Properties

Label 2415.2.a.v.1.2
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2415,2,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

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: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 83x^{6} - 137x^{5} - 164x^{4} + 208x^{3} + 108x^{2} - 83x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.13650\) of defining polynomial
Character \(\chi\) \(=\) 2415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.13650 q^{2} +1.00000 q^{3} +2.56465 q^{4} +1.00000 q^{5} -2.13650 q^{6} -1.00000 q^{7} -1.20638 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.13650 q^{2} +1.00000 q^{3} +2.56465 q^{4} +1.00000 q^{5} -2.13650 q^{6} -1.00000 q^{7} -1.20638 q^{8} +1.00000 q^{9} -2.13650 q^{10} +1.78663 q^{11} +2.56465 q^{12} +4.29660 q^{13} +2.13650 q^{14} +1.00000 q^{15} -2.55186 q^{16} -0.0943613 q^{17} -2.13650 q^{18} +6.77795 q^{19} +2.56465 q^{20} -1.00000 q^{21} -3.81715 q^{22} +1.00000 q^{23} -1.20638 q^{24} +1.00000 q^{25} -9.17970 q^{26} +1.00000 q^{27} -2.56465 q^{28} -2.21636 q^{29} -2.13650 q^{30} -0.792668 q^{31} +7.86483 q^{32} +1.78663 q^{33} +0.201603 q^{34} -1.00000 q^{35} +2.56465 q^{36} +6.35324 q^{37} -14.4811 q^{38} +4.29660 q^{39} -1.20638 q^{40} +7.49896 q^{41} +2.13650 q^{42} -8.14100 q^{43} +4.58209 q^{44} +1.00000 q^{45} -2.13650 q^{46} -10.7203 q^{47} -2.55186 q^{48} +1.00000 q^{49} -2.13650 q^{50} -0.0943613 q^{51} +11.0193 q^{52} -10.3783 q^{53} -2.13650 q^{54} +1.78663 q^{55} +1.20638 q^{56} +6.77795 q^{57} +4.73527 q^{58} +3.09238 q^{59} +2.56465 q^{60} -5.84740 q^{61} +1.69354 q^{62} -1.00000 q^{63} -11.6995 q^{64} +4.29660 q^{65} -3.81715 q^{66} +13.5810 q^{67} -0.242004 q^{68} +1.00000 q^{69} +2.13650 q^{70} +5.24064 q^{71} -1.20638 q^{72} +11.5241 q^{73} -13.5737 q^{74} +1.00000 q^{75} +17.3831 q^{76} -1.78663 q^{77} -9.17970 q^{78} +11.7510 q^{79} -2.55186 q^{80} +1.00000 q^{81} -16.0216 q^{82} -9.22806 q^{83} -2.56465 q^{84} -0.0943613 q^{85} +17.3933 q^{86} -2.21636 q^{87} -2.15536 q^{88} +5.11100 q^{89} -2.13650 q^{90} -4.29660 q^{91} +2.56465 q^{92} -0.792668 q^{93} +22.9040 q^{94} +6.77795 q^{95} +7.86483 q^{96} +5.36007 q^{97} -2.13650 q^{98} +1.78663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{5} + 2 q^{6} - 10 q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{5} + 2 q^{6} - 10 q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} - q^{11} + 16 q^{12} - 2 q^{13} - 2 q^{14} + 10 q^{15} + 36 q^{16} + 8 q^{17} + 2 q^{18} + 11 q^{19} + 16 q^{20} - 10 q^{21} + 10 q^{23} + 6 q^{24} + 10 q^{25} + 15 q^{26} + 10 q^{27} - 16 q^{28} + 6 q^{29} + 2 q^{30} + 16 q^{31} - q^{32} - q^{33} + 21 q^{34} - 10 q^{35} + 16 q^{36} - 6 q^{38} - 2 q^{39} + 6 q^{40} + 23 q^{41} - 2 q^{42} + 4 q^{43} - q^{44} + 10 q^{45} + 2 q^{46} + 21 q^{47} + 36 q^{48} + 10 q^{49} + 2 q^{50} + 8 q^{51} + 10 q^{52} - q^{53} + 2 q^{54} - q^{55} - 6 q^{56} + 11 q^{57} + 8 q^{58} + 15 q^{59} + 16 q^{60} + 29 q^{61} + 12 q^{62} - 10 q^{63} + 80 q^{64} - 2 q^{65} + 32 q^{67} - 28 q^{68} + 10 q^{69} - 2 q^{70} - 4 q^{71} + 6 q^{72} + 18 q^{73} - 49 q^{74} + 10 q^{75} + 49 q^{76} + q^{77} + 15 q^{78} + 8 q^{79} + 36 q^{80} + 10 q^{81} + 6 q^{82} + 2 q^{83} - 16 q^{84} + 8 q^{85} - 14 q^{86} + 6 q^{87} - 69 q^{88} + 6 q^{89} + 2 q^{90} + 2 q^{91} + 16 q^{92} + 16 q^{93} - 2 q^{94} + 11 q^{95} - q^{96} - 2 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.13650 −1.51074 −0.755369 0.655300i \(-0.772542\pi\)
−0.755369 + 0.655300i \(0.772542\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.56465 1.28233
\(5\) 1.00000 0.447214
\(6\) −2.13650 −0.872224
\(7\) −1.00000 −0.377964
\(8\) −1.20638 −0.426521
\(9\) 1.00000 0.333333
\(10\) −2.13650 −0.675622
\(11\) 1.78663 0.538690 0.269345 0.963044i \(-0.413193\pi\)
0.269345 + 0.963044i \(0.413193\pi\)
