Properties

Label 1450.2.b.i.349.2
Level $1450$
Weight $2$
Character 1450.349
Analytic conductor $11.578$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1450,2,Mod(349,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1450.349
Dual form 1450.2.b.i.349.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.44949i q^{3} -1.00000 q^{4} +2.44949 q^{6} +0.449490i q^{7} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.44949i q^{3} -1.00000 q^{4} +2.44949 q^{6} +0.449490i q^{7} +1.00000i q^{8} -3.00000 q^{9} +2.00000 q^{11} -2.44949i q^{12} +2.44949i q^{13} +0.449490 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} -1.55051 q^{19} -1.10102 q^{21} -2.00000i q^{22} +2.44949i q^{23} -2.44949 q^{24} +2.44949 q^{26} -0.449490i q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +4.89898i q^{33} +2.00000 q^{34} +3.00000 q^{36} -1.44949i q^{37} +1.55051i q^{38} -6.00000 q^{39} -3.34847 q^{41} +1.10102i q^{42} +0.898979i q^{43} -2.00000 q^{44} +2.44949 q^{46} +3.89898i q^{47} +2.44949i q^{48} +6.79796 q^{49} -4.89898 q^{51} -2.44949i q^{52} +5.55051i q^{53} -0.449490 q^{56} -3.79796i q^{57} -1.00000i q^{58} -13.4495 q^{59} -1.44949 q^{61} +3.00000i q^{62} -1.34847i q^{63} -1.00000 q^{64} +4.89898 q^{66} +8.55051i q^{67} -2.00000i q^{68} -6.00000 q^{69} -3.34847 q^{71} -3.00000i q^{72} -1.34847i q^{73} -1.44949 q^{74} +1.55051 q^{76} +0.898979i q^{77} +6.00000i q^{78} -6.89898 q^{79} -9.00000 q^{81} +3.34847i q^{82} -6.00000i q^{83} +1.10102 q^{84} +0.898979 q^{86} +2.44949i q^{87} +2.00000i q^{88} -8.44949 q^{89} -1.10102 q^{91} -2.44949i q^{92} -7.34847i q^{93} +3.89898 q^{94} +2.44949 q^{96} +7.34847i q^{97} -6.79796i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 12 q^{9} + 8 q^{11} - 8 q^{14} + 4 q^{16} - 16 q^{19} - 24 q^{21} + 4 q^{29} - 12 q^{31} + 8 q^{34} + 12 q^{36} - 24 q^{39} + 16 q^{41} - 8 q^{44} - 12 q^{49} + 8 q^{56} - 44 q^{59} + 4 q^{61} - 4 q^{64} - 24 q^{69} + 16 q^{71} + 4 q^{74} + 16 q^{76} - 8 q^{79} - 36 q^{81} + 24 q^{84} - 16 q^{86} - 24 q^{89} - 24 q^{91} - 4 q^{94} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 2.44949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.44949 1.00000
\(7\) 0.449490i 0.169891i 0.996386 + 0.0849456i \(0.0270716\pi\)
−0.996386 + 0.0849456i \(0.972928\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) − 2.44949i − 0.707107i
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0.449490 0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 3.00000i 0.707107i
\(19\) −1.55051 −0.355711 −0.177856 0.984057i \(-0.556916\pi\)
−0.177856 + 0.984057i \(0.556916\pi\)
\(20\) 0 0
\(21\) −1.10102 −0.240262
\(22\) − 2.00000i − 0.426401i
\(23\) 2.44949i 0.510754i 0.966842 + 0.255377i \(0.0821996\pi\)
−0.966842 + 0.255377i \(0.917800\pi\)
\(24\) −2.44949 −0.500000
\(25\) 0 0
\(26\) 2.44949 0.480384
\(27\) 0 0
\(28\) − 0.449490i − 0.0849456i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.89898i 0.852803i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) − 1.44949i − 0.238295i −0.992877 0.119147i \(-0.961984\pi\)
0.992877 0.119147i \(-0.0380161\pi\)
\(38\) 1.55051i 0.251526i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −3.34847 −0.522943 −0.261472 0.965211i \(-0.584208\pi\)
−0.261472 + 0.965211i \(0.584208\pi\)
\(42\) 1.10102i 0.169891i
\(43\) 0.898979i 0.137093i 0.997648 + 0.0685465i \(0.0218362\pi\)
−0.997648 + 0.0685465i \(0.978164\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.44949 0.361158
\(47\) 3.89898i 0.568725i 0.958717 + 0.284362i \(0.0917819\pi\)
−0.958717 + 0.284362i \(0.908218\pi\)
\(48\) 2.44949i 0.353553i
\(49\) 6.79796 0.971137
\(50\) 0 0
\(51\) −4.89898 −0.685994
\(52\) − 2.44949i − 0.339683i
\(53\) 5.55051i 0.762421i 0.924488 + 0.381211i \(0.124493\pi\)
−0.924488 + 0.381211i \(0.875507\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.449490 −0.0600656
\(57\) − 3.79796i − 0.503052i
\(58\) − 1.00000i − 0.131306i
\(59\) −13.4495 −1.75097 −0.875487 0.483241i \(-0.839459\pi\)
−0.875487 + 0.483241i \(0.839459\pi\)
\(60\) 0 0
\(61\) −1.44949 −0.185588 −0.0927941 0.995685i \(-0.529580\pi\)
−0.0927941 + 0.995685i \(0.529580\pi\)
\(62\) 3.00000i 0.381000i
\(63\) − 1.34847i − 0.169891i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.89898 0.603023
\(67\) 8.55051i 1.04461i 0.852758 + 0.522306i \(0.174928\pi\)
−0.852758 + 0.522306i \(0.825072\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −3.34847 −0.397390 −0.198695 0.980061i \(-0.563670\pi\)
−0.198695 + 0.980061i \(0.563670\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 1.34847i − 0.157826i −0.996881 0.0789132i \(-0.974855\pi\)
0.996881 0.0789132i \(-0.0251450\pi\)
\(74\) −1.44949 −0.168500
\(75\) 0 0
\(76\) 1.55051 0.177856
\(77\) 0.898979i 0.102448i
\(78\) 6.00000i 0.679366i
\(79\) −6.89898 −0.776196 −0.388098 0.921618i \(-0.626868\pi\)
