Properties

Label 124.4.d.a.123.2
Level $124$
Weight $4$
Character 124.123
Analytic conductor $7.316$
Analytic rank $0$
Dimension $2$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [124,4,Mod(123,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.123");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.d (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: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 123.2
Root \(0.500000 + 2.78388i\) of defining polynomial
Character \(\chi\) \(=\) 124.123
Dual form 124.4.d.a.123.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 2.78388i) q^{2} +(-7.50000 + 2.78388i) q^{4} -2.00000 q^{5} -33.4066i q^{7} +(-11.5000 - 19.4872i) q^{8} -27.0000 q^{9} +O(q^{10})\) \(q+(0.500000 + 2.78388i) q^{2} +(-7.50000 + 2.78388i) q^{4} -2.00000 q^{5} -33.4066i q^{7} +(-11.5000 - 19.4872i) q^{8} -27.0000 q^{9} +(-1.00000 - 5.56776i) q^{10} +(93.0000 - 16.7033i) q^{14} +(48.5000 - 41.7582i) q^{16} +(-13.5000 - 75.1648i) q^{18} -55.6776i q^{19} +(15.0000 - 5.56776i) q^{20} -121.000 q^{25} +(93.0000 + 250.549i) q^{28} -172.601i q^{31} +(140.500 + 114.139i) q^{32} +66.8132i q^{35} +(202.500 - 75.1648i) q^{36} +(155.000 - 27.8388i) q^{38} +(23.0000 + 38.9744i) q^{40} -278.000 q^{41} +54.0000 q^{45} +189.304i q^{47} -773.000 q^{49} +(-60.5000 - 336.850i) q^{50} +(-651.000 + 384.176i) q^{56} +523.370i q^{59} +(480.500 - 86.3003i) q^{62} +901.978i q^{63} +(-247.500 + 448.205i) q^{64} -857.436i q^{67} +(-186.000 + 33.4066i) q^{70} -656.996i q^{71} +(310.500 + 526.154i) q^{72} +(155.000 + 417.582i) q^{76} +(-97.0000 + 83.5165i) q^{80} +729.000 q^{81} +(-139.000 - 773.919i) q^{82} +(27.0000 + 150.330i) q^{90} +(-527.000 + 94.6520i) q^{94} +111.355i q^{95} +1906.00 q^{97} +(-386.500 - 2151.94i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9} - 2 q^{10} + 186 q^{14} + 97 q^{16} - 27 q^{18} + 30 q^{20} - 242 q^{25} + 186 q^{28} + 281 q^{32} + 405 q^{36} + 310 q^{38} + 46 q^{40} - 556 q^{41} + 108 q^{45} - 1546 q^{49} - 121 q^{50} - 1302 q^{56} + 961 q^{62} - 495 q^{64} - 372 q^{70} + 621 q^{72} + 310 q^{76} - 194 q^{80} + 1458 q^{81} - 278 q^{82} + 54 q^{90} - 1054 q^{94} + 3812 q^{97} - 773 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 2.78388i 0.176777 + 0.984251i
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −7.50000 + 2.78388i −0.937500 + 0.347985i
\(5\) −2.00000 −0.178885 −0.0894427 0.995992i \(-0.528509\pi\)
−0.0894427 + 0.995992i \(0.528509\pi\)
\(6\) 0 0
\(7\) 33.4066i 1.80379i −0.431959 0.901893i \(-0.642178\pi\)
0.431959 0.901893i \(-0.357822\pi\)
\(8\) −11.5000 19.4872i −0.508233 0.861220i
\(9\) −27.0000 −1.00000
\(10\) −1.00000 5.56776i −0.0316228 0.176068i
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 93.0000 16.7033i 1.77538 0.318867i
\(15\) 0 0
\(16\) 48.5000 41.7582i 0.757812 0.652472i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −13.5000 75.1648i −0.176777 0.984251i
\(19\) 55.6776i 0.672280i −0.941812 0.336140i \(-0.890878\pi\)
0.941812 0.336140i \(-0.109122\pi\)
\(20\) 15.0000 5.56776i 0.167705 0.0622495i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −121.000 −0.968000
\(26\) 0 0
\(27\) 0 0
\(28\) 93.0000 + 250.549i 0.627691 + 1.69105i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 172.601i 1.00000i
\(32\) 140.500 + 114.139i 0.776160 + 0.630536i
\(33\) 0 0
\(34\) 0 0
\(35\) 66.8132i 0.322671i
\(36\) 202.500 75.1648i 0.937500 0.347985i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 155.000 27.8388i 0.661693 0.118844i
\(39\) 0 0
\(40\) 23.0000 + 38.9744i 0.0909155 + 0.154060i
