Properties

Label 1450.2.b.i.349.4
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 349.4
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1450.349
Dual form 1450.2.b.i.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.44949i q^{3} -1.00000 q^{4} -2.44949 q^{6} +4.44949i q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.44949i q^{3} -1.00000 q^{4} -2.44949 q^{6} +4.44949i q^{7} -1.00000i q^{8} -3.00000 q^{9} +2.00000 q^{11} -2.44949i q^{12} +2.44949i q^{13} -4.44949 q^{14} +1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} -6.44949 q^{19} -10.8990 q^{21} +2.00000i q^{22} +2.44949i q^{23} +2.44949 q^{24} -2.44949 q^{26} -4.44949i q^{28} +1.00000 q^{29} -3.00000 q^{31} +1.00000i q^{32} +4.89898i q^{33} +2.00000 q^{34} +3.00000 q^{36} -3.44949i q^{37} -6.44949i q^{38} -6.00000 q^{39} +11.3485 q^{41} -10.8990i q^{42} +8.89898i q^{43} -2.00000 q^{44} -2.44949 q^{46} +5.89898i q^{47} +2.44949i q^{48} -12.7980 q^{49} +4.89898 q^{51} -2.44949i q^{52} -10.4495i q^{53} +4.44949 q^{56} -15.7980i q^{57} +1.00000i q^{58} -8.55051 q^{59} +3.44949 q^{61} -3.00000i q^{62} -13.3485i q^{63} -1.00000 q^{64} -4.89898 q^{66} -13.4495i q^{67} +2.00000i q^{68} -6.00000 q^{69} +11.3485 q^{71} +3.00000i q^{72} -13.3485i q^{73} +3.44949 q^{74} +6.44949 q^{76} +8.89898i q^{77} -6.00000i q^{78} +2.89898 q^{79} -9.00000 q^{81} +11.3485i q^{82} +6.00000i q^{83} +10.8990 q^{84} -8.89898 q^{86} +2.44949i q^{87} -2.00000i q^{88} -3.55051 q^{89} -10.8990 q^{91} -2.44949i q^{92} -7.34847i q^{93} -5.89898 q^{94} -2.44949 q^{96} +7.34847i q^{97} -12.7980i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 12 q^{9}+O(q^{10}) \) 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
\(7\) 4.44949i 1.68175i 0.541230 + 0.840875i \(0.317959\pi\)
−0.541230 + 0.840875i \(0.682041\pi\)
\(8\) − 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 \(0.889680\pi\)
\(14\) −4.44949 −1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) −6.44949 −1.47961 −0.739807 0.672819i \(-0.765083\pi\)
−0.739807 + 0.672819i \(0.765083\pi\)
\(20\) 0 0
\(21\) −10.8990 −2.37835
\(22\) 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
\(28\) − 4.44949i − 0.840875i
\(29\) 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
\(37\) − 3.44949i − 0.567093i −0.958959 0.283546i \(-0.908489\pi\)
0.958959 0.283546i \(-0.0915110\pi\)
\(38\) − 6.44949i − 1.04625i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 11.3485 1.77233 0.886167 0.463367i \(-0.153359\pi\)
0.886167 + 0.463367i \(0.153359\pi\)
\(42\) − 10.8990i − 1.68175i
\(43\) 8.89898i 1.35708i 0.734563 + 0.678541i \(0.237387\pi\)
−0.734563 + 0.678541i \(0.762613\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −2.44949 −0.361158
\(47\) 5.89898i 0.860455i 0.902721 + 0.430227i \(0.141567\pi\)
−0.902721 + 0.430227i \(0.858433\pi\)
\(48\) 2.44949i 0.353553i
\(49\) −12.7980 −1.82828
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) − 2.44949i − 0.339683i
\(53\) − 10.4495i − 1.43535i −0.696379 0.717674i \(-0.745207\pi\)
0.696379 0.717674i \(-0.254793\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) − 15.7980i − 2.09249i
\(58\) 1.00000i 0.131306i
\(59\) −8.55051 −1.11318 −0.556591 0.830787i \(-0.687891\pi\)
−0.556591 + 0.830787i \(0.687891\pi\)
\(60\) 0 0
\(61\) 3.44949 0.441662 0.220831 0.975312i \(-0.429123\pi\)
0.220831 + 0.975312i \(0.429123\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) − 13.3485i − 1.68175i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.89898 −0.603023
\(67\) − 13.4495i − 1.64312i −0.570124 0.821558i \(-0.693105\pi\)
0.570124 0.821558i \(-0.306895\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 11.3485 1.34682 0.673408 0.739271i \(-0.264830\pi\)
0.673408 + 0.739271i \(0.264830\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 13.3485i − 1.56232i −0.624331 0.781160i \(-0.714628\pi\)
0.624331 0.781160i \(-0.285372\pi\)
\(74\) 3.44949 0.400995
\(75\) 0 0
\(76\) 6.44949 0.739807
\(77\) 8.89898i 1.01413i
\(78\) − 6.00000i − 0.679366i
\(79\) 2.89898 0.326161 0.163080 0.986613i \(-0.447857\pi\)
0.163080 + 0.986613i \(0.447857\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 11.3485i 1.25323i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 10.8990 1.18918
\(85\) 0 0
\(86\) −8.89898 −0.959602
\(87\) 2.44949i 0.262613i
\(88\) − 2.00000i − 0.213201i
\(89\) −3.55051 −0.376353 −0.188177 0.982135i \(-0.560258\pi\)
−0.188177 + 0.982135i \(0.560258\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) − 2.44949i − 0.255377i
\(93\) − 7.34847i − 0.762001i
\(94\) −5.89898 −0.608433
\(95\) 0 0
\(96\) −2.44949 −0.250000
\(97\) 7.34847i 0.746124i 0.927806 + 0.373062i \(0.121692\pi\)
