Properties

Label 550.6.b.a.199.2
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 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 550.199
Dual form 550.6.b.a.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{2} +29.0000i q^{3} -16.0000 q^{4} -116.000 q^{6} -230.000i q^{7} -64.0000i q^{8} -598.000 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} +29.0000i q^{3} -16.0000 q^{4} -116.000 q^{6} -230.000i q^{7} -64.0000i q^{8} -598.000 q^{9} +121.000 q^{11} -464.000i q^{12} -112.000i q^{13} +920.000 q^{14} +256.000 q^{16} -1142.00i q^{17} -2392.00i q^{18} +612.000 q^{19} +6670.00 q^{21} +484.000i q^{22} +1941.00i q^{23} +1856.00 q^{24} +448.000 q^{26} -10295.0i q^{27} +3680.00i q^{28} -1192.00 q^{29} -1037.00 q^{31} +1024.00i q^{32} +3509.00i q^{33} +4568.00 q^{34} +9568.00 q^{36} +8083.00i q^{37} +2448.00i q^{38} +3248.00 q^{39} -10444.0 q^{41} +26680.0i q^{42} -58.0000i q^{43} -1936.00 q^{44} -7764.00 q^{46} +8656.00i q^{47} +7424.00i q^{48} -36093.0 q^{49} +33118.0 q^{51} +1792.00i q^{52} +20318.0i q^{53} +41180.0 q^{54} -14720.0 q^{56} +17748.0i q^{57} -4768.00i q^{58} +21351.0 q^{59} +47044.0 q^{61} -4148.00i q^{62} +137540. i q^{63} -4096.00 q^{64} -14036.0 q^{66} +48093.0i q^{67} +18272.0i q^{68} -56289.0 q^{69} -24967.0 q^{71} +38272.0i q^{72} +42288.0i q^{73} -32332.0 q^{74} -9792.00 q^{76} -27830.0i q^{77} +12992.0i q^{78} +72410.0 q^{79} +153241. q^{81} -41776.0i q^{82} +15806.0i q^{83} -106720. q^{84} +232.000 q^{86} -34568.0i q^{87} -7744.00i q^{88} +114761. q^{89} -25760.0 q^{91} -31056.0i q^{92} -30073.0i q^{93} -34624.0 q^{94} -29696.0 q^{96} -5159.00i q^{97} -144372. i q^{98} -72358.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 232 q^{6} - 1196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 232 q^{6} - 1196 q^{9} + 242 q^{11} + 1840 q^{14} + 512 q^{16} + 1224 q^{19} + 13340 q^{21} + 3712 q^{24} + 896 q^{26} - 2384 q^{29} - 2074 q^{31} + 9136 q^{34} + 19136 q^{36} + 6496 q^{39} - 20888 q^{41} - 3872 q^{44} - 15528 q^{46} - 72186 q^{49} + 66236 q^{51} + 82360 q^{54} - 29440 q^{56} + 42702 q^{59} + 94088 q^{61} - 8192 q^{64} - 28072 q^{66} - 112578 q^{69} - 49934 q^{71} - 64664 q^{74} - 19584 q^{76} + 144820 q^{79} + 306482 q^{81} - 213440 q^{84} + 464 q^{86} + 229522 q^{89} - 51520 q^{91} - 69248 q^{94} - 59392 q^{96} - 144716 q^{99}+O(q^{100}) \) 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
\(3\) 29.0000i 1.86035i 0.367115 + 0.930175i \(0.380345\pi\)
−0.367115 + 0.930175i \(0.619655\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) −116.000 −1.31547
\(7\) − 230.000i − 1.77412i −0.461655 0.887059i \(-0.652744\pi\)
0.461655 0.887059i \(-0.347256\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) −598.000 −2.46091
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) − 464.000i − 0.930175i
\(13\) − 112.000i − 0.183806i −0.995768 0.0919030i \(-0.970705\pi\)
0.995768 0.0919030i \(-0.0292950\pi\)
\(14\) 920.000 1.25449
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 1142.00i − 0.958393i −0.877708 0.479197i \(-0.840928\pi\)
0.877708 0.479197i \(-0.159072\pi\)
\(18\) − 2392.00i − 1.74012i
\(19\) 612.000 0.388926 0.194463 0.980910i \(-0.437704\pi\)
0.194463 + 0.980910i \(0.437704\pi\)
\(20\) 0 0
\(21\) 6670.00 3.30048
\(22\) 484.000i 0.213201i
\(23\) 1941.00i 0.765078i 0.923939 + 0.382539i \(0.124950\pi\)
−0.923939 + 0.382539i \(0.875050\pi\)
\(24\) 1856.00 0.657733
\(25\) 0 0
\(26\) 448.000 0.129970
\(27\) − 10295.0i − 2.71780i
\(28\) 3680.00i 0.887059i
\(29\) −1192.00 −0.263197 −0.131599 0.991303i \(-0.542011\pi\)
−0.131599 + 0.991303i \(0.542011\pi\)
\(30\) 0 0
\(31\) −1037.00 −0.193809 −0.0969046 0.995294i \(-0.530894\pi\)
−0.0969046 + 0.995294i \(0.530894\pi\)
\(32\) 1024.00i 0.176777i
\(33\) 3509.00i 0.560917i
\(34\) 4568.00 0.677686
\(35\) 0 0
\(36\) 9568.00 1.23045
\(37\) 8083.00i 0.970663i 0.874330 + 0.485331i \(0.161301\pi\)
−0.874330 + 0.485331i \(0.838699\pi\)
\(38\) 2448.00i 0.275012i
\(39\) 3248.00 0.341944
\(40\) 0 0
\(41\) −10444.0 −0.970303 −0.485151 0.874430i \(-0.661235\pi\)
−0.485151 + 0.874430i \(0.661235\pi\)
\(42\) 26680.0i 2.33379i
\(43\) − 58.0000i − 0.00478362i −0.999997 0.00239181i \(-0.999239\pi\)
0.999997 0.00239181i \(-0.000761338\pi\)
\(44\) −1936.00 −0.150756
\(45\) 0 0
\(46\) −7764.00 −0.540992
\(47\) 8656.00i 0.571574i 0.958293 + 0.285787i \(0.0922550\pi\)
−0.958293 + 0.285787i \(0.907745\pi\)
\(48\) 7424.00i 0.465088i
\(49\) −36093.0 −2.14750
\(50\) 0 0
\(51\) 33118.0 1.78295
\(52\) 1792.00i 0.0919030i
\(53\) 20318.0i 0.993554i 0.867878 + 0.496777i \(0.165483\pi\)
−0.867878 + 0.496777i \(0.834517\pi\)
\(54\) 41180.0 1.92177
\(55\) 0 0
\(56\) −14720.0 −0.627246
\(57\) 17748.0i 0.723540i
\(58\) − 4768.00i − 0.186109i
\(59\) 21351.0 0.798524 0.399262 0.916837i \(-0.369266\pi\)
0.399262 + 0.916837i \(0.369266\pi\)
\(60\) 0 0
\(61\) 47044.0 1.61875 0.809375 0.587293i \(-0.199806\pi\)
0.809375 + 0.587293i \(0.199806\pi\)
\(62\) − 4148.00i − 0.137044i
\(63\) 137540.i 4.36594i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) −14036.0 −0.396628