\(41\) −278.000 −1.05893 −0.529467 0.848330i \(-0.677608\pi\)
−0.529467 + 0.848330i \(0.677608\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 54.0000 0.178885
\(46\) 0 0
\(47\) 189.304i 0.587507i 0.955881 + 0.293753i \(0.0949045\pi\)
−0.955881 + 0.293753i \(0.905096\pi\)
\(48\) 0 0
\(49\) −773.000 −2.25364
\(50\) −60.5000 336.850i −0.171120 0.952755i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −651.000 + 384.176i −1.55346 + 0.916744i
\(57\) 0 0
\(58\) 0 0
\(59\) 523.370i 1.15486i 0.816439 + 0.577432i \(0.195945\pi\)
−0.816439 + 0.577432i \(0.804055\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 480.500 86.3003i 0.984251 0.176777i
\(63\) 901.978i 1.80379i
\(64\) −247.500 + 448.205i −0.483398 + 0.875400i
\(65\) 0 0
\(66\) 0 0
\(67\) 857.436i 1.56347i −0.623611 0.781735i \(-0.714335\pi\)
0.623611 0.781735i \(-0.285665\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −186.000 + 33.4066i −0.317589 + 0.0570407i
\(71\) 656.996i 1.09818i −0.835762 0.549092i \(-0.814974\pi\)
0.835762 0.549092i \(-0.185026\pi\)
\(72\) 310.500 + 526.154i 0.508233 + 0.861220i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 155.000 + 417.582i 0.233944 + 0.630263i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −97.0000 + 83.5165i −0.135562 + 0.116718i
\(81\) 729.000 1.00000
\(82\) −139.000 773.919i −0.187195 1.04226i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 27.0000 + 150.330i 0.0316228 + 0.176068i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −527.000 + 94.6520i −0.578254 + 0.103858i
\(95\) 111.355i 0.120261i
\(96\) 0 0
\(97\) 1906.00 1.99510 0.997551 0.0699366i \(-0.0222797\pi\)
0.997551 + 0.0699366i \(0.0222797\pi\)
\(98\) −386.500 2151.94i −0.398392 2.21815i
\(99\) 0 0
\(100\) 907.500 336.850i 0.907500 0.336850i
\(101\) 1790.00 1.76348 0.881741 0.471734i \(-0.156372\pi\)
0.881741 + 0.471734i \(0.156372\pi\)
\(102\) 0 0
\(103\) 1659.19i 1.58724i 0.608417 + 0.793618i \(0.291805\pi\)
−0.608417 + 0.793618i \(0.708195\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1525.57i 1.37834i −0.724601 0.689169i \(-0.757976\pi\)
0.724601 0.689169i \(-0.242024\pi\)
\(108\) 0 0
\(109\) −1546.00 −1.35853 −0.679266 0.733892i \(-0.737702\pi\)
−0.679266 + 0.733892i \(0.737702\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1395.00 1620.22i −1.17692 1.36693i
\(113\) 2402.00 1.99966 0.999828 0.0185406i \(-0.00590199\pi\)
0.999828 + 0.0185406i \(0.00590199\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −1457.00 + 261.685i −1.13668 + 0.204153i
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 480.500 + 1294.51i 0.347985 + 0.937500i
\(125\) 492.000 0.352047
\(126\) −2511.00 + 450.989i −1.77538 + 0.318867i
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1371.50 464.908i −0.947067 0.321035i
\(129\) 0 0
\(130\) 0 0
\(131\) 2995.46i 1.99782i −0.0466865 0.998910i \(-0.514866\pi\)
0.0466865 0.998910i \(-0.485134\pi\)
\(132\) 0 0
\(133\) −1860.00 −1.21265
\(134\) 2387.00 428.718i 1.53885 0.276385i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −186.000 501.099i −0.112285 0.302504i
\(141\) 0 0
\(142\) 1829.00 328.498i 1.08089 0.194133i
\(143\) 0 0
\(144\) −1309.50 + 1127.47i −0.757812 + 0.652472i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3470.00 1.90788 0.953938 0.300004i \(-0.0969881\pi\)
0.953938 + 0.300004i \(0.0969881\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −1085.00 + 640.293i −0.578981 + 0.341675i
\(153\) 0 0
\(154\) 0 0
\(155\) 345.201i 0.178885i
\(156\) 0 0
\(157\) −1146.00 −0.582553 −0.291276 0.956639i \(-0.594080\pi\)
