 2.39230 0.128611
\(347\) 21.4906i 1.15368i −0.816859 0.576838i \(-0.804286\pi\)
0.816859 0.576838i \(-0.195714\pi\)
\(348\) 0 0
\(349\) 22.7846 1.21963 0.609816 0.792543i \(-0.291243\pi\)
0.609816 + 0.792543i \(0.291243\pi\)
\(350\) −1.26795 6.21166i −0.0677747 0.332027i
\(351\) 0 0
\(352\) 6.51626i 0.347318i
\(353\) 31.9449i 1.70026i −0.526576 0.850128i \(-0.676525\pi\)
0.526576 0.850128i \(-0.323475\pi\)
\(354\) 0 0
\(355\) −6.00000 4.89898i −0.318447 0.260011i
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 1.31268i 0.0693772i
\(359\) −7.26795 −0.383588 −0.191794 0.981435i \(-0.561430\pi\)
−0.191794 + 0.981435i \(0.561430\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) 8.28221i 0.435303i
\(363\) 0 0
\(364\) 7.60770 0.398752
\(365\) −10.3923 + 12.7279i −0.543958 + 0.666210i
\(366\) 0 0
\(367\) 33.6365i 1.75581i −0.478833 0.877906i \(-0.658940\pi\)
0.478833 0.877906i \(-0.341060\pi\)
\(368\) 15.5563i 0.810931i
\(369\) 0 0
\(370\) 8.19615 + 6.69213i 0.426098 + 0.347907i
\(371\) 14.5359 0.754666
\(372\) 0 0
\(373\) 19.5959i 1.01464i 0.861758 + 0.507319i \(0.169363\pi\)
−0.861758 + 0.507319i \(0.830637\pi\)
\(374\) −0.928203 −0.0479962
\(375\) 0 0
\(376\) −2.73205 −0.140895
\(377\) 1.79315i 0.0923520i
\(378\) 0 0
\(379\) 16.1962 0.831940 0.415970 0.909378i \(-0.363442\pi\)
0.415970 + 0.909378i \(0.363442\pi\)
\(380\) 9.80385 + 8.00481i 0.502927 + 0.410638i
\(381\) 0 0
\(382\) 7.82894i 0.400564i
\(383\) 0.933740i 0.0477119i −0.999715 0.0238559i \(-0.992406\pi\)
0.999715 0.0238559i \(-0.00759430\pi\)
\(384\) 0 0
\(385\) 4.39230 5.37945i 0.223853 0.274162i
\(386\) 0.339746 0.0172926
\(387\) 0 0
\(388\) 18.9396i 0.961511i
\(389\) −20.7846 −1.05382 −0.526911 0.849921i \(-0.676650\pi\)
−0.526911 + 0.849921i \(0.676650\pi\)
\(390\) 0 0
\(391\) −8.92820 −0.451519
\(392\) 1.93185i 0.0975732i
\(393\) 0 0
\(394\) 11.6077 0.584787
\(395\) 7.26795 + 5.93426i 0.365690 + 0.298585i
\(396\) 0 0
\(397\) 4.89898i 0.245873i 0.992415 + 0.122936i \(0.0392311\pi\)
−0.992415 + 0.122936i \(0.960769\pi\)
\(398\) 7.24693i 0.363256i
\(399\) 0 0
\(400\) 2.46410 + 12.0716i 0.123205 + 0.603579i
\(401\) −37.8564 −1.89046 −0.945229 0.326407i \(-0.894162\pi\)
−0.945229 + 0.326407i \(0.894162\pi\)
\(402\) 0 0
\(403\) 15.6579i 0.779975i
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) −1.26795 −0.0629273
\(407\) 11.5911i 0.574550i
\(408\) 0 0
\(409\) −9.07180 −0.448571 −0.224286 0.974523i \(-0.572005\pi\)
−0.224286 + 0.974523i \(0.572005\pi\)
\(410\) 5.07180 6.21166i 0.250478 0.306772i
\(411\) 0 0
\(412\) 15.8338i 0.780073i
\(413\) 25.4558i 1.25260i
\(414\) 0 0
\(415\) −14.3923 + 17.6269i −0.706490 + 0.865271i
\(416\) 9.21539 0.451822
\(417\) 0 0
\(418\) 2.14488i 0.104910i
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 5.46410 0.266304 0.133152 0.991096i \(-0.457490\pi\)
0.133152 + 0.991096i \(0.457490\pi\)
\(422\) 7.62587i 0.371222i
\(423\) 0 0
\(424\) 11.4641 0.556746
\(425\) 6.92820 1.41421i 0.336067 0.0685994i
\(426\) 0 0
\(427\) 7.17260i 0.347107i