\(12\) 2.56465 0.740351
\(13\) 4.29660 1.19166 0.595831 0.803110i \(-0.296823\pi\)
0.595831 + 0.803110i \(0.296823\pi\)
\(14\) 2.13650 0.571005
\(15\) 1.00000 0.258199
\(16\) −2.55186 −0.637965
\(17\) −0.0943613 −0.0228860 −0.0114430 0.999935i \(-0.503642\pi\)
−0.0114430 + 0.999935i \(0.503642\pi\)
\(18\) −2.13650 −0.503579
\(19\) 6.77795 1.55497 0.777484 0.628903i \(-0.216496\pi\)
0.777484 + 0.628903i \(0.216496\pi\)
\(20\) 2.56465 0.573474
\(21\) −1.00000 −0.218218
\(22\) −3.81715 −0.813819
\(23\) 1.00000 0.208514
\(24\) −1.20638 −0.246252
\(25\) 1.00000 0.200000
\(26\) −9.17970 −1.80029
\(27\) 1.00000 0.192450
\(28\) −2.56465 −0.484674
\(29\) −2.21636 −0.411569 −0.205784 0.978597i \(-0.565975\pi\)
−0.205784 + 0.978597i \(0.565975\pi\)
\(30\) −2.13650 −0.390071
\(31\) −0.792668 −0.142367 −0.0711837 0.997463i \(-0.522678\pi\)
−0.0711837 + 0.997463i \(0.522678\pi\)
\(32\) 7.86483 1.39032
\(33\) 1.78663 0.311013
\(34\) 0.201603 0.0345747
\(35\) −1.00000 −0.169031
\(36\) 2.56465 0.427442
\(37\) 6.35324 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(38\) −14.4811 −2.34915
\(39\) 4.29660 0.688006
\(40\) −1.20638 −0.190746
\(41\) 7.49896 1.17114 0.585571 0.810621i \(-0.300871\pi\)
0.585571 + 0.810621i \(0.300871\pi\)
\(42\) 2.13650 0.329670
\(43\) −8.14100 −1.24149 −0.620745 0.784012i \(-0.713170\pi\)
−0.620745 + 0.784012i \(0.713170\pi\)
\(44\) 4.58209 0.690777
\(45\) 1.00000 0.149071
\(46\) −2.13650 −0.315010
\(47\) −10.7203 −1.56372 −0.781858 0.623456i \(-0.785728\pi\)
−0.781858 + 0.623456i \(0.785728\pi\)
\(48\) −2.55186 −0.368330
\(49\) 1.00000 0.142857
\(50\) −2.13650 −0.302147
\(51\) −0.0943613 −0.0132132
\(52\) 11.0193 1.52810
\(53\) −10.3783 −1.42557 −0.712785 0.701383i \(-0.752566\pi\)
−0.712785 + 0.701383i \(0.752566\pi\)
\(54\) −2.13650 −0.290741
\(55\) 1.78663 0.240910
\(56\) 1.20638 0.161210
\(57\) 6.77795 0.897761
\(58\) 4.73527 0.621772
\(59\) 3.09238 0.402594 0.201297 0.979530i \(-0.435484\pi\)
0.201297 + 0.979530i \(0.435484\pi\)
\(60\) 2.56465 0.331095
\(61\) −5.84740 −0.748683 −0.374341 0.927291i \(-0.622131\pi\)
−0.374341 + 0.927291i \(0.622131\pi\)
\(62\) 1.69354 0.215080
\(63\) −1.00000 −0.125988
\(64\) −11.6995 −1.46244
\(65\) 4.29660 0.532927
\(66\) −3.81715 −0.469859
\(67\) 13.5810 1.65918 0.829589 0.558374i \(-0.188575\pi\)
0.829589 + 0.558374i \(0.188575\pi\)
\(68\) −0.242004 −0.0293473
\(69\) 1.00000 0.120386
\(70\) 2.13650 0.255361
\(71\) 5.24064 0.621950 0.310975 0.950418i \(-0.399345\pi\)
0.310975 + 0.950418i \(0.399345\pi\)
\(72\) −1.20638 −0.142174
\(73\) 11.5241 1.34879 0.674397 0.738369i \(-0.264404\pi\)
0.674397 + 0.738369i \(0.264404\pi\)
\(74\) −13.5737 −1.57791
\(75\) 1.00000 0.115470
\(76\) 17.3831 1.99398
\(77\) −1.78663 −0.203606
\(78\) −9.17970 −1.03940
\(79\) 11.7510 1.32209 0.661047 0.750345i \(-0.270112\pi\)
0.661047 + 0.750345i \(0.270112\pi\)
\(80\) −2.55186 −0.285307
\(81\) 1.00000 0.111111
\(82\) −16.0216 −1.76929
\(83\) −9.22806 −1.01291 −0.506455 0.862266i \(-0.669045\pi\)
−0.506455 + 0.862266i \(0.669045\pi\)
\(84\) −2.56465 −0.279827
\(85\) −0.0943613 −0.0102349
\(86\) 17.3933 1.87557
\(87\) −2.21636 −0.237619
\(88\) −2.15536 −0.229763
\(89\) 5.11100 0.541765 0.270883 0.962612i \(-0.412684\pi\)
0.270883 + 0.962612i \(0.412684\pi\)
\(90\) −2.13650 −0.225207
\(91\) −4.29660 −0.450406
\(92\) 2.56465 0.267384
\(93\) −0.792668 −0.0821959
\(94\) 22.9040 2.36236
\(95\) 6.77795 0.695403
\(96\) 7.86483 0.802701
\(97\) 5.36007 0.544233 0.272116 0.962264i \(-0.412276\pi\)
0.272116 + 0.962264i \(0.412276\pi\)
\(98\) −2.13650 −0.215820
\(99\) 1.78663 0.179563
\(100\) 2.56465 0.256465
\(101\) −8.27803 −0.823695 −0.411848 0.911253i \(-0.635116\pi\)
−0.411848 + 0.911253i \(0.635116\pi\)
\(102\) 0.201603 0.0199617
\(103\) −17.5937 −1.73356 −0.866778 0.498695i \(-0.833813\pi\)