−0.291276 + 0.956639i \(0.594080\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −281.000 228.278i −0.138844 0.112794i
\(161\) 0 0
\(162\) 364.500 + 2029.45i 0.176777 + 0.984251i
\(163\) 3240.44i 1.55712i 0.627569 + 0.778561i \(0.284050\pi\)
−0.627569 + 0.778561i \(0.715950\pi\)
\(164\) 2085.00 773.919i 0.992751 0.368494i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) 1503.30i 0.672280i
\(172\) 0 0
\(173\) −4522.00 −1.98729 −0.993645 0.112556i \(-0.964096\pi\)
−0.993645 + 0.112556i \(0.964096\pi\)
\(174\) 0 0
\(175\) 4042.20i 1.74606i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −405.000 + 150.330i −0.167705 + 0.0622495i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −527.000 1419.78i −0.204444 0.550788i
\(189\) 0 0
\(190\) −310.000 + 55.6776i −0.118367 + 0.0212594i
\(191\) 278.388i 0.105463i 0.998609 + 0.0527316i \(0.0167928\pi\)
−0.998609 + 0.0527316i \(0.983207\pi\)
\(192\) 0 0
\(193\) −4542.00 −1.69399 −0.846996 0.531600i \(-0.821591\pi\)
−0.846996 + 0.531600i \(0.821591\pi\)
\(194\) 953.000 + 5306.08i 0.352688 + 1.96368i
\(195\) 0 0
\(196\) 5797.50 2151.94i 2.11279 0.784235i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1391.50 + 2357.95i 0.491970 + 0.833661i
\(201\) 0 0
\(202\) 895.000 + 4983.15i 0.311742 + 1.73571i
\(203\) 0 0
\(204\) 0 0
\(205\) 556.000 0.189428
\(206\) −4619.00 + 829.597i −1.56224 + 0.280586i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5578.90i 1.82022i −0.414362 0.910112i \(-0.635995\pi\)
0.414362 0.910112i \(-0.364005\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 4247.00 762.784i 1.35663 0.243658i
\(215\) 0 0
\(216\) 0 0
\(217\) −5766.00 −1.80379
\(218\) −773.000 4303.88i −0.240157 1.33714i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 3813.00 4693.63i 1.13735 1.40003i
\(225\) 3267.00 0.968000
\(226\) 1201.00 + 6686.88i 0.353493 + 1.96816i
\(227\) 6202.49i 1.81354i −0.421625 0.906770i \(-0.638540\pi\)
0.421625 0.906770i \(-0.361460\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5398.00 −1.51775 −0.758873 0.651239i \(-0.774250\pi\)
−0.758873 + 0.651239i \(0.774250\pi\)
\(234\) 0 0
\(235\) 378.608i 0.105096i
\(236\) −1457.00 3925.27i −0.401876 1.08268i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −665.500 3705.35i −0.176777 0.984251i
\(243\) 0 0
\(244\) 0 0
\(245\) 1546.00 0.403144
\(246\) 0 0
\(247\) 0 0
\(248\) −3363.50 + 1984.91i −0.861220 + 0.508233i
\(249\) 0 0
\(250\) 246.000 + 1369.67i 0.0622336 + 0.346502i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −2511.00 6764.83i −0.627691 1.69105i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 608.500 4050.55i 0.148560 0.988903i
\(257\) −4286.00 −1.04029 −0.520143 0.854079i \(-0.674121\pi\)
−0.520143 + 0.854079i \(0.674121\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 8339.00 1497.73i 1.96636 0.353168i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −930.000 5178.02i −0.214368 1.19355i
\(267\) 0 0
\(268\) 2387.00 + 6430.77i 0.544064 + 1.46575i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 4660.22i 1.00000i
\(280\) 1302.00 768.351i 0.277891 0.163992i
\(281\) 650.000 0.137992 0.0689960 0.997617i \(-0.478020\pi\)
0.0689960 + 0.997617i \(0.478020\pi\)
\(282\) 0 0
\(283\) 8919.56i 1.87354i −0.349941 0.936772i \(-0.613799\pi\)
0.349941 0.936772i \(-0.386201\pi\)
\(284\) 1829.00 + 4927.47i 0.382152 + 1.02955i
\(285\) 0 0
\(286\) 0 0
\(287\) 9287.03i 1.91009i
\(288\) −3793.50 3081.76i −0.776160 0.630536i
\(289\) 4913.00 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5278.00 1.05237 0.526184 0.850371i \(-0.323622\pi\)
