Properties

Label 225.10.b.i.199.4
Level $225$
Weight $10$
Character 225.199
Analytic conductor $115.883$
Analytic rank $0$
Dimension $4$
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] = [225,10,Mod(199,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), 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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(-7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.10.b.i.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+38.7863i q^{2} -992.374 q^{4} -12272.1i q^{7} -18631.9i q^{8} +O(q^{10})\) \(q+38.7863i q^{2} -992.374 q^{4} -12272.1i q^{7} -18631.9i q^{8} +65897.9 q^{11} -112300. i q^{13} +475990. q^{14} +214567. q^{16} +96634.7i q^{17} +181332. q^{19} +2.55593e6i q^{22} -244145. i q^{23} +4.35569e6 q^{26} +1.21785e7i q^{28} -5.26461e6 q^{29} +1.83457e6 q^{31} -1.21730e6i q^{32} -3.74810e6 q^{34} -6.36194e6i q^{37} +7.03318e6i q^{38} -1.57111e6 q^{41} -1.99504e7i q^{43} -6.53953e7 q^{44} +9.46946e6 q^{46} +3.00961e7i q^{47} -1.10251e8 q^{49} +1.11443e8i q^{52} +2.57306e6i q^{53} -2.28653e8 q^{56} -2.04195e8i q^{58} -1.19004e8 q^{59} +1.92875e8 q^{61} +7.11559e7i q^{62} +1.57073e8 q^{64} +1.20193e8i q^{67} -9.58978e7i q^{68} +699549. q^{71} -8.91287e7i q^{73} +2.46756e8 q^{74} -1.79949e8 q^{76} -8.08707e8i q^{77} -4.31205e8 q^{79} -6.09375e7i q^{82} +1.69761e7i q^{83} +7.73800e8 q^{86} -1.22780e9i q^{88} -3.09863e6 q^{89} -1.37816e9 q^{91} +2.42283e8i q^{92} -1.16732e9 q^{94} -5.72609e8i q^{97} -4.27624e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1082 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 1082 q^{4} + 43024 q^{11} + 923328 q^{14} + 774530 q^{16} + 191792 q^{19} + 10838332 q^{26} - 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34} - 52120744 q^{41} - 170859944 q^{44} + 34190784 q^{46} - 146565860 q^{49} - 484908480 q^{56} - 70989328 q^{59} + 682994680 q^{61} + 410240942 q^{64} - 420572128 q^{71} + 250480468 q^{74} - 437024728 q^{76} + 49510080 q^{79} + 1746292744 q^{86} - 855278232 q^{89} - 2253718656 q^{91} - 3295087792 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 38.7863i 1.71413i 0.515211 + 0.857063i \(0.327714\pi\)
−0.515211 + 0.857063i \(0.672286\pi\)
\(3\) 0 0
\(4\) −992.374 −1.93823
\(5\) 0 0
\(6\) 0 0
\(7\) − 12272.1i − 1.93187i −0.258780 0.965936i \(-0.583320\pi\)
0.258780 0.965936i \(-0.416680\pi\)
\(8\) − 18631.9i − 1.60825i
\(9\) 0 0
\(10\) 0 0
\(11\) 65897.9 1.35708 0.678538 0.734565i \(-0.262614\pi\)
0.678538 + 0.734565i \(0.262614\pi\)
\(12\) 0 0
\(13\) − 112300.i − 1.09052i −0.838267 0.545260i \(-0.816431\pi\)
0.838267 0.545260i \(-0.183569\pi\)
\(14\) 475990. 3.31147
\(15\) 0 0
\(16\) 214567. 0.818508
\(17\) 96634.7i 0.280616i 0.990108 + 0.140308i \(0.0448093\pi\)
−0.990108 + 0.140308i \(0.955191\pi\)
\(18\) 0 0
\(19\) 181332. 0.319214 0.159607 0.987181i \(-0.448977\pi\)
0.159607 + 0.987181i \(0.448977\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.55593e6i 2.32620i
\(23\) − 244145.i − 0.181916i −0.995855 0.0909582i \(-0.971007\pi\)
0.995855 0.0909582i \(-0.0289930\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.35569e6 1.86929
\(27\) 0 0
\(28\) 1.21785e7i 3.74442i
\(29\) −5.26461e6 −1.38221 −0.691107 0.722752i \(-0.742877\pi\)
−0.691107 + 0.722752i \(0.742877\pi\)
\(30\) 0 0
\(31\) 1.83457e6 0.356784 0.178392 0.983959i \(-0.442910\pi\)
0.178392 + 0.983959i \(0.442910\pi\)
\(32\) − 1.21730e6i − 0.205221i
\(33\) 0 0
\(34\) −3.74810e6 −0.481012
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.36194e6i − 0.558061i −0.960282 0.279030i \(-0.909987\pi\)
0.960282 0.279030i \(-0.0900130\pi\)
\(38\) 7.03318e6i 0.547174i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.57111e6 −0.0868319 −0.0434159 0.999057i \(-0.513824\pi\)
−0.0434159 + 0.999057i \(0.513824\pi\)
\(42\) 0 0
\(43\) − 1.99504e7i − 0.889903i −0.895555 0.444952i \(-0.853221\pi\)
0.895555 0.444952i \(-0.146779\pi\)
\(44\) −6.53953e7 −2.63033
\(45\) 0 0
\(46\) 9.46946e6 0.311828
\(47\) 3.00961e7i 0.899643i 0.893119 + 0.449821i \(0.148512\pi\)
−0.893119 + 0.449821i \(0.851488\pi\)
\(48\) 0 0
\(49\) −1.10251e8 −2.73213
\(50\) 0 0
\(51\) 0 0
\(52\) 1.11443e8i 2.11368i
\(53\) 2.57306e6i 0.0447929i 0.999749 + 0.0223964i \(0.00712960\pi\)
−0.999749 + 0.0223964i \(0.992870\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.28653e8 −3.10693
\(57\) 0 0
\(58\) − 2.04195e8i − 2.36929i
\(59\) −1.19004e8 −1.27858 −0.639290 0.768965i \(-0.720772\pi\)
−0.639290 + 0.768965i \(0.720772\pi\)
\(60\) 0 0
\(61\) 1.92875e8 1.78358 0.891790 0.452449i \(-0.149450\pi\)
0.891790 + 0.452449i \(0.149450\pi\)
\(62\) 7.11559e7i 0.611573i
\(63\) 0 0
\(64\) 1.57073e8 1.17028
\(65\) 0 0
\(66\) 0 0
\(67\) 1.20193e8i 0.728691i 0.931264 + 0.364345i \(0.118707\pi\)