−0.866778 + 0.498695i \(0.833813\pi\)
\(104\) −5.18334 −0.508269
\(105\) −1.00000 −0.0975900
\(106\) 22.1733 2.15366
\(107\) −12.9655 −1.25342 −0.626709 0.779253i \(-0.715599\pi\)
−0.626709 + 0.779253i \(0.715599\pi\)
\(108\) 2.56465 0.246784
\(109\) 13.0849 1.25331 0.626653 0.779299i \(-0.284424\pi\)
0.626653 + 0.779299i \(0.284424\pi\)
\(110\) −3.81715 −0.363951
\(111\) 6.35324 0.603023
\(112\) 2.55186 0.241128
\(113\) 3.26165 0.306830 0.153415 0.988162i \(-0.450973\pi\)
0.153415 + 0.988162i \(0.450973\pi\)
\(114\) −14.4811 −1.35628
\(115\) 1.00000 0.0932505
\(116\) −5.68421 −0.527765
\(117\) 4.29660 0.397221
\(118\) −6.60689 −0.608213
\(119\) 0.0943613 0.00865009
\(120\) −1.20638 −0.110127
\(121\) −7.80794 −0.709813
\(122\) 12.4930 1.13106
\(123\) 7.49896 0.676159
\(124\) −2.03292 −0.182561
\(125\) 1.00000 0.0894427
\(126\) 2.13650 0.190335
\(127\) 11.9605 1.06133 0.530664 0.847583i \(-0.321943\pi\)
0.530664 + 0.847583i \(0.321943\pi\)
\(128\) 9.26643 0.819045
\(129\) −8.14100 −0.716775
\(130\) −9.17970 −0.805113
\(131\) −10.3525 −0.904503 −0.452251 0.891891i \(-0.649379\pi\)
−0.452251 + 0.891891i \(0.649379\pi\)
\(132\) 4.58209 0.398820
\(133\) −6.77795 −0.587723
\(134\) −29.0158 −2.50658
\(135\) 1.00000 0.0860663
\(136\) 0.113836 0.00976135
\(137\) −7.09215 −0.605924 −0.302962 0.953003i \(-0.597975\pi\)
−0.302962 + 0.953003i \(0.597975\pi\)
\(138\) −2.13650 −0.181871
\(139\) 12.1235 1.02830 0.514149 0.857701i \(-0.328108\pi\)
0.514149 + 0.857701i \(0.328108\pi\)
\(140\) −2.56465 −0.216753
\(141\) −10.7203 −0.902812
\(142\) −11.1967 −0.939602
\(143\) 7.67644 0.641936
\(144\) −2.55186 −0.212655
\(145\) −2.21636 −0.184059
\(146\) −24.6213 −2.03767
\(147\) 1.00000 0.0824786
\(148\) 16.2939 1.33935
\(149\) −1.94223 −0.159114 −0.0795570 0.996830i \(-0.525351\pi\)
−0.0795570 + 0.996830i \(0.525351\pi\)
\(150\) −2.13650 −0.174445
\(151\) 7.56553 0.615674 0.307837 0.951439i \(-0.400395\pi\)
0.307837 + 0.951439i \(0.400395\pi\)
\(152\) −8.17680 −0.663226
\(153\) −0.0943613 −0.00762866
\(154\) 3.81715 0.307595
\(155\) −0.792668 −0.0636686
\(156\) 11.0193 0.882248
\(157\) −21.3972 −1.70768 −0.853840 0.520535i \(-0.825733\pi\)
−0.853840 + 0.520535i \(0.825733\pi\)
\(158\) −25.1061 −1.99734
\(159\) −10.3783 −0.823053
\(160\) 7.86483 0.621770
\(161\) −1.00000 −0.0788110
\(162\) −2.13650 −0.167860
\(163\) 0.103163 0.00808033 0.00404016 0.999992i \(-0.498714\pi\)
0.00404016 + 0.999992i \(0.498714\pi\)
\(164\) 19.2322 1.50179
\(165\) 1.78663 0.139089
\(166\) 19.7158 1.53024
\(167\) −15.7873 −1.22166 −0.610830 0.791762i \(-0.709164\pi\)
−0.610830 + 0.791762i \(0.709164\pi\)
\(168\) 1.20638 0.0930745
\(169\) 5.46074 0.420057
\(170\) 0.201603 0.0154623
\(171\) 6.77795 0.518323
\(172\) −20.8788 −1.59200
\(173\) −7.26997 −0.552726 −0.276363 0.961053i \(-0.589129\pi\)
−0.276363 + 0.961053i \(0.589129\pi\)
\(174\) 4.73527 0.358980
\(175\) −1.00000 −0.0755929
\(176\) −4.55924 −0.343666
\(177\) 3.09238 0.232438
\(178\) −10.9197 −0.818465
\(179\) 18.3495 1.37150 0.685752 0.727835i \(-0.259473\pi\)
0.685752 + 0.727835i \(0.259473\pi\)
\(180\) 2.56465 0.191158
\(181\) 21.1377 1.57115 0.785576 0.618765i \(-0.212367\pi\)
0.785576 + 0.618765i \(0.212367\pi\)
\(182\) 9.17970 0.680445
\(183\) −5.84740 −0.432252
\(184\) −1.20638 −0.0889358
\(185\) 6.35324 0.467100
\(186\) 1.69354 0.124176
\(187\) −0.168589 −0.0123285
\(188\) −27.4938 −2.00519
\(189\) −1.00000 −0.0727393
\(190\) −14.4811 −1.05057
\(191\) −4.75992 −0.344416 −0.172208 0.985061i \(-0.555090\pi\)
−0.172208 + 0.985061i \(0.555090\pi\)
\(192\) −11.6995 −0.844341
\(193\) 6.70344 0.482524 0.241262 0.970460i \(-0.422439\pi\)
0.241262 + 0.970460i \(0.422439\pi\)
\(194\) −11.4518 −0.822192
\(195\) 4.29660 0.307686
\(196\) 2.56465 0.183189