0.526184 + 0.850371i \(0.323622\pi\)
\(294\) 0 0
\(295\) 1046.74i 0.206588i
\(296\) 0 0
\(297\) 0 0
\(298\) 1735.00 + 9660.07i 0.337268 + 1.87783i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −2325.00 2700.37i −0.438644 0.509462i
\(305\) 0 0
\(306\) 0 0
\(307\) 4153.55i 0.772169i −0.922464 0.386084i \(-0.873827\pi\)
0.922464 0.386084i \(-0.126173\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −961.000 + 172.601i −0.176068 + 0.0316228i
\(311\) 3953.11i 0.720773i −0.932803 0.360387i \(-0.882645\pi\)
0.932803 0.360387i \(-0.117355\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −573.000 3190.33i −0.102982 0.573378i
\(315\) 1803.96i 0.322671i
\(316\) 0 0
\(317\) 11206.0 1.98546 0.992731 0.120352i \(-0.0384022\pi\)
0.992731 + 0.120352i \(0.0384022\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 495.000 896.410i 0.0864729 0.156596i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −5467.50 + 2029.45i −0.937500 + 0.347985i
\(325\) 0 0
\(326\) −9021.00 + 1620.22i −1.53260 + 0.275263i
\(327\) 0 0
\(328\) 3197.00 + 5417.43i 0.538185 + 0.911975i
\(329\) 6324.00 1.05974
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1714.87i 0.279682i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 1098.50 + 6116.19i 0.176777 + 0.984251i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −4185.00 + 751.648i −0.661693 + 0.118844i
\(343\) 14364.8i 2.26131i
\(344\) 0 0
\(345\) 0 0
\(346\) −2261.00 12588.7i −0.351307 1.95599i
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −4410.00 −0.676395 −0.338198 0.941075i \(-0.609817\pi\)
−0.338198 + 0.941075i \(0.609817\pi\)
\(350\) −11253.0 + 2021.10i −1.71857 + 0.308664i
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 1313.99i 0.196449i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12972.9i 1.90719i 0.301084 + 0.953597i \(0.402651\pi\)
−0.301084 + 0.953597i \(0.597349\pi\)
\(360\) −621.000 1052.31i −0.0909155 0.154060i
\(361\) 3759.00 0.548039
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 7506.00 1.05893
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14258.0 −1.97923 −0.989613 0.143758i \(-0.954081\pi\)
−0.989613 + 0.143758i \(0.954081\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3689.00 2177.00i 0.505972 0.298590i
\(377\) 0 0
\(378\) 0 0
\(379\) 701.538i 0.0950807i 0.998869 + 0.0475404i \(0.0151383\pi\)
−0.998869 + 0.0475404i \(0.984862\pi\)
\(380\) −310.000 835.165i −0.0418491 0.112745i
\(381\) 0 0
\(382\) −775.000 + 139.194i −0.103802 + 0.0186434i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2271.00 12644.4i −0.299458 1.66731i
\(387\) 0 0
\(388\) −14295.0 + 5306.08i −1.87041 + 0.694266i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8889.50 + 15063.6i 1.14538 + 1.94088i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15626.0 −1.97543 −0.987716 0.156260i \(-0.950056\pi\)
−0.987716 + 0.156260i \(0.950056\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5868.50 + 5052.75i −0.733563 + 0.631593i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −13425.0 + 4983.15i −1.65326 + 0.613666i
\(405\) −1458.00 −0.178885
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 278.000 + 1547.84i 0.0334864 + 0.186445i
\(411\) 0 0
\(412\) −4619.00 12444.0i −0.552334 1.48803i
\(413\) 17484.0 2.08313
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16892.6i 1.96959i −0.173726 0.984794i \(-0.555581\pi\)
0.173726 0.984794i \(-0.444419\pi\)
\(420\) 0 0
\(421\) 11838.0 1.37042 0.685212 0.728343i \(-0.259709\pi\)
0.685212 + 0.728343i \(0.259709\pi\)