−0.931264 + 0.364345i \(0.881293\pi\)
\(68\) − 9.58978e7i − 0.543899i
\(69\) 0 0
\(70\) 0 0
\(71\) 699549. 0.00326705 0.00163352 0.999999i \(-0.499480\pi\)
0.00163352 + 0.999999i \(0.499480\pi\)
\(72\) 0 0
\(73\) − 8.91287e7i − 0.367337i −0.982988 0.183669i \(-0.941203\pi\)
0.982988 0.183669i \(-0.0587973\pi\)
\(74\) 2.46756e8 0.956587
\(75\) 0 0
\(76\) −1.79949e8 −0.618711
\(77\) − 8.08707e8i − 2.62170i
\(78\) 0 0
\(79\) −4.31205e8 −1.24555 −0.622776 0.782400i \(-0.713995\pi\)
−0.622776 + 0.782400i \(0.713995\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.09375e7i − 0.148841i
\(83\) 1.69761e7i 0.0392633i 0.999807 + 0.0196316i \(0.00624935\pi\)
−0.999807 + 0.0196316i \(0.993751\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.73800e8 1.52541
\(87\) 0 0
\(88\) − 1.22780e9i − 2.18251i
\(89\) −3.09863e6 −0.00523498 −0.00261749 0.999997i \(-0.500833\pi\)
−0.00261749 + 0.999997i \(0.500833\pi\)
\(90\) 0 0
\(91\) −1.37816e9 −2.10675
\(92\) 2.42283e8i 0.352596i
\(93\) 0 0
\(94\) −1.16732e9 −1.54210
\(95\) 0 0
\(96\) 0 0
\(97\) − 5.72609e8i − 0.656727i −0.944551 0.328364i \(-0.893503\pi\)
0.944551 0.328364i \(-0.106497\pi\)
\(98\) − 4.27624e9i − 4.68322i
\(99\) 0 0
\(100\) 0 0
\(101\) −1.78445e9 −1.70631 −0.853156 0.521655i \(-0.825315\pi\)
−0.853156 + 0.521655i \(0.825315\pi\)
\(102\) 0 0
\(103\) 1.16632e9i 1.02106i 0.859861 + 0.510528i \(0.170550\pi\)
−0.859861 + 0.510528i \(0.829450\pi\)
\(104\) −2.09236e9 −1.75383
\(105\) 0 0
\(106\) −9.97995e7 −0.0767807
\(107\) − 1.72229e9i − 1.27022i −0.772422 0.635109i \(-0.780955\pi\)
0.772422 0.635109i \(-0.219045\pi\)
\(108\) 0 0
\(109\) −2.08468e9 −1.41456 −0.707279 0.706934i \(-0.750078\pi\)
−0.707279 + 0.706934i \(0.750078\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 2.63319e9i − 1.58125i
\(113\) 1.81995e9i 1.05004i 0.851090 + 0.525020i \(0.175942\pi\)
−0.851090 + 0.525020i \(0.824058\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.22446e9 2.67905
\(117\) 0 0
\(118\) − 4.61573e9i − 2.19165i
\(119\) 1.18591e9 0.542115
\(120\) 0 0
\(121\) 1.98458e9 0.841656
\(122\) 7.48092e9i 3.05728i
\(123\) 0 0
\(124\) −1.82058e9 −0.691530
\(125\) 0 0
\(126\) 0 0
\(127\) 3.74913e9i 1.27883i 0.768860 + 0.639417i \(0.220824\pi\)
−0.768860 + 0.639417i \(0.779176\pi\)
\(128\) 5.46900e9i 1.80079i
\(129\) 0 0
\(130\) 0 0
\(131\) 1.96598e9 0.583256 0.291628 0.956532i \(-0.405803\pi\)
0.291628 + 0.956532i \(0.405803\pi\)
\(132\) 0 0
\(133\) − 2.22532e9i − 0.616682i
\(134\) −4.66184e9 −1.24907
\(135\) 0 0
\(136\) 1.80049e9 0.451300
\(137\) − 3.63223e8i − 0.0880909i −0.999030 0.0440455i \(-0.985975\pi\)
0.999030 0.0440455i \(-0.0140246\pi\)
\(138\) 0 0
\(139\) −4.55580e9 −1.03514 −0.517569 0.855642i \(-0.673163\pi\)
−0.517569 + 0.855642i \(0.673163\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.71329e7i 0.00560014i
\(143\) − 7.40032e9i − 1.47992i
\(144\) 0 0
\(145\) 0 0
\(146\) 3.45697e9 0.629662
\(147\) 0 0
\(148\) 6.31342e9i 1.08165i
\(149\) 8.79830e9 1.46238 0.731191 0.682173i \(-0.238965\pi\)
0.731191 + 0.682173i \(0.238965\pi\)
\(150\) 0 0
\(151\) −2.70702e8 −0.0423736 −0.0211868 0.999776i \(-0.506744\pi\)
−0.0211868 + 0.999776i \(0.506744\pi\)
\(152\) − 3.37856e9i − 0.513376i
\(153\) 0 0
\(154\) 3.13667e10 4.49392
\(155\) 0 0
\(156\) 0 0
\(157\) − 9.73240e9i − 1.27841i −0.769035 0.639207i \(-0.779263\pi\)
0.769035 0.639207i \(-0.220737\pi\)
\(158\) − 1.67248e10i − 2.13504i
\(159\) 0 0
\(160\) 0 0
\(161\) −2.99617e9 −0.351439
\(162\) 0 0
\(163\) 7.76823e9i 0.861941i 0.902366 + 0.430971i \(0.141829\pi\)
−0.902366 + 0.430971i \(0.858171\pi\)
\(164\) 1.55913e9 0.168300
\(165\) 0 0
\(166\) −6.58440e8 −0.0673023
\(167\) − 8.41748e9i − 0.837448i −0.908114 0.418724i \(-0.862478\pi\)
0.908114 0.418724i \(-0.137522\pi\)
\(168\) 0 0
\(169\) −2.00674e9 −0.189235
\(170\) 0 0
\(171\) 0 0
\(172\) 1.97982e10i 1.72484i
\(173\) − 3.59838e8i − 0.0305422i −0.999883 0.0152711i \(-0.995139\pi\)
0.999883 0.0152711i \(-0.00486112\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41395e10 1.11078
\(177\) 0 0
\(178\) − 1.20184e8i − 0.00897342i
\(179\) −8.65338e9 −0.630009 −0.315005 0.949090i \(-0.602006\pi\)
−0.315005 + 0.949090i \(0.602006\pi\)
\(180\) 0 0
\(181\) 6.09713e9 0.422252 0.211126 0.977459i \(-0.432287\pi\)
0.211126 + 0.977459i \(0.432287\pi\)
\(182\) − 5.34535e10i − 3.61123i
\(183\) 0 0
\(184\) −4.54888e9 −0.292566
\(185\) 0 0
\(186\) 0 0
\(187\) 6.36802e9i 0.380818i
\(188\) − 2.98666e10i − 1.74371i
\(189\) 0 0
\(190\) 0 0
\(191\) 2.96238e10 1.61061 0.805306 0.592859i \(-0.202001\pi\)
0.805306 + 0.592859i \(0.202001\pi\)
\(192\) 0 0
\(193\) − 4.63438e8i − 0.0240427i −0.999928 0.0120214i \(-0.996173\pi\)
0.999928 0.0120214i \(-0.00382661\pi\)
