Properties

Label 550.6.b.f.199.1
Level $550$
Weight $6$
Character 550.199
Analytic conductor $88.211$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,6,Mod(199,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(88.2111008971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 550.199
Dual form 550.6.b.f.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} -1.00000i q^{3} -16.0000 q^{4} -4.00000 q^{6} -166.000i q^{7} +64.0000i q^{8} +242.000 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} -1.00000i q^{3} -16.0000 q^{4} -4.00000 q^{6} -166.000i q^{7} +64.0000i q^{8} +242.000 q^{9} -121.000 q^{11} +16.0000i q^{12} -692.000i q^{13} -664.000 q^{14} +256.000 q^{16} -738.000i q^{17} -968.000i q^{18} -1424.00 q^{19} -166.000 q^{21} +484.000i q^{22} +1779.00i q^{23} +64.0000 q^{24} -2768.00 q^{26} -485.000i q^{27} +2656.00i q^{28} +2064.00 q^{29} +6245.00 q^{31} -1024.00i q^{32} +121.000i q^{33} -2952.00 q^{34} -3872.00 q^{36} -14785.0i q^{37} +5696.00i q^{38} -692.000 q^{39} +5304.00 q^{41} +664.000i q^{42} -17798.0i q^{43} +1936.00 q^{44} +7116.00 q^{46} -17184.0i q^{47} -256.000i q^{48} -10749.0 q^{49} -738.000 q^{51} +11072.0i q^{52} +30726.0i q^{53} -1940.00 q^{54} +10624.0 q^{56} +1424.00i q^{57} -8256.00i q^{58} +34989.0 q^{59} -45940.0 q^{61} -24980.0i q^{62} -40172.0i q^{63} -4096.00 q^{64} +484.000 q^{66} +25343.0i q^{67} +11808.0i q^{68} +1779.00 q^{69} +13311.0 q^{71} +15488.0i q^{72} +53260.0i q^{73} -59140.0 q^{74} +22784.0 q^{76} +20086.0i q^{77} +2768.00i q^{78} -77234.0 q^{79} +58321.0 q^{81} -21216.0i q^{82} -55014.0i q^{83} +2656.00 q^{84} -71192.0 q^{86} -2064.00i q^{87} -7744.00i q^{88} -125415. q^{89} -114872. q^{91} -28464.0i q^{92} -6245.00i q^{93} -68736.0 q^{94} -1024.00 q^{96} -88807.0i q^{97} +42996.0i q^{98} -29282.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 8 q^{6} + 484 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 8 q^{6} + 484 q^{9} - 242 q^{11} - 1328 q^{14} + 512 q^{16} - 2848 q^{19} - 332 q^{21} + 128 q^{24} - 5536 q^{26} + 4128 q^{29} + 12490 q^{31} - 5904 q^{34} - 7744 q^{36} - 1384 q^{39} + 10608 q^{41} + 3872 q^{44} + 14232 q^{46} - 21498 q^{49} - 1476 q^{51} - 3880 q^{54} + 21248 q^{56} + 69978 q^{59} - 91880 q^{61} - 8192 q^{64} + 968 q^{66} + 3558 q^{69} + 26622 q^{71} - 118280 q^{74} + 45568 q^{76} - 154468 q^{79} + 116642 q^{81} + 5312 q^{84} - 142384 q^{86} - 250830 q^{89} - 229744 q^{91} - 137472 q^{94} - 2048 q^{96} - 58564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) − 1.00000i − 0.0641500i −0.999485 0.0320750i \(-0.989788\pi\)
0.999485 0.0320750i \(-0.0102115\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) −4.00000 −0.0453609
\(7\) − 166.000i − 1.28045i −0.768187 0.640226i \(-0.778841\pi\)
0.768187 0.640226i \(-0.221159\pi\)
\(8\) 64.0000i 0.353553i
\(9\) 242.000 0.995885
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 16.0000i 0.0320750i
\(13\) − 692.000i − 1.13566i −0.823146 0.567829i \(-0.807783\pi\)
0.823146 0.567829i \(-0.192217\pi\)
\(14\) −664.000 −0.905416
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 738.000i − 0.619347i −0.950843 0.309674i \(-0.899780\pi\)
0.950843 0.309674i \(-0.100220\pi\)
\(18\) − 968.000i − 0.704197i
\(19\) −1424.00 −0.904953 −0.452476 0.891776i \(-0.649459\pi\)
−0.452476 + 0.891776i \(0.649459\pi\)
\(20\) 0 0
\(21\) −166.000 −0.0821410
\(22\) 484.000i 0.213201i
\(23\) 1779.00i 0.701223i 0.936521 + 0.350612i \(0.114026\pi\)
−0.936521 + 0.350612i \(0.885974\pi\)
\(24\) 64.0000 0.0226805
\(25\) 0 0
\(26\) −2768.00 −0.803032
\(27\) − 485.000i − 0.128036i
\(28\) 2656.00i 0.640226i
\(29\) 2064.00 0.455737 0.227869 0.973692i \(-0.426824\pi\)
0.227869 + 0.973692i \(0.426824\pi\)
\(30\) 0 0
\(31\) 6245.00 1.16715 0.583577 0.812058i \(-0.301653\pi\)
0.583577 + 0.812058i \(0.301653\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 121.000i 0.0193420i
\(34\) −2952.00 −0.437944
\(35\) 0 0
\(36\) −3872.00 −0.497942
\(37\) − 14785.0i − 1.77549i −0.460340 0.887743i \(-0.652273\pi\)
0.460340 0.887743i \(-0.347727\pi\)
\(38\) 5696.00i 0.639898i
\(39\) −692.000 −0.0728525
\(40\) 0 0
\(41\) 5304.00 0.492770 0.246385 0.969172i \(-0.420757\pi\)
0.246385 + 0.969172i \(0.420757\pi\)
\(42\) 664.000i 0.0580824i
\(43\) − 17798.0i − 1.46791i −0.679197 0.733956i \(-0.737672\pi\)
0.679197 0.733956i \(-0.262328\pi\)
\(44\) 1936.00 0.150756
\(45\) 0 0
\(46\) 7116.00 0.495840
\(47\) − 17184.0i − 1.13470i −0.823478 0.567348i \(-0.807969\pi\)
0.823478 0.567348i \(-0.192031\pi\)
\(48\) − 256.000i − 0.0160375i
\(49\) −10749.0 −0.639555
\(50\) 0 0
\(51\) −738.000 −0.0397311
\(52\) 11072.0i 0.567829i
\(53\) 30726.0i 1.50251i 0.660014 + 0.751253i \(0.270550\pi\)
−0.660014 + 0.751253i \(0.729450\pi\)
\(54\) −1940.00 −0.0905352
\(55\) 0 0
\(56\) 10624.0 0.452708
\(57\) 1424.00i 0.0580528i
\(58\) − 8256.00i − 0.322255i
\(59\) 34989.0 1.30858 0.654292 0.756242i \(-0.272967\pi\)
0.654292 + 0.756242i \(0.272967\pi\)
\(60\) 0 0
\(61\) −45940.0 −1.58076 −0.790381 0.612616i \(-0.790117\pi\)
−0.790381 + 0.612616i \(0.790117\pi\)
\(62\) − 24980.0i − 0.825303i