\(428\) 32.3238i 1.56243i
\(429\) 0 0
\(430\) −8.19615 6.69213i −0.395254 0.322723i
\(431\) 37.1769 1.79075 0.895374 0.445314i \(-0.146908\pi\)
0.895374 + 0.445314i \(0.146908\pi\)
\(432\) 0 0
\(433\) 2.92996i 0.140805i 0.997519 + 0.0704025i \(0.0224284\pi\)
−0.997519 + 0.0704025i \(0.977572\pi\)
\(434\) 11.0718 0.531463
\(435\) 0 0
\(436\) −30.9282 −1.48119
\(437\) 20.6312i 0.986924i
\(438\) 0 0
\(439\) 7.07180 0.337518 0.168759 0.985657i \(-0.446024\pi\)
0.168759 + 0.985657i \(0.446024\pi\)
\(440\) 3.46410 4.24264i 0.165145 0.202260i
\(441\) 0 0
\(442\) 1.31268i 0.0624377i
\(443\) 18.1817i 0.863839i 0.901912 + 0.431919i \(0.142164\pi\)
−0.901912 + 0.431919i \(0.857836\pi\)
\(444\) 0 0
\(445\) 18.0000 + 14.6969i 0.853282 + 0.696702i
\(446\) 7.94744 0.376322
\(447\) 0 0
\(448\) 5.55532i 0.262464i
\(449\) −25.8564 −1.22024 −0.610120 0.792309i \(-0.708879\pi\)
−0.610120 + 0.792309i \(0.708879\pi\)
\(450\) 0 0
\(451\) 8.78461 0.413651
\(452\) 22.5259i 1.05953i
\(453\) 0 0
\(454\) 10.5885 0.496941
\(455\) 7.60770 + 6.21166i 0.356654 + 0.291207i
\(456\) 0 0
\(457\) 23.6627i 1.10689i −0.832884 0.553447i \(-0.813312\pi\)
0.832884 0.553447i \(-0.186688\pi\)
\(458\) 9.72363i 0.454355i
\(459\) 0 0
\(460\) 15.4641 18.9396i 0.721017 0.883062i
\(461\) −24.9282 −1.16102 −0.580511 0.814252i \(-0.697147\pi\)
−0.580511 + 0.814252i \(0.697147\pi\)
\(462\) 0 0
\(463\) 6.86800i 0.319183i −0.987183 0.159591i \(-0.948982\pi\)
0.987183 0.159591i \(-0.0510177\pi\)
\(464\) 2.46410 0.114393
\(465\) 0 0
\(466\) −1.75129 −0.0811269
\(467\) 21.0101i 0.972233i −0.873894 0.486116i \(-0.838413\pi\)
0.873894 0.486116i \(-0.161587\pi\)
\(468\) 0 0
\(469\) 10.3923 0.479872
\(470\) −1.26795 1.03528i −0.0584861 0.0477537i
\(471\) 0 0
\(472\) 20.0764i 0.924091i
\(473\) 11.5911i 0.532960i
\(474\) 0 0
\(475\) 3.26795 + 16.0096i 0.149944 + 0.734572i
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 5.37945i 0.246050i
\(479\) 5.66025 0.258624 0.129312 0.991604i \(-0.458723\pi\)
0.129312 + 0.991604i \(0.458723\pi\)
\(480\) 0 0
\(481\) −16.3923 −0.747425
\(482\) 5.93426i 0.270298i
\(483\) 0 0
\(484\) −16.2679 −0.739452
\(485\) 15.4641 18.9396i 0.702189 0.860002i
\(486\) 0 0
\(487\) 10.9348i 0.495502i −0.968824 0.247751i \(-0.920309\pi\)
0.968824 0.247751i \(-0.0796915\pi\)
\(488\) 5.65685i 0.256074i
\(489\) 0 0
\(490\) 0.732051 0.896575i 0.0330707 0.0405032i
\(491\) −7.26795 −0.327998 −0.163999 0.986461i \(-0.552439\pi\)
−0.163999 + 0.986461i \(0.552439\pi\)
\(492\) 0 0
\(493\) 1.41421i 0.0636930i
\(494\) 3.03332 0.136476
\(495\) 0 0
\(496\) −21.5167 −0.966127
\(497\) 8.48528i 0.380617i
\(498\) 0 0
\(499\) 1.07180 0.0479802 0.0239901 0.999712i \(-0.492363\pi\)
0.0239901 + 0.999712i \(0.492363\pi\)
\(500\) −9.00000 + 17.1464i −0.402492 + 0.766812i
\(501\) 0 0
\(502\) 7.82894i 0.349423i
\(503\) 12.1731i 0.542773i −0.962471 0.271386i \(-0.912518\pi\)
0.962471 0.271386i \(-0.0874821\pi\)
\(504\) 0 0
\(505\) 10.3923 + 8.48528i 0.462451 + 0.377590i
\(506\) −4.14359 −0.184205
\(507\) 0 0