\(194\) 2.22094e10 1.12571
\(195\) 0 0
\(196\) 1.09411e11 5.29550
\(197\) 1.23414e10i 0.583803i 0.956448 + 0.291902i \(0.0942880\pi\)
−0.956448 + 0.291902i \(0.905712\pi\)
\(198\) 0 0
\(199\) 1.00354e10 0.453624 0.226812 0.973939i \(-0.427170\pi\)
0.226812 + 0.973939i \(0.427170\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 6.92122e10i − 2.92484i
\(203\) 6.46080e10i 2.67026i
\(204\) 0 0
\(205\) 0 0
\(206\) −4.52371e10 −1.75022
\(207\) 0 0
\(208\) − 2.40958e10i − 0.892599i
\(209\) 1.19494e10 0.433198
\(210\) 0 0
\(211\) −3.56268e10 −1.23739 −0.618693 0.785633i \(-0.712338\pi\)
−0.618693 + 0.785633i \(0.712338\pi\)
\(212\) − 2.55344e9i − 0.0868189i
\(213\) 0 0
\(214\) 6.68011e10 2.17732
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.25140e10i − 0.689262i
\(218\) − 8.08571e10i − 2.42473i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.08521e10 0.306018
\(222\) 0 0
\(223\) − 3.41646e10i − 0.925133i −0.886585 0.462566i \(-0.846929\pi\)
0.886585 0.462566i \(-0.153071\pi\)
\(224\) −1.49388e10 −0.396460
\(225\) 0 0
\(226\) −7.05889e10 −1.79990
\(227\) 3.64936e10i 0.912221i 0.889923 + 0.456110i \(0.150758\pi\)
−0.889923 + 0.456110i \(0.849242\pi\)
\(228\) 0 0
\(229\) −3.87446e10 −0.931005 −0.465502 0.885047i \(-0.654126\pi\)
−0.465502 + 0.885047i \(0.654126\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.80898e10i 2.22294i
\(233\) 2.68717e10i 0.597302i 0.954362 + 0.298651i \(0.0965367\pi\)
−0.954362 + 0.298651i \(0.903463\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.18097e11 2.47818
\(237\) 0 0
\(238\) 4.59971e10i 0.929254i
\(239\) 1.34711e10 0.267063 0.133531 0.991045i \(-0.457368\pi\)
0.133531 + 0.991045i \(0.457368\pi\)
\(240\) 0 0
\(241\) −1.85860e10 −0.354902 −0.177451 0.984130i \(-0.556785\pi\)
−0.177451 + 0.984130i \(0.556785\pi\)
\(242\) 7.69745e10i 1.44271i
\(243\) 0 0
\(244\) −1.91405e11 −3.45699
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.03635e10i − 0.348110i
\(248\) − 3.41815e10i − 0.573797i
\(249\) 0 0
\(250\) 0 0
\(251\) −4.30327e10 −0.684332 −0.342166 0.939640i \(-0.611161\pi\)
−0.342166 + 0.939640i \(0.611161\pi\)
\(252\) 0 0
\(253\) − 1.60886e10i − 0.246875i
\(254\) −1.45415e11 −2.19208
\(255\) 0 0
\(256\) −1.31701e11 −1.91650
\(257\) − 6.26858e8i − 0.00896335i −0.999990 0.00448168i \(-0.998573\pi\)
0.999990 0.00448168i \(-0.00142657\pi\)
\(258\) 0 0
\(259\) −7.80745e10 −1.07810
\(260\) 0 0
\(261\) 0 0
\(262\) 7.62532e10i 0.999774i
\(263\) − 6.21446e10i − 0.800944i −0.916309 0.400472i \(-0.868846\pi\)
0.916309 0.400472i \(-0.131154\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.63120e10 1.05707
\(267\) 0 0
\(268\) − 1.19277e11i − 1.41237i
\(269\) −5.44876e10 −0.634472 −0.317236 0.948347i \(-0.602755\pi\)
−0.317236 + 0.948347i \(0.602755\pi\)
\(270\) 0 0
\(271\) −9.74930e10 −1.09802 −0.549012 0.835815i \(-0.684996\pi\)
−0.549012 + 0.835815i \(0.684996\pi\)
\(272\) 2.07346e10i 0.229687i
\(273\) 0 0
\(274\) 1.40881e10 0.150999
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.43195e11i − 1.46140i −0.682697 0.730702i \(-0.739193\pi\)
0.682697 0.730702i \(-0.260807\pi\)
\(278\) − 1.76702e11i − 1.77436i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.51280e10 −0.623146 −0.311573 0.950222i \(-0.600856\pi\)
−0.311573 + 0.950222i \(0.600856\pi\)
\(282\) 0 0
\(283\) − 1.65431e11i − 1.53312i −0.642171 0.766561i \(-0.721966\pi\)
0.642171 0.766561i \(-0.278034\pi\)
\(284\) −6.94214e8 −0.00633229
\(285\) 0 0
\(286\) 2.87031e11 2.53677
\(287\) 1.92808e10i 0.167748i
\(288\) 0 0
\(289\) 1.09250e11 0.921254
\(290\) 0 0
\(291\) 0 0
\(292\) 8.84490e10i 0.711984i
\(293\) − 5.64261e10i − 0.447277i −0.974672 0.223638i \(-0.928207\pi\)
0.974672 0.223638i \(-0.0717934\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.18535e11 −0.897499
\(297\) 0 0
\(298\) 3.41253e11i 2.50671i
\(299\) −2.74174e10 −0.198384
\(300\) 0 0
\(301\) −2.44833e11 −1.71918
\(302\) − 1.04995e10i − 0.0726338i
\(303\) 0 0
\(304\) 3.89078e10 0.261279
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.79360e10i − 0.115240i −0.998339 0.0576200i \(-0.981649\pi\)
0.998339 0.0576200i \(-0.0183512\pi\)
\(308\) 8.02540e11i 5.08146i
\(309\) 0 0
\(310\) 0 0
\(311\) −3.19953e11 −1.93938 −0.969692 0.244330i \(-0.921432\pi\)
−0.969692 + 0.244330i \(0.921432\pi\)
\(312\) 0 0
\(313\) − 3.16987e11i − 1.86677i −0.358872 0.933387i \(-0.616838\pi\)
0.358872 0.933387i \(-0.383162\pi\)
\(314\) 3.77483e11 2.19136
\(315\) 0 0
\(316\) 4.27917e11 2.41417
\(317\) − 3.08157e11i − 1.71398i −0.515334 0.856990i \(-0.672332\pi\)
0.515334 0.856990i \(-0.327668\pi\)
\(318\) 0 0
\(319\) −3.46927e11 −1.87577
\(320\) 0 0
\(321\) 0 0
\(322\) − 1.16210e11i − 0.602412i
\(323\) 1.75229e10i 0.0895768i
\(324\) 0 0
\(325\) 0 0