−0.927806 + 0.373062i \(0.878308\pi\)
\(98\) − 12.7980i − 1.29279i
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 13.4495 1.33827 0.669137 0.743139i \(-0.266664\pi\)
0.669137 + 0.743139i \(0.266664\pi\)
\(102\) 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
\(106\) 10.4495 1.01494
\(107\) 5.24745i 0.507290i 0.967297 + 0.253645i \(0.0816295\pi\)
−0.967297 + 0.253645i \(0.918370\pi\)
\(108\) 0 0
\(109\) 9.34847 0.895421 0.447710 0.894179i \(-0.352240\pi\)
0.447710 + 0.894179i \(0.352240\pi\)
\(110\) 0 0
\(111\) 8.44949 0.801990
\(112\) 4.44949i 0.420437i
\(113\) − 6.24745i − 0.587711i −0.955850 0.293855i \(-0.905062\pi\)
0.955850 0.293855i \(-0.0949384\pi\)
\(114\) 15.7980 1.47961
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 7.34847i − 0.679366i
\(118\) − 8.55051i − 0.787138i
\(119\) 8.89898 0.815768
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 3.44949i 0.312302i
\(123\) 27.7980i 2.50646i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 13.3485 1.18918
\(127\) 8.79796i 0.780693i 0.920668 + 0.390346i \(0.127645\pi\)
−0.920668 + 0.390346i \(0.872355\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −21.7980 −1.91920
\(130\) 0 0
\(131\) −7.34847 −0.642039 −0.321019 0.947073i \(-0.604025\pi\)
−0.321019 + 0.947073i \(0.604025\pi\)
\(132\) − 4.89898i − 0.426401i
\(133\) − 28.6969i − 2.48834i
\(134\) 13.4495 1.16186
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 0.898979i 0.0768050i 0.999262 + 0.0384025i \(0.0122269\pi\)
−0.999262 + 0.0384025i \(0.987773\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 11.4495 0.971133 0.485567 0.874200i \(-0.338613\pi\)
0.485567 + 0.874200i \(0.338613\pi\)
\(140\) 0 0
\(141\) −14.4495 −1.21687
\(142\) 11.3485i 0.952342i
\(143\) 4.89898i 0.409673i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 13.3485 1.10473
\(147\) − 31.3485i − 2.58558i
\(148\) 3.44949i 0.283546i
\(149\) −15.7980 −1.29422 −0.647110 0.762397i \(-0.724022\pi\)
−0.647110 + 0.762397i \(0.724022\pi\)
\(150\) 0 0
\(151\) 11.3485 0.923525 0.461763 0.887004i \(-0.347217\pi\)
0.461763 + 0.887004i \(0.347217\pi\)
\(152\) 6.44949i 0.523123i
\(153\) 6.00000i 0.485071i
\(154\) −8.89898 −0.717100
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 12.3485i 0.985515i 0.870167 + 0.492758i \(0.164011\pi\)
−0.870167 + 0.492758i \(0.835989\pi\)
\(158\) 2.89898i 0.230630i
\(159\) 25.5959 2.02989
\(160\) 0 0
\(161\) −10.8990 −0.858960
\(162\) − 9.00000i − 0.707107i
\(163\) 12.4495i 0.975119i 0.873090 + 0.487560i \(0.162113\pi\)
−0.873090 + 0.487560i \(0.837887\pi\)
\(164\) −11.3485 −0.886167
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) − 17.7980i − 1.37725i −0.725119 0.688624i \(-0.758215\pi\)
0.725119 0.688624i \(-0.241785\pi\)
\(168\) 10.8990i 0.840875i
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 19.3485 1.47961
\(172\) − 8.89898i − 0.678541i
\(173\) 16.0000i 1.21646i 0.793762 + 0.608229i \(0.208120\pi\)
−0.793762 + 0.608229i \(0.791880\pi\)
\(174\) −2.44949 −0.185695
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) − 20.9444i − 1.57428i
\(178\) − 3.55051i − 0.266122i
\(179\) 17.2474 1.28913 0.644567 0.764547i \(-0.277038\pi\)
0.644567 + 0.764547i \(0.277038\pi\)
\(180\) 0 0
\(181\) −26.0454 −1.93594 −0.967970 0.251066i \(-0.919219\pi\)
−0.967970 + 0.251066i \(0.919219\pi\)
\(182\) − 10.8990i − 0.807886i
\(183\) 8.44949i 0.624604i
\(184\) 2.44949 0.180579
\(185\) 0 0
\(186\) 7.34847 0.538816
\(187\) − 4.00000i − 0.292509i
\(188\) − 5.89898i − 0.430227i
\(189\) 0 0
\(190\) 0 0
\(191\) −11.6969 −0.846361 −0.423180 0.906045i \(-0.639086\pi\)
−0.423180 + 0.906045i \(0.639086\pi\)
\(192\) − 2.44949i − 0.176777i
\(193\) 5.34847i 0.384991i 0.981298 + 0.192496i \(0.0616581\pi\)
−0.981298 + 0.192496i \(0.938342\pi\)
\(194\) −7.34847 −0.527589
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) 5.10102i 0.363433i 0.983351 + 0.181716i \(0.0581653\pi\)
−0.983351 + 0.181716i \(0.941835\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) 0.651531 0.0461858 0.0230929 0.999733i \(-0.492649\pi\)
0.0230929 + 0.999733i \(0.492649\pi\)
\(200\) 0 0
\(201\) 32.9444 2.32372
\(202\) 13.4495i 0.946303i
\(203\) 4.44949i 0.312293i
\(204\) −4.89898 −0.342997
\(205\) 0 0
\(206\) −2.44949 −0.170664
\(207\) − 7.34847i − 0.510754i
\(208\) 2.44949i 0.169842i
\(209\) −12.8990 −0.892241
\(210\) 0 0
\(211\) −22.4949 −1.54861 −0.774306 0.632811i \(-0.781901\pi\)
−0.774306 + 0.632811i \(0.781901\pi\)
\(212\) 10.4495i 0.717674i
\(213\) 27.7980i 1.90468i
\(214\) −5.24745 −0.358708
\(215\) 0 0
\(216\) 0 0
\(217\) − 13.3485i − 0.906153i
\(218\) 9.34847i 0.633158i
\(219\) 32.6969 2.20945
\(220\) 0 0