\(422\) 15531.0 2789.45i 1.79156 0.321773i
\(423\) 5111.21i 0.587507i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 4247.00 + 11441.8i 0.479641 + 1.29219i
\(429\) 0 0
\(430\) 0 0
\(431\) 13017.4i 1.45482i 0.686203 + 0.727410i \(0.259276\pi\)
−0.686203 + 0.727410i \(0.740724\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −2883.00 16051.9i −0.318867 1.77538i
\(435\) 0 0
\(436\) 11595.0 4303.88i 1.27362 0.472749i
\(437\) 0 0
\(438\) 0 0
\(439\) 18295.7i 1.98908i −0.104370 0.994539i \(-0.533283\pi\)
0.104370 0.994539i \(-0.466717\pi\)
\(440\) 0 0
\(441\) 20871.0 2.25364
\(442\) 0 0
\(443\) 7137.87i 0.765532i −0.923845 0.382766i \(-0.874972\pi\)
0.923845 0.382766i \(-0.125028\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 14973.0 + 8268.13i 1.57904 + 0.871947i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 1633.50 + 9094.94i 0.171120 + 0.952755i
\(451\) 0 0
\(452\) −18015.0 + 6686.88i −1.87468 + 0.695851i
\(453\) 0 0
\(454\) 17267.0 3101.24i 1.78498 0.320592i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −2699.00 15027.4i −0.268302 1.49384i
\(467\) 18295.7i 1.81290i 0.422317 + 0.906448i \(0.361217\pi\)
−0.422317 + 0.906448i \(0.638783\pi\)
\(468\) 0 0
\(469\) −28644.0 −2.82016
\(470\) 1054.00 189.304i 0.103441 0.0185786i
\(471\) 0 0
\(472\) 10199.0 6018.75i 0.994591 0.586940i
\(473\) 0 0
\(474\) 0 0
\(475\) 6736.99i 0.650767i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12104.3i 1.15461i −0.816527 0.577307i \(-0.804103\pi\)
0.816527 0.577307i \(-0.195897\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 9982.50 3705.35i 0.937500 0.347985i
\(485\) −3812.00 −0.356895
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 773.000 + 4303.88i 0.0712665 + 0.396795i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −7207.50 8371.13i −0.652472 0.757812i
\(497\) −21948.0 −1.98089
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −3690.00 + 1369.67i −0.330044 + 0.122507i
\(501\) 0 0
\(502\) 0 0
\(503\) 679.267i 0.0602128i 0.999547 + 0.0301064i \(0.00958461\pi\)
−0.999547 + 0.0301064i \(0.990415\pi\)
\(504\) 17577.0 10372.7i 1.55346 0.916744i
\(505\) −3580.00 −0.315461
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11580.5 331.282i 0.999591 0.0285952i
\(513\) 0 0
\(514\) −2143.00 11931.7i −0.183898 1.02390i
\(515\) 3318.39i 0.283933i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14630.0 −1.23023 −0.615117 0.788436i \(-0.710891\pi\)
−0.615117 + 0.788436i \(0.710891\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 8339.00 + 22465.9i 0.695212 + 1.87296i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 14131.0i 1.15486i
\(532\) 13950.0 5178.02i 1.13686 0.421984i
\(533\) 0 0
\(534\) 0 0
\(535\) 3051.13i 0.246565i
\(536\) −16709.0 + 9860.51i −1.34649 + 0.794607i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24870.0 1.97642 0.988211 0.153095i \(-0.0489241\pi\)
0.988211 + 0.153095i \(0.0489241\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3092.00 0.243022
\(546\) 0 0
\(547\) 24910.2i 1.94713i −0.228402 0.973567i \(-0.573350\pi\)
0.228402 0.973567i \(-0.426650\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −12973.5 + 2330.11i −0.984251 + 0.176777i
\(559\) 0 0
\(560\) 2790.00 + 3240.44i 0.210534 + 0.244524i
\(561\) 0 0
\(562\) 325.000 + 1809.52i 0.0243938 + 0.135819i
\(563\) 22393.5i 1.67633i 0.545414 + 0.838167i \(0.316372\pi\)
−0.545414 + 0.838167i \(0.683628\pi\)
\(564\) 0 0
\(565\) −4804.00 −0.357709
\(566\) 24831.0 4459.78i 1.84404 0.331199i
\(567\) 24353.4i 1.80379i
\(568\) −12803.0 + 7555.46i −0.945778 + 0.558134i