\(63\) − 40172.0i − 1.27518i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 484.000 0.0136768
\(67\) 25343.0i 0.689717i 0.938655 + 0.344859i \(0.112073\pi\)
−0.938655 + 0.344859i \(0.887927\pi\)
\(68\) 11808.0i 0.309674i
\(69\) 1779.00 0.0449835
\(70\) 0 0
\(71\) 13311.0 0.313375 0.156688 0.987648i \(-0.449918\pi\)
0.156688 + 0.987648i \(0.449918\pi\)
\(72\) 15488.0i 0.352098i
\(73\) 53260.0i 1.16975i 0.811123 + 0.584876i \(0.198857\pi\)
−0.811123 + 0.584876i \(0.801143\pi\)
\(74\) −59140.0 −1.25546
\(75\) 0 0
\(76\) 22784.0 0.452476
\(77\) 20086.0i 0.386071i
\(78\) 2768.00i 0.0515145i
\(79\) −77234.0 −1.39233 −0.696163 0.717884i \(-0.745111\pi\)
−0.696163 + 0.717884i \(0.745111\pi\)
\(80\) 0 0
\(81\) 58321.0 0.987671
\(82\) − 21216.0i − 0.348441i
\(83\) − 55014.0i − 0.876553i −0.898840 0.438276i \(-0.855589\pi\)
0.898840 0.438276i \(-0.144411\pi\)
\(84\) 2656.00 0.0410705
\(85\) 0 0
\(86\) −71192.0 −1.03797
\(87\) − 2064.00i − 0.0292356i
\(88\) − 7744.00i − 0.106600i
\(89\) −125415. −1.67832 −0.839159 0.543886i \(-0.816953\pi\)
−0.839159 + 0.543886i \(0.816953\pi\)
\(90\) 0 0
\(91\) −114872. −1.45416
\(92\) − 28464.0i − 0.350612i
\(93\) − 6245.00i − 0.0748730i
\(94\) −68736.0 −0.802351
\(95\) 0 0
\(96\) −1024.00 −0.0113402
\(97\) − 88807.0i − 0.958336i −0.877723 0.479168i \(-0.840938\pi\)
0.877723 0.479168i \(-0.159062\pi\)
\(98\) 42996.0i 0.452234i
\(99\) −29282.0 −0.300271
\(100\) 0 0
\(101\) 1482.00 0.0144559 0.00722794 0.999974i \(-0.497699\pi\)
0.00722794 + 0.999974i \(0.497699\pi\)
\(102\) 2952.00i 0.0280942i
\(103\) 117496.i 1.09126i 0.838025 + 0.545632i \(0.183710\pi\)
−0.838025 + 0.545632i \(0.816290\pi\)
\(104\) 44288.0 0.401516
\(105\) 0 0
\(106\) 122904. 1.06243
\(107\) − 79362.0i − 0.670121i −0.942197 0.335060i \(-0.891243\pi\)
0.942197 0.335060i \(-0.108757\pi\)
\(108\) 7760.00i 0.0640180i
\(109\) −87842.0 −0.708167 −0.354084 0.935214i \(-0.615207\pi\)
−0.354084 + 0.935214i \(0.615207\pi\)
\(110\) 0 0
\(111\) −14785.0 −0.113897
\(112\) − 42496.0i − 0.320113i
\(113\) 47247.0i 0.348079i 0.984739 + 0.174040i \(0.0556821\pi\)
−0.984739 + 0.174040i \(0.944318\pi\)
\(114\) 5696.00 0.0410495
\(115\) 0 0
\(116\) −33024.0 −0.227869
\(117\) − 167464.i − 1.13098i
\(118\) − 139956.i − 0.925308i
\(119\) −122508. −0.793044
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 183760.i 1.11777i
\(123\) − 5304.00i − 0.0316112i
\(124\) −99920.0 −0.583577
\(125\) 0 0
\(126\) −160688. −0.901690
\(127\) − 239416.i − 1.31718i −0.752504 0.658588i \(-0.771154\pi\)
0.752504 0.658588i \(-0.228846\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) −17798.0 −0.0941666
\(130\) 0 0
\(131\) −98142.0 −0.499662 −0.249831 0.968289i \(-0.580375\pi\)
−0.249831 + 0.968289i \(0.580375\pi\)
\(132\) − 1936.00i − 0.00967098i
\(133\) 236384.i 1.15875i
\(134\) 101372. 0.487704
\(135\) 0 0
\(136\) 47232.0 0.218972
\(137\) 400137.i 1.82141i 0.413059 + 0.910704i \(0.364460\pi\)
−0.413059 + 0.910704i \(0.635540\pi\)
\(138\) − 7116.00i − 0.0318081i
\(139\) −205766. −0.903310 −0.451655 0.892193i \(-0.649166\pi\)
−0.451655 + 0.892193i \(0.649166\pi\)
\(140\) 0 0
\(141\) −17184.0 −0.0727908
\(142\) − 53244.0i − 0.221590i
\(143\) 83732.0i 0.342414i
\(144\) 61952.0 0.248971
\(145\) 0 0
\(146\) 213040. 0.827140
\(147\) 10749.0i 0.0410275i
\(148\) 236560.i 0.887743i
\(149\) −87726.0 −0.323715 −0.161857 0.986814i \(-0.551748\pi\)
−0.161857 + 0.986814i \(0.551748\pi\)
\(150\) 0 0
\(151\) −432778. −1.54462 −0.772312 0.635243i \(-0.780900\pi\)
−0.772312 + 0.635243i \(0.780900\pi\)
\(152\) − 91136.0i − 0.319949i
\(153\) − 178596.i − 0.616798i
\(154\) 80344.0 0.272993
\(155\) 0 0
\(156\) 11072.0 0.0364263
\(157\) − 34075.0i − 0.110328i −0.998477 0.0551641i \(-0.982432\pi\)
0.998477 0.0551641i \(-0.0175682\pi\)
\(158\) 308936.i 0.984523i
\(159\) 30726.0 0.0963858
\(160\) 0 0
\(161\) 295314. 0.897882
\(162\) − 233284.i − 0.698389i
\(163\) − 45020.0i − 0.132720i −0.997796 0.0663600i \(-0.978861\pi\)
0.997796 0.0663600i \(-0.0211386\pi\)
\(164\) −84864.0 −0.246385
\(165\) 0 0
\(166\) −220056. −0.619816
\(167\) 482556.i 1.33893i 0.742845 + 0.669463i \(0.233476\pi\)
−0.742845 + 0.669463i \(0.766524\pi\)
\(168\) − 10624.0i − 0.0290412i
\(169\) −107571. −0.289720
\(170\) 0 0
\(171\) −344608. −0.901229
\(172\) 284768.i 0.733956i
\(173\) 766254.i 1.94651i 0.229719 + 0.973257i \(0.426219\pi\)
−0.229719 + 0.973257i \(0.573781\pi\)
\(174\) −8256.00 −0.0206727
\(175\) 0 0
\(176\) −30976.0 −0.0753778
\(177\) − 34989.0i − 0.0839457i
\(178\) 501660.i 1.18675i
\(179\) −303399. −0.707753 −0.353876 0.935292i \(-0.615137\pi\)
−0.353876 + 0.935292i \(0.615137\pi\)
\(180\) 0 0
\(181\) −285181. −0.647030 −0.323515 0.946223i \(-0.604865\pi\)
−0.323515 + 0.946223i \(0.604865\pi\)
\(182\) 459488.i 1.02824i
\(183\) 45940.0i 0.101406i
\(184\) −113856. −0.247920
\(185\) 0 0
\(186\) −24980.0 −0.0529432
\(187\) 89298.0i 0.186740i
\(188\) 274944.i 0.567348i
\(189\) −80510.0 −0.163944
\(190\) 0 0
\(191\) 767067. 1.52142 0.760711 0.649090i \(-0.224850\pi\)
0.760711 + 0.649090i \(0.224850\pi\)
\(192\) 4096.00i 0.00801875i
\(193\) − 411668.i − 0.795525i −0.917488 0.397763i \(-0.869787\pi\)