\(326\) −3.01301e11 −1.47748
\(327\) 0 0
\(328\) 2.92728e10i 0.139647i
\(329\) 3.69343e11 1.73800
\(330\) 0 0
\(331\) −1.79184e11 −0.820490 −0.410245 0.911975i \(-0.634557\pi\)
−0.410245 + 0.911975i \(0.634557\pi\)
\(332\) − 1.68466e10i − 0.0761013i
\(333\) 0 0
\(334\) 3.26482e11 1.43549
\(335\) 0 0
\(336\) 0 0
\(337\) 9.73351e10i 0.411088i 0.978648 + 0.205544i \(0.0658964\pi\)
−0.978648 + 0.205544i \(0.934104\pi\)
\(338\) − 7.78340e10i − 0.324373i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.20894e11 0.484183
\(342\) 0 0
\(343\) 8.57794e11i 3.34626i
\(344\) −3.71713e11 −1.43118
\(345\) 0 0
\(346\) 1.39568e10 0.0523531
\(347\) − 3.47750e11i − 1.28761i −0.765189 0.643806i \(-0.777354\pi\)
0.765189 0.643806i \(-0.222646\pi\)
\(348\) 0 0
\(349\) 1.10131e11 0.397372 0.198686 0.980063i \(-0.436333\pi\)
0.198686 + 0.980063i \(0.436333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 8.02172e10i − 0.278500i
\(353\) 3.40052e11i 1.16563i 0.812606 + 0.582814i \(0.198048\pi\)
−0.812606 + 0.582814i \(0.801952\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.07500e9 0.0101466
\(357\) 0 0
\(358\) − 3.35632e11i − 1.07992i
\(359\) −2.75063e11 −0.873993 −0.436996 0.899463i \(-0.643958\pi\)
−0.436996 + 0.899463i \(0.643958\pi\)
\(360\) 0 0
\(361\) −2.89807e11 −0.898102
\(362\) 2.36485e11i 0.723793i
\(363\) 0 0
\(364\) 1.36765e12 4.08336
\(365\) 0 0
\(366\) 0 0
\(367\) 2.12218e11i 0.610639i 0.952250 + 0.305320i \(0.0987633\pi\)
−0.952250 + 0.305320i \(0.901237\pi\)
\(368\) − 5.23854e10i − 0.148900i
\(369\) 0 0
\(370\) 0 0
\(371\) 3.15769e10 0.0865341
\(372\) 0 0
\(373\) 6.08324e11i 1.62721i 0.581415 + 0.813607i \(0.302499\pi\)
−0.581415 + 0.813607i \(0.697501\pi\)
\(374\) −2.46992e11 −0.652770
\(375\) 0 0
\(376\) 5.60748e11 1.44685
\(377\) 5.91215e11i 1.50733i
\(378\) 0 0
\(379\) −4.64136e11 −1.15550 −0.577749 0.816214i \(-0.696069\pi\)
−0.577749 + 0.816214i \(0.696069\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.14900e12i 2.76079i
\(383\) − 5.22749e11i − 1.24136i −0.784063 0.620681i \(-0.786856\pi\)
0.784063 0.620681i \(-0.213144\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.79750e10 0.0412123
\(387\) 0 0
\(388\) 5.68242e11i 1.27289i
\(389\) −1.18899e11 −0.263271 −0.131636 0.991298i \(-0.542023\pi\)
−0.131636 + 0.991298i \(0.542023\pi\)
\(390\) 0 0
\(391\) 2.35928e10 0.0510487
\(392\) 2.05419e12i 4.39394i
\(393\) 0 0
\(394\) −4.78677e11 −1.00071
\(395\) 0 0
\(396\) 0 0
\(397\) 2.76954e11i 0.559564i 0.960064 + 0.279782i \(0.0902622\pi\)
−0.960064 + 0.279782i \(0.909738\pi\)
\(398\) 3.89236e11i 0.777569i
\(399\) 0 0
\(400\) 0 0
\(401\) 1.53482e11 0.296421 0.148210 0.988956i \(-0.452649\pi\)
0.148210 + 0.988956i \(0.452649\pi\)
\(402\) 0 0
\(403\) − 2.06021e11i − 0.389080i
\(404\) 1.77084e12 3.30723
\(405\) 0 0
\(406\) −2.50590e12 −4.57717
\(407\) − 4.19238e11i − 0.757331i
\(408\) 0 0
\(409\) 1.52364e11 0.269232 0.134616 0.990898i \(-0.457020\pi\)
0.134616 + 0.990898i \(0.457020\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 1.15742e12i − 1.97904i
\(413\) 1.46043e12i 2.47006i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.36702e11 −0.223797
\(417\) 0 0
\(418\) 4.63472e11i 0.742557i
\(419\) −9.09824e11 −1.44210 −0.721049 0.692885i \(-0.756340\pi\)
−0.721049 + 0.692885i \(0.756340\pi\)
\(420\) 0 0
\(421\) 9.88529e11 1.53363 0.766814 0.641869i \(-0.221841\pi\)
0.766814 + 0.641869i \(0.221841\pi\)
\(422\) − 1.38183e12i − 2.12104i
\(423\) 0 0
\(424\) 4.79411e10 0.0720380
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.36699e12i − 3.44565i
\(428\) 1.70915e12i 2.46198i
\(429\) 0 0
\(430\) 0 0
\(431\) −2.03278e11 −0.283755 −0.141878 0.989884i \(-0.545314\pi\)
−0.141878 + 0.989884i \(0.545314\pi\)
\(432\) 0 0
\(433\) − 2.10847e11i − 0.288251i −0.989559 0.144126i \(-0.953963\pi\)
0.989559 0.144126i \(-0.0460369\pi\)
\(434\) 8.73234e11 1.18148
\(435\) 0 0
\(436\) 2.06879e12 2.74174
\(437\) − 4.42712e10i − 0.0580704i
\(438\) 0 0
\(439\) −2.11413e11 −0.271670 −0.135835 0.990732i \(-0.543372\pi\)
−0.135835 + 0.990732i \(0.543372\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.20911e11i 0.524553i
\(443\) − 1.45517e12i − 1.79514i −0.440874 0.897569i \(-0.645331\pi\)
0.440874 0.897569i \(-0.354669\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.32512e12 1.58579
\(447\) 0 0
\(448\) − 1.92761e12i − 2.26084i
\(449\) −2.19470e11 −0.254840 −0.127420 0.991849i \(-0.540670\pi\)
−0.127420 + 0.991849i \(0.540670\pi\)
\(450\) 0 0
\(451\) −1.03533e11 −0.117837
\(452\) − 1.80607e12i − 2.03522i
\(453\) 0 0
\(454\) −1.41545e12 −1.56366
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.40816e11i − 0.794488i −0.917713 0.397244i \(-0.869967\pi\)
0.917713 0.397244i \(-0.130033\pi\)