\(197\) 21.1463 1.50661 0.753306 0.657670i \(-0.228458\pi\)
0.753306 + 0.657670i \(0.228458\pi\)
\(198\) −3.81715 −0.271273
\(199\) 24.0722 1.70643 0.853216 0.521557i \(-0.174649\pi\)
0.853216 + 0.521557i \(0.174649\pi\)
\(200\) −1.20638 −0.0853042
\(201\) 13.5810 0.957927
\(202\) 17.6861 1.24439
\(203\) 2.21636 0.155558
\(204\) −0.242004 −0.0169437
\(205\) 7.49896 0.523750
\(206\) 37.5890 2.61895
\(207\) 1.00000 0.0695048
\(208\) −10.9643 −0.760239
\(209\) 12.1097 0.837646
\(210\) 2.13650 0.147433
\(211\) −0.851389 −0.0586121 −0.0293060 0.999570i \(-0.509330\pi\)
−0.0293060 + 0.999570i \(0.509330\pi\)
\(212\) −26.6168 −1.82805
\(213\) 5.24064 0.359083
\(214\) 27.7008 1.89359
\(215\) −8.14100 −0.555211
\(216\) −1.20638 −0.0820840
\(217\) 0.792668 0.0538098
\(218\) −27.9559 −1.89341
\(219\) 11.5241 0.778726
\(220\) 4.58209 0.308925
\(221\) −0.405433 −0.0272724
\(222\) −13.5737 −0.911009
\(223\) 6.49348 0.434835 0.217418 0.976079i \(-0.430237\pi\)
0.217418 + 0.976079i \(0.430237\pi\)
\(224\) −7.86483 −0.525491
\(225\) 1.00000 0.0666667
\(226\) −6.96854 −0.463540
\(227\) 14.7385 0.978228 0.489114 0.872220i \(-0.337320\pi\)
0.489114 + 0.872220i \(0.337320\pi\)
\(228\) 17.3831 1.15122
\(229\) −17.0305 −1.12541 −0.562703 0.826659i \(-0.690239\pi\)
−0.562703 + 0.826659i \(0.690239\pi\)
\(230\) −2.13650 −0.140877
\(231\) −1.78663 −0.117552
\(232\) 2.67379 0.175543
\(233\) −4.08268 −0.267465 −0.133733 0.991017i \(-0.542696\pi\)
−0.133733 + 0.991017i \(0.542696\pi\)
\(234\) −9.17970 −0.600096
\(235\) −10.7203 −0.699315
\(236\) 7.93089 0.516257
\(237\) 11.7510 0.763311
\(238\) −0.201603 −0.0130680
\(239\) −23.2202 −1.50199 −0.750994 0.660309i \(-0.770425\pi\)
−0.750994 + 0.660309i \(0.770425\pi\)
\(240\) −2.55186 −0.164722
\(241\) 27.0592 1.74304 0.871520 0.490361i \(-0.163135\pi\)
0.871520 + 0.490361i \(0.163135\pi\)
\(242\) 16.6817 1.07234
\(243\) 1.00000 0.0641500
\(244\) −14.9965 −0.960056
\(245\) 1.00000 0.0638877
\(246\) −16.0216 −1.02150
\(247\) 29.1221 1.85300
\(248\) 0.956262 0.0607227
\(249\) −9.22806 −0.584804
\(250\) −2.13650 −0.135124
\(251\) 26.1630 1.65139 0.825695 0.564116i \(-0.190783\pi\)
0.825695 + 0.564116i \(0.190783\pi\)
\(252\) −2.56465 −0.161558
\(253\) 1.78663 0.112325
\(254\) −25.5538 −1.60339
\(255\) −0.0943613 −0.00590914
\(256\) 3.60128 0.225080
\(257\) 23.5375 1.46823 0.734113 0.679027i \(-0.237598\pi\)
0.734113 + 0.679027i \(0.237598\pi\)
\(258\) 17.3933 1.08286
\(259\) −6.35324 −0.394771
\(260\) 11.0193 0.683387
\(261\) −2.21636 −0.137190
\(262\) 22.1182 1.36647
\(263\) −18.4727 −1.13907 −0.569537 0.821966i \(-0.692877\pi\)
−0.569537 + 0.821966i \(0.692877\pi\)
\(264\) −2.15536 −0.132654
\(265\) −10.3783 −0.637534
\(266\) 14.4811 0.887894
\(267\) 5.11100 0.312788
\(268\) 34.8305 2.12761
\(269\) 14.3774 0.876606 0.438303 0.898827i \(-0.355580\pi\)
0.438303 + 0.898827i \(0.355580\pi\)
\(270\) −2.13650 −0.130024
\(271\) −18.0079 −1.09390 −0.546951 0.837165i \(-0.684211\pi\)
−0.546951 + 0.837165i \(0.684211\pi\)
\(272\) 0.240797 0.0146005
\(273\) −4.29660 −0.260042
\(274\) 15.1524 0.915391
\(275\) 1.78663 0.107738
\(276\) 2.56465 0.154374
\(277\) 5.94604 0.357263 0.178631 0.983916i \(-0.442833\pi\)
0.178631 + 0.983916i \(0.442833\pi\)
\(278\) −25.9018 −1.55349
\(279\) −0.792668 −0.0474558
\(280\) 1.20638 0.0720952
\(281\) −17.9816 −1.07269 −0.536347 0.843998i \(-0.680196\pi\)
−0.536347 + 0.843998i \(0.680196\pi\)
\(282\) 22.9040 1.36391
\(283\) −10.0103 −0.595049 −0.297524 0.954714i \(-0.596161\pi\)
−0.297524 + 0.954714i \(0.596161\pi\)
\(284\) 13.4404 0.797542
\(285\) 6.77795 0.401491
\(286\) −16.4008 −0.969797
\(287\) −7.49896 −0.442650
\(288\) 7.86483 0.463440
\(289\) −16.9911 −0.999476
\(290\) 4.73527 0.278065
\(291\) 5.36007 0.314213
\(292\) 29.5553 1.72959
\(293\) 28.8258 1.68402 0.842011 0.539460i \(-0.181371\pi\)