0.917488 0.397763i \(-0.130213\pi\)
\(194\) −355228. −0.677646
\(195\) 0 0
\(196\) 171984. 0.319777
\(197\) − 759258.i − 1.39387i −0.717132 0.696937i \(-0.754545\pi\)
0.717132 0.696937i \(-0.245455\pi\)
\(198\) 117128.i 0.212323i
\(199\) 46600.0 0.0834167 0.0417084 0.999130i \(-0.486720\pi\)
0.0417084 + 0.999130i \(0.486720\pi\)
\(200\) 0 0
\(201\) 25343.0 0.0442454
\(202\) − 5928.00i − 0.0102219i
\(203\) − 342624.i − 0.583549i
\(204\) 11808.0 0.0198656
\(205\) 0 0
\(206\) 469984. 0.771641
\(207\) 430518.i 0.698338i
\(208\) − 177152.i − 0.283915i
\(209\) 172304. 0.272854
\(210\) 0 0
\(211\) −932428. −1.44181 −0.720907 0.693032i \(-0.756274\pi\)
−0.720907 + 0.693032i \(0.756274\pi\)
\(212\) − 491616.i − 0.751253i
\(213\) − 13311.0i − 0.0201030i
\(214\) −317448. −0.473847
\(215\) 0 0
\(216\) 31040.0 0.0452676
\(217\) − 1.03667e6i − 1.49448i
\(218\) 351368.i 0.500750i
\(219\) 53260.0 0.0750397
\(220\) 0 0
\(221\) −510696. −0.703367
\(222\) 59140.0i 0.0805376i
\(223\) − 169745.i − 0.228578i −0.993448 0.114289i \(-0.963541\pi\)
0.993448 0.114289i \(-0.0364590\pi\)
\(224\) −169984. −0.226354
\(225\) 0 0
\(226\) 188988. 0.246129
\(227\) 198078.i 0.255136i 0.991830 + 0.127568i \(0.0407171\pi\)
−0.991830 + 0.127568i \(0.959283\pi\)
\(228\) − 22784.0i − 0.0290264i
\(229\) 849997. 1.07110 0.535548 0.844505i \(-0.320105\pi\)
0.535548 + 0.844505i \(0.320105\pi\)
\(230\) 0 0
\(231\) 20086.0 0.0247664
\(232\) 132096.i 0.161128i
\(233\) 401832.i 0.484903i 0.970164 + 0.242451i \(0.0779515\pi\)
−0.970164 + 0.242451i \(0.922048\pi\)
\(234\) −669856. −0.799727
\(235\) 0 0
\(236\) −559824. −0.654292
\(237\) 77234.0i 0.0893177i
\(238\) 490032.i 0.560766i
\(239\) −855174. −0.968411 −0.484206 0.874954i \(-0.660891\pi\)
−0.484206 + 0.874954i \(0.660891\pi\)
\(240\) 0 0
\(241\) 1.12546e6 1.24821 0.624107 0.781339i \(-0.285463\pi\)
0.624107 + 0.781339i \(0.285463\pi\)
\(242\) − 58564.0i − 0.0642824i
\(243\) − 176176.i − 0.191395i
\(244\) 735040. 0.790381
\(245\) 0 0
\(246\) −21216.0 −0.0223525
\(247\) 985408.i 1.02772i
\(248\) 399680.i 0.412651i
\(249\) −55014.0 −0.0562309
\(250\) 0 0
\(251\) −1.19751e6 −1.19976 −0.599882 0.800088i \(-0.704786\pi\)
−0.599882 + 0.800088i \(0.704786\pi\)
\(252\) 642752.i 0.637591i
\(253\) − 215259.i − 0.211427i
\(254\) −957664. −0.931384
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 37758.0i 0.0356596i 0.999841 + 0.0178298i \(0.00567570\pi\)
−0.999841 + 0.0178298i \(0.994324\pi\)
\(258\) 71192.0i 0.0665858i
\(259\) −2.45431e6 −2.27342
\(260\) 0 0
\(261\) 499488. 0.453862
\(262\) 392568.i 0.353315i
\(263\) 631254.i 0.562749i 0.959598 + 0.281375i \(0.0907903\pi\)
−0.959598 + 0.281375i \(0.909210\pi\)
\(264\) −7744.00 −0.00683842
\(265\) 0 0
\(266\) 945536. 0.819359
\(267\) 125415.i 0.107664i
\(268\) − 405488.i − 0.344859i
\(269\) 1.08034e6 0.910292 0.455146 0.890417i \(-0.349587\pi\)
0.455146 + 0.890417i \(0.349587\pi\)
\(270\) 0 0
\(271\) −816100. −0.675025 −0.337513 0.941321i \(-0.609586\pi\)
−0.337513 + 0.941321i \(0.609586\pi\)
\(272\) − 188928.i − 0.154837i
\(273\) 114872.i 0.0932841i
\(274\) 1.60055e6 1.28793
\(275\) 0 0
\(276\) −28464.0 −0.0224917
\(277\) 1.68820e6i 1.32198i 0.750396 + 0.660989i \(0.229863\pi\)
−0.750396 + 0.660989i \(0.770137\pi\)
\(278\) 823064.i 0.638736i
\(279\) 1.51129e6 1.16235
\(280\) 0 0
\(281\) −879042. −0.664116 −0.332058 0.943259i \(-0.607743\pi\)
−0.332058 + 0.943259i \(0.607743\pi\)
\(282\) 68736.0i 0.0514709i
\(283\) − 1.54027e6i − 1.14322i −0.820525 0.571611i \(-0.806319\pi\)
0.820525 0.571611i \(-0.193681\pi\)
\(284\) −212976. −0.156688
\(285\) 0 0
\(286\) 334928. 0.242123
\(287\) − 880464.i − 0.630967i
\(288\) − 247808.i − 0.176049i
\(289\) 875213. 0.616409
\(290\) 0 0
\(291\) −88807.0 −0.0614773
\(292\) − 852160.i − 0.584876i
\(293\) − 720840.i − 0.490535i −0.969455 0.245267i \(-0.921124\pi\)
0.969455 0.245267i \(-0.0788758\pi\)
\(294\) 42996.0 0.0290108
\(295\) 0 0
\(296\) 946240. 0.627729
\(297\) 58685.0i 0.0386043i
\(298\) 350904.i 0.228901i
\(299\) 1.23107e6 0.796350
\(300\) 0 0
\(301\) −2.95447e6 −1.87959
\(302\) 1.73111e6i 1.09221i
\(303\) − 1482.00i 0 0.000927346i
\(304\) −364544. −0.226238
\(305\) 0 0
\(306\) −714384. −0.436142
\(307\) − 1.03905e6i − 0.629201i −0.949224 0.314601i \(-0.898129\pi\)
0.949224 0.314601i \(-0.101871\pi\)
\(308\) − 321376.i − 0.193035i
\(309\) 117496. 0.0700046
\(310\) 0 0
\(311\) −1.25135e6 −0.733630 −0.366815 0.930294i \(-0.619552\pi\)
−0.366815 + 0.930294i \(0.619552\pi\)
\(312\) − 44288.0i − 0.0257573i
\(313\) 1.44336e6i 0.832749i 0.909193 + 0.416375i \(0.136699\pi\)
−0.909193 + 0.416375i \(0.863301\pi\)
\(314\) −136300. −0.0780139
\(315\) 0 0
\(316\) 1.23574e6 0.696163
\(317\) − 2.01208e6i − 1.12460i −0.826934 0.562298i \(-0.809917\pi\)
0.826934 0.562298i \(-0.190083\pi\)
\(318\) − 122904.i − 0.0681551i
\(319\) −249744. −0.137410
\(320\) 0 0
\(321\) −79362.0 −0.0429883
\(322\) − 1.18126e6i − 0.634899i
\(323\) 1.05091e6i 0.560480i
\(324\) −933136. −0.493836
\(325\) 0 0
\(326\) −180080. −0.0938472
\(327\) 87842.0i 0.0454290i
\(328\) 339456.i 0.174220i
\(329\) −2.85254e6 −1.45292
\(330\) 0 0