\(508\) 7.34847i 0.326036i
\(509\) 33.4641 1.48327 0.741635 0.670804i \(-0.234051\pi\)
0.741635 + 0.670804i \(0.234051\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 22.1841i 0.980408i
\(513\) 0 0
\(514\) 4.92820 0.217374
\(515\) 12.9282 15.8338i 0.569685 0.697718i
\(516\) 0 0
\(517\) 1.79315i 0.0788627i
\(518\) 11.5911i 0.509284i
\(519\) 0 0
\(520\) 6.00000 + 4.89898i 0.263117 + 0.214834i
\(521\) −4.39230 −0.192430 −0.0962152 0.995361i \(-0.530674\pi\)
−0.0962152 + 0.995361i \(0.530674\pi\)
\(522\) 0 0
\(523\) 1.61729i 0.0707190i −0.999375 0.0353595i \(-0.988742\pi\)
0.999375 0.0353595i \(-0.0112576\pi\)
\(524\) 8.19615 0.358051
\(525\) 0 0
\(526\) 2.48334 0.108279
\(527\) 12.3490i 0.537930i
\(528\) 0 0
\(529\) −16.8564 −0.732887
\(530\) 5.32051 + 4.34418i 0.231108 + 0.188699i
\(531\) 0 0
\(532\) 13.8647i 0.601112i
\(533\) 12.4233i 0.538113i
\(534\) 0 0
\(535\) −26.3923 + 32.3238i −1.14104 + 1.39748i
\(536\) 8.19615 0.354020
\(537\) 0 0
\(538\) 6.21166i 0.267804i
\(539\) 1.26795 0.0546144
\(540\) 0 0
\(541\) 42.3923 1.82259 0.911294 0.411757i \(-0.135085\pi\)
0.911294 + 0.411757i \(0.135085\pi\)
\(542\) 13.8375i 0.594373i
\(543\) 0 0
\(544\) −7.26795 −0.311611
\(545\) −30.9282 25.2528i −1.32482 1.08171i
\(546\) 0 0
\(547\) 2.44949i 0.104733i −0.998628 0.0523663i \(-0.983324\pi\)
0.998628 0.0523663i \(-0.0166763\pi\)
\(548\) 25.6317i 1.09493i
\(549\) 0 0
\(550\) 3.21539 0.656339i 0.137105 0.0279864i
\(551\) 3.26795 0.139219
\(552\) 0 0
\(553\) 10.2784i 0.437083i
\(554\) 9.71281 0.412658
\(555\) 0 0
\(556\) 13.8564 0.587643
\(557\) 0.554803i 0.0235078i 0.999931 + 0.0117539i \(0.00374146\pi\)
−0.999931 + 0.0117539i \(0.996259\pi\)
\(558\) 0 0
\(559\) 16.3923 0.693321
\(560\) 8.53590 10.4543i 0.360708 0.441775i
\(561\) 0 0
\(562\) 16.9706i 0.715860i
\(563\) 13.2827i 0.559800i 0.960029 + 0.279900i \(0.0903013\pi\)
−0.960029 + 0.279900i \(0.909699\pi\)
\(564\) 0 0
\(565\) 18.3923 22.5259i 0.773770 0.947671i
\(566\) 14.1962 0.596709
\(567\) 0 0
\(568\) 6.69213i 0.280796i
\(569\) 21.4641 0.899822 0.449911 0.893073i \(-0.351456\pi\)
0.449911 + 0.893073i \(0.351456\pi\)
\(570\) 0 0
\(571\) −39.3205 −1.64551 −0.822756 0.568395i \(-0.807565\pi\)
−0.822756 + 0.568395i \(0.807565\pi\)
\(572\) 3.93803i 0.164657i
\(573\) 0 0
\(574\) −8.78461 −0.366663
\(575\) 30.9282 6.31319i 1.28980 0.263278i
\(576\) 0 0
\(577\) 9.14162i 0.380571i 0.981729 + 0.190285i \(0.0609413\pi\)
−0.981729 + 0.190285i \(0.939059\pi\)
\(578\) 7.76457i 0.322964i
\(579\) 0 0
\(580\) 3.00000 + 2.44949i 0.124568 + 0.101710i
\(581\) 24.9282 1.03420
\(582\) 0 0
\(583\) 7.52433i 0.311626i
\(584\) 14.1962 0.587441
\(585\) 0 0
\(586\) 15.9090 0.657193
\(587\) 15.5563i 0.642079i 0.947066 + 0.321040i \(0.104032\pi\)
−0.947066 + 0.321040i \(0.895968\pi\)
\(588\) 0 0
\(589\) −28.5359 −1.17580
\(590\) −7.60770 + 9.31749i −0.313204 + 0.383595i
\(591\) 0 0
\(592\) 22.5259i 0.925808i
\(593\) 25.5302i 1.04840i 0.851596 + 0.524199i \(0.175635\pi\)
−0.851596 + 0.524199i \(0.824365\pi\)
\(594\) 0 0