\(458\) − 1.50276e12i − 1.59586i
\(459\) 0 0
\(460\) 0 0
\(461\) −2.80170e11 −0.288913 −0.144457 0.989511i \(-0.546143\pi\)
−0.144457 + 0.989511i \(0.546143\pi\)
\(462\) 0 0
\(463\) − 5.07092e10i − 0.0512828i −0.999671 0.0256414i \(-0.991837\pi\)
0.999671 0.0256414i \(-0.00816281\pi\)
\(464\) −1.12961e12 −1.13135
\(465\) 0 0
\(466\) −1.04225e12 −1.02385
\(467\) 1.77228e12i 1.72427i 0.506678 + 0.862135i \(0.330873\pi\)
−0.506678 + 0.862135i \(0.669127\pi\)
\(468\) 0 0
\(469\) 1.47503e12 1.40774
\(470\) 0 0
\(471\) 0 0
\(472\) 2.21728e12i 2.05627i
\(473\) − 1.31469e12i − 1.20767i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.17687e12 −1.05074
\(477\) 0 0
\(478\) 5.22495e11i 0.457779i
\(479\) −7.79555e11 −0.676608 −0.338304 0.941037i \(-0.609853\pi\)
−0.338304 + 0.941037i \(0.609853\pi\)
\(480\) 0 0
\(481\) −7.14444e11 −0.608577
\(482\) − 7.20880e11i − 0.608347i
\(483\) 0 0
\(484\) −1.96945e12 −1.63132
\(485\) 0 0
\(486\) 0 0
\(487\) 1.33958e12i 1.07916i 0.841933 + 0.539582i \(0.181418\pi\)
−0.841933 + 0.539582i \(0.818582\pi\)
\(488\) − 3.59364e12i − 2.86844i
\(489\) 0 0
\(490\) 0 0
\(491\) 9.76349e11 0.758121 0.379061 0.925372i \(-0.376247\pi\)
0.379061 + 0.925372i \(0.376247\pi\)
\(492\) 0 0
\(493\) − 5.08744e11i − 0.387872i
\(494\) 7.89825e11 0.596705
\(495\) 0 0
\(496\) 3.93637e11 0.292031
\(497\) − 8.58495e9i − 0.00631152i
\(498\) 0 0
\(499\) −1.32057e12 −0.953472 −0.476736 0.879046i \(-0.658180\pi\)
−0.476736 + 0.879046i \(0.658180\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 1.66908e12i − 1.17303i
\(503\) 1.16421e12i 0.810917i 0.914113 + 0.405459i \(0.132888\pi\)
−0.914113 + 0.405459i \(0.867112\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.24017e11 0.423174
\(507\) 0 0
\(508\) − 3.72054e12i − 2.47868i
\(509\) 8.32285e11 0.549594 0.274797 0.961502i \(-0.411389\pi\)
0.274797 + 0.961502i \(0.411389\pi\)
\(510\) 0 0
\(511\) −1.09380e12 −0.709649
\(512\) − 2.30806e12i − 1.48434i
\(513\) 0 0
\(514\) 2.43135e10 0.0153643
\(515\) 0 0
\(516\) 0 0
\(517\) 1.98327e12i 1.22088i
\(518\) − 3.02822e12i − 1.84800i
\(519\) 0 0
\(520\) 0 0
\(521\) −2.43459e12 −1.44763 −0.723813 0.689997i \(-0.757612\pi\)
−0.723813 + 0.689997i \(0.757612\pi\)
\(522\) 0 0
\(523\) − 1.07631e12i − 0.629043i −0.949250 0.314522i \(-0.898156\pi\)
0.949250 0.314522i \(-0.101844\pi\)
\(524\) −1.95099e12 −1.13048
\(525\) 0 0
\(526\) 2.41036e12 1.37292
\(527\) 1.77283e11i 0.100119i
\(528\) 0 0
\(529\) 1.74155e12 0.966906
\(530\) 0 0
\(531\) 0 0
\(532\) 2.20835e12i 1.19527i
\(533\) 1.76435e11i 0.0946919i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.23943e12 1.17191
\(537\) 0 0
\(538\) − 2.11337e12i − 1.08757i
\(539\) −7.26533e12 −3.70771
\(540\) 0 0
\(541\) −3.12532e12 −1.56858 −0.784291 0.620393i \(-0.786973\pi\)
−0.784291 + 0.620393i \(0.786973\pi\)
\(542\) − 3.78139e12i − 1.88215i
\(543\) 0 0
\(544\) 1.17633e11 0.0575883
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.59061e12i − 0.759663i −0.925056 0.379832i \(-0.875982\pi\)
0.925056 0.379832i \(-0.124018\pi\)
\(548\) 3.60453e11i 0.170741i
\(549\) 0 0
\(550\) 0 0
\(551\) −9.54641e11 −0.441223
\(552\) 0 0
\(553\) 5.29180e12i 2.40625i
\(554\) 5.55402e12 2.50503
\(555\) 0 0
\(556\) 4.52106e12 2.00633
\(557\) 3.76617e12i 1.65787i 0.559343 + 0.828937i \(0.311054\pi\)
−0.559343 + 0.828937i \(0.688946\pi\)
\(558\) 0 0
\(559\) −2.24042e12 −0.970458
\(560\) 0 0
\(561\) 0 0
\(562\) − 2.52607e12i − 1.06815i
\(563\) 2.32486e12i 0.975234i 0.873058 + 0.487617i \(0.162134\pi\)
−0.873058 + 0.487617i \(0.837866\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.41643e12 2.62797
\(567\) 0 0
\(568\) − 1.30339e10i − 0.00525422i
\(569\) 3.30244e11 0.132078 0.0660388 0.997817i \(-0.478964\pi\)
0.0660388 + 0.997817i \(0.478964\pi\)
\(570\) 0 0
\(571\) 3.83190e12 1.50852 0.754261 0.656575i \(-0.227995\pi\)
0.754261 + 0.656575i \(0.227995\pi\)
\(572\) 7.34388e12i 2.86843i
\(573\) 0 0
\(574\) −7.47832e11 −0.287542
\(575\) 0 0
\(576\) 0 0
\(577\) 1.66616e12i 0.625786i 0.949788 + 0.312893i \(0.101298\pi\)
−0.949788 + 0.312893i \(0.898702\pi\)
\(578\) 4.23738e12i 1.57915i
\(579\) 0 0
\(580\) 0 0
\(581\) 2.08333e11 0.0758517
\(582\) 0 0
\(583\) 1.69559e11i 0.0607874i
\(584\) −1.66064e12 −0.590769
\(585\) 0 0
\(586\) 2.18856e12 0.766689
\(587\) 2.00482e12i 0.696953i 0.937318 + 0.348476i \(0.113301\pi\)
−0.937318 + 0.348476i \(0.886699\pi\)
\(588\) 0 0
\(589\) 3.32665e11 0.113891
\(590\) 0 0
\(591\) 0 0
\(592\) − 1.36506e12i − 0.456777i
\(593\) − 1.46848e12i − 0.487666i −0.969817 0.243833i \(-0.921595\pi\)
0.969817 0.243833i \(-0.0784048\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8.73121e12 −2.83443
\(597\) 0 0
\(598\) − 1.06342e12i − 0.340055i