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −25854.0 + 4643.52i −1.88001 + 0.337660i
\(575\) 0 0
\(576\) 6682.50 12101.5i 0.483398 0.875400i
\(577\) −25326.0 −1.82727 −0.913635 0.406535i \(-0.866737\pi\)
−0.913635 + 0.406535i \(0.866737\pi\)
\(578\) 2456.50 + 13677.2i 0.176777 + 0.984251i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 2639.00 + 14693.3i 0.186034 + 1.03580i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −9610.00 −0.672280
\(590\) 2914.00 523.370i 0.203335 0.0365200i
\(591\) 0 0
\(592\) 0 0
\(593\) −25342.0 −1.75493 −0.877463 0.479644i \(-0.840766\pi\)
−0.877463 + 0.479644i \(0.840766\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −26025.0 + 9660.07i −1.78863 + 0.663913i
\(597\) 0 0
\(598\) 0 0
\(599\) 26780.9i 1.82678i 0.407088 + 0.913389i \(0.366544\pi\)
−0.407088 + 0.913389i \(0.633456\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 23150.8i 1.56347i
\(604\) 0 0
\(605\) 2662.00 0.178885
\(606\) 0 0
\(607\) 18897.0i 1.26360i 0.775131 + 0.631800i \(0.217684\pi\)
−0.775131 + 0.631800i \(0.782316\pi\)
\(608\) 6355.00 7822.71i 0.423897 0.521797i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 11563.0 2076.78i 0.760008 0.136501i
\(615\) 0 0
\(616\) 0 0
\(617\) 30074.0 1.96229 0.981146 0.193270i \(-0.0619094\pi\)
0.981146 + 0.193270i \(0.0619094\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −961.000 2589.01i −0.0622495 0.167705i
\(621\) 0 0
\(622\) 11005.0 1976.56i 0.709422 0.127416i
\(623\) 0 0
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) 0 0
\(627\) 0 0
\(628\) 8595.00 3190.33i 0.546143 0.202720i
\(629\) 0 0
\(630\) 5022.00 901.978i 0.317589 0.0570407i
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 5603.00 + 31196.2i 0.350984 + 1.95419i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 17738.9i 1.09818i
\(640\) 2743.00 + 929.817i 0.169417 + 0.0574285i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −8383.50 14206.2i −0.508233 0.861220i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −9021.00 24303.3i −0.541855 1.45980i
\(653\) 7222.00 0.432800 0.216400 0.976305i \(-0.430568\pi\)
0.216400 + 0.976305i \(0.430568\pi\)
\(654\) 0 0
\(655\) 5990.91i 0.357381i
\(656\) −13483.0 + 11608.8i −0.802474 + 0.690925i
\(657\) 0 0
\(658\) 3162.00 + 17605.3i 0.187337 + 1.04305i
\(659\) 33796.3i 1.99775i 0.0474074 + 0.998876i \(0.484904\pi\)
−0.0474074 + 0.998876i \(0.515096\pi\)
\(660\) 0 0
\(661\) 30350.0 1.78590 0.892949 0.450158i \(-0.148632\pi\)
0.892949 + 0.450158i \(0.148632\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3720.00 0.216925
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) −4774.00 + 857.436i −0.275277 + 0.0494412i
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −16477.5 + 6116.19i −0.937500 + 0.347985i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 63673.0i 3.59874i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25377.9i 1.42175i 0.703317 + 0.710876i \(0.251701\pi\)
−0.703317 + 0.710876i \(0.748299\pi\)
\(684\) −4185.00 11274.7i −0.233944 0.630263i
\(685\) 0 0
\(686\) −39990.0 + 7182.42i −2.22569 + 0.399746i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17872.5i 0.983940i −0.870612 0.491970i \(-0.836277\pi\)
0.870612 0.491970i \(-0.163723\pi\)
\(692\) 33915.0 12588.7i 1.86309 0.691548i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −2205.00 12276.9i −0.119571 0.665743i
\(699\) 0 0
\(700\) −11253.0 30316.5i −0.607605 1.63694i
\(701\) 17702.0 0.953774 0.476887 0.878965i \(-0.341765\pi\)