\(67\) 48093.0i 1.30887i 0.756120 + 0.654433i \(0.227092\pi\)
−0.756120 + 0.654433i \(0.772908\pi\)
\(68\) 18272.0i 0.479197i
\(69\) −56289.0 −1.42331
\(70\) 0 0
\(71\) −24967.0 −0.587788 −0.293894 0.955838i \(-0.594951\pi\)
−0.293894 + 0.955838i \(0.594951\pi\)
\(72\) 38272.0i 0.870061i
\(73\) 42288.0i 0.928774i 0.885632 + 0.464387i \(0.153725\pi\)
−0.885632 + 0.464387i \(0.846275\pi\)
\(74\) −32332.0 −0.686362
\(75\) 0 0
\(76\) −9792.00 −0.194463
\(77\) − 27830.0i − 0.534917i
\(78\) 12992.0i 0.241791i
\(79\) 72410.0 1.30536 0.652681 0.757633i \(-0.273644\pi\)
0.652681 + 0.757633i \(0.273644\pi\)
\(80\) 0 0
\(81\) 153241. 2.59515
\(82\) − 41776.0i − 0.686108i
\(83\) 15806.0i 0.251841i 0.992040 + 0.125921i \(0.0401885\pi\)
−0.992040 + 0.125921i \(0.959812\pi\)
\(84\) −106720. −1.65024
\(85\) 0 0
\(86\) 232.000 0.00338253
\(87\) − 34568.0i − 0.489639i
\(88\) − 7744.00i − 0.106600i
\(89\) 114761. 1.53575 0.767873 0.640602i \(-0.221315\pi\)
0.767873 + 0.640602i \(0.221315\pi\)
\(90\) 0 0
\(91\) −25760.0 −0.326094
\(92\) − 31056.0i − 0.382539i
\(93\) − 30073.0i − 0.360553i
\(94\) −34624.0 −0.404164
\(95\) 0 0
\(96\) −29696.0 −0.328867
\(97\) − 5159.00i − 0.0556719i −0.999613 0.0278360i \(-0.991138\pi\)
0.999613 0.0278360i \(-0.00886161\pi\)
\(98\) − 144372.i − 1.51851i
\(99\) −72358.0 −0.741991
\(100\) 0 0
\(101\) −61426.0 −0.599168 −0.299584 0.954070i \(-0.596848\pi\)
−0.299584 + 0.954070i \(0.596848\pi\)
\(102\) 132472.i 1.26073i
\(103\) − 185896.i − 1.72654i −0.504741 0.863271i \(-0.668412\pi\)
0.504741 0.863271i \(-0.331588\pi\)
\(104\) −7168.00 −0.0649852
\(105\) 0 0
\(106\) −81272.0 −0.702548
\(107\) − 23970.0i − 0.202399i −0.994866 0.101200i \(-0.967732\pi\)
0.994866 0.101200i \(-0.0322681\pi\)
\(108\) 164720.i 1.35890i
\(109\) 56326.0 0.454091 0.227045 0.973884i \(-0.427093\pi\)
0.227045 + 0.973884i \(0.427093\pi\)
\(110\) 0 0
\(111\) −234407. −1.80577
\(112\) − 58880.0i − 0.443530i
\(113\) 261903.i 1.92950i 0.263171 + 0.964749i \(0.415231\pi\)
−0.263171 + 0.964749i \(0.584769\pi\)
\(114\) −70992.0 −0.511620
\(115\) 0 0
\(116\) 19072.0 0.131599
\(117\) 66976.0i 0.452329i
\(118\) 85404.0i 0.564642i
\(119\) −262660. −1.70030
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 188176.i 1.14463i
\(123\) − 302876.i − 1.80510i
\(124\) 16592.0 0.0969046
\(125\) 0 0
\(126\) −550160. −3.08718
\(127\) 87404.0i 0.480864i 0.970666 + 0.240432i \(0.0772891\pi\)
−0.970666 + 0.240432i \(0.922711\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 1682.00 0.00889922
\(130\) 0 0
\(131\) 265122. 1.34979 0.674897 0.737912i \(-0.264188\pi\)
0.674897 + 0.737912i \(0.264188\pi\)
\(132\) − 56144.0i − 0.280458i
\(133\) − 140760.i − 0.690002i
\(134\) −192372. −0.925507
\(135\) 0 0
\(136\) −73088.0 −0.338843
\(137\) 245857.i 1.11913i 0.828786 + 0.559566i \(0.189032\pi\)
−0.828786 + 0.559566i \(0.810968\pi\)
\(138\) − 225156.i − 1.00644i
\(139\) 363594. 1.59617 0.798086 0.602544i \(-0.205846\pi\)
0.798086 + 0.602544i \(0.205846\pi\)
\(140\) 0 0
\(141\) −251024. −1.06333
\(142\) − 99868.0i − 0.415629i
\(143\) − 13552.0i − 0.0554196i
\(144\) −153088. −0.615226
\(145\) 0 0
\(146\) −169152. −0.656742
\(147\) − 1.04670e6i − 3.99510i
\(148\) − 129328.i − 0.485331i
\(149\) 55750.0 0.205721 0.102861 0.994696i \(-0.467200\pi\)
0.102861 + 0.994696i \(0.467200\pi\)
\(150\) 0 0
\(151\) 65642.0 0.234282 0.117141 0.993115i \(-0.462627\pi\)
0.117141 + 0.993115i \(0.462627\pi\)
\(152\) − 39168.0i − 0.137506i
\(153\) 682916.i 2.35852i
\(154\) 111320. 0.378243
\(155\) 0 0
\(156\) −51968.0 −0.170972
\(157\) − 275367.i − 0.891585i −0.895136 0.445793i \(-0.852922\pi\)
0.895136 0.445793i \(-0.147078\pi\)
\(158\) 289640.i 0.923030i
\(159\) −589222. −1.84836
\(160\) 0 0
\(161\) 446430. 1.35734
\(162\) 612964.i 1.83505i
\(163\) − 291940.i − 0.860646i −0.902675 0.430323i \(-0.858400\pi\)
0.902675 0.430323i \(-0.141600\pi\)
\(164\) 167104. 0.485151
\(165\) 0 0
\(166\) −63224.0 −0.178079
\(167\) − 337344.i − 0.936013i −0.883725 0.468006i \(-0.844972\pi\)
0.883725 0.468006i \(-0.155028\pi\)
\(168\) − 426880.i − 1.16690i
\(169\) 358749. 0.966215
\(170\) 0 0
\(171\) −365976. −0.957111
\(172\) 928.000i 0.00239181i
\(173\) 116742.i 0.296560i 0.988945 + 0.148280i \(0.0473736\pi\)
−0.988945 + 0.148280i \(0.952626\pi\)
\(174\) 138272. 0.346227
\(175\) 0 0
\(176\) 30976.0 0.0753778
\(177\) 619179.i 1.48554i
\(178\) 459044.i 1.08594i
\(179\) 19107.0 0.0445718 0.0222859 0.999752i \(-0.492906\pi\)
0.0222859 + 0.999752i \(0.492906\pi\)
\(180\) 0 0
\(181\) −16177.0 −0.0367030 −0.0183515 0.999832i \(-0.505842\pi\)
−0.0183515 + 0.999832i \(0.505842\pi\)
\(182\) − 103040.i − 0.230583i
\(183\) 1.36428e6i 3.01144i
\(184\) 124224. 0.270496
\(185\) 0 0
\(186\) 120292. 0.254950
\(187\) − 138182.i − 0.288966i
\(188\) − 138496.i − 0.285787i
\(189\) −2.36785e6 −4.82169
\(190\) 0 0
\(191\) 685333. 1.35931 0.679655 0.733532i \(-0.262130\pi\)
0.679655 + 0.733532i \(0.262130\pi\)
\(192\) − 118784.i − 0.232544i
\(193\) 309292.i 0.597689i 0.954302 + 0.298845i \(0.0966012\pi\)
−0.954302 + 0.298845i \(0.903399\pi\)
\(194\) 20636.0 0.0393660
\(195\) 0 0