−0.388098 + 0.921618i \(0.626868\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 3.34847i 0.369777i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 1.10102 0.120131
\(85\) 0 0
\(86\) 0.898979 0.0969395
\(87\) 2.44949i 0.262613i
\(88\) 2.00000i 0.213201i
\(89\) −8.44949 −0.895644 −0.447822 0.894123i \(-0.647800\pi\)
−0.447822 + 0.894123i \(0.647800\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) − 2.44949i − 0.255377i
\(93\) − 7.34847i − 0.762001i
\(94\) 3.89898 0.402149
\(95\) 0 0
\(96\) 2.44949 0.250000
\(97\) 7.34847i 0.746124i 0.927806 + 0.373062i \(0.121692\pi\)
−0.927806 + 0.373062i \(0.878308\pi\)
\(98\) − 6.79796i − 0.686698i
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 8.55051 0.850808 0.425404 0.905004i \(-0.360132\pi\)
0.425404 + 0.905004i \(0.360132\pi\)
\(102\) 4.89898i 0.485071i
\(103\) 2.44949i 0.241355i 0.992692 + 0.120678i \(0.0385068\pi\)
−0.992692 + 0.120678i \(0.961493\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) 5.55051 0.539113
\(107\) 19.2474i 1.86072i 0.366646 + 0.930361i \(0.380506\pi\)
−0.366646 + 0.930361i \(0.619494\pi\)
\(108\) 0 0
\(109\) −5.34847 −0.512290 −0.256145 0.966638i \(-0.582453\pi\)
−0.256145 + 0.966638i \(0.582453\pi\)
\(110\) 0 0
\(111\) 3.55051 0.337000
\(112\) 0.449490i 0.0424728i
\(113\) − 18.2474i − 1.71658i −0.513169 0.858288i \(-0.671528\pi\)
0.513169 0.858288i \(-0.328472\pi\)
\(114\) −3.79796 −0.355711
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 7.34847i − 0.679366i
\(118\) 13.4495i 1.23813i
\(119\) −0.898979 −0.0824093
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 1.44949i 0.131231i
\(123\) − 8.20204i − 0.739553i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −1.34847 −0.120131
\(127\) 10.7980i 0.958164i 0.877770 + 0.479082i \(0.159030\pi\)
−0.877770 + 0.479082i \(0.840970\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.20204 −0.193879
\(130\) 0 0
\(131\) 7.34847 0.642039 0.321019 0.947073i \(-0.395975\pi\)
0.321019 + 0.947073i \(0.395975\pi\)
\(132\) − 4.89898i − 0.426401i
\(133\) − 0.696938i − 0.0604322i
\(134\) 8.55051 0.738652
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 8.89898i 0.760291i 0.924927 + 0.380146i \(0.124126\pi\)
−0.924927 + 0.380146i \(0.875874\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 6.55051 0.555607 0.277804 0.960638i \(-0.410394\pi\)
0.277804 + 0.960638i \(0.410394\pi\)
\(140\) 0 0
\(141\) −9.55051 −0.804298
\(142\) 3.34847i 0.280997i
\(143\) 4.89898i 0.409673i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −1.34847 −0.111600
\(147\) 16.6515i 1.37340i
\(148\) 1.44949i 0.119147i
\(149\) 3.79796 0.311141 0.155570 0.987825i \(-0.450278\pi\)
0.155570 + 0.987825i \(0.450278\pi\)
\(150\) 0 0
\(151\) −3.34847 −0.272495 −0.136247 0.990675i \(-0.543504\pi\)
−0.136247 + 0.990675i \(0.543504\pi\)
\(152\) − 1.55051i − 0.125763i
\(153\) − 6.00000i − 0.485071i
\(154\) 0.898979 0.0724418
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 2.34847i 0.187428i 0.995599 + 0.0937141i \(0.0298740\pi\)
−0.995599 + 0.0937141i \(0.970126\pi\)
\(158\) 6.89898i 0.548853i
\(159\) −13.5959 −1.07823
\(160\) 0 0
\(161\) −1.10102 −0.0867726
\(162\) 9.00000i 0.707107i
\(163\) − 7.55051i − 0.591402i −0.955281 0.295701i \(-0.904447\pi\)
0.955281 0.295701i \(-0.0955531\pi\)
\(164\) 3.34847 0.261472
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) − 1.79796i − 0.139130i −0.997577 0.0695651i \(-0.977839\pi\)
0.997577 0.0695651i \(-0.0221612\pi\)
\(168\) − 1.10102i − 0.0849456i
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 4.65153 0.355711
\(172\) − 0.898979i − 0.0685465i
\(173\) − 16.0000i − 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(174\) 2.44949 0.185695
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) − 32.9444i − 2.47625i
\(178\) 8.44949i 0.633316i
\(179\) −7.24745 −0.541700 −0.270850 0.962622i \(-0.587305\pi\)
−0.270850 + 0.962622i \(0.587305\pi\)
\(180\) 0 0
\(181\) 18.0454 1.34130 0.670652 0.741772i \(-0.266014\pi\)
0.670652 + 0.741772i \(0.266014\pi\)
\(182\) 1.10102i 0.0816131i
\(183\) − 3.55051i − 0.262461i
\(184\) −2.44949 −0.180579
\(185\) 0 0
\(186\) −7.34847 −0.538816
\(187\) 4.00000i 0.292509i
\(188\) − 3.89898i − 0.284362i
\(189\) 0 0
\(190\) 0 0
\(191\) 17.6969 1.28051 0.640253 0.768164i \(-0.278830\pi\)
0.640253 + 0.768164i \(0.278830\pi\)
\(192\) − 2.44949i − 0.176777i
\(193\) 9.34847i 0.672918i 0.941698 + 0.336459i \(0.109229\pi\)
−0.941698 + 0.336459i \(0.890771\pi\)
\(194\) 7.34847 0.527589
\(195\) 0 0
\(196\) −6.79796 −0.485568
\(197\) − 14.8990i − 1.06151i −0.847526 0.530754i \(-0.821909\pi\)
0.847526 0.530754i \(-0.178091\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 15.3485 1.08802 0.544012 0.839077i \(-0.316905\pi\)
0.544012 + 0.839077i \(0.316905\pi\)