\(599\) 5.67302e12 1.80050 0.900251 0.435370i \(-0.143383\pi\)
0.900251 + 0.435370i \(0.143383\pi\)
\(600\) 0 0
\(601\) 2.23747e12 0.699555 0.349778 0.936833i \(-0.386257\pi\)
0.349778 + 0.936833i \(0.386257\pi\)
\(602\) − 9.49617e12i − 2.94689i
\(603\) 0 0
\(604\) 2.68638e11 0.0821299
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.87986e11i − 0.235597i −0.993038 0.117798i \(-0.962416\pi\)
0.993038 0.117798i \(-0.0375837\pi\)
\(608\) − 2.20734e11i − 0.0655094i
\(609\) 0 0
\(610\) 0 0
\(611\) 3.37979e12 0.981079
\(612\) 0 0
\(613\) − 4.35930e12i − 1.24694i −0.781848 0.623468i \(-0.785723\pi\)
0.781848 0.623468i \(-0.214277\pi\)
\(614\) 6.95671e11 0.197536
\(615\) 0 0
\(616\) −1.50678e13 −4.21634
\(617\) − 2.04791e12i − 0.568889i −0.958693 0.284444i \(-0.908191\pi\)
0.958693 0.284444i \(-0.0918091\pi\)
\(618\) 0 0
\(619\) 6.31079e12 1.72773 0.863865 0.503723i \(-0.168037\pi\)
0.863865 + 0.503723i \(0.168037\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 1.24098e13i − 3.32435i
\(623\) 3.80268e10i 0.0101133i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.22947e13 3.19989
\(627\) 0 0
\(628\) 9.65818e12i 2.47786i
\(629\) 6.14784e11 0.156601
\(630\) 0 0
\(631\) 4.94106e12 1.24076 0.620380 0.784301i \(-0.286978\pi\)
0.620380 + 0.784301i \(0.286978\pi\)
\(632\) 8.03418e12i 2.00316i
\(633\) 0 0
\(634\) 1.19523e13 2.93798
\(635\) 0 0
\(636\) 0 0
\(637\) 1.23812e13i 2.97945i
\(638\) − 1.34560e13i − 3.21531i
\(639\) 0 0
\(640\) 0 0
\(641\) 6.55819e11 0.153434 0.0767172 0.997053i \(-0.475556\pi\)
0.0767172 + 0.997053i \(0.475556\pi\)
\(642\) 0 0
\(643\) 5.76324e11i 0.132959i 0.997788 + 0.0664794i \(0.0211767\pi\)
−0.997788 + 0.0664794i \(0.978823\pi\)
\(644\) 2.97332e12 0.681171
\(645\) 0 0
\(646\) −6.79649e11 −0.153546
\(647\) − 3.52372e12i − 0.790556i −0.918562 0.395278i \(-0.870648\pi\)
0.918562 0.395278i \(-0.129352\pi\)
\(648\) 0 0
\(649\) −7.84212e12 −1.73513
\(650\) 0 0
\(651\) 0 0
\(652\) − 7.70899e12i − 1.67064i
\(653\) 4.37841e11i 0.0942338i 0.998889 + 0.0471169i \(0.0150033\pi\)
−0.998889 + 0.0471169i \(0.984997\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.37108e11 −0.0710725
\(657\) 0 0
\(658\) 1.43254e13i 2.97914i
\(659\) 1.98477e12 0.409945 0.204972 0.978768i \(-0.434290\pi\)
0.204972 + 0.978768i \(0.434290\pi\)
\(660\) 0 0
\(661\) 2.80945e12 0.572420 0.286210 0.958167i \(-0.407604\pi\)
0.286210 + 0.958167i \(0.407604\pi\)
\(662\) − 6.94988e12i − 1.40642i
\(663\) 0 0
\(664\) 3.16297e11 0.0631450
\(665\) 0 0
\(666\) 0 0
\(667\) 1.28533e12i 0.251447i
\(668\) 8.35328e12i 1.62317i
\(669\) 0 0
\(670\) 0 0
\(671\) 1.27101e13 2.42045
\(672\) 0 0
\(673\) − 2.08224e12i − 0.391258i −0.980678 0.195629i \(-0.937325\pi\)
0.980678 0.195629i \(-0.0626749\pi\)
\(674\) −3.77526e12 −0.704657
\(675\) 0 0
\(676\) 1.99144e12 0.366781
\(677\) − 6.44805e12i − 1.17972i −0.807505 0.589861i \(-0.799183\pi\)
0.807505 0.589861i \(-0.200817\pi\)
\(678\) 0 0
\(679\) −7.02712e12 −1.26871
\(680\) 0 0
\(681\) 0 0
\(682\) 4.68902e12i 0.829952i
\(683\) 4.02024e12i 0.706902i 0.935453 + 0.353451i \(0.114992\pi\)
−0.935453 + 0.353451i \(0.885008\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −3.32706e13 −5.73591
\(687\) 0 0
\(688\) − 4.28069e12i − 0.728392i
\(689\) 2.88954e11 0.0488476
\(690\) 0 0
\(691\) −8.43146e12 −1.40686 −0.703431 0.710763i \(-0.748350\pi\)
−0.703431 + 0.710763i \(0.748350\pi\)
\(692\) 3.57094e11i 0.0591978i
\(693\) 0 0
\(694\) 1.34879e13 2.20713
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.51824e11i − 0.0243664i
\(698\) 4.27159e12i 0.681145i
\(699\) 0 0
\(700\) 0 0
\(701\) −6.55956e12 −1.02599 −0.512995 0.858391i \(-0.671464\pi\)
−0.512995 + 0.858391i \(0.671464\pi\)
\(702\) 0 0
\(703\) − 1.15362e12i − 0.178141i
\(704\) 1.03507e13 1.58816
\(705\) 0 0
\(706\) −1.31894e13 −1.99803
\(707\) 2.18990e13i 3.29638i
\(708\) 0 0
\(709\) 5.30959e12 0.789138 0.394569 0.918866i \(-0.370894\pi\)
0.394569 + 0.918866i \(0.370894\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 5.77335e10i 0.00841914i
\(713\) − 4.47899e11i − 0.0649049i
\(714\) 0 0
\(715\) 0 0
\(716\) 8.58739e12 1.22110
\(717\) 0 0
\(718\) − 1.06687e13i − 1.49813i
\(719\) 1.02053e13 1.42411 0.712056 0.702123i \(-0.247764\pi\)
0.712056 + 0.702123i \(0.247764\pi\)
\(720\) 0 0
\(721\) 1.43132e13 1.97255
\(722\) − 1.12405e13i − 1.53946i
\(723\) 0 0
\(724\) −6.05063e12 −0.818422
\(725\) 0 0
\(726\) 0 0
\(727\) 7.45813e12i 0.990206i 0.868834 + 0.495103i \(0.164870\pi\)
−0.868834 + 0.495103i \(0.835130\pi\)
\(728\) 2.56777e13i 3.38817i
\(729\) 0 0
\(730\) 0 0
\(731\) 1.92790e12 0.249721
\(732\) 0 0
\(733\) − 1.06610e13i − 1.36405i −0.731329 0.682024i \(-0.761100\pi\)
0.731329 0.682024i \(-0.238900\pi\)
\(734\) −8.23114e12 −1.04671
\(735\) 0 0
\(736\) −2.97196e11 −0.0373330