\(221\) 4.89898 0.329541
\(222\) 8.44949i 0.567093i
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) −4.44949 −0.297294
\(225\) 0 0
\(226\) 6.24745 0.415574
\(227\) 15.2474i 1.01201i 0.862531 + 0.506004i \(0.168878\pi\)
−0.862531 + 0.506004i \(0.831122\pi\)
\(228\) 15.7980i 1.04625i
\(229\) 17.1010 1.13007 0.565034 0.825068i \(-0.308863\pi\)
0.565034 + 0.825068i \(0.308863\pi\)
\(230\) 0 0
\(231\) −21.7980 −1.43420
\(232\) − 1.00000i − 0.0656532i
\(233\) 11.0000i 0.720634i 0.932830 + 0.360317i \(0.117331\pi\)
−0.932830 + 0.360317i \(0.882669\pi\)
\(234\) 7.34847 0.480384
\(235\) 0 0
\(236\) 8.55051 0.556591
\(237\) 7.10102i 0.461261i
\(238\) 8.89898i 0.576835i
\(239\) −21.5959 −1.39692 −0.698462 0.715647i \(-0.746132\pi\)
−0.698462 + 0.715647i \(0.746132\pi\)
\(240\) 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
\(244\) −3.44949 −0.220831
\(245\) 0 0
\(246\) −27.7980 −1.77233
\(247\) − 15.7980i − 1.00520i
\(248\) 3.00000i 0.190500i
\(249\) −14.6969 −0.931381
\(250\) 0 0
\(251\) −3.14643 −0.198601 −0.0993004 0.995058i \(-0.531660\pi\)
−0.0993004 + 0.995058i \(0.531660\pi\)
\(252\) 13.3485i 0.840875i
\(253\) 4.89898i 0.307996i
\(254\) −8.79796 −0.552033
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 26.6969i 1.66531i 0.553793 + 0.832655i \(0.313180\pi\)
−0.553793 + 0.832655i \(0.686820\pi\)
\(258\) − 21.7980i − 1.35708i
\(259\) 15.3485 0.953707
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) − 7.34847i − 0.453990i
\(263\) 16.7980i 1.03581i 0.855439 + 0.517903i \(0.173287\pi\)
−0.855439 + 0.517903i \(0.826713\pi\)
\(264\) 4.89898 0.301511
\(265\) 0 0
\(266\) 28.6969 1.75952
\(267\) − 8.69694i − 0.532244i
\(268\) 13.4495i 0.821558i
\(269\) −24.3485 −1.48455 −0.742276 0.670094i \(-0.766254\pi\)
−0.742276 + 0.670094i \(0.766254\pi\)
\(270\) 0 0
\(271\) 15.6969 0.953521 0.476761 0.879033i \(-0.341811\pi\)
0.476761 + 0.879033i \(0.341811\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) − 26.6969i − 1.61577i
\(274\) −0.898979 −0.0543093
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 22.9444i − 1.37859i −0.724479 0.689297i \(-0.757919\pi\)
0.724479 0.689297i \(-0.242081\pi\)
\(278\) 11.4495i 0.686695i
\(279\) 9.00000 0.538816
\(280\) 0 0
\(281\) −28.7980 −1.71794 −0.858971 0.512024i \(-0.828896\pi\)
−0.858971 + 0.512024i \(0.828896\pi\)
\(282\) − 14.4495i − 0.860455i
\(283\) − 24.0000i − 1.42665i −0.700832 0.713326i \(-0.747188\pi\)
0.700832 0.713326i \(-0.252812\pi\)
\(284\) −11.3485 −0.673408
\(285\) 0 0
\(286\) −4.89898 −0.289683
\(287\) 50.4949i 2.98062i
\(288\) − 3.00000i − 0.176777i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 13.3485i 0.781160i
\(293\) 7.44949i 0.435204i 0.976038 + 0.217602i \(0.0698235\pi\)
−0.976038 + 0.217602i \(0.930177\pi\)
\(294\) 31.3485 1.82828
\(295\) 0 0
\(296\) −3.44949 −0.200498
\(297\) 0 0
\(298\) − 15.7980i − 0.915151i
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −39.5959 −2.28227
\(302\) 11.3485i 0.653031i
\(303\) 32.9444i 1.89261i
\(304\) −6.44949 −0.369904
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 25.3939i 1.44931i 0.689114 + 0.724653i \(0.258000\pi\)
−0.689114 + 0.724653i \(0.742000\pi\)
\(308\) − 8.89898i − 0.507066i
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −19.5959 −1.11118 −0.555591 0.831456i \(-0.687508\pi\)
−0.555591 + 0.831456i \(0.687508\pi\)
\(312\) 6.00000i 0.339683i
\(313\) 32.5959i 1.84243i 0.389054 + 0.921215i \(0.372802\pi\)
−0.389054 + 0.921215i \(0.627198\pi\)
\(314\) −12.3485 −0.696864
\(315\) 0 0
\(316\) −2.89898 −0.163080
\(317\) 13.6515i 0.766746i 0.923594 + 0.383373i \(0.125238\pi\)
−0.923594 + 0.383373i \(0.874762\pi\)
\(318\) 25.5959i 1.43535i
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −12.8536 −0.717416
\(322\) − 10.8990i − 0.607376i
\(323\) 12.8990i 0.717718i
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −12.4495 −0.689513
\(327\) 22.8990i 1.26632i
\(328\) − 11.3485i − 0.626614i
\(329\) −26.2474 −1.44707
\(330\) 0 0
\(331\) −30.2474 −1.66255 −0.831275 0.555861i \(-0.812389\pi\)
−0.831275 + 0.555861i \(0.812389\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) 10.3485i 0.567093i
\(334\) 17.7980 0.973861
\(335\) 0 0
\(336\) −10.8990 −0.594588
\(337\) 23.7980i 1.29636i 0.761488 + 0.648179i \(0.224469\pi\)
−0.761488 + 0.648179i \(0.775531\pi\)
\(338\) 7.00000i 0.380750i
\(339\) 15.3031 0.831148
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 19.3485i 1.04625i
\(343\) − 25.7980i − 1.39296i
\(344\) 8.89898 0.479801
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) − 25.0454i − 1.34451i −0.740321 0.672254i \(-0.765326\pi\)
0.740321 0.672254i \(-0.234674\pi\)
\(348\) − 2.44949i − 0.131306i