\(331\) 2.01734e6 1.01207 0.506033 0.862514i \(-0.331112\pi\)
0.506033 + 0.862514i \(0.331112\pi\)
\(332\) 880224.i 0.438276i
\(333\) − 3.57797e6i − 1.76818i
\(334\) 1.93022e6 0.946764
\(335\) 0 0
\(336\) −42496.0 −0.0205352
\(337\) 264122.i 0.126686i 0.997992 + 0.0633432i \(0.0201763\pi\)
−0.997992 + 0.0633432i \(0.979824\pi\)
\(338\) 430284.i 0.204863i
\(339\) 47247.0 0.0223293
\(340\) 0 0
\(341\) −755645. −0.351910
\(342\) 1.37843e6i 0.637265i
\(343\) − 1.00563e6i − 0.461532i
\(344\) 1.13907e6 0.518985
\(345\) 0 0
\(346\) 3.06502e6 1.37639
\(347\) − 1.71049e6i − 0.762601i −0.924451 0.381300i \(-0.875476\pi\)
0.924451 0.381300i \(-0.124524\pi\)
\(348\) 33024.0i 0.0146178i
\(349\) −218822. −0.0961673 −0.0480836 0.998843i \(-0.515311\pi\)
−0.0480836 + 0.998843i \(0.515311\pi\)
\(350\) 0 0
\(351\) −335620. −0.145405
\(352\) 123904.i 0.0533002i
\(353\) − 3.68192e6i − 1.57267i −0.617802 0.786334i \(-0.711977\pi\)
0.617802 0.786334i \(-0.288023\pi\)
\(354\) −139956. −0.0593586
\(355\) 0 0
\(356\) 2.00664e6 0.839159
\(357\) 122508.i 0.0508738i
\(358\) 1.21360e6i 0.500457i
\(359\) −1.88528e6 −0.772042 −0.386021 0.922490i \(-0.626151\pi\)
−0.386021 + 0.922490i \(0.626151\pi\)
\(360\) 0 0
\(361\) −448323. −0.181060
\(362\) 1.14072e6i 0.457519i
\(363\) − 14641.0i − 0.00583182i
\(364\) 1.83795e6 0.727078
\(365\) 0 0
\(366\) 183760. 0.0717048
\(367\) − 3.11666e6i − 1.20788i −0.797029 0.603940i \(-0.793596\pi\)
0.797029 0.603940i \(-0.206404\pi\)
\(368\) 455424.i 0.175306i
\(369\) 1.28357e6 0.490742
\(370\) 0 0
\(371\) 5.10052e6 1.92389
\(372\) 99920.0i 0.0374365i
\(373\) − 1.39441e6i − 0.518943i −0.965751 0.259471i \(-0.916452\pi\)
0.965751 0.259471i \(-0.0835484\pi\)
\(374\) 357192. 0.132045
\(375\) 0 0
\(376\) 1.09978e6 0.401176
\(377\) − 1.42829e6i − 0.517562i
\(378\) 322040.i 0.115926i
\(379\) 4.26036e6 1.52352 0.761759 0.647860i \(-0.224336\pi\)
0.761759 + 0.647860i \(0.224336\pi\)
\(380\) 0 0
\(381\) −239416. −0.0844969
\(382\) − 3.06827e6i − 1.07581i
\(383\) − 201765.i − 0.0702828i −0.999382 0.0351414i \(-0.988812\pi\)
0.999382 0.0351414i \(-0.0111882\pi\)
\(384\) 16384.0 0.00567012
\(385\) 0 0
\(386\) −1.64667e6 −0.562521
\(387\) − 4.30712e6i − 1.46187i
\(388\) 1.42091e6i 0.479168i
\(389\) −1.94882e6 −0.652977 −0.326489 0.945201i \(-0.605865\pi\)
−0.326489 + 0.945201i \(0.605865\pi\)
\(390\) 0 0
\(391\) 1.31290e6 0.434301
\(392\) − 687936.i − 0.226117i
\(393\) 98142.0i 0.0320534i
\(394\) −3.03703e6 −0.985618
\(395\) 0 0
\(396\) 468512. 0.150135
\(397\) − 1.46826e6i − 0.467548i −0.972291 0.233774i \(-0.924892\pi\)
0.972291 0.233774i \(-0.0751076\pi\)
\(398\) − 186400.i − 0.0589845i
\(399\) 236384. 0.0743337
\(400\) 0 0
\(401\) 2.24618e6 0.697563 0.348781 0.937204i \(-0.386596\pi\)
0.348781 + 0.937204i \(0.386596\pi\)
\(402\) − 101372.i − 0.0312862i
\(403\) − 4.32154e6i − 1.32549i
\(404\) −23712.0 −0.00722794
\(405\) 0 0
\(406\) −1.37050e6 −0.412632
\(407\) 1.78898e6i 0.535329i
\(408\) − 47232.0i − 0.0140471i
\(409\) 3.61488e6 1.06853 0.534263 0.845318i \(-0.320589\pi\)
0.534263 + 0.845318i \(0.320589\pi\)
\(410\) 0 0
\(411\) 400137. 0.116843
\(412\) − 1.87994e6i − 0.545632i
\(413\) − 5.80817e6i − 1.67558i
\(414\) 1.72207e6 0.493799
\(415\) 0 0
\(416\) −708608. −0.200758
\(417\) 205766.i 0.0579473i
\(418\) − 689216.i − 0.192937i
\(419\) 3.81239e6 1.06087 0.530435 0.847726i \(-0.322029\pi\)
0.530435 + 0.847726i \(0.322029\pi\)
\(420\) 0 0
\(421\) 1.97346e6 0.542655 0.271327 0.962487i \(-0.412537\pi\)
0.271327 + 0.962487i \(0.412537\pi\)
\(422\) 3.72971e6i 1.01952i
\(423\) − 4.15853e6i − 1.13003i
\(424\) −1.96646e6 −0.531216
\(425\) 0 0
\(426\) −53244.0 −0.0142150
\(427\) 7.62604e6i 2.02409i
\(428\) 1.26979e6i 0.335060i
\(429\) 83732.0 0.0219659
\(430\) 0 0
\(431\) −2.08359e6 −0.540280 −0.270140 0.962821i \(-0.587070\pi\)
−0.270140 + 0.962821i \(0.587070\pi\)
\(432\) − 124160.i − 0.0320090i
\(433\) 72691.0i 0.0186321i 0.999957 + 0.00931603i \(0.00296543\pi\)
−0.999957 + 0.00931603i \(0.997035\pi\)
\(434\) −4.14668e6 −1.05676
\(435\) 0 0
\(436\) 1.40547e6 0.354084
\(437\) − 2.53330e6i − 0.634574i
\(438\) − 213040.i − 0.0530611i
\(439\) −594392. −0.147201 −0.0736007 0.997288i \(-0.523449\pi\)
−0.0736007 + 0.997288i \(0.523449\pi\)
\(440\) 0 0
\(441\) −2.60126e6 −0.636923
\(442\) 2.04278e6i 0.497355i
\(443\) − 4.56651e6i − 1.10554i −0.833334 0.552770i \(-0.813571\pi\)
0.833334 0.552770i \(-0.186429\pi\)
\(444\) 236560. 0.0569487
\(445\) 0 0
\(446\) −678980. −0.161629
\(447\) 87726.0i 0.0207663i
\(448\) 679936.i 0.160056i
\(449\) 5.44382e6 1.27435 0.637174 0.770720i \(-0.280103\pi\)
0.637174 + 0.770720i \(0.280103\pi\)
\(450\) 0 0
\(451\) −641784. −0.148576
\(452\) − 755952.i − 0.174040i
\(453\) 432778.i 0.0990877i
\(454\) 792312. 0.180408
\(455\) 0 0
\(456\) −91136.0 −0.0205247
\(457\) 6.70312e6i 1.50137i 0.660662 + 0.750683i \(0.270276\pi\)
−0.660662 + 0.750683i \(0.729724\pi\)
\(458\) − 3.39999e6i − 0.757380i
\(459\) −357930. −0.0792988
\(460\) 0 0
\(461\) −1.25994e6 −0.276120 −0.138060 0.990424i \(-0.544087\pi\)
−0.138060 + 0.990424i \(0.544087\pi\)
\(462\) − 80344.0i − 0.0175125i