\(200\) 0 0
\(201\) −20.9444 −1.47730
\(202\) − 8.55051i − 0.601612i
\(203\) 0.449490i 0.0315480i
\(204\) 4.89898 0.342997
\(205\) 0 0
\(206\) 2.44949 0.170664
\(207\) − 7.34847i − 0.510754i
\(208\) 2.44949i 0.169842i
\(209\) −3.10102 −0.214502
\(210\) 0 0
\(211\) 26.4949 1.82398 0.911992 0.410208i \(-0.134544\pi\)
0.911992 + 0.410208i \(0.134544\pi\)
\(212\) − 5.55051i − 0.381211i
\(213\) − 8.20204i − 0.561995i
\(214\) 19.2474 1.31573
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.34847i − 0.0915401i
\(218\) 5.34847i 0.362244i
\(219\) 3.30306 0.223200
\(220\) 0 0
\(221\) −4.89898 −0.329541
\(222\) − 3.55051i − 0.238295i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0.449490 0.0300328
\(225\) 0 0
\(226\) −18.2474 −1.21380
\(227\) 9.24745i 0.613775i 0.951746 + 0.306887i \(0.0992875\pi\)
−0.951746 + 0.306887i \(0.900713\pi\)
\(228\) 3.79796i 0.251526i
\(229\) 26.8990 1.77753 0.888767 0.458359i \(-0.151562\pi\)
0.888767 + 0.458359i \(0.151562\pi\)
\(230\) 0 0
\(231\) −2.20204 −0.144884
\(232\) 1.00000i 0.0656532i
\(233\) − 11.0000i − 0.720634i −0.932830 0.360317i \(-0.882669\pi\)
0.932830 0.360317i \(-0.117331\pi\)
\(234\) −7.34847 −0.480384
\(235\) 0 0
\(236\) 13.4495 0.875487
\(237\) − 16.8990i − 1.09771i
\(238\) 0.898979i 0.0582722i
\(239\) 17.5959 1.13819 0.569093 0.822273i \(-0.307295\pi\)
0.569093 + 0.822273i \(0.307295\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 7.00000i 0.449977i
\(243\) − 22.0454i − 1.41421i
\(244\) 1.44949 0.0927941
\(245\) 0 0
\(246\) −8.20204 −0.522943
\(247\) − 3.79796i − 0.241658i
\(248\) − 3.00000i − 0.190500i
\(249\) 14.6969 0.931381
\(250\) 0 0
\(251\) 31.1464 1.96595 0.982973 0.183752i \(-0.0588245\pi\)
0.982973 + 0.183752i \(0.0588245\pi\)
\(252\) 1.34847i 0.0849456i
\(253\) 4.89898i 0.307996i
\(254\) 10.7980 0.677524
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.69694i 0.168230i 0.996456 + 0.0841152i \(0.0268064\pi\)
−0.996456 + 0.0841152i \(0.973194\pi\)
\(258\) 2.20204i 0.137093i
\(259\) 0.651531 0.0404842
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) − 7.34847i − 0.453990i
\(263\) 2.79796i 0.172530i 0.996272 + 0.0862648i \(0.0274931\pi\)
−0.996272 + 0.0862648i \(0.972507\pi\)
\(264\) −4.89898 −0.301511
\(265\) 0 0
\(266\) −0.696938 −0.0427320
\(267\) − 20.6969i − 1.26663i
\(268\) − 8.55051i − 0.522306i
\(269\) −9.65153 −0.588464 −0.294232 0.955734i \(-0.595064\pi\)
−0.294232 + 0.955734i \(0.595064\pi\)
\(270\) 0 0
\(271\) −13.6969 −0.832030 −0.416015 0.909358i \(-0.636574\pi\)
−0.416015 + 0.909358i \(0.636574\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 2.69694i − 0.163226i
\(274\) 8.89898 0.537607
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 30.9444i − 1.85927i −0.368484 0.929634i \(-0.620123\pi\)
0.368484 0.929634i \(-0.379877\pi\)
\(278\) − 6.55051i − 0.392873i
\(279\) 9.00000 0.538816
\(280\) 0 0
\(281\) −9.20204 −0.548948 −0.274474 0.961595i \(-0.588504\pi\)
−0.274474 + 0.961595i \(0.588504\pi\)
\(282\) 9.55051i 0.568725i
\(283\) 24.0000i 1.42665i 0.700832 + 0.713326i \(0.252812\pi\)
−0.700832 + 0.713326i \(0.747188\pi\)
\(284\) 3.34847 0.198695
\(285\) 0 0
\(286\) 4.89898 0.289683
\(287\) − 1.50510i − 0.0888434i
\(288\) 3.00000i 0.176777i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 1.34847i 0.0789132i
\(293\) − 2.55051i − 0.149002i −0.997221 0.0745012i \(-0.976264\pi\)
0.997221 0.0745012i \(-0.0237365\pi\)
\(294\) 16.6515 0.971137
\(295\) 0 0
\(296\) 1.44949 0.0842499
\(297\) 0 0
\(298\) − 3.79796i − 0.220010i
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −0.404082 −0.0232909
\(302\) 3.34847i 0.192683i
\(303\) 20.9444i 1.20322i
\(304\) −1.55051 −0.0889279
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 33.3939i 1.90589i 0.303144 + 0.952945i \(0.401964\pi\)
−0.303144 + 0.952945i \(0.598036\pi\)
\(308\) − 0.898979i − 0.0512241i
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 19.5959 1.11118 0.555591 0.831456i \(-0.312492\pi\)
0.555591 + 0.831456i \(0.312492\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 6.59592i 0.372823i 0.982472 + 0.186412i \(0.0596858\pi\)
−0.982472 + 0.186412i \(0.940314\pi\)
\(314\) 2.34847 0.132532
\(315\) 0 0
\(316\) 6.89898 0.388098
\(317\) − 28.3485i − 1.59221i −0.605159 0.796104i \(-0.706891\pi\)
0.605159 0.796104i \(-0.293109\pi\)
\(318\) 13.5959i 0.762421i
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −47.1464 −2.63146
\(322\) 1.10102i 0.0613575i
\(323\) − 3.10102i − 0.172545i
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −7.55051 −0.418184
\(327\) − 13.1010i − 0.724488i
\(328\) − 3.34847i − 0.184888i
\(329\) −1.75255 −0.0966213
\(330\) 0 0
\(331\) −5.75255 −0.316189 −0.158094 0.987424i \(-0.550535\pi\)
−0.158094 + 0.987424i \(0.550535\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 4.34847i 0.238295i