0.476887 + 0.878965i \(0.341765\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 59797.8i 3.18094i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −3658.00 + 656.996i −0.193355 + 0.0347276i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −36115.0 + 6486.45i −1.87716 + 0.337148i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 2619.00 2254.94i 0.135562 0.116718i
\(721\) 55428.0 2.86303
\(722\) 1879.50 + 10464.6i 0.0968805 + 0.539408i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23017.1i 1.17422i −0.809507 0.587110i \(-0.800265\pi\)
0.809507 0.587110i \(-0.199735\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 27078.0 1.36446 0.682230 0.731138i \(-0.261010\pi\)
0.682230 + 0.731138i \(0.261010\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 3753.00 + 20895.8i 0.187195 + 1.04226i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −6940.00 −0.341291
\(746\) −7129.00 39692.6i −0.349881 1.94806i
\(747\) 0 0
\(748\) 0 0
\(749\) −50964.0 −2.48623
\(750\) 0 0
\(751\) 7672.38i 0.372795i 0.982474 + 0.186398i \(0.0596812\pi\)
−0.982474 + 0.186398i \(0.940319\pi\)
\(752\) 7905.00 + 9181.24i 0.383332 + 0.445220i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −1953.00 + 350.769i −0.0935833 + 0.0168081i
\(759\) 0 0
\(760\) 2170.00 1280.59i 0.103571 0.0611207i
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 51646.6i 2.45050i
\(764\) −775.000 2087.91i −0.0366996 0.0988717i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 25170.0 1.18030 0.590152 0.807292i \(-0.299068\pi\)
0.590152 + 0.807292i \(0.299068\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 34065.0 12644.4i 1.58812 0.589484i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 20884.7i 0.968000i
\(776\) −21919.0 37142.6i −1.01398 1.71822i
\(777\) 0 0
\(778\) 0 0
\(779\) 15478.4i 0.711901i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −37490.5 + 32279.1i −1.70784 + 1.47044i
\(785\) 2292.00 0.104210
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 80242.6i 3.60695i
\(792\) 0 0
\(793\) 0 0
\(794\) −7813.00 43500.9i −0.349210 1.94432i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −17000.5 13810.8i −0.751323 0.610359i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −20585.0 34882.0i −0.896260 1.51875i
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −729.000 4058.90i −0.0316228 0.176068i
\(811\) 37137.0i 1.60796i 0.594656 + 0.803980i \(0.297288\pi\)
−0.594656 + 0.803980i \(0.702712\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6480.88i 0.278546i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −4170.00 + 1547.84i −0.177589 + 0.0659181i
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 32333.0 19080.7i 1.36696 0.806685i
\(825\) 0 0
\(826\) 8742.00 + 48673.4i 0.368248 + 2.05032i
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 47027.0 8446.30i 1.93857 0.348177i
\(839\) 32460.1i 1.33569i −0.744299 0.667846i \(-0.767216\pi\)
0.744299 0.667846i \(-0.232784\pi\)
\(840\) 0 0
\(841\) 24389.0 1.00000
\(842\) 5919.00 + 32955.6i 0.242259 + 1.34884i
\(843\) 0 0
\(844\) 15531.0 + 41841.7i 0.633411 + 1.70646i
\(845\) −4394.00 −0.178885
\(846\) 14229.0 2555.60i 0.578254 0.103858i
\(847\) 44464.2i 1.80379i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 19278.0 0.773817 0.386909 0.922118i \(-0.373543\pi\)
0.386909 + 0.922118i \(0.373543\pi\)
\(854\) 0 0
\(855\) 3006.59i 0.120261i
\(856\) −29729.0 + 17544.0i −1.18705 + 0.700517i
\(857\) 42826.0 1.70701 0.853505 0.521084i \(-0.174472\pi\)
0.853505 + 0.521084i \(0.174472\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −36239.0 + 6508.72i −1.43191 + 0.257178i