\(463\) 5.02308e6i 1.08897i 0.838769 + 0.544487i \(0.183276\pi\)
−0.838769 + 0.544487i \(0.816724\pi\)
\(464\) 528384. 0.113934
\(465\) 0 0
\(466\) 1.60733e6 0.342878
\(467\) − 2.35660e6i − 0.500028i −0.968242 0.250014i \(-0.919565\pi\)
0.968242 0.250014i \(-0.0804353\pi\)
\(468\) 2.67942e6i 0.565492i
\(469\) 4.20694e6 0.883149
\(470\) 0 0
\(471\) −34075.0 −0.00707756
\(472\) 2.23930e6i 0.462654i
\(473\) 2.15356e6i 0.442592i
\(474\) 308936. 0.0631572
\(475\) 0 0
\(476\) 1.96013e6 0.396522
\(477\) 7.43569e6i 1.49632i
\(478\) 3.42070e6i 0.684770i
\(479\) 6.72258e6 1.33874 0.669371 0.742928i \(-0.266563\pi\)
0.669371 + 0.742928i \(0.266563\pi\)
\(480\) 0 0
\(481\) −1.02312e7 −2.01634
\(482\) − 4.50186e6i − 0.882620i
\(483\) − 295314.i − 0.0575992i
\(484\) −234256. −0.0454545
\(485\) 0 0
\(486\) −704704. −0.135337
\(487\) 1.96001e6i 0.374487i 0.982314 + 0.187243i \(0.0599553\pi\)
−0.982314 + 0.187243i \(0.940045\pi\)
\(488\) − 2.94016e6i − 0.558884i
\(489\) −45020.0 −0.00851399
\(490\) 0 0
\(491\) −579624. −0.108503 −0.0542516 0.998527i \(-0.517277\pi\)
−0.0542516 + 0.998527i \(0.517277\pi\)
\(492\) 84864.0i 0.0158056i
\(493\) − 1.52323e6i − 0.282260i
\(494\) 3.94163e6 0.726706
\(495\) 0 0
\(496\) 1.59872e6 0.291789
\(497\) − 2.20963e6i − 0.401262i
\(498\) 220056.i 0.0397612i
\(499\) −1.36905e6 −0.246132 −0.123066 0.992398i \(-0.539273\pi\)
−0.123066 + 0.992398i \(0.539273\pi\)
\(500\) 0 0
\(501\) 482556. 0.0858921
\(502\) 4.79005e6i 0.848361i
\(503\) − 1.83343e6i − 0.323105i −0.986864 0.161552i \(-0.948350\pi\)
0.986864 0.161552i \(-0.0516501\pi\)
\(504\) 2.57101e6 0.450845
\(505\) 0 0
\(506\) −861036. −0.149501
\(507\) 107571.i 0.0185855i
\(508\) 3.83066e6i 0.658588i
\(509\) 1.71266e6 0.293006 0.146503 0.989210i \(-0.453198\pi\)
0.146503 + 0.989210i \(0.453198\pi\)
\(510\) 0 0
\(511\) 8.84116e6 1.49781
\(512\) − 262144.i − 0.0441942i
\(513\) 690640.i 0.115867i
\(514\) 151032. 0.0252151
\(515\) 0 0
\(516\) 284768. 0.0470833
\(517\) 2.07926e6i 0.342124i
\(518\) 9.81724e6i 1.60755i
\(519\) 766254. 0.124869
\(520\) 0 0
\(521\) −789435. −0.127415 −0.0637077 0.997969i \(-0.520293\pi\)
−0.0637077 + 0.997969i \(0.520293\pi\)
\(522\) − 1.99795e6i − 0.320929i
\(523\) − 627392.i − 0.100296i −0.998742 0.0501481i \(-0.984031\pi\)
0.998742 0.0501481i \(-0.0159693\pi\)
\(524\) 1.57027e6 0.249831
\(525\) 0 0
\(526\) 2.52502e6 0.397924
\(527\) − 4.60881e6i − 0.722873i
\(528\) 30976.0i 0.00483549i
\(529\) 3.27150e6 0.508286
\(530\) 0 0
\(531\) 8.46734e6 1.30320
\(532\) − 3.78214e6i − 0.579374i
\(533\) − 3.67037e6i − 0.559618i
\(534\) 501660. 0.0761301
\(535\) 0 0
\(536\) −1.62195e6 −0.243852
\(537\) 303399.i 0.0454024i
\(538\) − 4.32137e6i − 0.643673i
\(539\) 1.30063e6 0.192833
\(540\) 0 0
\(541\) 3.20895e6 0.471379 0.235689 0.971828i \(-0.424265\pi\)
0.235689 + 0.971828i \(0.424265\pi\)
\(542\) 3.26440e6i 0.477315i
\(543\) 285181.i 0.0415070i
\(544\) −755712. −0.109486
\(545\) 0 0
\(546\) 459488. 0.0659618
\(547\) 3.42658e6i 0.489658i 0.969566 + 0.244829i \(0.0787319\pi\)
−0.969566 + 0.244829i \(0.921268\pi\)
\(548\) − 6.40219e6i − 0.910704i
\(549\) −1.11175e7 −1.57426
\(550\) 0 0
\(551\) −2.93914e6 −0.412421
\(552\) 113856.i 0.0159041i
\(553\) 1.28208e7i 1.78280i
\(554\) 6.75279e6 0.934779
\(555\) 0 0
\(556\) 3.29226e6 0.451655
\(557\) 1.05198e7i 1.43672i 0.695674 + 0.718358i \(0.255106\pi\)
−0.695674 + 0.718358i \(0.744894\pi\)
\(558\) − 6.04516e6i − 0.821906i
\(559\) −1.23162e7 −1.66705
\(560\) 0 0
\(561\) 89298.0 0.0119794
\(562\) 3.51617e6i 0.469601i
\(563\) − 5.47288e6i − 0.727687i −0.931460 0.363844i \(-0.881464\pi\)
0.931460 0.363844i \(-0.118536\pi\)
\(564\) 274944. 0.0363954
\(565\) 0 0
\(566\) −6.16107e6 −0.808379
\(567\) − 9.68129e6i − 1.26466i
\(568\) 851904.i 0.110795i
\(569\) 1.17787e7 1.52516 0.762580 0.646893i \(-0.223932\pi\)
0.762580 + 0.646893i \(0.223932\pi\)
\(570\) 0 0
\(571\) −8.35628e6 −1.07256 −0.536281 0.844039i \(-0.680171\pi\)
−0.536281 + 0.844039i \(0.680171\pi\)
\(572\) − 1.33971e6i − 0.171207i
\(573\) − 767067.i − 0.0975993i
\(574\) −3.52186e6 −0.446161
\(575\) 0 0
\(576\) −991232. −0.124486
\(577\) − 1.37758e7i − 1.72258i −0.508117 0.861288i \(-0.669658\pi\)
0.508117 0.861288i \(-0.330342\pi\)
\(578\) − 3.50085e6i − 0.435867i
\(579\) −411668. −0.0510330
\(580\) 0 0
\(581\) −9.13232e6 −1.12238
\(582\) 355228.i 0.0434710i
\(583\) − 3.71785e6i − 0.453023i
\(584\) −3.40864e6 −0.413570
\(585\) 0 0
\(586\) −2.88336e6 −0.346860
\(587\) − 1.27093e7i − 1.52239i −0.648522 0.761196i \(-0.724612\pi\)
0.648522 0.761196i \(-0.275388\pi\)
\(588\) − 171984.i − 0.0205137i
\(589\) −8.89288e6 −1.05622
\(590\) 0 0
\(591\) −759258. −0.0894171
\(592\) − 3.78496e6i − 0.443871i
\(593\) − 1.00825e6i − 0.117742i −0.998266 0.0588711i \(-0.981250\pi\)
0.998266 0.0588711i \(-0.0187501\pi\)
\(594\) 234740. 0.0272974
\(595\) 0 0
\(596\) 1.40362e6 0.161857
\(597\) − 46600.0i − 0.00535119i
\(598\) − 4.92427e6i − 0.563105i
\(599\) −1.05100e7 −1.19684 −0.598421 0.801182i \(-0.704205\pi\)
−0.598421 + 0.801182i \(0.704205\pi\)
\(600\) 0 0
\(601\) −199390. −0.0225173 −0.0112587 0.999937i \(-0.503584\pi\)
−0.0112587 + 0.999937i \(0.503584\pi\)