\(196\) 577488. 1.07375
\(197\) − 120930.i − 0.222008i −0.993820 0.111004i \(-0.964593\pi\)
0.993820 0.111004i \(-0.0354066\pi\)
\(198\) − 289432.i − 0.524667i
\(199\) −915536. −1.63886 −0.819432 0.573177i \(-0.805711\pi\)
−0.819432 + 0.573177i \(0.805711\pi\)
\(200\) 0 0
\(201\) −1.39470e6 −2.43495
\(202\) − 245704.i − 0.423676i
\(203\) 274160.i 0.466943i
\(204\) −529888. −0.891474
\(205\) 0 0
\(206\) 743584. 1.22085
\(207\) − 1.16072e6i − 1.88279i
\(208\) − 28672.0i − 0.0459515i
\(209\) 74052.0 0.117266
\(210\) 0 0
\(211\) −134580. −0.208101 −0.104051 0.994572i \(-0.533180\pi\)
−0.104051 + 0.994572i \(0.533180\pi\)
\(212\) − 325088.i − 0.496777i
\(213\) − 724043.i − 1.09349i
\(214\) 95880.0 0.143118
\(215\) 0 0
\(216\) −658880. −0.960886
\(217\) 238510.i 0.343841i
\(218\) 225304.i 0.321091i
\(219\) −1.22635e6 −1.72785
\(220\) 0 0
\(221\) −127904. −0.176158
\(222\) − 937628.i − 1.27687i
\(223\) − 468839.i − 0.631337i −0.948869 0.315669i \(-0.897771\pi\)
0.948869 0.315669i \(-0.102229\pi\)
\(224\) 235520. 0.313623
\(225\) 0 0
\(226\) −1.04761e6 −1.36436
\(227\) 275022.i 0.354244i 0.984189 + 0.177122i \(0.0566788\pi\)
−0.984189 + 0.177122i \(0.943321\pi\)
\(228\) − 283968.i − 0.361770i
\(229\) 642281. 0.809350 0.404675 0.914461i \(-0.367385\pi\)
0.404675 + 0.914461i \(0.367385\pi\)
\(230\) 0 0
\(231\) 807070. 0.995133
\(232\) 76288.0i 0.0930543i
\(233\) 1.50485e6i 1.81595i 0.419029 + 0.907973i \(0.362371\pi\)
−0.419029 + 0.907973i \(0.637629\pi\)
\(234\) −267904. −0.319845
\(235\) 0 0
\(236\) −341616. −0.399262
\(237\) 2.09989e6i 2.42843i
\(238\) − 1.05064e6i − 1.20230i
\(239\) 304694. 0.345040 0.172520 0.985006i \(-0.444809\pi\)
0.172520 + 0.985006i \(0.444809\pi\)
\(240\) 0 0
\(241\) 1.27181e6 1.41052 0.705260 0.708949i \(-0.250830\pi\)
0.705260 + 0.708949i \(0.250830\pi\)
\(242\) 58564.0i 0.0642824i
\(243\) 1.94230e6i 2.11009i
\(244\) −752704. −0.809375
\(245\) 0 0
\(246\) 1.21150e6 1.27640
\(247\) − 68544.0i − 0.0714870i
\(248\) 66368.0i 0.0685219i
\(249\) −458374. −0.468513
\(250\) 0 0
\(251\) 629965. 0.631149 0.315575 0.948901i \(-0.397803\pi\)
0.315575 + 0.948901i \(0.397803\pi\)
\(252\) − 2.20064e6i − 2.18297i
\(253\) 234861.i 0.230680i
\(254\) −349616. −0.340022
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 544086.i 0.513848i 0.966432 + 0.256924i \(0.0827090\pi\)
−0.966432 + 0.256924i \(0.917291\pi\)
\(258\) 6728.00i 0.00629270i
\(259\) 1.85909e6 1.72207
\(260\) 0 0
\(261\) 712816. 0.647703
\(262\) 1.06049e6i 0.954449i
\(263\) 1.98933e6i 1.77345i 0.462300 + 0.886724i \(0.347024\pi\)
−0.462300 + 0.886724i \(0.652976\pi\)
\(264\) 224576. 0.198314
\(265\) 0 0
\(266\) 563040. 0.487905
\(267\) 3.32807e6i 2.85703i
\(268\) − 769488.i − 0.654433i
\(269\) −1.75446e6 −1.47830 −0.739149 0.673541i \(-0.764772\pi\)
−0.739149 + 0.673541i \(0.764772\pi\)
\(270\) 0 0
\(271\) −1.65824e6 −1.37159 −0.685795 0.727795i \(-0.740545\pi\)
−0.685795 + 0.727795i \(0.740545\pi\)
\(272\) − 292352.i − 0.239598i
\(273\) − 747040.i − 0.606649i
\(274\) −983428. −0.791346
\(275\) 0 0
\(276\) 900624. 0.711657
\(277\) − 42634.0i − 0.0333854i −0.999861 0.0166927i \(-0.994686\pi\)
0.999861 0.0166927i \(-0.00531370\pi\)
\(278\) 1.45438e6i 1.12866i
\(279\) 620126. 0.476946
\(280\) 0 0
\(281\) 319510. 0.241390 0.120695 0.992690i \(-0.461488\pi\)
0.120695 + 0.992690i \(0.461488\pi\)
\(282\) − 1.00410e6i − 0.751887i
\(283\) 2.02735e6i 1.50474i 0.658739 + 0.752371i \(0.271090\pi\)
−0.658739 + 0.752371i \(0.728910\pi\)
\(284\) 399472. 0.293894
\(285\) 0 0
\(286\) 54208.0 0.0391876
\(287\) 2.40212e6i 1.72143i
\(288\) − 612352.i − 0.435031i
\(289\) 115693. 0.0814821
\(290\) 0 0
\(291\) 149611. 0.103569
\(292\) − 676608.i − 0.464387i
\(293\) − 718844.i − 0.489177i −0.969627 0.244588i \(-0.921347\pi\)
0.969627 0.244588i \(-0.0786528\pi\)
\(294\) 4.18679e6 2.82496
\(295\) 0 0
\(296\) 517312. 0.343181
\(297\) − 1.24570e6i − 0.819446i
\(298\) 223000.i 0.145467i
\(299\) 217392. 0.140626
\(300\) 0 0
\(301\) −13340.0 −0.00848671
\(302\) 262568.i 0.165663i
\(303\) − 1.78135e6i − 1.11466i
\(304\) 156672. 0.0972316
\(305\) 0 0
\(306\) −2.73166e6 −1.66772
\(307\) − 1.98142e6i − 1.19986i −0.800052 0.599930i \(-0.795195\pi\)
0.800052 0.599930i \(-0.204805\pi\)
\(308\) 445280.i 0.267458i
\(309\) 5.39098e6 3.21197
\(310\) 0 0
\(311\) −1.51030e6 −0.885446 −0.442723 0.896658i \(-0.645988\pi\)
−0.442723 + 0.896658i \(0.645988\pi\)
\(312\) − 207872.i − 0.120895i
\(313\) − 2.00092e6i − 1.15443i −0.816591 0.577216i \(-0.804139\pi\)
0.816591 0.577216i \(-0.195861\pi\)
\(314\) 1.10147e6 0.630446
\(315\) 0 0
\(316\) −1.15856e6 −0.652681
\(317\) − 259331.i − 0.144946i −0.997370 0.0724730i \(-0.976911\pi\)
0.997370 0.0724730i \(-0.0230891\pi\)
\(318\) − 2.35689e6i − 1.30699i
\(319\) −144232. −0.0793569
\(320\) 0 0
\(321\) 695130. 0.376533
\(322\) 1.78572e6i 0.959784i
\(323\) − 698904.i − 0.372744i
\(324\) −2.45186e6 −1.29757
\(325\) 0 0
\(326\) 1.16776e6 0.608569
\(327\) 1.63345e6i 0.844768i
\(328\) 668416.i 0.343054i
\(329\) 1.99088e6 1.01404
\(330\) 0 0
\(331\) 51203.0 0.0256877 0.0128439 0.999918i \(-0.495912\pi\)