\(349\) 30.9444 1.65642 0.828208 0.560422i \(-0.189361\pi\)
0.828208 + 0.560422i \(0.189361\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000i 0.106600i
\(353\) 16.7980i 0.894065i 0.894518 + 0.447033i \(0.147519\pi\)
−0.894518 + 0.447033i \(0.852481\pi\)
\(354\) 20.9444 1.11318
\(355\) 0 0
\(356\) 3.55051 0.188177
\(357\) 21.7980i 1.15367i
\(358\) 17.2474i 0.911556i
\(359\) 16.5959 0.875899 0.437950 0.899000i \(-0.355705\pi\)
0.437950 + 0.899000i \(0.355705\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) − 26.0454i − 1.36892i
\(363\) − 17.1464i − 0.899954i
\(364\) 10.8990 0.571262
\(365\) 0 0
\(366\) −8.44949 −0.441662
\(367\) − 1.20204i − 0.0627460i −0.999508 0.0313730i \(-0.990012\pi\)
0.999508 0.0313730i \(-0.00998798\pi\)
\(368\) 2.44949i 0.127688i
\(369\) −34.0454 −1.77233
\(370\) 0 0
\(371\) 46.4949 2.41389
\(372\) 7.34847i 0.381000i
\(373\) − 28.4949i − 1.47541i −0.675123 0.737705i \(-0.735910\pi\)
0.675123 0.737705i \(-0.264090\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 5.89898 0.304217
\(377\) 2.44949i 0.126155i
\(378\) 0 0
\(379\) −25.1464 −1.29169 −0.645843 0.763471i \(-0.723494\pi\)
−0.645843 + 0.763471i \(0.723494\pi\)
\(380\) 0 0
\(381\) −21.5505 −1.10407
\(382\) − 11.6969i − 0.598467i
\(383\) 11.7980i 0.602848i 0.953490 + 0.301424i \(0.0974619\pi\)
−0.953490 + 0.301424i \(0.902538\pi\)
\(384\) 2.44949 0.125000
\(385\) 0 0
\(386\) −5.34847 −0.272230
\(387\) − 26.6969i − 1.35708i
\(388\) − 7.34847i − 0.373062i
\(389\) 24.3485 1.23452 0.617258 0.786761i \(-0.288243\pi\)
0.617258 + 0.786761i \(0.288243\pi\)
\(390\) 0 0
\(391\) 4.89898 0.247752
\(392\) 12.7980i 0.646395i
\(393\) − 18.0000i − 0.907980i
\(394\) −5.10102 −0.256986
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 13.1464i 0.659800i 0.944016 + 0.329900i \(0.107015\pi\)
−0.944016 + 0.329900i \(0.892985\pi\)
\(398\) 0.651531i 0.0326583i
\(399\) 70.2929 3.51904
\(400\) 0 0
\(401\) 38.5959 1.92739 0.963694 0.267009i \(-0.0860353\pi\)
0.963694 + 0.267009i \(0.0860353\pi\)
\(402\) 32.9444i 1.64312i
\(403\) − 7.34847i − 0.366053i
\(404\) −13.4495 −0.669137
\(405\) 0 0
\(406\) −4.44949 −0.220824
\(407\) − 6.89898i − 0.341970i
\(408\) − 4.89898i − 0.242536i
\(409\) 5.79796 0.286691 0.143345 0.989673i \(-0.454214\pi\)
0.143345 + 0.989673i \(0.454214\pi\)
\(410\) 0 0
\(411\) −2.20204 −0.108619
\(412\) − 2.44949i − 0.120678i
\(413\) − 38.0454i − 1.87209i
\(414\) 7.34847 0.361158
\(415\) 0 0
\(416\) −2.44949 −0.120096
\(417\) 28.0454i 1.37339i
\(418\) − 12.8990i − 0.630910i
\(419\) 10.1464 0.495685 0.247843 0.968800i \(-0.420278\pi\)
0.247843 + 0.968800i \(0.420278\pi\)
\(420\) 0 0
\(421\) 13.4495 0.655488 0.327744 0.944767i \(-0.393712\pi\)
0.327744 + 0.944767i \(0.393712\pi\)
\(422\) − 22.4949i − 1.09503i
\(423\) − 17.6969i − 0.860455i
\(424\) −10.4495 −0.507472
\(425\) 0 0
\(426\) −27.7980 −1.34682
\(427\) 15.3485i 0.742764i
\(428\) − 5.24745i − 0.253645i
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 18.4495 0.888681 0.444340 0.895858i \(-0.353438\pi\)
0.444340 + 0.895858i \(0.353438\pi\)
\(432\) 0 0
\(433\) − 30.4495i − 1.46331i −0.681676 0.731655i \(-0.738748\pi\)
0.681676 0.731655i \(-0.261252\pi\)
\(434\) 13.3485 0.640747
\(435\) 0 0
\(436\) −9.34847 −0.447710
\(437\) − 15.7980i − 0.755719i
\(438\) 32.6969i 1.56232i
\(439\) −18.6969 −0.892356 −0.446178 0.894944i \(-0.647215\pi\)
−0.446178 + 0.894944i \(0.647215\pi\)
\(440\) 0 0
\(441\) 38.3939 1.82828
\(442\) 4.89898i 0.233021i
\(443\) 13.1010i 0.622448i 0.950337 + 0.311224i \(0.100739\pi\)
−0.950337 + 0.311224i \(0.899261\pi\)
\(444\) −8.44949 −0.400995
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) − 38.6969i − 1.83030i
\(448\) − 4.44949i − 0.210219i
\(449\) −25.7980 −1.21748 −0.608740 0.793369i \(-0.708325\pi\)
−0.608740 + 0.793369i \(0.708325\pi\)
\(450\) 0 0
\(451\) 22.6969 1.06876
\(452\) 6.24745i 0.293855i
\(453\) 27.7980i 1.30606i
\(454\) −15.2474 −0.715598
\(455\) 0 0
\(456\) −15.7980 −0.739807
\(457\) − 17.7980i − 0.832553i −0.909238 0.416277i \(-0.863335\pi\)
0.909238 0.416277i \(-0.136665\pi\)
\(458\) 17.1010i 0.799078i
\(459\) 0 0
\(460\) 0 0
\(461\) −28.2929 −1.31773 −0.658865 0.752261i \(-0.728963\pi\)
−0.658865 + 0.752261i \(0.728963\pi\)
\(462\) − 21.7980i − 1.01413i
\(463\) 3.10102i 0.144117i 0.997400 + 0.0720583i \(0.0229568\pi\)
−0.997400 + 0.0720583i \(0.977043\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −11.0000 −0.509565
\(467\) − 22.9444i − 1.06174i −0.847453 0.530870i \(-0.821865\pi\)
0.847453 0.530870i \(-0.178135\pi\)
\(468\) 7.34847i 0.339683i
\(469\) 59.8434 2.76331
\(470\) 0 0
\(471\) −30.2474 −1.39373
\(472\) 8.55051i 0.393569i