\(602\) 1.18179e7i 1.32907i
\(603\) 6.13301e6i 0.686879i
\(604\) 6.92445e6 0.772312
\(605\) 0 0
\(606\) −5928.00 −0.000655732 0
\(607\) 16190.0i 0.00178351i 1.00000 0.000891754i \(0.000283854\pi\)
−1.00000 0.000891754i \(0.999716\pi\)
\(608\) 1.45818e6i 0.159975i
\(609\) −342624. −0.0374347
\(610\) 0 0
\(611\) −1.18913e7 −1.28863
\(612\) 2.85754e6i 0.308399i
\(613\) 1.15253e7i 1.23880i 0.785074 + 0.619402i \(0.212625\pi\)
−0.785074 + 0.619402i \(0.787375\pi\)
\(614\) −4.15619e6 −0.444913
\(615\) 0 0
\(616\) −1.28550e6 −0.136497
\(617\) 1.69974e7i 1.79750i 0.438459 + 0.898751i \(0.355524\pi\)
−0.438459 + 0.898751i \(0.644476\pi\)
\(618\) − 469984.i − 0.0495008i
\(619\) 1.84875e7 1.93933 0.969663 0.244445i \(-0.0786058\pi\)
0.969663 + 0.244445i \(0.0786058\pi\)
\(620\) 0 0
\(621\) 862815. 0.0897819
\(622\) 5.00539e6i 0.518755i
\(623\) 2.08189e7i 2.14901i
\(624\) −177152. −0.0182131
\(625\) 0 0
\(626\) 5.77344e6 0.588842
\(627\) − 172304.i − 0.0175036i
\(628\) 545200.i 0.0551641i
\(629\) −1.09113e7 −1.09964
\(630\) 0 0
\(631\) −4.54281e6 −0.454204 −0.227102 0.973871i \(-0.572925\pi\)
−0.227102 + 0.973871i \(0.572925\pi\)
\(632\) − 4.94298e6i − 0.492261i
\(633\) 932428.i 0.0924924i
\(634\) −8.04832e6 −0.795210
\(635\) 0 0
\(636\) −491616. −0.0481929
\(637\) 7.43831e6i 0.726316i
\(638\) 998976.i 0.0971635i
\(639\) 3.22126e6 0.312086
\(640\) 0 0
\(641\) 1.84286e7 1.77153 0.885764 0.464136i \(-0.153635\pi\)
0.885764 + 0.464136i \(0.153635\pi\)
\(642\) 317448.i 0.0303973i
\(643\) − 9.66604e6i − 0.921979i −0.887406 0.460989i \(-0.847495\pi\)
0.887406 0.460989i \(-0.152505\pi\)
\(644\) −4.72502e6 −0.448941
\(645\) 0 0
\(646\) 4.20365e6 0.396319
\(647\) − 4.51430e6i − 0.423965i −0.977273 0.211982i \(-0.932008\pi\)
0.977273 0.211982i \(-0.0679920\pi\)
\(648\) 3.73254e6i 0.349195i
\(649\) −4.23367e6 −0.394553
\(650\) 0 0
\(651\) −1.03667e6 −0.0958712
\(652\) 720320.i 0.0663600i
\(653\) 5.37235e6i 0.493039i 0.969138 + 0.246519i \(0.0792869\pi\)
−0.969138 + 0.246519i \(0.920713\pi\)
\(654\) 351368. 0.0321231
\(655\) 0 0
\(656\) 1.35782e6 0.123192
\(657\) 1.28889e7i 1.16494i
\(658\) 1.14102e7i 1.02737i
\(659\) −9.87956e6 −0.886184 −0.443092 0.896476i \(-0.646119\pi\)
−0.443092 + 0.896476i \(0.646119\pi\)
\(660\) 0 0
\(661\) 1.08052e7 0.961898 0.480949 0.876748i \(-0.340292\pi\)
0.480949 + 0.876748i \(0.340292\pi\)
\(662\) − 8.06935e6i − 0.715638i
\(663\) 510696.i 0.0451210i
\(664\) 3.52090e6 0.309908
\(665\) 0 0
\(666\) −1.43119e7 −1.25029
\(667\) 3.67186e6i 0.319574i
\(668\) − 7.72090e6i − 0.669463i
\(669\) −169745. −0.0146633
\(670\) 0 0
\(671\) 5.55874e6 0.476618
\(672\) 169984.i 0.0145206i
\(673\) − 1.13275e7i − 0.964042i −0.876160 0.482021i \(-0.839903\pi\)
0.876160 0.482021i \(-0.160097\pi\)
\(674\) 1.05649e6 0.0895808
\(675\) 0 0
\(676\) 1.72114e6 0.144860
\(677\) − 1.20595e7i − 1.01125i −0.862754 0.505624i \(-0.831262\pi\)
0.862754 0.505624i \(-0.168738\pi\)
\(678\) − 188988.i − 0.0157892i
\(679\) −1.47420e7 −1.22710
\(680\) 0 0
\(681\) 198078. 0.0163670
\(682\) 3.02258e6i 0.248838i
\(683\) 5.14166e6i 0.421747i 0.977513 + 0.210873i \(0.0676308\pi\)
−0.977513 + 0.210873i \(0.932369\pi\)
\(684\) 5.51373e6 0.450614
\(685\) 0 0
\(686\) −4.02251e6 −0.326353
\(687\) − 849997.i − 0.0687109i
\(688\) − 4.55629e6i − 0.366978i
\(689\) 2.12624e7 1.70633
\(690\) 0 0
\(691\) 1.31243e7 1.04563 0.522817 0.852445i \(-0.324881\pi\)
0.522817 + 0.852445i \(0.324881\pi\)
\(692\) − 1.22601e7i − 0.973257i
\(693\) 4.86081e6i 0.384482i
\(694\) −6.84197e6 −0.539240
\(695\) 0 0
\(696\) 132096. 0.0103363
\(697\) − 3.91435e6i − 0.305195i
\(698\) 875288.i 0.0680005i
\(699\) 401832. 0.0311065
\(700\) 0 0
\(701\) 3.65956e6 0.281277 0.140638 0.990061i \(-0.455084\pi\)
0.140638 + 0.990061i \(0.455084\pi\)
\(702\) 1.34248e6i 0.102817i
\(703\) 2.10538e7i 1.60673i
\(704\) 495616. 0.0376889
\(705\) 0 0
\(706\) −1.47277e7 −1.11204
\(707\) − 246012.i − 0.0185101i
\(708\) 559824.i 0.0419728i
\(709\) −1.02252e7 −0.763935 −0.381968 0.924176i \(-0.624753\pi\)
−0.381968 + 0.924176i \(0.624753\pi\)
\(710\) 0 0
\(711\) −1.86906e7 −1.38660
\(712\) − 8.02656e6i − 0.593375i
\(713\) 1.11099e7i 0.818436i
\(714\) 490032. 0.0359732
\(715\) 0 0
\(716\) 4.85438e6 0.353876
\(717\) 855174.i 0.0621236i
\(718\) 7.54114e6i 0.545916i
\(719\) −2.41683e7 −1.74351 −0.871753 0.489945i \(-0.837017\pi\)
−0.871753 + 0.489945i \(0.837017\pi\)
\(720\) 0 0
\(721\) 1.95043e7 1.39731
\(722\) 1.79329e6i 0.128029i
\(723\) − 1.12546e6i − 0.0800730i
\(724\) 4.56290e6 0.323515
\(725\) 0 0
\(726\) −58564.0 −0.00412372
\(727\) 1.68246e7i 1.18062i 0.807177 + 0.590310i \(0.200994\pi\)
−0.807177 + 0.590310i \(0.799006\pi\)
\(728\) − 7.35181e6i − 0.514121i
\(729\) 1.39958e7 0.975393
\(730\) 0 0
\(731\) −1.31349e7 −0.909147
\(732\) − 735040.i − 0.0507030i
\(733\) 5.04168e6i 0.346590i 0.984870 + 0.173295i \(0.0554414\pi\)
−0.984870 + 0.173295i \(0.944559\pi\)
\(734\) −1.24666e7 −0.854101
\(735\) 0 0
\(736\) 1.82170e6 0.123960
\(737\) − 3.06650e6i − 0.207958i
\(738\) − 5.13427e6i − 0.347007i
\(739\) 6.26375e6 0.421913 0.210957 0.977495i \(-0.432342\pi\)