\(737\) 7.92047e12i 0.988889i
\(738\) 0 0
\(739\) −2.33179e12 −0.287601 −0.143800 0.989607i \(-0.545932\pi\)
−0.143800 + 0.989607i \(0.545932\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.22475e12i 0.148330i
\(743\) 7.80869e12i 0.940001i 0.882666 + 0.470001i \(0.155746\pi\)
−0.882666 + 0.470001i \(0.844254\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.35946e13 −2.78925
\(747\) 0 0
\(748\) − 6.31946e12i − 0.738113i
\(749\) −2.11361e13 −2.45390
\(750\) 0 0
\(751\) 4.20664e12 0.482565 0.241283 0.970455i \(-0.422432\pi\)
0.241283 + 0.970455i \(0.422432\pi\)
\(752\) 6.45763e12i 0.736364i
\(753\) 0 0
\(754\) −2.29310e13 −2.58376
\(755\) 0 0
\(756\) 0 0
\(757\) 1.54834e13i 1.71370i 0.515570 + 0.856848i \(0.327580\pi\)
−0.515570 + 0.856848i \(0.672420\pi\)
\(758\) − 1.80021e13i − 1.98067i
\(759\) 0 0
\(760\) 0 0
\(761\) −9.77615e12 −1.05666 −0.528332 0.849038i \(-0.677182\pi\)
−0.528332 + 0.849038i \(0.677182\pi\)
\(762\) 0 0
\(763\) 2.55835e13i 2.73275i
\(764\) −2.93979e13 −3.12174
\(765\) 0 0
\(766\) 2.02755e13 2.12785
\(767\) 1.33641e13i 1.39432i
\(768\) 0 0
\(769\) −3.35411e12 −0.345867 −0.172933 0.984934i \(-0.555325\pi\)
−0.172933 + 0.984934i \(0.555325\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.59904e11i 0.0466003i
\(773\) 6.04474e12i 0.608933i 0.952523 + 0.304467i \(0.0984782\pi\)
−0.952523 + 0.304467i \(0.901522\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.06688e13 −1.05618
\(777\) 0 0
\(778\) − 4.61163e12i − 0.451280i
\(779\) −2.84892e11 −0.0277180
\(780\) 0 0
\(781\) 4.60988e10 0.00443364
\(782\) 9.15078e11i 0.0875040i
\(783\) 0 0
\(784\) −2.36563e13 −2.23627
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.16503e12i − 0.387019i −0.981098 0.193509i \(-0.938013\pi\)
0.981098 0.193509i \(-0.0619870\pi\)
\(788\) − 1.22473e13i − 1.13155i
\(789\) 0 0
\(790\) 0 0
\(791\) 2.23346e13 2.02854
\(792\) 0 0
\(793\) − 2.16599e13i − 1.94503i
\(794\) −1.07420e13 −0.959164
\(795\) 0 0
\(796\) −9.95887e12 −0.879228
\(797\) 2.14334e13i 1.88160i 0.338957 + 0.940802i \(0.389926\pi\)
−0.338957 + 0.940802i \(0.610074\pi\)
\(798\) 0 0
\(799\) −2.90833e12 −0.252454
\(800\) 0 0
\(801\) 0 0
\(802\) 5.95301e12i 0.508103i
\(803\) − 5.87339e12i − 0.498504i
\(804\) 0 0
\(805\) 0 0
\(806\) 7.99080e12 0.666933
\(807\) 0 0
\(808\) 3.32478e13i 2.74417i
\(809\) −1.82139e11 −0.0149498 −0.00747490 0.999972i \(-0.502379\pi\)
−0.00747490 + 0.999972i \(0.502379\pi\)
\(810\) 0 0
\(811\) −1.32191e13 −1.07302 −0.536511 0.843894i \(-0.680258\pi\)
−0.536511 + 0.843894i \(0.680258\pi\)
\(812\) − 6.41153e13i − 5.17558i
\(813\) 0 0
\(814\) 1.62607e13 1.29816
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.61763e12i − 0.284070i
\(818\) 5.90963e12i 0.461498i
\(819\) 0 0
\(820\) 0 0
\(821\) 1.15652e13 0.888398 0.444199 0.895928i \(-0.353488\pi\)
0.444199 + 0.895928i \(0.353488\pi\)
\(822\) 0 0
\(823\) 9.35914e12i 0.711110i 0.934655 + 0.355555i \(0.115708\pi\)
−0.934655 + 0.355555i \(0.884292\pi\)
\(824\) 2.17307e13 1.64211
\(825\) 0 0
\(826\) −5.66448e13 −4.23399
\(827\) − 2.25652e13i − 1.67751i −0.544512 0.838753i \(-0.683285\pi\)
0.544512 0.838753i \(-0.316715\pi\)
\(828\) 0 0
\(829\) −1.22484e13 −0.900705 −0.450353 0.892851i \(-0.648702\pi\)
−0.450353 + 0.892851i \(0.648702\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.76392e13i − 1.27622i
\(833\) − 1.06541e13i − 0.766681i
\(834\) 0 0
\(835\) 0 0
\(836\) −1.18582e13 −0.839638
\(837\) 0 0
\(838\) − 3.52887e13i − 2.47194i
\(839\) 4.67277e12 0.325571 0.162786 0.986661i \(-0.447952\pi\)
0.162786 + 0.986661i \(0.447952\pi\)
\(840\) 0 0
\(841\) 1.32090e13 0.910516
\(842\) 3.83414e13i 2.62883i
\(843\) 0 0
\(844\) 3.53551e13 2.39834
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.43550e13i − 1.62597i
\(848\) 5.52094e11i 0.0366633i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.55323e12 −0.101520
\(852\) 0 0
\(853\) 4.04678e12i 0.261721i 0.991401 + 0.130861i \(0.0417740\pi\)
−0.991401 + 0.130861i \(0.958226\pi\)
\(854\) 9.18067e13 5.90628
\(855\) 0 0
\(856\) −3.20895e13 −2.04282
\(857\) − 2.35372e13i − 1.49053i −0.666766 0.745267i \(-0.732322\pi\)
0.666766 0.745267i \(-0.267678\pi\)
\(858\) 0 0
\(859\) −2.19727e13 −1.37694 −0.688470 0.725265i \(-0.741717\pi\)
−0.688470 + 0.725265i \(0.741717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 7.88441e12i − 0.486392i
\(863\) 2.06939e13i 1.26997i 0.772525 + 0.634984i \(0.218993\pi\)
−0.772525 + 0.634984i \(0.781007\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.17796e12 0.494099
\(867\) 0 0
\(868\) 2.23423e13i 1.33595i
\(869\) −2.84155e13 −1.69031
\(870\) 0 0
\(871\) 1.34977e13 0.794652
\(872\) 3.88416e13i 2.27496i
\(873\) 0 0
\(874\) 1.71711e12 0.0995400
\(875\) 0 0