0.0128439 + 0.999918i \(0.495912\pi\)
\(332\) − 252896.i − 0.125921i
\(333\) − 4.83363e6i − 2.38871i
\(334\) 1.34938e6 0.661861
\(335\) 0 0
\(336\) 1.70752e6 0.825121
\(337\) 266870.i 0.128004i 0.997950 + 0.0640022i \(0.0203865\pi\)
−0.997950 + 0.0640022i \(0.979614\pi\)
\(338\) 1.43500e6i 0.683217i
\(339\) −7.59519e6 −3.58954
\(340\) 0 0
\(341\) −125477. −0.0584357
\(342\) − 1.46390e6i − 0.676780i
\(343\) 4.43578e6i 2.03580i
\(344\) −3712.00 −0.00169127
\(345\) 0 0
\(346\) −466968. −0.209699
\(347\) 622800.i 0.277667i 0.990316 + 0.138834i \(0.0443354\pi\)
−0.990316 + 0.138834i \(0.955665\pi\)
\(348\) 553088.i 0.244820i
\(349\) −2.43649e6 −1.07078 −0.535391 0.844604i \(-0.679836\pi\)
−0.535391 + 0.844604i \(0.679836\pi\)
\(350\) 0 0
\(351\) −1.15304e6 −0.499547
\(352\) 123904.i 0.0533002i
\(353\) − 1.55957e6i − 0.666144i −0.942901 0.333072i \(-0.891915\pi\)
0.942901 0.333072i \(-0.108085\pi\)
\(354\) −2.47672e6 −1.05043
\(355\) 0 0
\(356\) −1.83618e6 −0.767873
\(357\) − 7.61714e6i − 3.16316i
\(358\) 76428.0i 0.0315170i
\(359\) −1.91961e6 −0.786098 −0.393049 0.919518i \(-0.628580\pi\)
−0.393049 + 0.919518i \(0.628580\pi\)
\(360\) 0 0
\(361\) −2.10156e6 −0.848736
\(362\) − 64708.0i − 0.0259529i
\(363\) 424589.i 0.169123i
\(364\) 412160. 0.163047
\(365\) 0 0
\(366\) −5.45710e6 −2.12941
\(367\) 3.61225e6i 1.39995i 0.714167 + 0.699975i \(0.246806\pi\)
−0.714167 + 0.699975i \(0.753194\pi\)
\(368\) 496896.i 0.191270i
\(369\) 6.24551e6 2.38782
\(370\) 0 0
\(371\) 4.67314e6 1.76268
\(372\) 481168.i 0.180277i
\(373\) − 3.93968e6i − 1.46619i −0.680128 0.733093i \(-0.738076\pi\)
0.680128 0.733093i \(-0.261924\pi\)
\(374\) 552728. 0.204330
\(375\) 0 0
\(376\) 553984. 0.202082
\(377\) 133504.i 0.0483772i
\(378\) − 9.47140e6i − 3.40945i
\(379\) 2.18829e6 0.782540 0.391270 0.920276i \(-0.372036\pi\)
0.391270 + 0.920276i \(0.372036\pi\)
\(380\) 0 0
\(381\) −2.53472e6 −0.894575
\(382\) 2.74133e6i 0.961177i
\(383\) − 768387.i − 0.267660i −0.991004 0.133830i \(-0.957272\pi\)
0.991004 0.133830i \(-0.0427276\pi\)
\(384\) 475136. 0.164433
\(385\) 0 0
\(386\) −1.23717e6 −0.422630
\(387\) 34684.0i 0.0117720i
\(388\) 82544.0i 0.0278360i
\(389\) −324313. −0.108665 −0.0543326 0.998523i \(-0.517303\pi\)
−0.0543326 + 0.998523i \(0.517303\pi\)
\(390\) 0 0
\(391\) 2.21662e6 0.733246
\(392\) 2.30995e6i 0.759255i
\(393\) 7.68854e6i 2.51109i
\(394\) 483720. 0.156983
\(395\) 0 0
\(396\) 1.15773e6 0.370995
\(397\) 334758.i 0.106599i 0.998579 + 0.0532997i \(0.0169739\pi\)
−0.998579 + 0.0532997i \(0.983026\pi\)
\(398\) − 3.66214e6i − 1.15885i
\(399\) 4.08204e6 1.28365
\(400\) 0 0
\(401\) −902022. −0.280128 −0.140064 0.990142i \(-0.544731\pi\)
−0.140064 + 0.990142i \(0.544731\pi\)
\(402\) − 5.57879e6i − 1.72177i
\(403\) 116144.i 0.0356233i
\(404\) 982816. 0.299584
\(405\) 0 0
\(406\) −1.09664e6 −0.330179
\(407\) 978043.i 0.292666i
\(408\) − 2.11955e6i − 0.630367i
\(409\) 5.00457e6 1.47931 0.739654 0.672987i \(-0.234989\pi\)
0.739654 + 0.672987i \(0.234989\pi\)
\(410\) 0 0
\(411\) −7.12985e6 −2.08198
\(412\) 2.97434e6i 0.863271i
\(413\) − 4.91073e6i − 1.41668i
\(414\) 4.64287e6 1.33133
\(415\) 0 0
\(416\) 114688. 0.0324926
\(417\) 1.05442e7i 2.96944i
\(418\) 296208.i 0.0829194i
\(419\) 3.00124e6 0.835151 0.417576 0.908642i \(-0.362880\pi\)
0.417576 + 0.908642i \(0.362880\pi\)
\(420\) 0 0
\(421\) 4.56224e6 1.25451 0.627253 0.778816i \(-0.284179\pi\)
0.627253 + 0.778816i \(0.284179\pi\)
\(422\) − 538320.i − 0.147150i
\(423\) − 5.17629e6i − 1.40659i
\(424\) 1.30035e6 0.351274
\(425\) 0 0
\(426\) 2.89617e6 0.773215
\(427\) − 1.08201e7i − 2.87185i
\(428\) 383520.i 0.101200i
\(429\) 393008. 0.103100
\(430\) 0 0
\(431\) 4.89783e6 1.27002 0.635009 0.772504i \(-0.280996\pi\)
0.635009 + 0.772504i \(0.280996\pi\)
\(432\) − 2.63552e6i − 0.679449i
\(433\) − 6.72876e6i − 1.72471i −0.506307 0.862353i \(-0.668990\pi\)
0.506307 0.862353i \(-0.331010\pi\)
\(434\) −954040. −0.243132
\(435\) 0 0
\(436\) −901216. −0.227045
\(437\) 1.18789e6i 0.297559i
\(438\) − 4.90541e6i − 1.22177i
\(439\) 3.35034e6 0.829711 0.414856 0.909887i \(-0.363832\pi\)
0.414856 + 0.909887i \(0.363832\pi\)
\(440\) 0 0
\(441\) 2.15836e7 5.28479
\(442\) − 511616.i − 0.124563i
\(443\) 7.12434e6i 1.72479i 0.506238 + 0.862394i \(0.331036\pi\)
−0.506238 + 0.862394i \(0.668964\pi\)
\(444\) 3.75051e6 0.902886
\(445\) 0 0
\(446\) 1.87536e6 0.446423
\(447\) 1.61675e6i 0.382714i
\(448\) 942080.i 0.221765i
\(449\) 2.70928e6 0.634218 0.317109 0.948389i \(-0.397288\pi\)
0.317109 + 0.948389i \(0.397288\pi\)
\(450\) 0 0
\(451\) −1.26372e6 −0.292557
\(452\) − 4.19045e6i − 0.964749i
\(453\) 1.90362e6i 0.435847i
\(454\) −1.10009e6 −0.250489
\(455\) 0 0
\(456\) 1.13587e6 0.255810
\(457\) 2.41361e6i 0.540601i 0.962776 + 0.270301i \(0.0871231\pi\)
−0.962776 + 0.270301i \(0.912877\pi\)
\(458\) 2.56912e6i 0.572297i
\(459\) −1.17569e7 −2.60472
\(460\) 0 0
\(461\) −6.56065e6 −1.43779 −0.718894 0.695120i \(-0.755351\pi\)
−0.718894 + 0.695120i \(0.755351\pi\)
\(462\) 3.22828e6i 0.703665i
\(463\) 4.72421e6i 1.02418i 0.858932 + 0.512090i \(0.171129\pi\)
−0.858932 + 0.512090i \(0.828871\pi\)