0.842011 + 0.539460i \(0.181371\pi\)
\(294\) −2.13650 −0.124603
\(295\) 3.09238 0.180045
\(296\) −7.66444 −0.445487
\(297\) 1.78663 0.103671
\(298\) 4.14959 0.240379
\(299\) 4.29660 0.248479
\(300\) 2.56465 0.148070
\(301\) 8.14100 0.469239
\(302\) −16.1638 −0.930122
\(303\) −8.27803 −0.475561
\(304\) −17.2964 −0.992016
\(305\) −5.84740 −0.334821
\(306\) 0.201603 0.0115249
\(307\) −21.9082 −1.25037 −0.625184 0.780478i \(-0.714976\pi\)
−0.625184 + 0.780478i \(0.714976\pi\)
\(308\) −4.58209 −0.261089
\(309\) −17.5937 −1.00087
\(310\) 1.69354 0.0961866
\(311\) 31.9210 1.81008 0.905038 0.425331i \(-0.139842\pi\)
0.905038 + 0.425331i \(0.139842\pi\)
\(312\) −5.18334 −0.293449
\(313\) −0.0950302 −0.00537142 −0.00268571 0.999996i \(-0.500855\pi\)
−0.00268571 + 0.999996i \(0.500855\pi\)
\(314\) 45.7152 2.57986
\(315\) −1.00000 −0.0563436
\(316\) 30.1373 1.69536
\(317\) 9.58935 0.538591 0.269296 0.963058i \(-0.413209\pi\)
0.269296 + 0.963058i \(0.413209\pi\)
\(318\) 22.1733 1.24342
\(319\) −3.95983 −0.221708
\(320\) −11.6995 −0.654023
\(321\) −12.9655 −0.723662
\(322\) 2.13650 0.119063
\(323\) −0.639576 −0.0355870
\(324\) 2.56465 0.142481
\(325\) 4.29660 0.238332
\(326\) −0.220408 −0.0122072
\(327\) 13.0849 0.723596
\(328\) −9.04662 −0.499516
\(329\) 10.7203 0.591029
\(330\) −3.81715 −0.210127
\(331\) 4.28108 0.235309 0.117655 0.993055i \(-0.462462\pi\)
0.117655 + 0.993055i \(0.462462\pi\)
\(332\) −23.6668 −1.29888
\(333\) 6.35324 0.348155
\(334\) 33.7297 1.84561
\(335\) 13.5810 0.742007
\(336\) 2.55186 0.139215
\(337\) 36.2564 1.97501 0.987507 0.157576i \(-0.0503680\pi\)
0.987507 + 0.157576i \(0.0503680\pi\)
\(338\) −11.6669 −0.634596
\(339\) 3.26165 0.177149
\(340\) −0.242004 −0.0131245
\(341\) −1.41621 −0.0766919
\(342\) −14.4811 −0.783049
\(343\) −1.00000 −0.0539949
\(344\) 9.82116 0.529522
\(345\) 1.00000 0.0538382
\(346\) 15.5323 0.835023
\(347\) 18.9656 1.01812 0.509062 0.860730i \(-0.329992\pi\)
0.509062 + 0.860730i \(0.329992\pi\)
\(348\) −5.68421 −0.304705
\(349\) −4.03031 −0.215738 −0.107869 0.994165i \(-0.534403\pi\)
−0.107869 + 0.994165i \(0.534403\pi\)
\(350\) 2.13650 0.114201
\(351\) 4.29660 0.229335
\(352\) 14.0516 0.748951
\(353\) 3.50110 0.186345 0.0931723 0.995650i \(-0.470299\pi\)
0.0931723 + 0.995650i \(0.470299\pi\)
\(354\) −6.60689 −0.351152
\(355\) 5.24064 0.278144
\(356\) 13.1079 0.694720
\(357\) 0.0943613 0.00499413
\(358\) −39.2037 −2.07198
\(359\) 7.71988 0.407440 0.203720 0.979029i \(-0.434697\pi\)
0.203720 + 0.979029i \(0.434697\pi\)
\(360\) −1.20638 −0.0635820
\(361\) 26.9406 1.41793
\(362\) −45.1608 −2.37360
\(363\) −7.80794 −0.409811
\(364\) −11.0193 −0.577567
\(365\) 11.5241 0.603199
\(366\) 12.4930 0.653019
\(367\) −3.63482 −0.189736 −0.0948680 0.995490i \(-0.530243\pi\)
−0.0948680 + 0.995490i \(0.530243\pi\)
\(368\) −2.55186 −0.133025
\(369\) 7.49896 0.390380
\(370\) −13.5737 −0.705665
\(371\) 10.3783 0.538815
\(372\) −2.03292 −0.105402
\(373\) 9.22651 0.477730 0.238865 0.971053i \(-0.423225\pi\)
0.238865 + 0.971053i \(0.423225\pi\)
\(374\) 0.360192 0.0186251
\(375\) 1.00000 0.0516398
\(376\) 12.9328 0.666958
\(377\) −9.52283 −0.490451
\(378\) 2.13650 0.109890
\(379\) 22.1036 1.13539 0.567693 0.823240i \(-0.307836\pi\)
0.567693 + 0.823240i \(0.307836\pi\)
\(380\) 17.3831 0.891733
\(381\) 11.9605 0.612758
\(382\) 10.1696 0.520322
\(383\) −22.7321 −1.16156 −0.580778 0.814062i \(-0.697252\pi\)
−0.580778 + 0.814062i \(0.697252\pi\)
\(384\) 9.26643 0.472876
\(385\) −1.78663 −0.0910553
\(386\) −14.3219 −0.728967
\(387\) −8.14100 −0.413830
\(388\) 13.7467 0.697884
\(389\) 35.4091 1.79531 0.897655 0.440698i \(-0.145269\pi\)
0.897655 + 0.440698i \(0.145269\pi\)
\(390\) −9.17970 −0.464832
\(391\) −0.0943613 −0.00477206
\(392\) −1.20638 −0.0609316
\(393\) −10.3525 −0.522215