\(595\) −6.00000 4.89898i −0.245976 0.200839i
\(596\) −1.60770 −0.0658538
\(597\) 0 0
\(598\) 5.85993i 0.239630i
\(599\) 1.26795 0.0518070 0.0259035 0.999664i \(-0.491754\pi\)
0.0259035 + 0.999664i \(0.491754\pi\)
\(600\) 0 0
\(601\) 16.7846 0.684659 0.342329 0.939580i \(-0.388784\pi\)
0.342329 + 0.939580i \(0.388784\pi\)
\(602\) 11.5911i 0.472418i
\(603\) 0 0
\(604\) −32.5359 −1.32387
\(605\) −16.2679 13.2827i −0.661386 0.540020i
\(606\) 0 0
\(607\) 16.3142i 0.662174i 0.943600 + 0.331087i \(0.107415\pi\)
−0.943600 + 0.331087i \(0.892585\pi\)
\(608\) 16.7947i 0.681115i
\(609\) 0 0
\(610\) 2.14359 2.62536i 0.0867916 0.106298i
\(611\) 2.53590 0.102591
\(612\) 0 0
\(613\) 18.2832i 0.738453i −0.929339 0.369227i \(-0.879623\pi\)
0.929339 0.369227i \(-0.120377\pi\)
\(614\) 5.41154 0.218392
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 32.0464i 1.29014i 0.764123 + 0.645071i \(0.223172\pi\)
−0.764123 + 0.645071i \(0.776828\pi\)
\(618\) 0 0
\(619\) 23.8038 0.956757 0.478379 0.878154i \(-0.341225\pi\)
0.478379 + 0.878154i \(0.341225\pi\)
\(620\) −26.1962 21.3891i −1.05206 0.859006i
\(621\) 0 0
\(622\) 3.41044i 0.136746i
\(623\) 25.4558i 1.01987i
\(624\) 0 0
\(625\) −23.0000 + 9.79796i −0.920000 + 0.391918i
\(626\) 12.2487 0.489557
\(627\) 0 0
\(628\) 18.9396i 0.755771i
\(629\) 12.9282 0.515481
\(630\) 0 0
\(631\) 15.6077 0.621333 0.310666 0.950519i \(-0.399448\pi\)
0.310666 + 0.950519i \(0.399448\pi\)
\(632\) 8.10634i 0.322453i
\(633\) 0 0
\(634\) 2.98076 0.118381
\(635\) −6.00000 + 7.34847i −0.238103 + 0.291615i
\(636\) 0 0
\(637\) 1.79315i 0.0710472i
\(638\) 0.656339i 0.0259847i
\(639\) 0 0
\(640\) 16.1962 19.8362i 0.640209 0.784093i
\(641\) −34.6410 −1.36824 −0.684119 0.729370i \(-0.739813\pi\)
−0.684119 + 0.729370i \(0.739813\pi\)
\(642\) 0 0
\(643\) 10.4543i 0.412277i 0.978523 + 0.206139i \(0.0660898\pi\)
−0.978523 + 0.206139i \(0.933910\pi\)
\(644\) −26.7846 −1.05546
\(645\) 0 0
\(646\) −2.39230 −0.0941240
\(647\) 14.3180i 0.562899i −0.959576 0.281449i \(-0.909185\pi\)
0.959576 0.281449i \(-0.0908151\pi\)
\(648\) 0 0
\(649\) −13.1769 −0.517239
\(650\) 0.928203 + 4.54725i 0.0364071 + 0.178358i
\(651\) 0 0
\(652\) 24.3190i 0.952407i
\(653\) 13.0053i 0.508938i −0.967081 0.254469i \(-0.918099\pi\)
0.967081 0.254469i \(-0.0819006\pi\)
\(654\) 0 0
\(655\) 8.19615 + 6.69213i 0.320250 + 0.261483i
\(656\) 17.0718 0.666542
\(657\) 0 0
\(658\) 1.79315i 0.0699043i
\(659\) 32.4449 1.26387 0.631936 0.775020i \(-0.282260\pi\)
0.631936 + 0.775020i \(0.282260\pi\)
\(660\) 0 0
\(661\) −0.784610 −0.0305178 −0.0152589 0.999884i \(-0.504857\pi\)
−0.0152589 + 0.999884i \(0.504857\pi\)
\(662\) 9.41902i 0.366081i
\(663\) 0 0
\(664\) 19.6603 0.762966
\(665\) 11.3205 13.8647i 0.438990 0.537651i
\(666\) 0 0
\(667\) 6.31319i 0.244448i
\(668\) 19.4201i 0.751384i
\(669\) 0 0
\(670\) 3.80385 + 3.10583i 0.146955 + 0.119989i
\(671\) 3.71281 0.143332
\(672\) 0 0
\(673\) 46.3644i 1.78722i −0.448846 0.893609i \(-0.648165\pi\)
0.448846 0.893609i \(-0.351835\pi\)
\(674\) −14.4449 −0.556395
\(675\) 0 0