\(334\) −1.79796 −0.0983799
\(335\) 0 0
\(336\) −1.10102 −0.0600656
\(337\) − 4.20204i − 0.228900i −0.993429 0.114450i \(-0.963489\pi\)
0.993429 0.114450i \(-0.0365105\pi\)
\(338\) − 7.00000i − 0.380750i
\(339\) 44.6969 2.42760
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) − 4.65153i − 0.251526i
\(343\) 6.20204i 0.334879i
\(344\) −0.898979 −0.0484697
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) − 19.0454i − 1.02241i −0.859458 0.511206i \(-0.829199\pi\)
0.859458 0.511206i \(-0.170801\pi\)
\(348\) − 2.44949i − 0.131306i
\(349\) −22.9444 −1.22818 −0.614092 0.789234i \(-0.710478\pi\)
−0.614092 + 0.789234i \(0.710478\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) 2.79796i 0.148920i 0.997224 + 0.0744602i \(0.0237234\pi\)
−0.997224 + 0.0744602i \(0.976277\pi\)
\(354\) −32.9444 −1.75097
\(355\) 0 0
\(356\) 8.44949 0.447822
\(357\) − 2.20204i − 0.116544i
\(358\) 7.24745i 0.383040i
\(359\) −22.5959 −1.19257 −0.596283 0.802774i \(-0.703357\pi\)
−0.596283 + 0.802774i \(0.703357\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) − 18.0454i − 0.948446i
\(363\) − 17.1464i − 0.899954i
\(364\) 1.10102 0.0577092
\(365\) 0 0
\(366\) −3.55051 −0.185588
\(367\) 20.7980i 1.08564i 0.839848 + 0.542822i \(0.182644\pi\)
−0.839848 + 0.542822i \(0.817356\pi\)
\(368\) 2.44949i 0.127688i
\(369\) 10.0454 0.522943
\(370\) 0 0
\(371\) −2.49490 −0.129529
\(372\) 7.34847i 0.381000i
\(373\) − 20.4949i − 1.06119i −0.847627 0.530593i \(-0.821969\pi\)
0.847627 0.530593i \(-0.178031\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −3.89898 −0.201075
\(377\) 2.44949i 0.126155i
\(378\) 0 0
\(379\) 9.14643 0.469820 0.234910 0.972017i \(-0.424520\pi\)
0.234910 + 0.972017i \(0.424520\pi\)
\(380\) 0 0
\(381\) −26.4495 −1.35505
\(382\) − 17.6969i − 0.905454i
\(383\) 7.79796i 0.398457i 0.979953 + 0.199229i \(0.0638436\pi\)
−0.979953 + 0.199229i \(0.936156\pi\)
\(384\) −2.44949 −0.125000
\(385\) 0 0
\(386\) 9.34847 0.475825
\(387\) − 2.69694i − 0.137093i
\(388\) − 7.34847i − 0.373062i
\(389\) 9.65153 0.489352 0.244676 0.969605i \(-0.421318\pi\)
0.244676 + 0.969605i \(0.421318\pi\)
\(390\) 0 0
\(391\) −4.89898 −0.247752
\(392\) 6.79796i 0.343349i
\(393\) 18.0000i 0.907980i
\(394\) −14.8990 −0.750600
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 21.1464i 1.06131i 0.847588 + 0.530654i \(0.178054\pi\)
−0.847588 + 0.530654i \(0.821946\pi\)
\(398\) − 15.3485i − 0.769349i
\(399\) 1.70714 0.0854641
\(400\) 0 0
\(401\) −0.595918 −0.0297587 −0.0148794 0.999889i \(-0.504736\pi\)
−0.0148794 + 0.999889i \(0.504736\pi\)
\(402\) 20.9444i 1.04461i
\(403\) − 7.34847i − 0.366053i
\(404\) −8.55051 −0.425404
\(405\) 0 0
\(406\) 0.449490 0.0223078
\(407\) − 2.89898i − 0.143697i
\(408\) − 4.89898i − 0.242536i
\(409\) −13.7980 −0.682265 −0.341133 0.940015i \(-0.610811\pi\)
−0.341133 + 0.940015i \(0.610811\pi\)
\(410\) 0 0
\(411\) −21.7980 −1.07521
\(412\) − 2.44949i − 0.120678i
\(413\) − 6.04541i − 0.297475i
\(414\) −7.34847 −0.361158
\(415\) 0 0
\(416\) 2.44949 0.120096
\(417\) 16.0454i 0.785747i
\(418\) 3.10102i 0.151676i
\(419\) −24.1464 −1.17963 −0.589815 0.807538i \(-0.700799\pi\)
−0.589815 + 0.807538i \(0.700799\pi\)
\(420\) 0 0
\(421\) 8.55051 0.416726 0.208363 0.978052i \(-0.433186\pi\)
0.208363 + 0.978052i \(0.433186\pi\)
\(422\) − 26.4949i − 1.28975i
\(423\) − 11.6969i − 0.568725i
\(424\) −5.55051 −0.269557
\(425\) 0 0
\(426\) −8.20204 −0.397390
\(427\) − 0.651531i − 0.0315298i
\(428\) − 19.2474i − 0.930361i
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 13.5505 0.652705 0.326353 0.945248i \(-0.394180\pi\)
0.326353 + 0.945248i \(0.394180\pi\)
\(432\) 0 0
\(433\) 25.5505i 1.22788i 0.789353 + 0.613940i \(0.210416\pi\)
−0.789353 + 0.613940i \(0.789584\pi\)
\(434\) −1.34847 −0.0647286
\(435\) 0 0
\(436\) 5.34847 0.256145
\(437\) − 3.79796i − 0.181681i
\(438\) − 3.30306i − 0.157826i
\(439\) 10.6969 0.510537 0.255269 0.966870i \(-0.417836\pi\)
0.255269 + 0.966870i \(0.417836\pi\)
\(440\) 0 0
\(441\) −20.3939 −0.971137
\(442\) 4.89898i 0.233021i
\(443\) − 22.8990i − 1.08796i −0.839097 0.543982i \(-0.816916\pi\)
0.839097 0.543982i \(-0.183084\pi\)
\(444\) −3.55051 −0.168500
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 9.30306i 0.440020i
\(448\) − 0.449490i − 0.0212364i
\(449\) −6.20204 −0.292692 −0.146346 0.989233i \(-0.546751\pi\)
−0.146346 + 0.989233i \(0.546751\pi\)
\(450\) 0 0
\(451\) −6.69694 −0.315347
\(452\) 18.2474i 0.858288i
\(453\) − 8.20204i − 0.385366i
\(454\) 9.24745 0.434004
\(455\) 0 0
\(456\) 3.79796 0.177856
\(457\) − 1.79796i − 0.0841050i −0.999115 0.0420525i \(-0.986610\pi\)
0.999115 0.0420525i \(-0.0133897\pi\)
\(458\) − 26.8990i − 1.25691i
\(459\) 0 0
\(460\) 0 0
\(461\) 40.2929 1.87663 0.938313 0.345788i \(-0.112388\pi\)
0.938313 + 0.345788i \(0.112388\pi\)