\(473\) 17.7980i 0.818351i
\(474\) −7.10102 −0.326161
\(475\) 0 0
\(476\) −8.89898 −0.407884
\(477\) 31.3485i 1.43535i
\(478\) − 21.5959i − 0.987774i
\(479\) 4.20204 0.191996 0.0959981 0.995382i \(-0.469396\pi\)
0.0959981 + 0.995382i \(0.469396\pi\)
\(480\) 0 0
\(481\) 8.44949 0.385264
\(482\) 17.0000i 0.774329i
\(483\) − 26.6969i − 1.21475i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 22.0454 1.00000
\(487\) − 9.10102i − 0.412407i −0.978509 0.206203i \(-0.933889\pi\)
0.978509 0.206203i \(-0.0661108\pi\)
\(488\) − 3.44949i − 0.156151i
\(489\) −30.4949 −1.37903
\(490\) 0 0
\(491\) 26.4949 1.19570 0.597849 0.801609i \(-0.296022\pi\)
0.597849 + 0.801609i \(0.296022\pi\)
\(492\) − 27.7980i − 1.25323i
\(493\) − 2.00000i − 0.0900755i
\(494\) 15.7980 0.710784
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 50.4949i 2.26501i
\(498\) − 14.6969i − 0.658586i
\(499\) 24.3485 1.08999 0.544994 0.838440i \(-0.316532\pi\)
0.544994 + 0.838440i \(0.316532\pi\)
\(500\) 0 0
\(501\) 43.5959 1.94772
\(502\) − 3.14643i − 0.140432i
\(503\) − 4.79796i − 0.213930i −0.994263 0.106965i \(-0.965887\pi\)
0.994263 0.106965i \(-0.0341133\pi\)
\(504\) −13.3485 −0.594588
\(505\) 0 0
\(506\) −4.89898 −0.217786
\(507\) 17.1464i 0.761500i
\(508\) − 8.79796i − 0.390346i
\(509\) 17.3939 0.770970 0.385485 0.922714i \(-0.374034\pi\)
0.385485 + 0.922714i \(0.374034\pi\)
\(510\) 0 0
\(511\) 59.3939 2.62743
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −26.6969 −1.17755
\(515\) 0 0
\(516\) 21.7980 0.959602
\(517\) 11.7980i 0.518874i
\(518\) 15.3485i 0.674373i
\(519\) −39.1918 −1.72033
\(520\) 0 0
\(521\) −10.1010 −0.442534 −0.221267 0.975213i \(-0.571019\pi\)
−0.221267 + 0.975213i \(0.571019\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) 27.4495i 1.20028i 0.799894 + 0.600141i \(0.204889\pi\)
−0.799894 + 0.600141i \(0.795111\pi\)
\(524\) 7.34847 0.321019
\(525\) 0 0
\(526\) −16.7980 −0.732426
\(527\) 6.00000i 0.261364i
\(528\) 4.89898i 0.213201i
\(529\) 17.0000 0.739130
\(530\) 0 0
\(531\) 25.6515 1.11318
\(532\) 28.6969i 1.24417i
\(533\) 27.7980i 1.20406i
\(534\) 8.69694 0.376353
\(535\) 0 0
\(536\) −13.4495 −0.580929
\(537\) 42.2474i 1.82311i
\(538\) − 24.3485i − 1.04974i
\(539\) −25.5959 −1.10249
\(540\) 0 0
\(541\) −21.0454 −0.904813 −0.452406 0.891812i \(-0.649434\pi\)
−0.452406 + 0.891812i \(0.649434\pi\)
\(542\) 15.6969i 0.674241i
\(543\) − 63.7980i − 2.73783i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 26.6969 1.14252
\(547\) − 22.0000i − 0.940652i −0.882493 0.470326i \(-0.844136\pi\)
0.882493 0.470326i \(-0.155864\pi\)
\(548\) − 0.898979i − 0.0384025i
\(549\) −10.3485 −0.441662
\(550\) 0 0
\(551\) −6.44949 −0.274758
\(552\) 6.00000i 0.255377i
\(553\) 12.8990i 0.548520i
\(554\) 22.9444 0.974814
\(555\) 0 0
\(556\) −11.4495 −0.485567
\(557\) − 6.49490i − 0.275198i −0.990488 0.137599i \(-0.956062\pi\)
0.990488 0.137599i \(-0.0439385\pi\)
\(558\) 9.00000i 0.381000i
\(559\) −21.7980 −0.921955
\(560\) 0 0
\(561\) 9.79796 0.413670
\(562\) − 28.7980i − 1.21477i
\(563\) 21.7980i 0.918674i 0.888262 + 0.459337i \(0.151913\pi\)
−0.888262 + 0.459337i \(0.848087\pi\)
\(564\) 14.4495 0.608433
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) − 40.0454i − 1.68175i
\(568\) − 11.3485i − 0.476171i
\(569\) 8.69694 0.364595 0.182297 0.983243i \(-0.441647\pi\)
0.182297 + 0.983243i \(0.441647\pi\)
\(570\) 0 0
\(571\) 25.0454 1.04812 0.524059 0.851682i \(-0.324417\pi\)
0.524059 + 0.851682i \(0.324417\pi\)
\(572\) − 4.89898i − 0.204837i
\(573\) − 28.6515i − 1.19693i
\(574\) −50.4949 −2.10762
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) 28.9444i 1.20497i 0.798130 + 0.602485i \(0.205823\pi\)
−0.798130 + 0.602485i \(0.794177\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −13.1010 −0.544460
\(580\) 0 0
\(581\) −26.6969 −1.10758
\(582\) − 18.0000i − 0.746124i
\(583\) − 20.8990i − 0.865547i
\(584\) −13.3485 −0.552364
\(585\) 0 0
\(586\) −7.44949 −0.307736
\(587\) − 14.7526i − 0.608903i −0.952528 0.304451i \(-0.901527\pi\)
0.952528 0.304451i \(-0.0984731\pi\)
\(588\) 31.3485i 1.29279i
\(589\) 19.3485 0.797240
\(590\) 0 0
\(591\) −12.4949 −0.513971
\(592\) − 3.44949i − 0.141773i
\(593\) 18.3939i 0.755346i 0.925939 + 0.377673i \(0.123276\pi\)
−0.925939 + 0.377673i \(0.876724\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.7980 0.647110
\(597\) 1.59592i 0.0653166i
\(598\) − 6.00000i − 0.245358i
\(599\) 12.1010 0.494434 0.247217 0.968960i \(-0.420484\pi\)
0.247217 + 0.968960i \(0.420484\pi\)
\(600\) 0 0
\(601\) −6.69694 −0.273174 −0.136587 0.990628i \(-0.543613\pi\)
−0.136587 + 0.990628i \(0.543613\pi\)
\(602\) − 39.5959i − 1.61381i