0.210957 + 0.977495i \(0.432342\pi\)
\(740\) 0 0
\(741\) 985408. 0.0659281
\(742\) − 2.04021e7i − 1.36039i
\(743\) − 3.63976e6i − 0.241880i −0.992660 0.120940i \(-0.961409\pi\)
0.992660 0.120940i \(-0.0385909\pi\)
\(744\) 399680. 0.0264716
\(745\) 0 0
\(746\) −5.57766e6 −0.366948
\(747\) − 1.33134e7i − 0.872945i
\(748\) − 1.42877e6i − 0.0933701i
\(749\) −1.31741e7 −0.858057
\(750\) 0 0
\(751\) −1.87370e7 −1.21227 −0.606135 0.795362i \(-0.707281\pi\)
−0.606135 + 0.795362i \(0.707281\pi\)
\(752\) − 4.39910e6i − 0.283674i
\(753\) 1.19751e6i 0.0769649i
\(754\) −5.71315e6 −0.365972
\(755\) 0 0
\(756\) 1.28816e6 0.0819720
\(757\) 489242.i 0.0310302i 0.999880 + 0.0155151i \(0.00493880\pi\)
−0.999880 + 0.0155151i \(0.995061\pi\)
\(758\) − 1.70414e7i − 1.07729i
\(759\) −215259. −0.0135630
\(760\) 0 0
\(761\) 1.46969e7 0.919952 0.459976 0.887931i \(-0.347858\pi\)
0.459976 + 0.887931i \(0.347858\pi\)
\(762\) 957664.i 0.0597483i
\(763\) 1.45818e7i 0.906774i
\(764\) −1.22731e7 −0.760711
\(765\) 0 0
\(766\) −807060. −0.0496974
\(767\) − 2.42124e7i − 1.48610i
\(768\) − 65536.0i − 0.00400938i
\(769\) −2.42072e7 −1.47615 −0.738073 0.674721i \(-0.764264\pi\)
−0.738073 + 0.674721i \(0.764264\pi\)
\(770\) 0 0
\(771\) 37758.0 0.00228756
\(772\) 6.58669e6i 0.397763i
\(773\) − 1.35260e7i − 0.814181i −0.913388 0.407091i \(-0.866543\pi\)
0.913388 0.407091i \(-0.133457\pi\)
\(774\) −1.72285e7 −1.03370
\(775\) 0 0
\(776\) 5.68365e6 0.338823
\(777\) 2.45431e6i 0.145840i
\(778\) 7.79528e6i 0.461725i
\(779\) −7.55290e6 −0.445933
\(780\) 0 0
\(781\) −1.61063e6 −0.0944862
\(782\) − 5.25161e6i − 0.307097i
\(783\) − 1.00104e6i − 0.0583508i
\(784\) −2.75174e6 −0.159889
\(785\) 0 0
\(786\) 392568. 0.0226651
\(787\) 1.42094e7i 0.817786i 0.912582 + 0.408893i \(0.134085\pi\)
−0.912582 + 0.408893i \(0.865915\pi\)
\(788\) 1.21481e7i 0.696937i
\(789\) 631254. 0.0361004
\(790\) 0 0
\(791\) 7.84300e6 0.445698
\(792\) − 1.87405e6i − 0.106162i
\(793\) 3.17905e7i 1.79521i
\(794\) −5.87303e6 −0.330606
\(795\) 0 0
\(796\) −745600. −0.0417084
\(797\) − 7.93333e6i − 0.442395i −0.975229 0.221197i \(-0.929003\pi\)
0.975229 0.221197i \(-0.0709965\pi\)
\(798\) − 945536.i − 0.0525619i
\(799\) −1.26818e7 −0.702771
\(800\) 0 0
\(801\) −3.03504e7 −1.67141
\(802\) − 8.98471e6i − 0.493251i
\(803\) − 6.44446e6i − 0.352694i
\(804\) −405488. −0.0221227
\(805\) 0 0
\(806\) −1.72862e7 −0.937262
\(807\) − 1.08034e6i − 0.0583952i
\(808\) 94848.0i 0.00511093i
\(809\) 1.04685e7 0.562359 0.281180 0.959655i \(-0.409274\pi\)
0.281180 + 0.959655i \(0.409274\pi\)
\(810\) 0 0
\(811\) 1.19147e7 0.636110 0.318055 0.948072i \(-0.396970\pi\)
0.318055 + 0.948072i \(0.396970\pi\)
\(812\) 5.48198e6i 0.291775i
\(813\) 816100.i 0.0433029i
\(814\) 7.15594e6 0.378535
\(815\) 0 0
\(816\) −188928. −0.00993278
\(817\) 2.53444e7i 1.32839i
\(818\) − 1.44595e7i − 0.755562i
\(819\) −2.77990e7 −1.44817
\(820\) 0 0
\(821\) −1.86112e6 −0.0963645 −0.0481822 0.998839i \(-0.515343\pi\)
−0.0481822 + 0.998839i \(0.515343\pi\)
\(822\) − 1.60055e6i − 0.0826208i
\(823\) − 2.30153e7i − 1.18445i −0.805773 0.592225i \(-0.798250\pi\)
0.805773 0.592225i \(-0.201750\pi\)
\(824\) −7.51974e6 −0.385820
\(825\) 0 0
\(826\) −2.32327e7 −1.18481
\(827\) − 1.68351e7i − 0.855959i −0.903788 0.427980i \(-0.859225\pi\)
0.903788 0.427980i \(-0.140775\pi\)
\(828\) − 6.88829e6i − 0.349169i
\(829\) 2.35299e7 1.18914 0.594570 0.804044i \(-0.297322\pi\)
0.594570 + 0.804044i \(0.297322\pi\)
\(830\) 0 0
\(831\) 1.68820e6 0.0848049
\(832\) 2.83443e6i 0.141957i
\(833\) 7.93276e6i 0.396106i
\(834\) 823064. 0.0409750
\(835\) 0 0
\(836\) −2.75686e6 −0.136427
\(837\) − 3.02882e6i − 0.149438i
\(838\) − 1.52496e7i − 0.750148i
\(839\) −2.91549e7 −1.42990 −0.714952 0.699173i \(-0.753552\pi\)
−0.714952 + 0.699173i \(0.753552\pi\)
\(840\) 0 0
\(841\) −1.62511e7 −0.792303
\(842\) − 7.89385e6i − 0.383715i
\(843\) 879042.i 0.0426030i
\(844\) 1.49188e7 0.720907
\(845\) 0 0
\(846\) −1.66341e7 −0.799050
\(847\) − 2.43041e6i − 0.116405i
\(848\) 7.86586e6i 0.375627i
\(849\) −1.54027e6 −0.0733377
\(850\) 0 0
\(851\) 2.63025e7 1.24501
\(852\) 212976.i 0.0100515i
\(853\) 9.49052e6i 0.446599i 0.974750 + 0.223299i \(0.0716828\pi\)
−0.974750 + 0.223299i \(0.928317\pi\)
\(854\) 3.05042e7 1.43125
\(855\) 0 0
\(856\) 5.07917e6 0.236924
\(857\) − 1.81553e6i − 0.0844405i −0.999108 0.0422203i \(-0.986557\pi\)
0.999108 0.0422203i \(-0.0134431\pi\)
\(858\) − 334928.i − 0.0155322i
\(859\) 1.07812e7 0.498522 0.249261 0.968436i \(-0.419812\pi\)
0.249261 + 0.968436i \(0.419812\pi\)
\(860\) 0 0
\(861\) −880464. −0.0404766
\(862\) 8.33436e6i 0.382036i
\(863\) 2.83355e7i 1.29510i 0.762023 + 0.647550i \(0.224206\pi\)
−0.762023 + 0.647550i \(0.775794\pi\)
\(864\) −496640. −0.0226338
\(865\) 0 0
\(866\) 290764. 0.0131749
\(867\) − 875213.i − 0.0395427i
\(868\) 1.65867e7i 0.747242i
\(869\) 9.34531e6 0.419802
\(870\) 0 0
\(871\) 1.75374e7 0.783283
\(872\) − 5.62189e6i − 0.250375i
\(873\) − 2.14913e7i − 0.954392i
\(874\) −1.01332e7 −0.448712
\(875\) 0 0
\(876\) −852160. −0.0375198
\(877\) − 2.68919e7i − 1.18065i −0.807165 0.590326i \(-0.798999\pi\)
0.807165 0.590326i \(-0.201001\pi\)