\(394\) −45.1792 −2.27610
\(395\) 11.7510 0.591258
\(396\) 4.58209 0.230259
\(397\) −35.9923 −1.80640 −0.903200 0.429219i \(-0.858789\pi\)
−0.903200 + 0.429219i \(0.858789\pi\)
\(398\) −51.4304 −2.57797
\(399\) −6.77795 −0.339322
\(400\) −2.55186 −0.127593
\(401\) 16.8014 0.839024 0.419512 0.907750i \(-0.362201\pi\)
0.419512 + 0.907750i \(0.362201\pi\)
\(402\) −29.0158 −1.44718
\(403\) −3.40578 −0.169654
\(404\) −21.2303 −1.05625
\(405\) 1.00000 0.0496904
\(406\) −4.73527 −0.235008
\(407\) 11.3509 0.562644
\(408\) 0.113836 0.00563572
\(409\) −15.2505 −0.754087 −0.377044 0.926196i \(-0.623059\pi\)
−0.377044 + 0.926196i \(0.623059\pi\)
\(410\) −16.0216 −0.791249
\(411\) −7.09215 −0.349830
\(412\) −45.1216 −2.22298
\(413\) −3.09238 −0.152166
\(414\) −2.13650 −0.105003
\(415\) −9.22806 −0.452987
\(416\) 33.7920 1.65679
\(417\) 12.1235 0.593688
\(418\) −25.8725 −1.26546
\(419\) 20.7184 1.01216 0.506080 0.862486i \(-0.331094\pi\)
0.506080 + 0.862486i \(0.331094\pi\)
\(420\) −2.56465 −0.125142
\(421\) −2.34883 −0.114475 −0.0572374 0.998361i \(-0.518229\pi\)
−0.0572374 + 0.998361i \(0.518229\pi\)
\(422\) 1.81900 0.0885474
\(423\) −10.7203 −0.521239
\(424\) 12.5202 0.608035
\(425\) −0.0943613 −0.00457720
\(426\) −11.1967 −0.542480
\(427\) 5.84740 0.282975
\(428\) −33.2519 −1.60729
\(429\) 7.67644 0.370622
\(430\) 17.3933 0.838778
\(431\) −16.4999 −0.794774 −0.397387 0.917651i \(-0.630083\pi\)
−0.397387 + 0.917651i \(0.630083\pi\)
\(432\) −2.55186 −0.122777
\(433\) −21.5192 −1.03415 −0.517074 0.855941i \(-0.672979\pi\)
−0.517074 + 0.855941i \(0.672979\pi\)
\(434\) −1.69354 −0.0812925
\(435\) −2.21636 −0.106267
\(436\) 33.5582 1.60715
\(437\) 6.77795 0.324233
\(438\) −24.6213 −1.17645
\(439\) −18.2234 −0.869754 −0.434877 0.900490i \(-0.643208\pi\)
−0.434877 + 0.900490i \(0.643208\pi\)
\(440\) −2.15536 −0.102753
\(441\) 1.00000 0.0476190
\(442\) 0.866209 0.0412014
\(443\) 18.3345 0.871099 0.435550 0.900165i \(-0.356554\pi\)
0.435550 + 0.900165i \(0.356554\pi\)
\(444\) 16.2939 0.773272
\(445\) 5.11100 0.242285
\(446\) −13.8733 −0.656922
\(447\) −1.94223 −0.0918645
\(448\) 11.6995 0.552751
\(449\) −25.7028 −1.21299 −0.606495 0.795087i \(-0.707425\pi\)
−0.606495 + 0.795087i \(0.707425\pi\)
\(450\) −2.13650 −0.100716
\(451\) 13.3979 0.630882
\(452\) 8.36501 0.393457
\(453\) 7.56553 0.355460
\(454\) −31.4889 −1.47785
\(455\) −4.29660 −0.201428
\(456\) −8.17680 −0.382914
\(457\) −31.0069 −1.45044 −0.725220 0.688517i \(-0.758262\pi\)
−0.725220 + 0.688517i \(0.758262\pi\)
\(458\) 36.3857 1.70019
\(459\) −0.0943613 −0.00440441
\(460\) 2.56465 0.119578
\(461\) −4.01677 −0.187079 −0.0935397 0.995616i \(-0.529818\pi\)
−0.0935397 + 0.995616i \(0.529818\pi\)
\(462\) 3.81715 0.177590
\(463\) −2.90641 −0.135072 −0.0675361 0.997717i \(-0.521514\pi\)
−0.0675361 + 0.997717i \(0.521514\pi\)
\(464\) 5.65586 0.262567
\(465\) −0.792668 −0.0367591
\(466\) 8.72267 0.404070
\(467\) −13.8420 −0.640532 −0.320266 0.947328i \(-0.603772\pi\)
−0.320266 + 0.947328i \(0.603772\pi\)
\(468\) 11.0193 0.509366
\(469\) −13.5810 −0.627111
\(470\) 22.9040 1.05648
\(471\) −21.3972 −0.985930
\(472\) −3.73060 −0.171715
\(473\) −14.5450 −0.668779
\(474\) −25.1061 −1.15316
\(475\) 6.77795 0.310994
\(476\) 0.242004 0.0110922
\(477\) −10.3783 −0.475190
\(478\) 49.6100 2.26911
\(479\) −24.5487 −1.12166 −0.560829 0.827931i \(-0.689518\pi\)
−0.560829 + 0.827931i \(0.689518\pi\)
\(480\) 7.86483 0.358979
\(481\) 27.2973 1.24465
\(482\) −57.8122 −2.63327
\(483\) −1.00000 −0.0455016
\(484\) −20.0247 −0.910212
\(485\) 5.36007 0.243388
\(486\) −2.13650 −0.0969138
\(487\) −25.2374 −1.14362 −0.571808 0.820388i \(-0.693758\pi\)
−0.571808 + 0.820388i \(0.693758\pi\)
\(488\) 7.05420 0.319329
\(489\) 0.103163 0.00466518
\(490\) −2.13650 −0.0965174