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 9044.00 0.355497
\(866\) 0 0
\(867\) 0 0
\(868\) 43245.0 16051.9i 1.69105 0.627691i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 17779.0 + 30127.2i 0.690451 + 1.16999i
\(873\) −51462.0 −1.99510
\(874\) 0 0
\(875\) 16436.0i 0.635017i
\(876\) 0 0
\(877\) −38634.0 −1.48755 −0.743773 0.668433i \(-0.766966\pi\)
−0.743773 + 0.668433i \(0.766966\pi\)
\(878\) 50933.0 9147.84i 1.95775 0.351622i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 10435.5 + 58102.4i 0.398392 + 2.21815i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 19871.0 3568.94i 0.753476 0.135328i
\(887\) 11280.3i 0.427007i 0.976942 + 0.213503i \(0.0684875\pi\)
−0.976942 + 0.213503i \(0.931513\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10540.0 0.394969
\(894\) 0 0
\(895\) 0 0
\(896\) −15531.0 + 45817.1i −0.579078 + 1.70831i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −24502.5 + 9094.94i −0.907500 + 0.336850i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −27623.0 46808.2i −1.01629 1.72214i
\(905\) 0 0
\(906\) 0 0
\(907\) 50856.0i 1.86179i 0.365286 + 0.930895i \(0.380971\pi\)
−0.365286 + 0.930895i \(0.619029\pi\)
\(908\) 17267.0 + 46518.7i 0.631085 + 1.70019i
\(909\) −48330.0 −1.76348
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −100068. −3.60364
\(918\) 0 0
\(919\) 19097.4i 0.685491i −0.939428 0.342745i \(-0.888643\pi\)
0.939428 0.342745i \(-0.111357\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 44798.2i 1.58724i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 43038.8i 1.51508i
\(932\) 40485.0 15027.4i 1.42289 0.528153i
\(933\) 0 0
\(934\) −50933.0 + 9147.84i −1.78435 + 0.320478i
\(935\) 0 0
\(936\) 0 0
\(937\) −43974.0 −1.53316 −0.766578 0.642151i \(-0.778042\pi\)
−0.766578 + 0.642151i \(0.778042\pi\)
\(938\) −14322.0 79741.5i −0.498539 2.77575i
\(939\) 0 0
\(940\) 1054.00 + 2839.56i 0.0365720 + 0.0985279i
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 21855.0 + 25383.4i 0.753517 + 0.875170i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −18755.0 + 3368.50i −0.640518 + 0.115041i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 556.776i 0.0188658i
\(956\) 0 0
\(957\) 0 0
\(958\) 33697.0 6052.16i 1.13643 0.204109i
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 41190.3i 1.37834i
\(964\) 0 0
\(965\) 9084.00 0.303030
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 15306.5 + 25937.4i 0.508233 + 0.861220i
\(969\) 0 0
\(970\) −1906.00 10612.2i −0.0630907 0.351274i
\(971\) 10411.7i 0.344107i 0.985088 + 0.172054i \(0.0550402\pi\)
−0.985088 + 0.172054i \(0.944960\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23246.0 −0.761213 −0.380607 0.924737i \(-0.624285\pi\)
−0.380607 + 0.924737i \(0.624285\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −11595.0 + 4303.88i −0.377948 + 0.140288i
\(981\) 41742.0 1.35853
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 19700.5 24250.4i 0.630536 0.776160i
\(993\) 0 0
\(994\) −10974.0 61100.6i −0.350175 1.94969i
\(995\) 0 0
\(996\) 0 0
\(997\) 54686.0 1.73713 0.868567 0.495571i \(-0.165041\pi\)
0.868567 + 0.495571i \(0.165041\pi\)
\(998\) 0 0
\(999\) 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 124.4.d.a.123.2 yes 2
4.3 odd 2 inner 124.4.d.a.123.1 2
31.30 odd 2 CM 124.4.d.a.123.2 yes 2
124.123 even 2 inner 124.4.d.a.123.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.4.d.a.123.1 2 4.3 odd 2 inner
124.4.d.a.123.1 2 124.123 even 2 inner
124.4.d.a.123.2 yes 2 1.1 even 1 trivial
124.4.d.a.123.2 yes 2 31.30 odd 2 CM