\(464\) −305152. −0.0657993
\(465\) 0 0
\(466\) −6.01939e6 −1.28407
\(467\) − 2.28444e6i − 0.484716i −0.970187 0.242358i \(-0.922079\pi\)
0.970187 0.242358i \(-0.0779209\pi\)
\(468\) − 1.07162e6i − 0.226165i
\(469\) 1.10614e7 2.32208
\(470\) 0 0
\(471\) 7.98564e6 1.65866
\(472\) − 1.36646e6i − 0.282321i
\(473\) − 7018.00i − 0.00144232i
\(474\) −8.39956e6 −1.71716
\(475\) 0 0
\(476\) 4.20256e6 0.850152
\(477\) − 1.21502e7i − 2.44504i
\(478\) 1.21878e6i 0.243980i
\(479\) −951544. −0.189492 −0.0947458 0.995501i \(-0.530204\pi\)
−0.0947458 + 0.995501i \(0.530204\pi\)
\(480\) 0 0
\(481\) 905296. 0.178414
\(482\) 5.08723e6i 0.997388i
\(483\) 1.29465e7i 2.52513i
\(484\) −234256. −0.0454545
\(485\) 0 0
\(486\) −7.76922e6 −1.49206
\(487\) 3.51484e6i 0.671558i 0.941941 + 0.335779i \(0.109000\pi\)
−0.941941 + 0.335779i \(0.891000\pi\)
\(488\) − 3.01082e6i − 0.572314i
\(489\) 8.46626e6 1.60110
\(490\) 0 0
\(491\) −5.78719e6 −1.08334 −0.541669 0.840592i \(-0.682207\pi\)
−0.541669 + 0.840592i \(0.682207\pi\)
\(492\) 4.84602e6i 0.902552i
\(493\) 1.36126e6i 0.252246i
\(494\) 274176. 0.0505489
\(495\) 0 0
\(496\) −265472. −0.0484523
\(497\) 5.74241e6i 1.04281i
\(498\) − 1.83350e6i − 0.331289i
\(499\) −1.02912e6 −0.185019 −0.0925095 0.995712i \(-0.529489\pi\)
−0.0925095 + 0.995712i \(0.529489\pi\)
\(500\) 0 0
\(501\) 9.78298e6 1.74131
\(502\) 2.51986e6i 0.446290i
\(503\) 727370.i 0.128184i 0.997944 + 0.0640922i \(0.0204152\pi\)
−0.997944 + 0.0640922i \(0.979585\pi\)
\(504\) 8.80256e6 1.54359
\(505\) 0 0
\(506\) −939444. −0.163115
\(507\) 1.04037e7i 1.79750i
\(508\) − 1.39846e6i − 0.240432i
\(509\) 1.94630e6 0.332977 0.166489 0.986043i \(-0.446757\pi\)
0.166489 + 0.986043i \(0.446757\pi\)
\(510\) 0 0
\(511\) 9.72624e6 1.64776
\(512\) 262144.i 0.0441942i
\(513\) − 6.30054e6i − 1.05702i
\(514\) −2.17634e6 −0.363345
\(515\) 0 0
\(516\) −26912.0 −0.00444961
\(517\) 1.04738e6i 0.172336i
\(518\) 7.43636e6i 1.21769i
\(519\) −3.38552e6 −0.551705
\(520\) 0 0
\(521\) −1.03133e7 −1.66457 −0.832286 0.554346i \(-0.812968\pi\)
−0.832286 + 0.554346i \(0.812968\pi\)
\(522\) 2.85126e6i 0.457995i
\(523\) 6.86840e6i 1.09800i 0.835823 + 0.548998i \(0.184991\pi\)
−0.835823 + 0.548998i \(0.815009\pi\)
\(524\) −4.24195e6 −0.674897
\(525\) 0 0
\(526\) −7.95734e6 −1.25402
\(527\) 1.18425e6i 0.185746i
\(528\) 898304.i 0.140229i
\(529\) 2.66886e6 0.414655
\(530\) 0 0
\(531\) −1.27679e7 −1.96509
\(532\) 2.25216e6i 0.345001i
\(533\) 1.16973e6i 0.178347i
\(534\) −1.33123e7 −2.02022
\(535\) 0 0
\(536\) 3.07795e6 0.462754
\(537\) 554103.i 0.0829191i
\(538\) − 7.01783e6i − 1.04532i
\(539\) −4.36725e6 −0.647495
\(540\) 0 0
\(541\) 1.00545e7 1.47695 0.738476 0.674280i \(-0.235546\pi\)
0.738476 + 0.674280i \(0.235546\pi\)
\(542\) − 6.63296e6i − 0.969860i
\(543\) − 469133.i − 0.0682805i
\(544\) 1.16941e6 0.169422
\(545\) 0 0
\(546\) 2.98816e6 0.428965
\(547\) 9.85725e6i 1.40860i 0.709903 + 0.704299i \(0.248739\pi\)
−0.709903 + 0.704299i \(0.751261\pi\)
\(548\) − 3.93371e6i − 0.559566i
\(549\) −2.81323e7 −3.98359
\(550\) 0 0
\(551\) −729504. −0.102364
\(552\) 3.60250e6i 0.503218i
\(553\) − 1.66543e7i − 2.31587i
\(554\) 170536. 0.0236070
\(555\) 0 0
\(556\) −5.81750e6 −0.798086
\(557\) − 1.45892e7i − 1.99247i −0.0866757 0.996237i \(-0.527624\pi\)
0.0866757 0.996237i \(-0.472376\pi\)
\(558\) 2.48050e6i 0.337252i
\(559\) −6496.00 −0.000879258 0
\(560\) 0 0
\(561\) 4.00728e6 0.537579
\(562\) 1.27804e6i 0.170688i
\(563\) 1.02413e7i 1.36171i 0.732418 + 0.680855i \(0.238392\pi\)
−0.732418 + 0.680855i \(0.761608\pi\)
\(564\) 4.01638e6 0.531664
\(565\) 0 0
\(566\) −8.10939e6 −1.06401
\(567\) − 3.52454e7i − 4.60410i
\(568\) 1.59789e6i 0.207814i
\(569\) 751816. 0.0973489 0.0486744 0.998815i \(-0.484500\pi\)
0.0486744 + 0.998815i \(0.484500\pi\)
\(570\) 0 0
\(571\) −7.01854e6 −0.900858 −0.450429 0.892812i \(-0.648729\pi\)
−0.450429 + 0.892812i \(0.648729\pi\)
\(572\) 216832.i 0.0277098i
\(573\) 1.98747e7i 2.52879i
\(574\) −9.60848e6 −1.21724
\(575\) 0 0
\(576\) 2.44941e6 0.307613
\(577\) − 3.36377e6i − 0.420617i −0.977635 0.210308i \(-0.932553\pi\)
0.977635 0.210308i \(-0.0674468\pi\)
\(578\) 462772.i 0.0576166i
\(579\) −8.96947e6 −1.11191
\(580\) 0 0
\(581\) 3.63538e6 0.446796
\(582\) 598444.i 0.0732346i
\(583\) 2.45848e6i 0.299568i
\(584\) 2.70643e6 0.328371
\(585\) 0 0
\(586\) 2.87538e6 0.345900
\(587\) 1.40585e7i 1.68401i 0.539468 + 0.842006i \(0.318625\pi\)
−0.539468 + 0.842006i \(0.681375\pi\)
\(588\) 1.67472e7i 1.99755i
\(589\) −634644. −0.0753775
\(590\) 0 0
\(591\) 3.50697e6 0.413013
\(592\) 2.06925e6i 0.242666i
\(593\) 5.39420e6i 0.629927i 0.949104 + 0.314963i \(0.101992\pi\)
−0.949104 + 0.314963i \(0.898008\pi\)
\(594\) 4.98278e6 0.579436
\(595\) 0 0
\(596\) −892000. −0.102861
\(597\) − 2.65505e7i − 3.04886i
\(598\) 869568.i 0.0994376i
\(599\) 1.20204e7 1.36883 0.684417 0.729090i \(-0.260057\pi\)
0.684417 + 0.729090i \(0.260057\pi\)
\(600\) 0 0
\(601\) 1.64636e6 0.185925 0.0929626 0.995670i \(-0.470366\pi\)
0.0929626 + 0.995670i \(0.470366\pi\)
\(602\) − 53360.0i − 0.00600101i
\(603\) − 2.87596e7i − 3.22099i