\(491\) −20.3759 −0.919553 −0.459777 0.888035i \(-0.652071\pi\)
−0.459777 + 0.888035i \(0.652071\pi\)
\(492\) 19.2322 0.867056
\(493\) 0.209139 0.00941915
\(494\) −62.2195 −2.79939
\(495\) 1.78663 0.0803032
\(496\) 2.02278 0.0908255
\(497\) −5.24064 −0.235075
\(498\) 19.7158 0.883486
\(499\) 1.90564 0.0853081 0.0426541 0.999090i \(-0.486419\pi\)
0.0426541 + 0.999090i \(0.486419\pi\)
\(500\) 2.56465 0.114695
\(501\) −15.7873 −0.705325
\(502\) −55.8973 −2.49482
\(503\) −34.3617 −1.53211 −0.766057 0.642773i \(-0.777784\pi\)
−0.766057 + 0.642773i \(0.777784\pi\)
\(504\) 1.20638 0.0537366
\(505\) −8.27803 −0.368368
\(506\) −3.81715 −0.169693
\(507\) 5.46074 0.242520
\(508\) 30.6747 1.36097
\(509\) 21.8470 0.968353 0.484176 0.874970i \(-0.339119\pi\)
0.484176 + 0.874970i \(0.339119\pi\)
\(510\) 0.201603 0.00892715
\(511\) −11.5241 −0.509796
\(512\) −26.2270 −1.15908
\(513\) 6.77795 0.299254
\(514\) −50.2879 −2.21810
\(515\) −17.5937 −0.775270
\(516\) −20.8788 −0.919139
\(517\) −19.1532 −0.842359
\(518\) 13.5737 0.596395
\(519\) −7.26997 −0.319116
\(520\) −5.18334 −0.227305
\(521\) 10.5180 0.460804 0.230402 0.973096i \(-0.425996\pi\)
0.230402 + 0.973096i \(0.425996\pi\)
\(522\) 4.73527 0.207257
\(523\) −12.0486 −0.526849 −0.263425 0.964680i \(-0.584852\pi\)
−0.263425 + 0.964680i \(0.584852\pi\)
\(524\) −26.5506 −1.15987
\(525\) −1.00000 −0.0436436
\(526\) 39.4669 1.72084
\(527\) 0.0747972 0.00325822
\(528\) −4.55924 −0.198416
\(529\) 1.00000 0.0434783
\(530\) 22.1733 0.963147
\(531\) 3.09238 0.134198
\(532\) −17.3831 −0.753652
\(533\) 32.2200 1.39560
\(534\) −10.9197 −0.472541
\(535\) −12.9655 −0.560546
\(536\) −16.3838 −0.707674
\(537\) 18.3495 0.791838
\(538\) −30.7174 −1.32432
\(539\) 1.78663 0.0769558
\(540\) 2.56465 0.110365
\(541\) 31.6201 1.35945 0.679726 0.733466i \(-0.262099\pi\)
0.679726 + 0.733466i \(0.262099\pi\)
\(542\) 38.4739 1.65260
\(543\) 21.1377 0.907105
\(544\) −0.742136 −0.0318188
\(545\) 13.0849 0.560495
\(546\) 9.17970 0.392855
\(547\) −36.1537 −1.54582 −0.772911 0.634514i \(-0.781200\pi\)
−0.772911 + 0.634514i \(0.781200\pi\)
\(548\) −18.1889 −0.776992
\(549\) −5.84740 −0.249561
\(550\) −3.81715 −0.162764
\(551\) −15.0224 −0.639976
\(552\) −1.20638 −0.0513471
\(553\) −11.7510 −0.499705
\(554\) −12.7037 −0.539730
\(555\) 6.35324 0.269680
\(556\) 31.0925 1.31861
\(557\) 15.4602 0.655070 0.327535 0.944839i \(-0.393782\pi\)
0.327535 + 0.944839i \(0.393782\pi\)
\(558\) 1.69354 0.0716932
\(559\) −34.9786 −1.47944
\(560\) 2.55186 0.107836
\(561\) −0.168589 −0.00711784
\(562\) 38.4178 1.62056
\(563\) −30.2491 −1.27485 −0.637425 0.770513i \(-0.720000\pi\)
−0.637425 + 0.770513i \(0.720000\pi\)
\(564\) −27.4938 −1.15770
\(565\) 3.26165 0.137219
\(566\) 21.3870 0.898962
\(567\) −1.00000 −0.0419961
\(568\) −6.32222 −0.265275
\(569\) −3.54282 −0.148523 −0.0742613 0.997239i \(-0.523660\pi\)
−0.0742613 + 0.997239i \(0.523660\pi\)
\(570\) −14.4811 −0.606547
\(571\) −17.3304 −0.725256 −0.362628 0.931934i \(-0.618121\pi\)
−0.362628 + 0.931934i \(0.618121\pi\)
\(572\) 19.6874 0.823172
\(573\) −4.75992 −0.198849
\(574\) 16.0216 0.668727
\(575\) 1.00000 0.0417029
\(576\) −11.6995 −0.487480
\(577\) −12.0064 −0.499833 −0.249917 0.968267i \(-0.580403\pi\)
−0.249917 + 0.968267i \(0.580403\pi\)
\(578\) 36.3016 1.50995
\(579\) 6.70344 0.278585
\(580\) −5.68421 −0.236024
\(581\) 9.22806 0.382844
\(582\) −11.4518 −0.474693
\(583\) −18.5422 −0.767941
\(584\) −13.9025 −0.575289
\(585\) 4.29660 0.177642
\(586\) −61.5865 −2.54412
\(587\) 3.28838 0.135726 0.0678629 0.997695i \(-0.478382\pi\)
0.0678629 + 0.997695i \(0.478382\pi\)
\(588\) 2.56465 0.105764
\(589\) −5.37266 −0.221377
\(590\) −6.60689 −0.272001
\(591\) 21.1463 0.869843
\(592\) −16.2126 −0.666334
\(593\) 32.4005 1.33053 0.665265 0.746607i \(-0.268318\pi\)
0.665265 + 0.746607i \(0.268318\pi\)