\(603\) 40.3485i 1.64312i
\(604\) −11.3485 −0.461763
\(605\) 0 0
\(606\) −32.9444 −1.33827
\(607\) − 34.3939i − 1.39600i −0.716096 0.698002i \(-0.754073\pi\)
0.716096 0.698002i \(-0.245927\pi\)
\(608\) − 6.44949i − 0.261561i
\(609\) −10.8990 −0.441649
\(610\) 0 0
\(611\) −14.4495 −0.584564
\(612\) − 6.00000i − 0.242536i
\(613\) − 46.8990i − 1.89423i −0.320891 0.947116i \(-0.603982\pi\)
0.320891 0.947116i \(-0.396018\pi\)
\(614\) −25.3939 −1.02481
\(615\) 0 0
\(616\) 8.89898 0.358550
\(617\) 14.4495i 0.581715i 0.956766 + 0.290857i \(0.0939405\pi\)
−0.956766 + 0.290857i \(0.906060\pi\)
\(618\) − 6.00000i − 0.241355i
\(619\) −2.89898 −0.116520 −0.0582599 0.998301i \(-0.518555\pi\)
−0.0582599 + 0.998301i \(0.518555\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 19.5959i − 0.785725i
\(623\) − 15.7980i − 0.632932i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −32.5959 −1.30279
\(627\) − 31.5959i − 1.26182i
\(628\) − 12.3485i − 0.492758i
\(629\) −6.89898 −0.275080
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) − 2.89898i − 0.115315i
\(633\) − 55.1010i − 2.19007i
\(634\) −13.6515 −0.542172
\(635\) 0 0
\(636\) −25.5959 −1.01494
\(637\) − 31.3485i − 1.24207i
\(638\) 2.00000i 0.0791808i
\(639\) −34.0454 −1.34682
\(640\) 0 0
\(641\) 24.2474 0.957717 0.478858 0.877892i \(-0.341051\pi\)
0.478858 + 0.877892i \(0.341051\pi\)
\(642\) − 12.8536i − 0.507290i
\(643\) 24.8434i 0.979727i 0.871799 + 0.489863i \(0.162953\pi\)
−0.871799 + 0.489863i \(0.837047\pi\)
\(644\) 10.8990 0.429480
\(645\) 0 0
\(646\) −12.8990 −0.507504
\(647\) 39.5959i 1.55668i 0.627845 + 0.778338i \(0.283937\pi\)
−0.627845 + 0.778338i \(0.716063\pi\)
\(648\) 9.00000i 0.353553i
\(649\) −17.1010 −0.671274
\(650\) 0 0
\(651\) 32.6969 1.28149
\(652\) − 12.4495i − 0.487560i
\(653\) − 11.2474i − 0.440147i −0.975483 0.220073i \(-0.929370\pi\)
0.975483 0.220073i \(-0.0706297\pi\)
\(654\) −22.8990 −0.895421
\(655\) 0 0
\(656\) 11.3485 0.443083
\(657\) 40.0454i 1.56232i
\(658\) − 26.2474i − 1.02323i
\(659\) 38.6969 1.50742 0.753709 0.657208i \(-0.228263\pi\)
0.753709 + 0.657208i \(0.228263\pi\)
\(660\) 0 0
\(661\) 9.75255 0.379330 0.189665 0.981849i \(-0.439260\pi\)
0.189665 + 0.981849i \(0.439260\pi\)
\(662\) − 30.2474i − 1.17560i
\(663\) 12.0000i 0.466041i
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −10.3485 −0.400995
\(667\) 2.44949i 0.0948446i
\(668\) 17.7980i 0.688624i
\(669\) 9.79796 0.378811
\(670\) 0 0
\(671\) 6.89898 0.266332
\(672\) − 10.8990i − 0.420437i
\(673\) 16.7980i 0.647514i 0.946140 + 0.323757i \(0.104946\pi\)
−0.946140 + 0.323757i \(0.895054\pi\)
\(674\) −23.7980 −0.916663
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) − 25.0454i − 0.962573i −0.876563 0.481287i \(-0.840170\pi\)
0.876563 0.481287i \(-0.159830\pi\)
\(678\) 15.3031i 0.587711i
\(679\) −32.6969 −1.25479
\(680\) 0 0
\(681\) −37.3485 −1.43120
\(682\) − 6.00000i − 0.229752i
\(683\) − 11.1010i − 0.424769i −0.977186 0.212384i \(-0.931877\pi\)
0.977186 0.212384i \(-0.0681229\pi\)
\(684\) −19.3485 −0.739807
\(685\) 0 0
\(686\) 25.7980 0.984971
\(687\) 41.8888i 1.59816i
\(688\) 8.89898i 0.339270i
\(689\) 25.5959 0.975127
\(690\) 0 0
\(691\) −43.9444 −1.67172 −0.835862 0.548940i \(-0.815031\pi\)
−0.835862 + 0.548940i \(0.815031\pi\)
\(692\) − 16.0000i − 0.608229i
\(693\) − 26.6969i − 1.01413i
\(694\) 25.0454 0.950711
\(695\) 0 0
\(696\) 2.44949 0.0928477
\(697\) − 22.6969i − 0.859708i
\(698\) 30.9444i 1.17126i
\(699\) −26.9444 −1.01913
\(700\) 0 0
\(701\) −25.3939 −0.959113 −0.479557 0.877511i \(-0.659203\pi\)
−0.479557 + 0.877511i \(0.659203\pi\)
\(702\) 0 0
\(703\) 22.2474i 0.839078i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −16.7980 −0.632200
\(707\) 59.8434i 2.25064i
\(708\) 20.9444i 0.787138i
\(709\) 27.3939 1.02880 0.514399 0.857551i \(-0.328015\pi\)
0.514399 + 0.857551i \(0.328015\pi\)
\(710\) 0 0
\(711\) −8.69694 −0.326161
\(712\) 3.55051i 0.133061i
\(713\) − 7.34847i − 0.275202i
\(714\) −21.7980 −0.815768
\(715\) 0 0
\(716\) −17.2474 −0.644567
\(717\) − 52.8990i − 1.97555i
\(718\) 16.5959i 0.619354i
\(719\) 36.7423 1.37026 0.685129 0.728422i \(-0.259746\pi\)
0.685129 + 0.728422i \(0.259746\pi\)
\(720\) 0 0
\(721\) −10.8990 −0.405899
\(722\) 22.5959i 0.840933i
\(723\) 41.6413i 1.54866i
\(724\) 26.0454 0.967970
\(725\) 0 0
\(726\) 17.1464 0.636364
\(727\) − 33.5959i − 1.24600i −0.782220 0.623002i \(-0.785913\pi\)
0.782220 0.623002i \(-0.214087\pi\)
\(728\) 10.8990i 0.403943i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 17.7980 0.658281
\(732\) − 8.44949i − 0.312302i
\(733\) − 1.10102i − 0.0406671i −0.999793 0.0203336i \(-0.993527\pi\)
0.999793 0.0203336i \(-0.00647282\pi\)