\(876\) 0 0
\(877\) 8.18344e12i 0.467130i 0.972341 + 0.233565i \(0.0750392\pi\)
−0.972341 + 0.233565i \(0.924961\pi\)
\(878\) − 8.19991e12i − 0.465676i
\(879\) 0 0
\(880\) 0 0
\(881\) −2.43747e13 −1.36316 −0.681581 0.731743i \(-0.738707\pi\)
−0.681581 + 0.731743i \(0.738707\pi\)
\(882\) 0 0
\(883\) − 1.17064e13i − 0.648039i −0.946050 0.324019i \(-0.894966\pi\)
0.946050 0.324019i \(-0.105034\pi\)
\(884\) −1.07693e13 −0.593133
\(885\) 0 0
\(886\) 5.64407e13 3.07709
\(887\) 2.83247e13i 1.53642i 0.640198 + 0.768210i \(0.278852\pi\)
−0.640198 + 0.768210i \(0.721148\pi\)
\(888\) 0 0
\(889\) 4.60098e13 2.47055
\(890\) 0 0
\(891\) 0 0
\(892\) 3.39040e13i 1.79312i
\(893\) 5.45738e12i 0.287179i
\(894\) 0 0
\(895\) 0 0
\(896\) 6.71163e13 3.47890
\(897\) 0 0
\(898\) − 8.51244e12i − 0.436828i
\(899\) −9.65827e12 −0.493152
\(900\) 0 0
\(901\) −2.48647e11 −0.0125696
\(902\) − 4.01565e12i − 0.201988i
\(903\) 0 0
\(904\) 3.39091e13 1.68872
\(905\) 0 0
\(906\) 0 0
\(907\) − 5.36313e12i − 0.263139i −0.991307 0.131570i \(-0.957998\pi\)
0.991307 0.131570i \(-0.0420017\pi\)
\(908\) − 3.62153e13i − 1.76809i
\(909\) 0 0
\(910\) 0 0
\(911\) −3.43721e13 −1.65338 −0.826691 0.562656i \(-0.809780\pi\)
−0.826691 + 0.562656i \(0.809780\pi\)
\(912\) 0 0
\(913\) 1.11869e12i 0.0532833i
\(914\) 2.87335e13 1.36185
\(915\) 0 0
\(916\) 3.84492e13 1.80450
\(917\) − 2.41268e13i − 1.12678i
\(918\) 0 0
\(919\) 2.98037e13 1.37832 0.689160 0.724609i \(-0.257980\pi\)
0.689160 + 0.724609i \(0.257980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.08668e13i − 0.495234i
\(923\) − 7.85592e10i − 0.00356278i
\(924\) 0 0
\(925\) 0 0
\(926\) 1.96682e12 0.0879053
\(927\) 0 0
\(928\) 6.40859e12i 0.283659i
\(929\) −2.03150e13 −0.894843 −0.447421 0.894323i \(-0.647658\pi\)
−0.447421 + 0.894323i \(0.647658\pi\)
\(930\) 0 0
\(931\) −1.99921e13 −0.872136
\(932\) − 2.66668e13i − 1.15771i
\(933\) 0 0
\(934\) −6.87399e13 −2.95562
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.73553e13i − 1.15935i −0.814848 0.579674i \(-0.803180\pi\)
0.814848 0.579674i \(-0.196820\pi\)
\(938\) 5.72107e13i 2.41304i
\(939\) 0 0
\(940\) 0 0
\(941\) 4.24123e13 1.76335 0.881675 0.471857i \(-0.156416\pi\)
0.881675 + 0.471857i \(0.156416\pi\)
\(942\) 0 0
\(943\) 3.83578e11i 0.0157961i
\(944\) −2.55344e13 −1.04653
\(945\) 0 0
\(946\) 5.09918e13 2.07009
\(947\) 9.23741e12i 0.373229i 0.982433 + 0.186614i \(0.0597515\pi\)
−0.982433 + 0.186614i \(0.940248\pi\)
\(948\) 0 0
\(949\) −1.00091e13 −0.400589
\(950\) 0 0
\(951\) 0 0
\(952\) − 2.20958e13i − 0.871855i
\(953\) − 1.11343e13i − 0.437265i −0.975807 0.218633i \(-0.929840\pi\)
0.975807 0.218633i \(-0.0701596\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.33684e13 −0.517629
\(957\) 0 0
\(958\) − 3.02360e13i − 1.15979i
\(959\) −4.45752e12 −0.170180
\(960\) 0 0
\(961\) −2.30740e13 −0.872705
\(962\) − 2.77106e13i − 1.04318i
\(963\) 0 0
\(964\) 1.84442e13 0.687882
\(965\) 0 0
\(966\) 0 0
\(967\) 5.48630e12i 0.201772i 0.994898 + 0.100886i \(0.0321677\pi\)
−0.994898 + 0.100886i \(0.967832\pi\)
\(968\) − 3.69766e13i − 1.35359i
\(969\) 0 0
\(970\) 0 0
\(971\) 2.07887e13 0.750484 0.375242 0.926927i \(-0.377560\pi\)
0.375242 + 0.926927i \(0.377560\pi\)
\(972\) 0 0
\(973\) 5.59093e13i 1.99975i
\(974\) −5.19572e13 −1.84982
\(975\) 0 0
\(976\) 4.13847e13 1.45987
\(977\) 2.25372e13i 0.791362i 0.918388 + 0.395681i \(0.129491\pi\)
−0.918388 + 0.395681i \(0.870509\pi\)
\(978\) 0 0
\(979\) −2.04193e11 −0.00710427
\(980\) 0 0
\(981\) 0 0
\(982\) 3.78689e13i 1.29952i
\(983\) 2.20727e13i 0.753989i 0.926216 + 0.376994i \(0.123042\pi\)
−0.926216 + 0.376994i \(0.876958\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.97323e13 0.664861
\(987\) 0 0
\(988\) 2.02082e13i 0.674717i
\(989\) −4.87077e12 −0.161888
\(990\) 0 0
\(991\) −5.27788e13 −1.73831 −0.869157 0.494536i \(-0.835338\pi\)
−0.869157 + 0.494536i \(0.835338\pi\)
\(992\) − 2.23321e12i − 0.0732195i
\(993\) 0 0
\(994\) 3.32978e11 0.0108188
\(995\) 0 0
\(996\) 0 0
\(997\) 4.19157e13i 1.34353i 0.740763 + 0.671767i \(0.234464\pi\)
−0.740763 + 0.671767i \(0.765536\pi\)
\(998\) − 5.12199e13i − 1.63437i
\(999\) 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 225.10.b.i.199.4 4
3.2 odd 2 75.10.b.f.49.1 4
5.2 odd 4 45.10.a.d.1.1 2
5.3 odd 4 225.10.a.k.1.2 2
5.4 even 2 inner 225.10.b.i.199.1 4
15.2 even 4 15.10.a.d.1.2 2
15.8 even 4 75.10.a.f.1.1 2
15.14 odd 2 75.10.b.f.49.4 4
60.47 odd 4 240.10.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.2 2 15.2 even 4
45.10.a.d.1.1 2 5.2 odd 4
75.10.a.f.1.1 2 15.8 even 4
75.10.b.f.49.1 4 3.2 odd 2
75.10.b.f.49.4 4 15.14 odd 2
225.10.a.k.1.2 2 5.3 odd 4
225.10.b.i.199.1 4 5.4 even 2 inner
225.10.b.i.199.4 4 1.1 even 1 trivial
240.10.a.r.1.1 2 60.47 odd 4