\(676\) 16.9474 0.651825
\(677\) 32.2495i 1.23945i −0.784819 0.619725i \(-0.787244\pi\)
0.784819 0.619725i \(-0.212756\pi\)
\(678\) 0 0
\(679\) −26.7846 −1.02790
\(680\) −4.73205 3.86370i −0.181466 0.148166i
\(681\) 0 0
\(682\) 5.73118i 0.219458i
\(683\) 21.9711i 0.840700i −0.907362 0.420350i \(-0.861907\pi\)
0.907362 0.420350i \(-0.138093\pi\)
\(684\) 0 0
\(685\) 20.9282 25.6317i 0.799626 0.979337i
\(686\) −10.1436 −0.387284
\(687\) 0 0
\(688\) 22.5259i 0.858791i
\(689\) −10.6410 −0.405390
\(690\) 0 0
\(691\) 41.7128 1.58683 0.793415 0.608681i \(-0.208301\pi\)
0.793415 + 0.608681i \(0.208301\pi\)
\(692\) 8.00481i 0.304297i
\(693\) 0 0
\(694\) −11.1244 −0.422275
\(695\) 13.8564 + 11.3137i 0.525603 + 0.429153i
\(696\) 0 0
\(697\) 9.79796i 0.371124i
\(698\) 11.7942i 0.446416i
\(699\) 0 0
\(700\) 20.7846 4.24264i 0.785584 0.160357i
\(701\) −27.7128 −1.04670 −0.523349 0.852118i \(-0.675318\pi\)
−0.523349 + 0.852118i \(0.675318\pi\)
\(702\) 0 0
\(703\) 29.8744i 1.12673i
\(704\) 2.87564 0.108380
\(705\) 0 0
\(706\) −16.5359 −0.622337
\(707\) 14.6969i 0.552735i
\(708\) 0 0
\(709\) −20.3923 −0.765849 −0.382925 0.923780i \(-0.625083\pi\)
−0.382925 + 0.923780i \(0.625083\pi\)
\(710\) −2.53590 + 3.10583i −0.0951706 + 0.116560i
\(711\) 0 0
\(712\) 20.0764i 0.752395i
\(713\) 55.1271i 2.06453i
\(714\) 0 0
\(715\) −3.21539 + 3.93803i −0.120249 + 0.147274i
\(716\) 4.39230 0.164148
\(717\) 0 0
\(718\) 3.76217i 0.140403i
\(719\) −19.8564 −0.740519 −0.370260 0.928928i \(-0.620731\pi\)
−0.370260 + 0.928928i \(0.620731\pi\)
\(720\) 0 0
\(721\) −22.3923 −0.833933
\(722\) 4.30701i 0.160290i
\(723\) 0 0
\(724\) −27.7128 −1.02994
\(725\) 1.00000 + 4.89898i 0.0371391 + 0.181944i
\(726\) 0 0
\(727\) 46.6690i 1.73086i −0.501031 0.865430i \(-0.667046\pi\)
0.501031 0.865430i \(-0.332954\pi\)
\(728\) 8.48528i 0.314485i
\(729\) 0 0
\(730\) 6.58846 + 5.37945i 0.243850 + 0.199102i
\(731\) −12.9282 −0.478167
\(732\) 0 0
\(733\) 12.7279i 0.470117i −0.971981 0.235058i \(-0.924472\pi\)
0.971981 0.235058i \(-0.0755281\pi\)
\(734\) −17.4115 −0.642672
\(735\) 0 0
\(736\) −32.4449 −1.19593
\(737\) 5.37945i 0.198155i
\(738\) 0 0
\(739\) 18.9808 0.698219 0.349109 0.937082i \(-0.386484\pi\)
0.349109 + 0.937082i \(0.386484\pi\)
\(740\) −22.3923 + 27.4249i −0.823157 + 1.00816i
\(741\) 0 0
\(742\) 7.52433i 0.276227i
\(743\) 9.89949i 0.363177i −0.983375 0.181589i \(-0.941876\pi\)
0.983375 0.181589i \(-0.0581239\pi\)
\(744\) 0 0
\(745\) −1.60770 1.31268i −0.0589014 0.0480928i
\(746\) 10.1436 0.371383
\(747\) 0 0
\(748\) 3.10583i 0.113560i
\(749\) 45.7128 1.67031
\(750\) 0 0
\(751\) 49.9090 1.82120 0.910602 0.413284i \(-0.135618\pi\)
0.910602 + 0.413284i \(0.135618\pi\)
\(752\) 3.48477i 0.127076i
\(753\) 0 0
\(754\) 0.928203 0.0338032
\(755\) −32.5359 26.5654i −1.18410 0.966816i
\(756\) 0 0
\(757\) 10.9348i 0.397431i 0.980057 + 0.198716i \(0.0636770\pi\)
−0.980057 + 0.198716i \(0.936323\pi\)
\(758\) 8.38375i 0.304511i
\(759\) 0 0
\(760\) 8.92820 10.9348i 0.323860 0.396646i