\(734\) 1.20204 0.0443681
\(735\) 0 0
\(736\) −2.44949 −0.0902894
\(737\) − 26.8990i − 0.990837i
\(738\) − 34.0454i − 1.25323i
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 38.6969 1.42157
\(742\) 46.4949i 1.70688i
\(743\) − 12.6969i − 0.465805i −0.972500 0.232903i \(-0.925178\pi\)
0.972500 0.232903i \(-0.0748224\pi\)
\(744\) −7.34847 −0.269408
\(745\) 0 0
\(746\) 28.4949 1.04327
\(747\) − 18.0000i − 0.658586i
\(748\) 4.00000i 0.146254i
\(749\) −23.3485 −0.853134
\(750\) 0 0
\(751\) 24.1010 0.879459 0.439729 0.898130i \(-0.355074\pi\)
0.439729 + 0.898130i \(0.355074\pi\)
\(752\) 5.89898i 0.215114i
\(753\) − 7.70714i − 0.280864i
\(754\) −2.44949 −0.0892052
\(755\) 0 0
\(756\) 0 0
\(757\) − 19.3939i − 0.704882i −0.935834 0.352441i \(-0.885352\pi\)
0.935834 0.352441i \(-0.114648\pi\)
\(758\) − 25.1464i − 0.913359i
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 14.1010 0.511162 0.255581 0.966788i \(-0.417733\pi\)
0.255581 + 0.966788i \(0.417733\pi\)
\(762\) − 21.5505i − 0.780693i
\(763\) 41.5959i 1.50587i
\(764\) 11.6969 0.423180
\(765\) 0 0
\(766\) −11.7980 −0.426278
\(767\) − 20.9444i − 0.756258i
\(768\) 2.44949i 0.0883883i
\(769\) 3.55051 0.128035 0.0640173 0.997949i \(-0.479609\pi\)
0.0640173 + 0.997949i \(0.479609\pi\)
\(770\) 0 0
\(771\) −65.3939 −2.35510
\(772\) − 5.34847i − 0.192496i
\(773\) 17.4495i 0.627615i 0.949487 + 0.313807i \(0.101605\pi\)
−0.949487 + 0.313807i \(0.898395\pi\)
\(774\) 26.6969 0.959602
\(775\) 0 0
\(776\) 7.34847 0.263795
\(777\) 37.5959i 1.34875i
\(778\) 24.3485i 0.872935i
\(779\) −73.1918 −2.62237
\(780\) 0 0
\(781\) 22.6969 0.812160
\(782\) 4.89898i 0.175187i
\(783\) 0 0
\(784\) −12.7980 −0.457070
\(785\) 0 0
\(786\) 18.0000 0.642039
\(787\) − 40.5505i − 1.44547i −0.691125 0.722735i \(-0.742885\pi\)
0.691125 0.722735i \(-0.257115\pi\)
\(788\) − 5.10102i − 0.181716i
\(789\) −41.1464 −1.46485
\(790\) 0 0
\(791\) 27.7980 0.988382
\(792\) 6.00000i 0.213201i
\(793\) 8.44949i 0.300050i
\(794\) −13.1464 −0.466549
\(795\) 0 0
\(796\) −0.651531 −0.0230929
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 70.2929i 2.48834i
\(799\) 11.7980 0.417382
\(800\) 0 0
\(801\) 10.6515 0.376353
\(802\) 38.5959i 1.36287i
\(803\) − 26.6969i − 0.942114i
\(804\) −32.9444 −1.16186
\(805\) 0 0
\(806\) 7.34847 0.258839
\(807\) − 59.6413i − 2.09947i
\(808\) − 13.4495i − 0.473151i
\(809\) −31.5959 −1.11085 −0.555427 0.831566i \(-0.687445\pi\)
−0.555427 + 0.831566i \(0.687445\pi\)
\(810\) 0 0
\(811\) −29.4495 −1.03411 −0.517056 0.855952i \(-0.672972\pi\)
−0.517056 + 0.855952i \(0.672972\pi\)
\(812\) − 4.44949i − 0.156146i
\(813\) 38.4495i 1.34848i
\(814\) 6.89898 0.241809
\(815\) 0 0
\(816\) 4.89898 0.171499
\(817\) − 57.3939i − 2.00796i
\(818\) 5.79796i 0.202721i
\(819\) 32.6969 1.14252
\(820\) 0 0
\(821\) 19.7526 0.689369 0.344684 0.938719i \(-0.387986\pi\)
0.344684 + 0.938719i \(0.387986\pi\)
\(822\) − 2.20204i − 0.0768050i
\(823\) 11.7980i 0.411251i 0.978631 + 0.205625i \(0.0659229\pi\)
−0.978631 + 0.205625i \(0.934077\pi\)
\(824\) 2.44949 0.0853320
\(825\) 0 0
\(826\) 38.0454 1.32377
\(827\) 19.3031i 0.671233i 0.941999 + 0.335617i \(0.108945\pi\)
−0.941999 + 0.335617i \(0.891055\pi\)
\(828\) 7.34847i 0.255377i
\(829\) −14.4949 −0.503429 −0.251714 0.967802i \(-0.580994\pi\)
−0.251714 + 0.967802i \(0.580994\pi\)
\(830\) 0 0
\(831\) 56.2020 1.94963
\(832\) − 2.44949i − 0.0849208i
\(833\) 25.5959i 0.886846i
\(834\) −28.0454 −0.971133
\(835\) 0 0
\(836\) 12.8990 0.446121
\(837\) 0 0
\(838\) 10.1464i 0.350503i
\(839\) 30.7980 1.06326 0.531632 0.846976i \(-0.321579\pi\)
0.531632 + 0.846976i \(0.321579\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 13.4495i 0.463500i
\(843\) − 70.5403i − 2.42954i
\(844\) 22.4949 0.774306
\(845\) 0 0
\(846\) 17.6969 0.608433
\(847\) − 31.1464i − 1.07020i
\(848\) − 10.4495i − 0.358837i
\(849\) 58.7878 2.01759
\(850\) 0 0
\(851\) 8.44949 0.289645
\(852\) − 27.7980i − 0.952342i
\(853\) − 8.20204i − 0.280833i −0.990093 0.140416i \(-0.955156\pi\)
0.990093 0.140416i \(-0.0448441\pi\)
\(854\) −15.3485 −0.525214
\(855\) 0 0
\(856\) 5.24745 0.179354
\(857\) − 39.3939i − 1.34567i −0.739793 0.672835i \(-0.765077\pi\)
0.739793 0.672835i \(-0.234923\pi\)
\(858\) − 12.0000i − 0.409673i
\(859\) −22.8990 −0.781303 −0.390652 0.920539i \(-0.627750\pi\)
−0.390652 + 0.920539i \(0.627750\pi\)
\(860\) 0 0
\(861\) −123.687 −4.21523
\(862\) 18.4495i 0.628392i
\(863\) − 11.7526i − 0.400061i −0.979790 0.200031i \(-0.935896\pi\)
0.979790 0.200031i \(-0.0641042\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 30.4495 1.03472
\(867\) 31.8434i 1.08146i
\(868\) 13.3485i 0.453077i
\(869\) 5.79796 0.196682