\(462\) 2.20204i 0.102448i
\(463\) − 12.8990i − 0.599466i −0.954023 0.299733i \(-0.903102\pi\)
0.954023 0.299733i \(-0.0968977\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −11.0000 −0.509565
\(467\) − 30.9444i − 1.43194i −0.698133 0.715968i \(-0.745986\pi\)
0.698133 0.715968i \(-0.254014\pi\)
\(468\) 7.34847i 0.339683i
\(469\) −3.84337 −0.177470
\(470\) 0 0
\(471\) −5.75255 −0.265064
\(472\) − 13.4495i − 0.619063i
\(473\) 1.79796i 0.0826702i
\(474\) −16.8990 −0.776196
\(475\) 0 0
\(476\) 0.898979 0.0412047
\(477\) − 16.6515i − 0.762421i
\(478\) − 17.5959i − 0.804819i
\(479\) 23.7980 1.08736 0.543678 0.839294i \(-0.317031\pi\)
0.543678 + 0.839294i \(0.317031\pi\)
\(480\) 0 0
\(481\) 3.55051 0.161889
\(482\) − 17.0000i − 0.774329i
\(483\) − 2.69694i − 0.122715i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −22.0454 −1.00000
\(487\) 18.8990i 0.856395i 0.903685 + 0.428197i \(0.140851\pi\)
−0.903685 + 0.428197i \(0.859149\pi\)
\(488\) − 1.44949i − 0.0656153i
\(489\) 18.4949 0.836368
\(490\) 0 0
\(491\) −22.4949 −1.01518 −0.507590 0.861599i \(-0.669464\pi\)
−0.507590 + 0.861599i \(0.669464\pi\)
\(492\) 8.20204i 0.369777i
\(493\) 2.00000i 0.0900755i
\(494\) −3.79796 −0.170878
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) − 1.50510i − 0.0675131i
\(498\) − 14.6969i − 0.658586i
\(499\) 9.65153 0.432062 0.216031 0.976387i \(-0.430689\pi\)
0.216031 + 0.976387i \(0.430689\pi\)
\(500\) 0 0
\(501\) 4.40408 0.196760
\(502\) − 31.1464i − 1.39013i
\(503\) − 14.7980i − 0.659808i −0.944014 0.329904i \(-0.892984\pi\)
0.944014 0.329904i \(-0.107016\pi\)
\(504\) 1.34847 0.0600656
\(505\) 0 0
\(506\) 4.89898 0.217786
\(507\) 17.1464i 0.761500i
\(508\) − 10.7980i − 0.479082i
\(509\) −41.3939 −1.83475 −0.917376 0.398022i \(-0.869697\pi\)
−0.917376 + 0.398022i \(0.869697\pi\)
\(510\) 0 0
\(511\) 0.606123 0.0268133
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 2.69694 0.118957
\(515\) 0 0
\(516\) 2.20204 0.0969395
\(517\) 7.79796i 0.342954i
\(518\) − 0.651531i − 0.0286266i
\(519\) 39.1918 1.72033
\(520\) 0 0
\(521\) −19.8990 −0.871790 −0.435895 0.899997i \(-0.643568\pi\)
−0.435895 + 0.899997i \(0.643568\pi\)
\(522\) 3.00000i 0.131306i
\(523\) − 22.5505i − 0.986065i −0.870011 0.493032i \(-0.835888\pi\)
0.870011 0.493032i \(-0.164112\pi\)
\(524\) −7.34847 −0.321019
\(525\) 0 0
\(526\) 2.79796 0.121997
\(527\) − 6.00000i − 0.261364i
\(528\) 4.89898i 0.213201i
\(529\) 17.0000 0.739130
\(530\) 0 0
\(531\) 40.3485 1.75097
\(532\) 0.696938i 0.0302161i
\(533\) − 8.20204i − 0.355270i
\(534\) −20.6969 −0.895644
\(535\) 0 0
\(536\) −8.55051 −0.369326
\(537\) − 17.7526i − 0.766079i
\(538\) 9.65153i 0.416107i
\(539\) 13.5959 0.585618
\(540\) 0 0
\(541\) 23.0454 0.990799 0.495400 0.868665i \(-0.335022\pi\)
0.495400 + 0.868665i \(0.335022\pi\)
\(542\) 13.6969i 0.588334i
\(543\) 44.2020i 1.89689i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −2.69694 −0.115418
\(547\) 22.0000i 0.940652i 0.882493 + 0.470326i \(0.155864\pi\)
−0.882493 + 0.470326i \(0.844136\pi\)
\(548\) − 8.89898i − 0.380146i
\(549\) 4.34847 0.185588
\(550\) 0 0
\(551\) −1.55051 −0.0660540
\(552\) − 6.00000i − 0.255377i
\(553\) − 3.10102i − 0.131869i
\(554\) −30.9444 −1.31470
\(555\) 0 0
\(556\) −6.55051 −0.277804
\(557\) − 42.4949i − 1.80057i −0.435304 0.900283i \(-0.643359\pi\)
0.435304 0.900283i \(-0.356641\pi\)
\(558\) − 9.00000i − 0.381000i
\(559\) −2.20204 −0.0931364
\(560\) 0 0
\(561\) −9.79796 −0.413670
\(562\) 9.20204i 0.388165i
\(563\) − 2.20204i − 0.0928050i −0.998923 0.0464025i \(-0.985224\pi\)
0.998923 0.0464025i \(-0.0147757\pi\)
\(564\) 9.55051 0.402149
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) − 4.04541i − 0.169891i
\(568\) − 3.34847i − 0.140499i
\(569\) −20.6969 −0.867661 −0.433830 0.900995i \(-0.642838\pi\)
−0.433830 + 0.900995i \(0.642838\pi\)
\(570\) 0 0
\(571\) −19.0454 −0.797026 −0.398513 0.917163i \(-0.630474\pi\)
−0.398513 + 0.917163i \(0.630474\pi\)
\(572\) − 4.89898i − 0.204837i
\(573\) 43.3485i 1.81091i
\(574\) −1.50510 −0.0628218
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) 24.9444i 1.03845i 0.854638 + 0.519224i \(0.173779\pi\)
−0.854638 + 0.519224i \(0.826221\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −22.8990 −0.951649
\(580\) 0 0
\(581\) 2.69694 0.111888
\(582\) 18.0000i 0.746124i
\(583\) 11.1010i 0.459757i
\(584\) 1.34847 0.0558001
\(585\) 0 0
\(586\) −2.55051 −0.105361
\(587\) 39.2474i 1.61992i 0.586488 + 0.809958i \(0.300510\pi\)
−0.586488 + 0.809958i \(0.699490\pi\)
\(588\) − 16.6515i − 0.686698i
\(589\) 4.65153 0.191663
\(590\) 0 0
\(591\) 36.4949 1.50120
\(592\) − 1.44949i − 0.0595737i
\(593\) 40.3939i 1.65878i 0.558672 + 0.829389i \(0.311311\pi\)
−0.558672 + 0.829389i \(0.688689\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.79796 −0.155570