\(761\) 35.5692 1.28938 0.644692 0.764443i \(-0.276986\pi\)
0.644692 + 0.764443i \(0.276986\pi\)
\(762\) 0 0
\(763\) 43.7391i 1.58346i
\(764\) −26.1962 −0.947744
\(765\) 0 0
\(766\) −0.483340 −0.0174638
\(767\) 18.6350i 0.672870i
\(768\) 0 0
\(769\) 19.0718 0.687747 0.343873 0.939016i \(-0.388261\pi\)
0.343873 + 0.939016i \(0.388261\pi\)
\(770\) −2.78461 2.27362i −0.100350 0.0819357i
\(771\) 0 0
\(772\) 1.13681i 0.0409148i
\(773\) 11.2122i 0.403274i −0.979460 0.201637i \(-0.935374\pi\)
0.979460 0.201637i \(-0.0646261\pi\)
\(774\) 0 0
\(775\) −8.73205 42.7781i −0.313665 1.53664i
\(776\) −21.1244 −0.758320
\(777\) 0 0
\(778\) 10.7589i 0.385725i
\(779\) 22.6410 0.811199
\(780\) 0 0
\(781\) −4.39230 −0.157169
\(782\) 4.62158i 0.165267i
\(783\) 0 0
\(784\) 2.46410 0.0880036
\(785\) −15.4641 + 18.9396i −0.551937 + 0.675983i
\(786\) 0 0
\(787\) 35.0779i 1.25039i 0.780467 + 0.625197i \(0.214981\pi\)
−0.780467 + 0.625197i \(0.785019\pi\)
\(788\) 38.8401i 1.38362i
\(789\) 0 0
\(790\) 3.07180 3.76217i 0.109290 0.133852i
\(791\) −31.8564 −1.13268
\(792\) 0 0
\(793\) 5.25071i 0.186458i
\(794\) 2.53590 0.0899957
\(795\) 0 0
\(796\) 24.2487 0.859473
\(797\) 29.0149i 1.02776i −0.857862 0.513881i \(-0.828207\pi\)
0.857862 0.513881i \(-0.171793\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 25.1769 5.13922i 0.890138 0.181699i
\(801\) 0 0
\(802\) 19.5959i 0.691956i
\(803\) 9.31749i 0.328807i
\(804\) 0 0
\(805\) −26.7846 21.8695i −0.944033 0.770800i
\(806\) −8.10512 −0.285491
\(807\) 0 0
\(808\) 11.5911i 0.407774i
\(809\) 13.6077 0.478421 0.239211 0.970968i \(-0.423111\pi\)
0.239211 + 0.970968i \(0.423111\pi\)
\(810\) 0 0
\(811\) −36.7846 −1.29168 −0.645841 0.763472i \(-0.723493\pi\)
−0.645841 + 0.763472i \(0.723493\pi\)
\(812\) 4.24264i 0.148888i
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 19.8564 24.3190i 0.695540 0.851859i
\(816\) 0 0
\(817\) 29.8744i 1.04517i
\(818\) 4.69591i 0.164189i
\(819\) 0 0
\(820\) 20.7846 + 16.9706i 0.725830 + 0.592638i
\(821\) −9.21539 −0.321619 −0.160810 0.986985i \(-0.551411\pi\)
−0.160810 + 0.986985i \(0.551411\pi\)
\(822\) 0 0
\(823\) 40.8091i 1.42252i 0.702931 + 0.711258i \(0.251874\pi\)
−0.702931 + 0.711258i \(0.748126\pi\)
\(824\) −17.6603 −0.615224
\(825\) 0 0
\(826\) 13.1769 0.458483
\(827\) 3.68784i 0.128239i −0.997942 0.0641193i \(-0.979576\pi\)
0.997942 0.0641193i \(-0.0204238\pi\)
\(828\) 0 0
\(829\) 16.7846 0.582954 0.291477 0.956578i \(-0.405853\pi\)
0.291477 + 0.956578i \(0.405853\pi\)
\(830\) 9.12436 + 7.45001i 0.316711 + 0.258593i
\(831\) 0 0
\(832\) 4.06678i 0.140990i
\(833\) 1.41421i 0.0489996i
\(834\) 0 0
\(835\) 15.8564 19.4201i 0.548734 0.672059i
\(836\) 7.17691 0.248219
\(837\) 0 0
\(838\) 3.10583i 0.107289i
\(839\) 40.9808 1.41481 0.707407 0.706807i \(-0.249865\pi\)
0.707407 + 0.706807i \(0.249865\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 2.82843i 0.0974740i
\(843\) 0 0
\(844\) −25.5167 −0.878320
\(845\) 16.9474 + 13.8375i 0.583010 + 0.476025i
\(846\) 0 0
\(847\) 23.0064i 0.790508i
\(848\) 14.6226i 0.502142i