\(878\) 2.37757e6i 0.104087i
\(879\) −720840. −0.0314678
\(880\) 0 0
\(881\) −1.92132e7 −0.833989 −0.416995 0.908909i \(-0.636917\pi\)
−0.416995 + 0.908909i \(0.636917\pi\)
\(882\) 1.04050e7i 0.450373i
\(883\) − 1.15931e7i − 0.500378i −0.968197 0.250189i \(-0.919507\pi\)
0.968197 0.250189i \(-0.0804927\pi\)
\(884\) 8.17114e6 0.351683
\(885\) 0 0
\(886\) −1.82660e7 −0.781735
\(887\) 1.31857e7i 0.562721i 0.959602 + 0.281361i \(0.0907857\pi\)
−0.959602 + 0.281361i \(0.909214\pi\)
\(888\) − 946240.i − 0.0402688i
\(889\) −3.97431e7 −1.68658
\(890\) 0 0
\(891\) −7.05684e6 −0.297794
\(892\) 2.71592e6i 0.114289i
\(893\) 2.44700e7i 1.02685i
\(894\) 350904. 0.0146840
\(895\) 0 0
\(896\) 2.71974e6 0.113177
\(897\) − 1.23107e6i − 0.0510859i
\(898\) − 2.17753e7i − 0.901100i
\(899\) 1.28897e7 0.531916
\(900\) 0 0
\(901\) 2.26758e7 0.930573
\(902\) 2.56714e6i 0.105059i
\(903\) 2.95447e6i 0.120576i
\(904\) −3.02381e6 −0.123065
\(905\) 0 0
\(906\) 1.73111e6 0.0700656
\(907\) 2.98195e6i 0.120360i 0.998188 + 0.0601800i \(0.0191675\pi\)
−0.998188 + 0.0601800i \(0.980833\pi\)
\(908\) − 3.16925e6i − 0.127568i
\(909\) 358644. 0.0143964
\(910\) 0 0
\(911\) 2.96579e7 1.18398 0.591989 0.805946i \(-0.298343\pi\)
0.591989 + 0.805946i \(0.298343\pi\)
\(912\) 364544.i 0.0145132i
\(913\) 6.65669e6i 0.264291i
\(914\) 2.68125e7 1.06163
\(915\) 0 0
\(916\) −1.36000e7 −0.535548
\(917\) 1.62916e7i 0.639793i
\(918\) 1.43172e6i 0.0560727i
\(919\) −3.18057e7 −1.24227 −0.621135 0.783704i \(-0.713328\pi\)
−0.621135 + 0.783704i \(0.713328\pi\)
\(920\) 0 0
\(921\) −1.03905e6 −0.0403633
\(922\) 5.03976e6i 0.195246i
\(923\) − 9.21121e6i − 0.355887i
\(924\) −321376. −0.0123832
\(925\) 0 0
\(926\) 2.00923e7 0.770021
\(927\) 2.84340e7i 1.08677i
\(928\) − 2.11354e6i − 0.0805638i
\(929\) 2.33444e7 0.887451 0.443725 0.896163i \(-0.353657\pi\)
0.443725 + 0.896163i \(0.353657\pi\)
\(930\) 0 0
\(931\) 1.53066e7 0.578767
\(932\) − 6.42931e6i − 0.242451i
\(933\) 1.25135e6i 0.0470624i
\(934\) −9.42642e6 −0.353573
\(935\) 0 0
\(936\) 1.07177e7 0.399864
\(937\) 2.07372e7i 0.771616i 0.922579 + 0.385808i \(0.126077\pi\)
−0.922579 + 0.385808i \(0.873923\pi\)
\(938\) − 1.68278e7i − 0.624481i
\(939\) 1.44336e6 0.0534209
\(940\) 0 0
\(941\) 2.69193e7 0.991036 0.495518 0.868598i \(-0.334978\pi\)
0.495518 + 0.868598i \(0.334978\pi\)
\(942\) 136300.i 0.00500459i
\(943\) 9.43582e6i 0.345542i
\(944\) 8.95718e6 0.327146
\(945\) 0 0
\(946\) 8.61423e6 0.312960
\(947\) 1.01896e7i 0.369216i 0.982812 + 0.184608i \(0.0591016\pi\)
−0.982812 + 0.184608i \(0.940898\pi\)
\(948\) − 1.23574e6i − 0.0446589i
\(949\) 3.68559e7 1.32844
\(950\) 0 0
\(951\) −2.01208e6 −0.0721429
\(952\) − 7.84051e6i − 0.280383i
\(953\) − 1.03924e7i − 0.370665i −0.982676 0.185333i \(-0.940664\pi\)
0.982676 0.185333i \(-0.0593362\pi\)
\(954\) 2.97428e7 1.05806
\(955\) 0 0
\(956\) 1.36828e7 0.484206
\(957\) 249744.i 0.00881486i
\(958\) − 2.68903e7i − 0.946634i
\(959\) 6.64227e7 2.33222
\(960\) 0 0
\(961\) 1.03709e7 0.362249
\(962\) 4.09249e7i 1.42577i
\(963\) − 1.92056e7i − 0.667363i
\(964\) −1.80074e7 −0.624107
\(965\) 0 0
\(966\) −1.18126e6 −0.0407288
\(967\) − 8.18877e6i − 0.281613i −0.990037 0.140806i \(-0.955030\pi\)
0.990037 0.140806i \(-0.0449695\pi\)
\(968\) 937024.i 0.0321412i
\(969\) 1.05091e6 0.0359548
\(970\) 0 0
\(971\) −1.73274e7 −0.589775 −0.294887 0.955532i \(-0.595282\pi\)
−0.294887 + 0.955532i \(0.595282\pi\)
\(972\) 2.81882e6i 0.0956976i
\(973\) 3.41572e7i 1.15664i
\(974\) 7.84005e6 0.264802
\(975\) 0 0
\(976\) −1.17606e7 −0.395190
\(977\) − 438963.i − 0.0147127i −0.999973 0.00735634i \(-0.997658\pi\)
0.999973 0.00735634i \(-0.00234162\pi\)
\(978\) 180080.i 0.00602030i
\(979\) 1.51752e7 0.506032
\(980\) 0 0
\(981\) −2.12578e7 −0.705253
\(982\) 2.31850e6i 0.0767234i
\(983\) 2.79124e7i 0.921326i 0.887575 + 0.460663i \(0.152388\pi\)
−0.887575 + 0.460663i \(0.847612\pi\)
\(984\) 339456. 0.0111762
\(985\) 0 0
\(986\) −6.09293e6 −0.199588
\(987\) 2.85254e6i 0.0932051i
\(988\) − 1.57665e7i − 0.513859i
\(989\) 3.16626e7 1.02933
\(990\) 0 0
\(991\) −4.26846e7 −1.38066 −0.690331 0.723494i \(-0.742535\pi\)
−0.690331 + 0.723494i \(0.742535\pi\)
\(992\) − 6.39488e6i − 0.206326i
\(993\) − 2.01734e6i − 0.0649240i
\(994\) −8.83850e6 −0.283735
\(995\) 0 0
\(996\) 880224. 0.0281154
\(997\) − 2.21044e7i − 0.704273i −0.935949 0.352137i \(-0.885455\pi\)
0.935949 0.352137i \(-0.114545\pi\)
\(998\) 5.47621e6i 0.174042i
\(999\) −7.17072e6 −0.227326
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 550.6.b.f.199.1 2
5.2 odd 4 550.6.a.f.1.1 1
5.3 odd 4 22.6.a.b.1.1 1
5.4 even 2 inner 550.6.b.f.199.2 2
15.8 even 4 198.6.a.i.1.1 1
20.3 even 4 176.6.a.b.1.1 1
35.13 even 4 1078.6.a.a.1.1 1
40.3 even 4 704.6.a.f.1.1 1
40.13 odd 4 704.6.a.e.1.1 1
55.43 even 4 242.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.b.1.1 1 5.3 odd 4
176.6.a.b.1.1 1 20.3 even 4
198.6.a.i.1.1 1 15.8 even 4
242.6.a.d.1.1 1 55.43 even 4
550.6.a.f.1.1 1 5.2 odd 4
550.6.b.f.199.1 2 1.1 even 1 trivial
550.6.b.f.199.2 2 5.4 even 2 inner
704.6.a.e.1.1 1 40.13 odd 4
704.6.a.f.1.1 1 40.3 even 4
1078.6.a.a.1.1 1 35.13 even 4