\(597\) 37.5959i 1.53870i
\(598\) 6.00000i 0.245358i
\(599\) 21.8990 0.894768 0.447384 0.894342i \(-0.352356\pi\)
0.447384 + 0.894342i \(0.352356\pi\)
\(600\) 0 0
\(601\) 22.6969 0.925827 0.462914 0.886403i \(-0.346804\pi\)
0.462914 + 0.886403i \(0.346804\pi\)
\(602\) 0.404082i 0.0164692i
\(603\) − 25.6515i − 1.04461i
\(604\) 3.34847 0.136247
\(605\) 0 0
\(606\) 20.9444 0.850808
\(607\) − 24.3939i − 0.990117i −0.868860 0.495058i \(-0.835147\pi\)
0.868860 0.495058i \(-0.164853\pi\)
\(608\) 1.55051i 0.0628815i
\(609\) −1.10102 −0.0446156
\(610\) 0 0
\(611\) −9.55051 −0.386372
\(612\) 6.00000i 0.242536i
\(613\) 37.1010i 1.49850i 0.662289 + 0.749248i \(0.269585\pi\)
−0.662289 + 0.749248i \(0.730415\pi\)
\(614\) 33.3939 1.34767
\(615\) 0 0
\(616\) −0.898979 −0.0362209
\(617\) − 9.55051i − 0.384489i −0.981347 0.192245i \(-0.938423\pi\)
0.981347 0.192245i \(-0.0615767\pi\)
\(618\) 6.00000i 0.241355i
\(619\) 6.89898 0.277293 0.138647 0.990342i \(-0.455725\pi\)
0.138647 + 0.990342i \(0.455725\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 19.5959i − 0.785725i
\(623\) − 3.79796i − 0.152162i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 6.59592 0.263626
\(627\) − 7.59592i − 0.303352i
\(628\) − 2.34847i − 0.0937141i
\(629\) 2.89898 0.115590
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) − 6.89898i − 0.274427i
\(633\) 64.8990i 2.57950i
\(634\) −28.3485 −1.12586
\(635\) 0 0
\(636\) 13.5959 0.539113
\(637\) 16.6515i 0.659758i
\(638\) − 2.00000i − 0.0791808i
\(639\) 10.0454 0.397390
\(640\) 0 0
\(641\) −0.247449 −0.00977364 −0.00488682 0.999988i \(-0.501556\pi\)
−0.00488682 + 0.999988i \(0.501556\pi\)
\(642\) 47.1464i 1.86072i
\(643\) 38.8434i 1.53183i 0.642940 + 0.765916i \(0.277714\pi\)
−0.642940 + 0.765916i \(0.722286\pi\)
\(644\) 1.10102 0.0433863
\(645\) 0 0
\(646\) −3.10102 −0.122008
\(647\) − 0.404082i − 0.0158861i −0.999968 0.00794305i \(-0.997472\pi\)
0.999968 0.00794305i \(-0.00252838\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) −26.8990 −1.05588
\(650\) 0 0
\(651\) 3.30306 0.129457
\(652\) 7.55051i 0.295701i
\(653\) − 13.2474i − 0.518413i −0.965822 0.259206i \(-0.916539\pi\)
0.965822 0.259206i \(-0.0834610\pi\)
\(654\) −13.1010 −0.512290
\(655\) 0 0
\(656\) −3.34847 −0.130736
\(657\) 4.04541i 0.157826i
\(658\) 1.75255i 0.0683216i
\(659\) 9.30306 0.362396 0.181198 0.983447i \(-0.442003\pi\)
0.181198 + 0.983447i \(0.442003\pi\)
\(660\) 0 0
\(661\) 34.2474 1.33207 0.666036 0.745920i \(-0.267990\pi\)
0.666036 + 0.745920i \(0.267990\pi\)
\(662\) 5.75255i 0.223579i
\(663\) − 12.0000i − 0.466041i
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.34847 0.168500
\(667\) 2.44949i 0.0948446i
\(668\) 1.79796i 0.0695651i
\(669\) −9.79796 −0.378811
\(670\) 0 0
\(671\) −2.89898 −0.111914
\(672\) 1.10102i 0.0424728i
\(673\) 2.79796i 0.107853i 0.998545 + 0.0539267i \(0.0171737\pi\)
−0.998545 + 0.0539267i \(0.982826\pi\)
\(674\) −4.20204 −0.161857
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) − 19.0454i − 0.731974i −0.930620 0.365987i \(-0.880731\pi\)
0.930620 0.365987i \(-0.119269\pi\)
\(678\) − 44.6969i − 1.71658i
\(679\) −3.30306 −0.126760
\(680\) 0 0
\(681\) −22.6515 −0.868009
\(682\) 6.00000i 0.229752i
\(683\) 20.8990i 0.799677i 0.916586 + 0.399839i \(0.130934\pi\)
−0.916586 + 0.399839i \(0.869066\pi\)
\(684\) −4.65153 −0.177856
\(685\) 0 0
\(686\) 6.20204 0.236795
\(687\) 65.8888i 2.51381i
\(688\) 0.898979i 0.0342733i
\(689\) −13.5959 −0.517963
\(690\) 0 0
\(691\) 9.94439 0.378302 0.189151 0.981948i \(-0.439426\pi\)
0.189151 + 0.981948i \(0.439426\pi\)
\(692\) 16.0000i 0.608229i
\(693\) − 2.69694i − 0.102448i
\(694\) −19.0454 −0.722954
\(695\) 0 0
\(696\) −2.44949 −0.0928477
\(697\) − 6.69694i − 0.253665i
\(698\) 22.9444i 0.868458i
\(699\) 26.9444 1.01913
\(700\) 0 0
\(701\) 33.3939 1.26127 0.630635 0.776080i \(-0.282795\pi\)
0.630635 + 0.776080i \(0.282795\pi\)
\(702\) 0 0
\(703\) 2.24745i 0.0847641i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 2.79796 0.105303
\(707\) 3.84337i 0.144545i
\(708\) 32.9444i 1.23813i
\(709\) −31.3939 −1.17902 −0.589511 0.807760i \(-0.700679\pi\)
−0.589511 + 0.807760i \(0.700679\pi\)
\(710\) 0 0
\(711\) 20.6969 0.776196
\(712\) − 8.44949i − 0.316658i
\(713\) − 7.34847i − 0.275202i
\(714\) −2.20204 −0.0824093
\(715\) 0 0
\(716\) 7.24745 0.270850
\(717\) 43.1010i 1.60964i
\(718\) 22.5959i 0.843272i
\(719\) −36.7423 −1.37026 −0.685129 0.728422i \(-0.740254\pi\)
−0.685129 + 0.728422i \(0.740254\pi\)
\(720\) 0 0
\(721\) −1.10102 −0.0410041
\(722\) 16.5959i 0.617636i
\(723\) 41.6413i 1.54866i
\(724\) −18.0454 −0.670652
\(725\) 0 0
\(726\) −17.1464 −0.636364
\(727\) − 5.59592i − 0.207541i −0.994601 0.103771i \(-0.966909\pi\)
0.994601 0.103771i \(-0.0330908\pi\)
\(728\) − 1.10102i − 0.0408065i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −1.79796 −0.0664999