\(849\) 0 0
\(850\) −0.732051 3.58630i −0.0251091 0.123009i
\(851\) 57.7128 1.97837
\(852\) 0 0
\(853\) 32.3238i 1.10675i −0.832933 0.553374i \(-0.813340\pi\)
0.832933 0.553374i \(-0.186660\pi\)
\(854\) −3.71281 −0.127050
\(855\) 0 0
\(856\) 36.0526 1.23225
\(857\) 33.2576i 1.13606i −0.823009 0.568029i \(-0.807706\pi\)
0.823009 0.568029i \(-0.192294\pi\)
\(858\) 0 0
\(859\) 18.7321 0.639129 0.319565 0.947564i \(-0.396463\pi\)
0.319565 + 0.947564i \(0.396463\pi\)
\(860\) 22.3923 27.4249i 0.763571 0.935180i
\(861\) 0 0
\(862\) 19.2442i 0.655460i
\(863\) 1.69161i 0.0575832i 0.999585 + 0.0287916i \(0.00916592\pi\)
−0.999585 + 0.0287916i \(0.990834\pi\)
\(864\) 0 0
\(865\) −6.53590 + 8.00481i −0.222227 + 0.272172i
\(866\) 1.51666 0.0515382
\(867\) 0 0
\(868\) 37.0470i 1.25746i
\(869\) 5.32051 0.180486
\(870\) 0 0
\(871\) −7.60770 −0.257777
\(872\) 34.4959i 1.16818i
\(873\) 0 0
\(874\) −10.6795 −0.361239
\(875\) 24.2487 + 12.7279i 0.819756 + 0.430282i
\(876\) 0 0
\(877\) 24.9754i 0.843358i 0.906745 + 0.421679i \(0.138559\pi\)
−0.906745 + 0.421679i \(0.861441\pi\)
\(878\) 3.66063i 0.123540i
\(879\) 0 0
\(880\) 5.41154 + 4.41851i 0.182423 + 0.148948i
\(881\) 50.5359 1.70260 0.851299 0.524681i \(-0.175815\pi\)
0.851299 + 0.524681i \(0.175815\pi\)
\(882\) 0 0
\(883\) 1.48854i 0.0500935i −0.999686 0.0250467i \(-0.992027\pi\)
0.999686 0.0250467i \(-0.00797346\pi\)
\(884\) 4.39230 0.147729
\(885\) 0 0
\(886\) 9.41154 0.316187
\(887\) 21.0101i 0.705451i −0.935727 0.352726i \(-0.885255\pi\)
0.935727 0.352726i \(-0.114745\pi\)
\(888\) 0 0
\(889\) 10.3923 0.348547
\(890\) 7.60770 9.31749i 0.255011 0.312323i
\(891\) 0 0
\(892\) 26.5927i 0.890388i
\(893\) 4.62158i 0.154655i
\(894\) 0 0
\(895\) 4.39230 + 3.58630i 0.146819 + 0.119877i
\(896\) −28.0526 −0.937170
\(897\) 0 0
\(898\) 13.3843i 0.446639i
\(899\) −8.73205 −0.291230
\(900\) 0 0
\(901\) 8.39230 0.279588
\(902\) 4.54725i 0.151407i
\(903\) 0 0
\(904\) −25.1244 −0.835624
\(905\) −27.7128 22.6274i −0.921205 0.752161i
\(906\) 0 0
\(907\) 45.7081i 1.51771i −0.651258 0.758856i \(-0.725758\pi\)
0.651258 0.758856i \(-0.274242\pi\)
\(908\) 35.4297i 1.17577i
\(909\) 0 0
\(910\) 3.21539 3.93803i 0.106589 0.130545i
\(911\) 9.80385 0.324816 0.162408 0.986724i \(-0.448074\pi\)
0.162408 + 0.986724i \(0.448074\pi\)
\(912\) 0 0
\(913\) 12.9038i 0.427053i
\(914\) −12.2487 −0.405151
\(915\) 0 0
\(916\) −32.5359 −1.07502
\(917\) 11.5911i 0.382772i
\(918\) 0 0
\(919\) −43.9615 −1.45016 −0.725078 0.688666i \(-0.758197\pi\)
−0.725078 + 0.688666i \(0.758197\pi\)
\(920\) −21.1244 17.2480i −0.696449 0.568649i
\(921\) 0 0
\(922\) 12.9038i 0.424964i
\(923\) 6.21166i 0.204459i
\(924\) 0 0
\(925\) −44.7846 + 9.14162i −1.47251 + 0.300575i
\(926\) −3.55514 −0.116829
\(927\) 0 0
\(928\) 5.13922i 0.168703i
\(929\) 55.8564 1.83259 0.916295 0.400505i \(-0.131165\pi\)
0.916295 + 0.400505i \(0.131165\pi\)
\(930\) 0 0
\(931\) 3.26795 0.107103
\(932\) 5.85993i 0.191948i
\(933\) 0 0
\(934\) −10.8756 −0.355862
\(935\) 2.53590 3.10583i 0.0829327 0.101571i