\(604\) −1.05027e6 −0.117141
\(605\) 0 0
\(606\) 7.12542e6 0.788186
\(607\) 4.88451e6i 0.538083i 0.963129 + 0.269041i \(0.0867068\pi\)
−0.963129 + 0.269041i \(0.913293\pi\)
\(608\) 626688.i 0.0687531i
\(609\) −7.95064e6 −0.868678
\(610\) 0 0
\(611\) 969472. 0.105059
\(612\) − 1.09267e7i − 1.17926i
\(613\) − 3.49011e6i − 0.375136i −0.982252 0.187568i \(-0.939940\pi\)
0.982252 0.187568i \(-0.0600604\pi\)
\(614\) 7.92568e6 0.848429
\(615\) 0 0
\(616\) −1.78112e6 −0.189122
\(617\) 9.12072e6i 0.964531i 0.876025 + 0.482266i \(0.160186\pi\)
−0.876025 + 0.482266i \(0.839814\pi\)
\(618\) 2.15639e7i 2.27121i
\(619\) −1.46635e7 −1.53820 −0.769098 0.639131i \(-0.779294\pi\)
−0.769098 + 0.639131i \(0.779294\pi\)
\(620\) 0 0
\(621\) 1.99826e7 2.07933
\(622\) − 6.04120e6i − 0.626105i
\(623\) − 2.63950e7i − 2.72460i
\(624\) 831488. 0.0854859
\(625\) 0 0
\(626\) 8.00368e6 0.816307
\(627\) 2.14751e6i 0.218155i
\(628\) 4.40587e6i 0.445793i
\(629\) 9.23079e6 0.930277
\(630\) 0 0
\(631\) −1.63870e7 −1.63842 −0.819212 0.573491i \(-0.805589\pi\)
−0.819212 + 0.573491i \(0.805589\pi\)
\(632\) − 4.63424e6i − 0.461515i
\(633\) − 3.90282e6i − 0.387141i
\(634\) 1.03732e6 0.102492
\(635\) 0 0
\(636\) 9.42755e6 0.924179
\(637\) 4.04242e6i 0.394723i
\(638\) − 576928.i − 0.0561138i
\(639\) 1.49303e7 1.44649
\(640\) 0 0
\(641\) −3.26835e6 −0.314184 −0.157092 0.987584i \(-0.550212\pi\)
−0.157092 + 0.987584i \(0.550212\pi\)
\(642\) 2.78052e6i 0.266249i
\(643\) − 8.32842e6i − 0.794393i −0.917734 0.397197i \(-0.869983\pi\)
0.917734 0.397197i \(-0.130017\pi\)
\(644\) −7.14288e6 −0.678670
\(645\) 0 0
\(646\) 2.79562e6 0.263570
\(647\) 2.49694e6i 0.234503i 0.993102 + 0.117251i \(0.0374083\pi\)
−0.993102 + 0.117251i \(0.962592\pi\)
\(648\) − 9.80742e6i − 0.917524i
\(649\) 2.58347e6 0.240764
\(650\) 0 0
\(651\) −6.91679e6 −0.639664
\(652\) 4.67104e6i 0.430323i
\(653\) 789105.i 0.0724189i 0.999344 + 0.0362094i \(0.0115283\pi\)
−0.999344 + 0.0362094i \(0.988472\pi\)
\(654\) −6.53382e6 −0.597341
\(655\) 0 0
\(656\) −2.67366e6 −0.242576
\(657\) − 2.52882e7i − 2.28562i
\(658\) 7.96352e6i 0.717035i
\(659\) 8.31393e6 0.745749 0.372874 0.927882i \(-0.378372\pi\)
0.372874 + 0.927882i \(0.378372\pi\)
\(660\) 0 0
\(661\) −4.33517e6 −0.385925 −0.192962 0.981206i \(-0.561810\pi\)
−0.192962 + 0.981206i \(0.561810\pi\)
\(662\) 204812.i 0.0181640i
\(663\) − 3.70922e6i − 0.327716i
\(664\) 1.01158e6 0.0890393
\(665\) 0 0
\(666\) 1.93345e7 1.68907
\(667\) − 2.31367e6i − 0.201366i
\(668\) 5.39750e6i 0.468006i
\(669\) 1.35963e7 1.17451
\(670\) 0 0
\(671\) 5.69232e6 0.488071
\(672\) 6.83008e6i 0.583449i
\(673\) − 7.29313e6i − 0.620693i −0.950624 0.310346i \(-0.899555\pi\)
0.950624 0.310346i \(-0.100445\pi\)
\(674\) −1.06748e6 −0.0905128
\(675\) 0 0
\(676\) −5.73998e6 −0.483108
\(677\) 1.55814e7i 1.30658i 0.757109 + 0.653288i \(0.226611\pi\)
−0.757109 + 0.653288i \(0.773389\pi\)
\(678\) − 3.03807e7i − 2.53819i
\(679\) −1.18657e6 −0.0987686
\(680\) 0 0
\(681\) −7.97564e6 −0.659019
\(682\) − 501908.i − 0.0413203i
\(683\) 2.16930e6i 0.177938i 0.996034 + 0.0889690i \(0.0283572\pi\)
−0.996034 + 0.0889690i \(0.971643\pi\)
\(684\) 5.85562e6 0.478556
\(685\) 0 0
\(686\) −1.77431e7 −1.43953
\(687\) 1.86261e7i 1.50567i
\(688\) − 14848.0i − 0.00119591i
\(689\) 2.27562e6 0.182621
\(690\) 0 0
\(691\) −1.32195e7 −1.05322 −0.526610 0.850107i \(-0.676537\pi\)
−0.526610 + 0.850107i \(0.676537\pi\)
\(692\) − 1.86787e6i − 0.148280i
\(693\) 1.66423e7i 1.31638i
\(694\) −2.49120e6 −0.196341
\(695\) 0 0
\(696\) −2.21235e6 −0.173114
\(697\) 1.19270e7i 0.929932i
\(698\) − 9.74596e6i − 0.757157i
\(699\) −4.36406e7 −3.37830
\(700\) 0 0
\(701\) −2.59395e7 −1.99373 −0.996866 0.0791122i \(-0.974791\pi\)
−0.996866 + 0.0791122i \(0.974791\pi\)
\(702\) − 4.61216e6i − 0.353233i
\(703\) 4.94680e6i 0.377516i
\(704\) −495616. −0.0376889
\(705\) 0 0
\(706\) 6.23828e6 0.471035
\(707\) 1.41280e7i 1.06300i
\(708\) − 9.90686e6i − 0.742768i
\(709\) −3.57531e6 −0.267115 −0.133557 0.991041i \(-0.542640\pi\)
−0.133557 + 0.991041i \(0.542640\pi\)
\(710\) 0 0
\(711\) −4.33012e7 −3.21237
\(712\) − 7.34470e6i − 0.542968i
\(713\) − 2.01282e6i − 0.148279i
\(714\) 3.04686e7 2.23669
\(715\) 0 0
\(716\) −305712. −0.0222859
\(717\) 8.83613e6i 0.641895i
\(718\) − 7.67843e6i − 0.555855i
\(719\) 1.95814e6 0.141261 0.0706304 0.997503i \(-0.477499\pi\)
0.0706304 + 0.997503i \(0.477499\pi\)
\(720\) 0 0
\(721\) −4.27561e7 −3.06309
\(722\) − 8.40622e6i − 0.600147i
\(723\) 3.68824e7i 2.62406i
\(724\) 258832. 0.0183515
\(725\) 0 0
\(726\) −1.69836e6 −0.119588
\(727\) 1.55360e7i 1.09019i 0.838374 + 0.545095i \(0.183507\pi\)
−0.838374 + 0.545095i \(0.816493\pi\)
\(728\) 1.64864e6i 0.115292i
\(729\) −1.90893e7 −1.33036
\(730\) 0 0
\(731\) −66236.0 −0.00458459
\(732\) − 2.18284e7i − 1.50572i
\(733\) 1.46002e7i 1.00369i 0.864958 + 0.501844i \(0.167345\pi\)
−0.864958 + 0.501844i \(0.832655\pi\)
\(734\) −1.44490e7 −0.989915
\(735\) 0 0
\(736\) −1.98758e6 −0.135248
\(737\) 5.81925e6i 0.394638i
\(738\) 2.49820e7i 1.68845i
\(739\) 2.06682e7 1.39217 0.696085 0.717959i \(-0.254923\pi\)