\(732\) 3.55051i 0.131231i
\(733\) 10.8990i 0.402563i 0.979533 + 0.201281i \(0.0645106\pi\)
−0.979533 + 0.201281i \(0.935489\pi\)
\(734\) 20.7980 0.767667
\(735\) 0 0
\(736\) 2.44949 0.0902894
\(737\) 17.1010i 0.629924i
\(738\) − 10.0454i − 0.369777i
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 9.30306 0.341757
\(742\) 2.49490i 0.0915906i
\(743\) − 16.6969i − 0.612551i −0.951943 0.306276i \(-0.900917\pi\)
0.951943 0.306276i \(-0.0990829\pi\)
\(744\) 7.34847 0.269408
\(745\) 0 0
\(746\) −20.4949 −0.750372
\(747\) 18.0000i 0.658586i
\(748\) − 4.00000i − 0.146254i
\(749\) −8.65153 −0.316120
\(750\) 0 0
\(751\) 33.8990 1.23699 0.618496 0.785788i \(-0.287742\pi\)
0.618496 + 0.785788i \(0.287742\pi\)
\(752\) 3.89898i 0.142181i
\(753\) 76.2929i 2.78027i
\(754\) 2.44949 0.0892052
\(755\) 0 0
\(756\) 0 0
\(757\) − 39.3939i − 1.43179i −0.698205 0.715897i \(-0.746018\pi\)
0.698205 0.715897i \(-0.253982\pi\)
\(758\) − 9.14643i − 0.332213i
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 23.8990 0.866337 0.433169 0.901313i \(-0.357395\pi\)
0.433169 + 0.901313i \(0.357395\pi\)
\(762\) 26.4495i 0.958164i
\(763\) − 2.40408i − 0.0870336i
\(764\) −17.6969 −0.640253
\(765\) 0 0
\(766\) 7.79796 0.281752
\(767\) − 32.9444i − 1.18955i
\(768\) 2.44949i 0.0883883i
\(769\) 8.44949 0.304696 0.152348 0.988327i \(-0.451316\pi\)
0.152348 + 0.988327i \(0.451316\pi\)
\(770\) 0 0
\(771\) −6.60612 −0.237914
\(772\) − 9.34847i − 0.336459i
\(773\) − 12.5505i − 0.451410i −0.974196 0.225705i \(-0.927531\pi\)
0.974196 0.225705i \(-0.0724686\pi\)
\(774\) −2.69694 −0.0969395
\(775\) 0 0
\(776\) −7.34847 −0.263795
\(777\) 1.59592i 0.0572532i
\(778\) − 9.65153i − 0.346024i
\(779\) 5.19184 0.186017
\(780\) 0 0
\(781\) −6.69694 −0.239635
\(782\) 4.89898i 0.175187i
\(783\) 0 0
\(784\) 6.79796 0.242784
\(785\) 0 0
\(786\) 18.0000 0.642039
\(787\) 45.4495i 1.62010i 0.586361 + 0.810050i \(0.300560\pi\)
−0.586361 + 0.810050i \(0.699440\pi\)
\(788\) 14.8990i 0.530754i
\(789\) −6.85357 −0.243994
\(790\) 0 0
\(791\) 8.20204 0.291631
\(792\) − 6.00000i − 0.213201i
\(793\) − 3.55051i − 0.126082i
\(794\) 21.1464 0.750459
\(795\) 0 0
\(796\) −15.3485 −0.544012
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) − 1.70714i − 0.0604322i
\(799\) −7.79796 −0.275872
\(800\) 0 0
\(801\) 25.3485 0.895644
\(802\) 0.595918i 0.0210426i
\(803\) − 2.69694i − 0.0951729i
\(804\) 20.9444 0.738652
\(805\) 0 0
\(806\) −7.34847 −0.258839
\(807\) − 23.6413i − 0.832214i
\(808\) 8.55051i 0.300806i
\(809\) 7.59592 0.267058 0.133529 0.991045i \(-0.457369\pi\)
0.133529 + 0.991045i \(0.457369\pi\)
\(810\) 0 0
\(811\) −24.5505 −0.862085 −0.431043 0.902332i \(-0.641854\pi\)
−0.431043 + 0.902332i \(0.641854\pi\)
\(812\) − 0.449490i − 0.0157740i
\(813\) − 33.5505i − 1.17667i
\(814\) −2.89898 −0.101609
\(815\) 0 0
\(816\) −4.89898 −0.171499
\(817\) − 1.39388i − 0.0487656i
\(818\) 13.7980i 0.482434i
\(819\) 3.30306 0.115418
\(820\) 0 0
\(821\) 44.2474 1.54425 0.772123 0.635473i \(-0.219195\pi\)
0.772123 + 0.635473i \(0.219195\pi\)
\(822\) 21.7980i 0.760291i
\(823\) 7.79796i 0.271820i 0.990721 + 0.135910i \(0.0433958\pi\)
−0.990721 + 0.135910i \(0.956604\pi\)
\(824\) −2.44949 −0.0853320
\(825\) 0 0
\(826\) −6.04541 −0.210347
\(827\) − 48.6969i − 1.69336i −0.532104 0.846679i \(-0.678598\pi\)
0.532104 0.846679i \(-0.321402\pi\)
\(828\) 7.34847i 0.255377i
\(829\) 34.4949 1.19806 0.599029 0.800728i \(-0.295554\pi\)
0.599029 + 0.800728i \(0.295554\pi\)
\(830\) 0 0
\(831\) 75.7980 2.62940
\(832\) − 2.44949i − 0.0849208i
\(833\) 13.5959i 0.471071i
\(834\) 16.0454 0.555607
\(835\) 0 0
\(836\) 3.10102 0.107251
\(837\) 0 0
\(838\) 24.1464i 0.834125i
\(839\) 11.2020 0.386737 0.193369 0.981126i \(-0.438059\pi\)
0.193369 + 0.981126i \(0.438059\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 8.55051i − 0.294670i
\(843\) − 22.5403i − 0.776329i
\(844\) −26.4949 −0.911992
\(845\) 0 0
\(846\) −11.6969 −0.402149
\(847\) − 3.14643i − 0.108113i
\(848\) 5.55051i 0.190605i
\(849\) −58.7878 −2.01759
\(850\) 0 0
\(851\) 3.55051 0.121710
\(852\) 8.20204i 0.280997i
\(853\) 27.7980i 0.951784i 0.879504 + 0.475892i \(0.157875\pi\)
−0.879504 + 0.475892i \(0.842125\pi\)
\(854\) −0.651531 −0.0222949
\(855\) 0 0
\(856\) −19.2474 −0.657864
\(857\) − 19.3939i − 0.662482i −0.943546 0.331241i \(-0.892533\pi\)
0.943546 0.331241i \(-0.107467\pi\)
\(858\) 12.0000i 0.409673i
\(859\) −13.1010 −0.447001 −0.223501 0.974704i \(-0.571748\pi\)
−0.223501 + 0.974704i \(0.571748\pi\)
\(860\) 0 0
\(861\) 3.68673 0.125644
\(862\) − 13.5505i − 0.461532i
\(863\) 36.2474i 1.23388i 0.787011 + 0.616939i \(0.211627\pi\)
−0.787011 + 0.616939i \(0.788373\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25.5505 0.868242
\(867\) 31.8434i 1.08146i