\(870\) 0 0
\(871\) 32.9444 1.11628
\(872\) − 9.34847i − 0.316579i
\(873\) − 22.0454i − 0.746124i
\(874\) 15.7980 0.534374
\(875\) 0 0
\(876\) −32.6969 −1.10473
\(877\) 8.00000i 0.270141i 0.990836 + 0.135070i \(0.0431261\pi\)
−0.990836 + 0.135070i \(0.956874\pi\)
\(878\) − 18.6969i − 0.630991i
\(879\) −18.2474 −0.615471
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 38.3939i 1.29279i
\(883\) − 29.9444i − 1.00771i −0.863789 0.503854i \(-0.831915\pi\)
0.863789 0.503854i \(-0.168085\pi\)
\(884\) −4.89898 −0.164771
\(885\) 0 0
\(886\) −13.1010 −0.440137
\(887\) − 47.7980i − 1.60490i −0.596720 0.802449i \(-0.703530\pi\)
0.596720 0.802449i \(-0.296470\pi\)
\(888\) − 8.44949i − 0.283546i
\(889\) −39.1464 −1.31293
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 4.00000i 0.133930i
\(893\) − 38.0454i − 1.27314i
\(894\) 38.6969 1.29422
\(895\) 0 0
\(896\) 4.44949 0.148647
\(897\) − 14.6969i − 0.490716i
\(898\) − 25.7980i − 0.860889i
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −20.8990 −0.696246
\(902\) 22.6969i 0.755725i
\(903\) − 96.9898i − 3.22762i
\(904\) −6.24745 −0.207787
\(905\) 0 0
\(906\) −27.7980 −0.923525
\(907\) − 21.3485i − 0.708864i −0.935082 0.354432i \(-0.884674\pi\)
0.935082 0.354432i \(-0.115326\pi\)
\(908\) − 15.2474i − 0.506004i
\(909\) −40.3485 −1.33827
\(910\) 0 0
\(911\) 51.4949 1.70610 0.853051 0.521827i \(-0.174750\pi\)
0.853051 + 0.521827i \(0.174750\pi\)
\(912\) − 15.7980i − 0.523123i
\(913\) 12.0000i 0.397142i
\(914\) 17.7980 0.588704
\(915\) 0 0
\(916\) −17.1010 −0.565034
\(917\) − 32.6969i − 1.07975i
\(918\) 0 0
\(919\) 24.8536 0.819844 0.409922 0.912121i \(-0.365556\pi\)
0.409922 + 0.912121i \(0.365556\pi\)
\(920\) 0 0
\(921\) −62.2020 −2.04963
\(922\) − 28.2929i − 0.931776i
\(923\) 27.7980i 0.914981i
\(924\) 21.7980 0.717100
\(925\) 0 0
\(926\) −3.10102 −0.101906
\(927\) − 7.34847i − 0.241355i
\(928\) 1.00000i 0.0328266i
\(929\) 39.4949 1.29579 0.647893 0.761732i \(-0.275650\pi\)
0.647893 + 0.761732i \(0.275650\pi\)
\(930\) 0 0
\(931\) 82.5403 2.70515
\(932\) − 11.0000i − 0.360317i
\(933\) − 48.0000i − 1.57145i
\(934\) 22.9444 0.750763
\(935\) 0 0
\(936\) −7.34847 −0.240192
\(937\) 53.2929i 1.74100i 0.492167 + 0.870501i \(0.336205\pi\)
−0.492167 + 0.870501i \(0.663795\pi\)
\(938\) 59.8434i 1.95396i
\(939\) −79.8434 −2.60559
\(940\) 0 0
\(941\) 8.44949 0.275445 0.137723 0.990471i \(-0.456022\pi\)
0.137723 + 0.990471i \(0.456022\pi\)
\(942\) − 30.2474i − 0.985515i
\(943\) 27.7980i 0.905226i
\(944\) −8.55051 −0.278295
\(945\) 0 0
\(946\) −17.7980 −0.578662
\(947\) 37.3485i 1.21366i 0.794831 + 0.606831i \(0.207560\pi\)
−0.794831 + 0.606831i \(0.792440\pi\)
\(948\) − 7.10102i − 0.230630i
\(949\) 32.6969 1.06139
\(950\) 0 0
\(951\) −33.4393 −1.08434
\(952\) − 8.89898i − 0.288418i
\(953\) − 34.2929i − 1.11085i −0.831565 0.555427i \(-0.812555\pi\)
0.831565 0.555427i \(-0.187445\pi\)
\(954\) −31.3485 −1.01494
\(955\) 0 0
\(956\) 21.5959 0.698462
\(957\) 4.89898i 0.158362i
\(958\) 4.20204i 0.135762i
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 8.44949i 0.272422i
\(963\) − 15.7423i − 0.507290i
\(964\) −17.0000 −0.547533
\(965\) 0 0
\(966\) 26.6969 0.858960
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 7.00000i 0.224989i
\(969\) −31.5959 −1.01501
\(970\) 0 0
\(971\) 22.9444 0.736320 0.368160 0.929762i \(-0.379988\pi\)
0.368160 + 0.929762i \(0.379988\pi\)
\(972\) 22.0454i 0.707107i
\(973\) 50.9444i 1.63320i
\(974\) 9.10102 0.291616
\(975\) 0 0
\(976\) 3.44949 0.110415
\(977\) 0.101021i 0.00323193i 0.999999 + 0.00161597i \(0.000514378\pi\)
−0.999999 + 0.00161597i \(0.999486\pi\)
\(978\) − 30.4949i − 0.975119i
\(979\) −7.10102 −0.226950
\(980\) 0 0
\(981\) −28.0454 −0.895421
\(982\) 26.4949i 0.845486i
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 27.7980 0.886167
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) − 64.2929i − 2.04646i
\(988\) 15.7980i 0.502600i
\(989\) −21.7980 −0.693135
\(990\) 0 0
\(991\) −3.50510 −0.111343 −0.0556716 0.998449i \(-0.517730\pi\)
−0.0556716 + 0.998449i \(0.517730\pi\)
\(992\) − 3.00000i − 0.0952501i
\(993\) − 74.0908i − 2.35120i
\(994\) −50.4949 −1.60160
\(995\) 0 0
\(996\) 14.6969 0.465690
\(997\) 29.7423i 0.941950i 0.882147 + 0.470975i \(0.156098\pi\)
−0.882147 + 0.470975i \(0.843902\pi\)
\(998\) 24.3485i 0.770737i
\(999\) 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.349.4 4
5.2 odd 4 1450.2.a.j.1.2 2
5.3 odd 4 1450.2.a.o.1.1 yes 2
5.4 even 2 inner 1450.2.b.i.349.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.j.1.2 2 5.2 odd 4
1450.2.a.o.1.1 yes 2 5.3 odd 4
1450.2.b.i.349.1 4 5.4 even 2 inner
1450.2.b.i.349.4 4 1.1 even 1 trivial