0.696085 + 0.717959i \(0.254923\pi\)
\(740\) 0 0
\(741\) 1.98778e6 0.132991
\(742\) 1.86926e7i 1.24640i
\(743\) − 1.17065e7i − 0.777953i −0.921248 0.388976i \(-0.872829\pi\)
0.921248 0.388976i \(-0.127171\pi\)
\(744\) −1.92467e6 −0.127475
\(745\) 0 0
\(746\) 1.57587e7 1.03675
\(747\) − 9.45199e6i − 0.619757i
\(748\) 2.21091e6i 0.144483i
\(749\) −5.51310e6 −0.359080
\(750\) 0 0
\(751\) −1.27607e7 −0.825610 −0.412805 0.910819i \(-0.635451\pi\)
−0.412805 + 0.910819i \(0.635451\pi\)
\(752\) 2.21594e6i 0.142894i
\(753\) 1.82690e7i 1.17416i
\(754\) −534016. −0.0342079
\(755\) 0 0
\(756\) 3.78856e7 2.41085
\(757\) − 1.40869e7i − 0.893458i −0.894669 0.446729i \(-0.852589\pi\)
0.894669 0.446729i \(-0.147411\pi\)
\(758\) 8.75316e6i 0.553339i
\(759\) −6.81097e6 −0.429145
\(760\) 0 0
\(761\) 2.33822e7 1.46360 0.731801 0.681518i \(-0.238680\pi\)
0.731801 + 0.681518i \(0.238680\pi\)
\(762\) − 1.01389e7i − 0.632560i
\(763\) − 1.29550e7i − 0.805611i
\(764\) −1.09653e7 −0.679655
\(765\) 0 0
\(766\) 3.07355e6 0.189264
\(767\) − 2.39131e6i − 0.146774i
\(768\) 1.90054e6i 0.116272i
\(769\) 1.09575e7 0.668185 0.334092 0.942540i \(-0.391570\pi\)
0.334092 + 0.942540i \(0.391570\pi\)
\(770\) 0 0
\(771\) −1.57785e7 −0.955938
\(772\) − 4.94867e6i − 0.298845i
\(773\) − 1.69336e7i − 1.01930i −0.860383 0.509648i \(-0.829776\pi\)
0.860383 0.509648i \(-0.170224\pi\)
\(774\) −138736. −0.00832409
\(775\) 0 0
\(776\) −330176. −0.0196830
\(777\) 5.39136e7i 3.20366i
\(778\) − 1.29725e6i − 0.0768379i
\(779\) −6.39173e6 −0.377376
\(780\) 0 0
\(781\) −3.02101e6 −0.177225
\(782\) 8.86649e6i 0.518483i
\(783\) 1.22716e7i 0.715316i
\(784\) −9.23981e6 −0.536875
\(785\) 0 0
\(786\) −3.07542e7 −1.77561
\(787\) − 7.51655e6i − 0.432595i −0.976327 0.216298i \(-0.930602\pi\)
0.976327 0.216298i \(-0.0693982\pi\)
\(788\) 1.93488e6i 0.111004i
\(789\) −5.76907e7 −3.29923
\(790\) 0 0
\(791\) 6.02377e7 3.42316
\(792\) 4.63091e6i 0.262333i
\(793\) − 5.26893e6i − 0.297536i
\(794\) −1.33903e6 −0.0753771
\(795\) 0 0
\(796\) 1.46486e7 0.819432
\(797\) 3.93788e6i 0.219592i 0.993954 + 0.109796i \(0.0350198\pi\)
−0.993954 + 0.109796i \(0.964980\pi\)
\(798\) 1.63282e7i 0.907674i
\(799\) 9.88515e6 0.547793
\(800\) 0 0
\(801\) −6.86271e7 −3.77932
\(802\) − 3.60809e6i − 0.198080i
\(803\) 5.11685e6i 0.280036i
\(804\) 2.23152e7 1.21747
\(805\) 0 0
\(806\) −464576. −0.0251895
\(807\) − 5.08793e7i − 2.75015i
\(808\) 3.93126e6i 0.211838i
\(809\) 1.73609e7 0.932612 0.466306 0.884624i \(-0.345585\pi\)
0.466306 + 0.884624i \(0.345585\pi\)
\(810\) 0 0
\(811\) 2.70850e7 1.44603 0.723014 0.690833i \(-0.242756\pi\)
0.723014 + 0.690833i \(0.242756\pi\)
\(812\) − 4.38656e6i − 0.233472i
\(813\) − 4.80890e7i − 2.55164i
\(814\) −3.91217e6 −0.206946
\(815\) 0 0
\(816\) 8.47821e6 0.445737
\(817\) − 35496.0i − 0.00186048i
\(818\) 2.00183e7i 1.04603i
\(819\) 1.54045e7 0.802486
\(820\) 0 0
\(821\) 3.59384e7 1.86080 0.930402 0.366540i \(-0.119458\pi\)
0.930402 + 0.366540i \(0.119458\pi\)
\(822\) − 2.85194e7i − 1.47218i
\(823\) 505509.i 0.0260153i 0.999915 + 0.0130077i \(0.00414058\pi\)
−0.999915 + 0.0130077i \(0.995859\pi\)
\(824\) −1.18973e7 −0.610425
\(825\) 0 0
\(826\) 1.96429e7 1.00174
\(827\) − 2.99955e7i − 1.52508i −0.646942 0.762539i \(-0.723952\pi\)
0.646942 0.762539i \(-0.276048\pi\)
\(828\) 1.85715e7i 0.941393i
\(829\) −2.96942e7 −1.50067 −0.750334 0.661059i \(-0.770107\pi\)
−0.750334 + 0.661059i \(0.770107\pi\)
\(830\) 0 0
\(831\) 1.23639e6 0.0621086
\(832\) 458752.i 0.0229757i
\(833\) 4.12182e7i 2.05815i
\(834\) −4.21769e7 −2.09971
\(835\) 0 0
\(836\) −1.18483e6 −0.0586329
\(837\) 1.06759e7i 0.526734i
\(838\) 1.20049e7i 0.590541i
\(839\) 1.41371e7 0.693356 0.346678 0.937984i \(-0.387310\pi\)
0.346678 + 0.937984i \(0.387310\pi\)
\(840\) 0 0
\(841\) −1.90903e7 −0.930727
\(842\) 1.82490e7i 0.887070i
\(843\) 9.26579e6i 0.449069i
\(844\) 2.15328e6 0.104051
\(845\) 0 0
\(846\) 2.07052e7 0.994609
\(847\) − 3.36743e6i − 0.161284i
\(848\) 5.20141e6i 0.248388i
\(849\) −5.87931e7 −2.79935
\(850\) 0 0
\(851\) −1.56891e7 −0.742633
\(852\) 1.15847e7i 0.546746i
\(853\) 4.68539e6i 0.220482i 0.993905 + 0.110241i \(0.0351623\pi\)
−0.993905 + 0.110241i \(0.964838\pi\)
\(854\) 4.32805e7 2.03071
\(855\) 0 0
\(856\) −1.53408e6 −0.0715589
\(857\) − 4.12846e7i − 1.92015i −0.279740 0.960076i \(-0.590248\pi\)
0.279740 0.960076i \(-0.409752\pi\)
\(858\) 1.57203e6i 0.0729026i
\(859\) 3.54805e6 0.164062 0.0820308 0.996630i \(-0.473859\pi\)
0.0820308 + 0.996630i \(0.473859\pi\)
\(860\) 0 0
\(861\) −6.96615e7 −3.20247
\(862\) 1.95913e7i 0.898039i
\(863\) 3.07605e7i 1.40594i 0.711219 + 0.702970i \(0.248143\pi\)
−0.711219 + 0.702970i \(0.751857\pi\)
\(864\) 1.05421e7 0.480443
\(865\) 0 0
\(866\) 2.69150e7 1.21955
\(867\) 3.35510e6i 0.151585i
\(868\) − 3.81616e6i − 0.171920i
\(869\) 8.76161e6 0.393581
\(870\) 0 0
\(871\) 5.38642e6 0.240577
\(872\) − 3.60486e6i − 0.160545i
\(873\) 3.08508e6i 0.137003i
\(874\) −4.75157e6 −0.210406
\(875\) 0 0
\(876\) 1.96216e7 0.863923
\(877\) 3.05535e7i 1.34141i 0.741723 + 0.670706i \(0.234009\pi\)
−0.741723 + 0.670706i \(0.765991\pi\)
\(878\) 1.34013e7i 0.586695i