\(868\) 1.34847i 0.0457700i
\(869\) −13.7980 −0.468064
\(870\) 0 0
\(871\) −20.9444 −0.709673
\(872\) − 5.34847i − 0.181122i
\(873\) − 22.0454i − 0.746124i
\(874\) −3.79796 −0.128468
\(875\) 0 0
\(876\) −3.30306 −0.111600
\(877\) − 8.00000i − 0.270141i −0.990836 0.135070i \(-0.956874\pi\)
0.990836 0.135070i \(-0.0431261\pi\)
\(878\) − 10.6969i − 0.361004i
\(879\) 6.24745 0.210721
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 20.3939i 0.686698i
\(883\) − 23.9444i − 0.805793i −0.915246 0.402896i \(-0.868003\pi\)
0.915246 0.402896i \(-0.131997\pi\)
\(884\) 4.89898 0.164771
\(885\) 0 0
\(886\) −22.8990 −0.769306
\(887\) 28.2020i 0.946932i 0.880812 + 0.473466i \(0.156997\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(888\) 3.55051i 0.119147i
\(889\) −4.85357 −0.162784
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) − 4.00000i − 0.133930i
\(893\) − 6.04541i − 0.202302i
\(894\) 9.30306 0.311141
\(895\) 0 0
\(896\) −0.449490 −0.0150164
\(897\) − 14.6969i − 0.490716i
\(898\) 6.20204i 0.206965i
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −11.1010 −0.369829
\(902\) 6.69694i 0.222984i
\(903\) − 0.989795i − 0.0329383i
\(904\) 18.2474 0.606901
\(905\) 0 0
\(906\) −8.20204 −0.272495
\(907\) 6.65153i 0.220860i 0.993884 + 0.110430i \(0.0352229\pi\)
−0.993884 + 0.110430i \(0.964777\pi\)
\(908\) − 9.24745i − 0.306887i
\(909\) −25.6515 −0.850808
\(910\) 0 0
\(911\) 2.50510 0.0829978 0.0414989 0.999139i \(-0.486787\pi\)
0.0414989 + 0.999139i \(0.486787\pi\)
\(912\) − 3.79796i − 0.125763i
\(913\) − 12.0000i − 0.397142i
\(914\) −1.79796 −0.0594712
\(915\) 0 0
\(916\) −26.8990 −0.888767
\(917\) 3.30306i 0.109077i
\(918\) 0 0
\(919\) 59.1464 1.95106 0.975530 0.219865i \(-0.0705617\pi\)
0.975530 + 0.219865i \(0.0705617\pi\)
\(920\) 0 0
\(921\) −81.7980 −2.69533
\(922\) − 40.2929i − 1.32697i
\(923\) − 8.20204i − 0.269973i
\(924\) 2.20204 0.0724418
\(925\) 0 0
\(926\) −12.8990 −0.423887
\(927\) − 7.34847i − 0.241355i
\(928\) − 1.00000i − 0.0328266i
\(929\) −9.49490 −0.311517 −0.155759 0.987795i \(-0.549782\pi\)
−0.155759 + 0.987795i \(0.549782\pi\)
\(930\) 0 0
\(931\) −10.5403 −0.345445
\(932\) 11.0000i 0.360317i
\(933\) 48.0000i 1.57145i
\(934\) −30.9444 −1.01253
\(935\) 0 0
\(936\) 7.34847 0.240192
\(937\) 15.2929i 0.499596i 0.968298 + 0.249798i \(0.0803642\pi\)
−0.968298 + 0.249798i \(0.919636\pi\)
\(938\) 3.84337i 0.125490i
\(939\) −16.1566 −0.527252
\(940\) 0 0
\(941\) 3.55051 0.115743 0.0578717 0.998324i \(-0.481569\pi\)
0.0578717 + 0.998324i \(0.481569\pi\)
\(942\) 5.75255i 0.187428i
\(943\) − 8.20204i − 0.267095i
\(944\) −13.4495 −0.437744
\(945\) 0 0
\(946\) 1.79796 0.0584567
\(947\) − 22.6515i − 0.736076i −0.929811 0.368038i \(-0.880030\pi\)
0.929811 0.368038i \(-0.119970\pi\)
\(948\) 16.8990i 0.548853i
\(949\) 3.30306 0.107222
\(950\) 0 0
\(951\) 69.4393 2.25172
\(952\) − 0.898979i − 0.0291361i
\(953\) − 34.2929i − 1.11085i −0.831565 0.555427i \(-0.812555\pi\)
0.831565 0.555427i \(-0.187445\pi\)
\(954\) −16.6515 −0.539113
\(955\) 0 0
\(956\) −17.5959 −0.569093
\(957\) 4.89898i 0.158362i
\(958\) − 23.7980i − 0.768877i
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 3.55051i − 0.114473i
\(963\) − 57.7423i − 1.86072i
\(964\) −17.0000 −0.547533
\(965\) 0 0
\(966\) −2.69694 −0.0867726
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) 7.59592 0.244016
\(970\) 0 0
\(971\) −30.9444 −0.993053 −0.496526 0.868022i \(-0.665391\pi\)
−0.496526 + 0.868022i \(0.665391\pi\)
\(972\) 22.0454i 0.707107i
\(973\) 2.94439i 0.0943927i
\(974\) 18.8990 0.605562
\(975\) 0 0
\(976\) −1.44949 −0.0463970
\(977\) − 9.89898i − 0.316696i −0.987383 0.158348i \(-0.949383\pi\)
0.987383 0.158348i \(-0.0506169\pi\)
\(978\) − 18.4949i − 0.591402i
\(979\) −16.8990 −0.540094
\(980\) 0 0
\(981\) 16.0454 0.512290
\(982\) 22.4949i 0.717841i
\(983\) 39.0000i 1.24391i 0.783054 + 0.621953i \(0.213661\pi\)
−0.783054 + 0.621953i \(0.786339\pi\)
\(984\) 8.20204 0.261472
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) − 4.29286i − 0.136643i
\(988\) 3.79796i 0.120829i
\(989\) −2.20204 −0.0700208
\(990\) 0 0
\(991\) −52.4949 −1.66756 −0.833778 0.552100i \(-0.813827\pi\)
−0.833778 + 0.552100i \(0.813827\pi\)
\(992\) 3.00000i 0.0952501i
\(993\) − 14.0908i − 0.447159i
\(994\) −1.50510 −0.0477390
\(995\) 0 0
\(996\) −14.6969 −0.465690
\(997\) 43.7423i 1.38533i 0.721258 + 0.692667i \(0.243564\pi\)
−0.721258 + 0.692667i \(0.756436\pi\)
\(998\) − 9.65153i − 0.305514i
\(999\) 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 1450.2.b.i.349.2 4
5.2 odd 4 1450.2.a.o.1.2 yes 2
5.3 odd 4 1450.2.a.j.1.1 2
5.4 even 2 inner 1450.2.b.i.349.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.j.1.1 2 5.3 odd 4
1450.2.a.o.1.2 yes 2 5.2 odd 4
1450.2.b.i.349.2 4 1.1 even 1 trivial
1450.2.b.i.349.3 4 5.4 even 2 inner