\(936\) 0 0
\(937\) 2.75410i 0.0899724i 0.998988 + 0.0449862i \(0.0143244\pi\)
−0.998988 + 0.0449862i \(0.985676\pi\)
\(938\) 5.37945i 0.175645i
\(939\) 0 0
\(940\) 3.46410 4.24264i 0.112987 0.138380i
\(941\) 7.85641 0.256112 0.128056 0.991767i \(-0.459126\pi\)
0.128056 + 0.991767i \(0.459126\pi\)
\(942\) 0 0
\(943\) 43.7391i 1.42434i
\(944\) −25.6077 −0.833459
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 27.9797i 0.909217i 0.890691 + 0.454608i \(0.150221\pi\)
−0.890691 + 0.454608i \(0.849779\pi\)
\(948\) 0 0
\(949\) −13.1769 −0.427741
\(950\) 8.28719 1.69161i 0.268872 0.0548832i
\(951\) 0 0
\(952\) 6.69213i 0.216893i
\(953\) 5.65685i 0.183243i −0.995794 0.0916217i \(-0.970795\pi\)
0.995794 0.0916217i \(-0.0292051\pi\)
\(954\) 0 0
\(955\) −26.1962 21.3891i −0.847688 0.692134i
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 2.92996i 0.0946628i
\(959\) −36.2487 −1.17053
\(960\) 0 0
\(961\) 45.2487 1.45964
\(962\) 8.48528i 0.273576i
\(963\) 0 0
\(964\) 19.8564 0.639532
\(965\) −0.928203 + 1.13681i −0.0298799 + 0.0365953i
\(966\) 0 0
\(967\) 1.96902i 0.0633193i −0.999499 0.0316596i \(-0.989921\pi\)
0.999499 0.0316596i \(-0.0100793\pi\)
\(968\) 18.1445i 0.583188i
\(969\) 0 0
\(970\) −9.80385 8.00481i −0.314783 0.257019i
\(971\) 49.5167 1.58907 0.794533 0.607221i \(-0.207716\pi\)
0.794533 + 0.607221i \(0.207716\pi\)
\(972\) 0 0
\(973\) 19.5959i 0.628216i
\(974\) −5.66025 −0.181366
\(975\) 0 0
\(976\) 7.21539 0.230959
\(977\) 22.9048i 0.732790i 0.930459 + 0.366395i \(0.119408\pi\)
−0.930459 + 0.366395i \(0.880592\pi\)
\(978\) 0 0
\(979\) 13.1769 0.421136
\(980\) 3.00000 + 2.44949i 0.0958315 + 0.0782461i
\(981\) 0 0
\(982\) 3.76217i 0.120056i
\(983\) 30.2533i 0.964930i 0.875915 + 0.482465i \(0.160258\pi\)
−0.875915 + 0.482465i \(0.839742\pi\)
\(984\) 0 0
\(985\) −31.7128 + 38.8401i −1.01045 + 1.23755i
\(986\) −0.732051 −0.0233132
\(987\) 0 0
\(988\) 10.1497i 0.322905i
\(989\) −57.7128 −1.83516
\(990\) 0 0
\(991\) 34.7846 1.10497 0.552485 0.833523i \(-0.313680\pi\)
0.552485 + 0.833523i \(0.313680\pi\)
\(992\) 44.8759i 1.42481i
\(993\) 0 0
\(994\) 4.39230 0.139315
\(995\) 24.2487 + 19.7990i 0.768736 + 0.627670i
\(996\) 0 0
\(997\) 26.5927i 0.842198i 0.907015 + 0.421099i \(0.138355\pi\)
−0.907015 + 0.421099i \(0.861645\pi\)
\(998\) 0.554803i 0.0175620i
\(999\) 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 1305.2.c.f.784.2 4
3.2 odd 2 145.2.b.b.59.3 yes 4
5.2 odd 4 6525.2.a.bj.1.3 4
5.3 odd 4 6525.2.a.bj.1.2 4
5.4 even 2 inner 1305.2.c.f.784.3 4
12.11 even 2 2320.2.d.f.929.3 4
15.2 even 4 725.2.a.f.1.2 4
15.8 even 4 725.2.a.f.1.3 4
15.14 odd 2 145.2.b.b.59.2 4
60.59 even 2 2320.2.d.f.929.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.b.59.2 4 15.14 odd 2
145.2.b.b.59.3 yes 4 3.2 odd 2
725.2.a.f.1.2 4 15.2 even 4
725.2.a.f.1.3 4 15.8 even 4
1305.2.c.f.784.2 4 1.1 even 1 trivial
1305.2.c.f.784.3 4 5.4 even 2 inner
2320.2.d.f.929.1 4 60.59 even 2
2320.2.d.f.929.3 4 12.11 even 2
6525.2.a.bj.1.2 4 5.3 odd 4
6525.2.a.bj.1.3 4 5.2 odd 4