\(879\) 2.08465e7 0.910040
\(880\) 0 0
\(881\) 4.21018e7 1.82751 0.913757 0.406262i \(-0.133168\pi\)
0.913757 + 0.406262i \(0.133168\pi\)
\(882\) 8.63345e7i 3.73691i
\(883\) − 57164.0i − 0.00246729i −0.999999 0.00123365i \(-0.999607\pi\)
0.999999 0.00123365i \(-0.000392682\pi\)
\(884\) 2.04646e6 0.0880792
\(885\) 0 0
\(886\) −2.84974e7 −1.21961
\(887\) − 1.16106e7i − 0.495504i −0.968824 0.247752i \(-0.920308\pi\)
0.968824 0.247752i \(-0.0796917\pi\)
\(888\) 1.50020e7i 0.638437i
\(889\) 2.01029e7 0.853109
\(890\) 0 0
\(891\) 1.85422e7 0.782467
\(892\) 7.50142e6i 0.315669i
\(893\) 5.29747e6i 0.222300i
\(894\) −6.46700e6 −0.270619
\(895\) 0 0
\(896\) −3.76832e6 −0.156811
\(897\) 6.30437e6i 0.261614i
\(898\) 1.08371e7i 0.448460i
\(899\) 1.23610e6 0.0510101
\(900\) 0 0
\(901\) 2.32032e7 0.952215
\(902\) − 5.05490e6i − 0.206869i
\(903\) − 386860.i − 0.0157883i
\(904\) 1.67618e7 0.682181
\(905\) 0 0
\(906\) −7.61447e6 −0.308191
\(907\) 2.09855e7i 0.847034i 0.905888 + 0.423517i \(0.139205\pi\)
−0.905888 + 0.423517i \(0.860795\pi\)
\(908\) − 4.40035e6i − 0.177122i
\(909\) 3.67327e7 1.47450
\(910\) 0 0
\(911\) 4.74125e7 1.89277 0.946383 0.323047i \(-0.104707\pi\)
0.946383 + 0.323047i \(0.104707\pi\)
\(912\) 4.54349e6i 0.180885i
\(913\) 1.91253e6i 0.0759330i
\(914\) −9.65445e6 −0.382263
\(915\) 0 0
\(916\) −1.02765e7 −0.404675
\(917\) − 6.09781e7i − 2.39470i
\(918\) − 4.70276e7i − 1.84181i
\(919\) −4.04326e7 −1.57922 −0.789610 0.613609i \(-0.789717\pi\)
−0.789610 + 0.613609i \(0.789717\pi\)
\(920\) 0 0
\(921\) 5.74612e7 2.23216
\(922\) − 2.62426e7i − 1.01667i
\(923\) 2.79630e6i 0.108039i
\(924\) −1.29131e7 −0.497567
\(925\) 0 0
\(926\) −1.88968e7 −0.724205
\(927\) 1.11166e8i 4.24885i
\(928\) − 1.22061e6i − 0.0465271i
\(929\) −3.30757e7 −1.25739 −0.628694 0.777652i \(-0.716410\pi\)
−0.628694 + 0.777652i \(0.716410\pi\)
\(930\) 0 0
\(931\) −2.20889e7 −0.835219
\(932\) − 2.40776e7i − 0.907973i
\(933\) − 4.37987e7i − 1.64724i
\(934\) 9.13776e6 0.342746
\(935\) 0 0
\(936\) 4.28646e6 0.159922
\(937\) − 3.15132e7i − 1.17258i −0.810100 0.586292i \(-0.800587\pi\)
0.810100 0.586292i \(-0.199413\pi\)
\(938\) 4.42456e7i 1.64196i
\(939\) 5.80267e7 2.14765
\(940\) 0 0
\(941\) −9.54147e6 −0.351270 −0.175635 0.984455i \(-0.556198\pi\)
−0.175635 + 0.984455i \(0.556198\pi\)
\(942\) 3.19426e7i 1.17285i
\(943\) − 2.02718e7i − 0.742358i
\(944\) 5.46586e6 0.199631
\(945\) 0 0
\(946\) 28072.0 0.00101987
\(947\) 2.24208e7i 0.812410i 0.913782 + 0.406205i \(0.133148\pi\)
−0.913782 + 0.406205i \(0.866852\pi\)
\(948\) − 3.35982e7i − 1.21422i
\(949\) 4.73626e6 0.170714
\(950\) 0 0
\(951\) 7.52060e6 0.269650
\(952\) 1.68102e7i 0.601148i
\(953\) − 1.68985e7i − 0.602720i −0.953510 0.301360i \(-0.902559\pi\)
0.953510 0.301360i \(-0.0974405\pi\)
\(954\) 4.86007e7 1.72891
\(955\) 0 0
\(956\) −4.87510e6 −0.172520
\(957\) − 4.18273e6i − 0.147632i
\(958\) − 3.80618e6i − 0.133991i
\(959\) 5.65471e7 1.98547
\(960\) 0 0
\(961\) −2.75538e7 −0.962438
\(962\) 3.62118e6i 0.126157i
\(963\) 1.43341e7i 0.498085i
\(964\) −2.03489e7 −0.705260
\(965\) 0 0
\(966\) −5.17859e7 −1.78554
\(967\) − 3.06946e7i − 1.05559i −0.849371 0.527796i \(-0.823018\pi\)
0.849371 0.527796i \(-0.176982\pi\)
\(968\) − 937024.i − 0.0321412i
\(969\) 2.02682e7 0.693436
\(970\) 0 0
\(971\) 3.35664e7 1.14250 0.571251 0.820776i \(-0.306458\pi\)
0.571251 + 0.820776i \(0.306458\pi\)
\(972\) − 3.10769e7i − 1.05505i
\(973\) − 8.36266e7i − 2.83180i
\(974\) −1.40594e7 −0.474863
\(975\) 0 0
\(976\) 1.20433e7 0.404687
\(977\) 2.47897e7i 0.830873i 0.909622 + 0.415436i \(0.136371\pi\)
−0.909622 + 0.415436i \(0.863629\pi\)
\(978\) 3.38650e7i 1.13215i
\(979\) 1.38861e7 0.463045
\(980\) 0 0
\(981\) −3.36829e7 −1.11747
\(982\) − 2.31488e7i − 0.766036i
\(983\) − 5.22606e6i − 0.172501i −0.996274 0.0862503i \(-0.972512\pi\)
0.996274 0.0862503i \(-0.0274885\pi\)
\(984\) −1.93841e7 −0.638200
\(985\) 0 0
\(986\) −5.44506e6 −0.178365
\(987\) 5.77355e7i 1.88647i
\(988\) 1.09670e6i 0.0357435i
\(989\) 112578. 0.00365985
\(990\) 0 0
\(991\) 2.40826e7 0.778967 0.389484 0.921033i \(-0.372653\pi\)
0.389484 + 0.921033i \(0.372653\pi\)
\(992\) − 1.06189e6i − 0.0342610i
\(993\) 1.48489e6i 0.0477882i
\(994\) −2.29696e7 −0.737375
\(995\) 0 0
\(996\) 7.33398e6 0.234256
\(997\) − 1.32606e7i − 0.422499i −0.977432 0.211249i \(-0.932247\pi\)
0.977432 0.211249i \(-0.0677532\pi\)
\(998\) − 4.11650e6i − 0.130828i
\(999\) 8.32145e7 2.63806
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.a.199.2 2
5.2 odd 4 550.6.a.c.1.1 1
5.3 odd 4 22.6.a.c.1.1 1
5.4 even 2 inner 550.6.b.a.199.1 2
15.8 even 4 198.6.a.b.1.1 1
20.3 even 4 176.6.a.e.1.1 1
35.13 even 4 1078.6.a.f.1.1 1
40.3 even 4 704.6.a.a.1.1 1
40.13 odd 4 704.6.a.j.1.1 1
55.43 even 4 242.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.c.1.1 1 5.3 odd 4
176.6.a.e.1.1 1 20.3 even 4
198.6.a.b.1.1 1 15.8 even 4
242.6.a.a.1.1 1 55.43 even 4
550.6.a.c.1.1 1 5.2 odd 4
550.6.b.a.199.1 2 5.4 even 2 inner
550.6.b.a.199.2 2 1.1 even 1 trivial
704.6.a.a.1.1 1 40.3 even 4
704.6.a.j.1.1 1 40.13 odd 4
1078.6.a.f.1.1 1 35.13 even 4