\(594\) −3.81715 −0.156620
\(595\) 0.0943613 0.00386844
\(596\) −4.98115 −0.204036
\(597\) 24.0722 0.985209
\(598\) −9.17970 −0.375386
\(599\) 8.30020 0.339137 0.169568 0.985518i \(-0.445763\pi\)
0.169568 + 0.985518i \(0.445763\pi\)
\(600\) −1.20638 −0.0492504
\(601\) −6.31789 −0.257712 −0.128856 0.991663i \(-0.541131\pi\)
−0.128856 + 0.991663i \(0.541131\pi\)
\(602\) −17.3933 −0.708897
\(603\) 13.5810 0.553060
\(604\) 19.4030 0.789495
\(605\) −7.80794 −0.317438
\(606\) 17.6861 0.718447
\(607\) 32.3279 1.31215 0.656075 0.754695i \(-0.272215\pi\)
0.656075 + 0.754695i \(0.272215\pi\)
\(608\) 53.3074 2.16190
\(609\) 2.21636 0.0898116
\(610\) 12.4930 0.505827
\(611\) −46.0608 −1.86342
\(612\) −0.242004 −0.00978244
\(613\) 1.22597 0.0495165 0.0247583 0.999693i \(-0.492118\pi\)
0.0247583 + 0.999693i \(0.492118\pi\)
\(614\) 46.8070 1.88898
\(615\) 7.49896 0.302387
\(616\) 2.15536 0.0868421
\(617\) −38.3230 −1.54282 −0.771412 0.636336i \(-0.780449\pi\)
−0.771412 + 0.636336i \(0.780449\pi\)
\(618\) 37.5890 1.51205
\(619\) −14.6750 −0.589838 −0.294919 0.955522i \(-0.595293\pi\)
−0.294919 + 0.955522i \(0.595293\pi\)
\(620\) −2.03292 −0.0816440
\(621\) 1.00000 0.0401286
\(622\) −68.1994 −2.73455
\(623\) −5.11100 −0.204768
\(624\) −10.9643 −0.438924
\(625\) 1.00000 0.0400000
\(626\) 0.203032 0.00811481
\(627\) 12.1097 0.483615
\(628\) −54.8763 −2.18980
\(629\) −0.599500 −0.0239036
\(630\) 2.13650 0.0851204
\(631\) −6.56327 −0.261280 −0.130640 0.991430i \(-0.541703\pi\)
−0.130640 + 0.991430i \(0.541703\pi\)
\(632\) −14.1762 −0.563901
\(633\) −0.851389 −0.0338397
\(634\) −20.4877 −0.813670
\(635\) 11.9605 0.474640
\(636\) −26.6168 −1.05542
\(637\) 4.29660 0.170237
\(638\) 8.46020 0.334943
\(639\) 5.24064 0.207317
\(640\) 9.26643 0.366288
\(641\) −26.0757 −1.02993 −0.514965 0.857211i \(-0.672195\pi\)
−0.514965 + 0.857211i \(0.672195\pi\)
\(642\) 27.7008 1.09326
\(643\) 3.55563 0.140220 0.0701102 0.997539i \(-0.477665\pi\)
0.0701102 + 0.997539i \(0.477665\pi\)
\(644\) −2.56465 −0.101061
\(645\) −8.14100 −0.320551
\(646\) 1.36646 0.0537626
\(647\) −8.95614 −0.352102 −0.176051 0.984381i \(-0.556332\pi\)
−0.176051 + 0.984381i \(0.556332\pi\)
\(648\) −1.20638 −0.0473912
\(649\) 5.52495 0.216873
\(650\) −9.17970 −0.360057
\(651\) 0.792668 0.0310671
\(652\) 0.264576 0.0103616
\(653\) 18.6879 0.731312 0.365656 0.930750i \(-0.380845\pi\)
0.365656 + 0.930750i \(0.380845\pi\)
\(654\) −27.9559 −1.09316
\(655\) −10.3525 −0.404506
\(656\) −19.1363 −0.747148
\(657\) 11.5241 0.449598
\(658\) −22.9040 −0.892890
\(659\) −40.3803 −1.57299 −0.786496 0.617596i \(-0.788107\pi\)
−0.786496 + 0.617596i \(0.788107\pi\)
\(660\) 4.58209 0.178358
\(661\) −19.6916 −0.765916 −0.382958 0.923766i \(-0.625095\pi\)
−0.382958 + 0.923766i \(0.625095\pi\)
\(662\) −9.14655 −0.355491
\(663\) −0.405433 −0.0157457
\(664\) 11.1326 0.432028
\(665\) −6.77795 −0.262838
\(666\) −13.5737 −0.525971
\(667\) −2.21636 −0.0858180
\(668\) −40.4890 −1.56657
\(669\) 6.49348 0.251052
\(670\) −29.0158 −1.12098
\(671\) −10.4472 −0.403308
\(672\) −7.86483 −0.303392
\(673\) −23.6121 −0.910180 −0.455090 0.890445i \(-0.650393\pi\)
−0.455090 + 0.890445i \(0.650393\pi\)
\(674\) −77.4620 −2.98373
\(675\) 1.00000 0.0384900
\(676\) 14.0049 0.538650
\(677\) 12.1823 0.468205 0.234103 0.972212i \(-0.424785\pi\)
0.234103 + 0.972212i \(0.424785\pi\)
\(678\) −6.96854 −0.267625
\(679\) −5.36007 −0.205701
\(680\) 0.113836 0.00436541
\(681\) 14.7385 0.564780
\(682\) 3.02573 0.115861
\(683\) 11.7118 0.448139 0.224069 0.974573i \(-0.428066\pi\)
0.224069 + 0.974573i \(0.428066\pi\)
\(684\) 17.3831 0.664659
\(685\) −7.09215 −0.270977
\(686\) 2.13650 0.0815721
\(687\) −17.0305 −0.649754
\(688\) 20.7747 0.792028
\(689\) −44.5914 −1.69880
\(690\) −2.13650 −0.0813353
\(691\) −0.455045 −0.0173107 −0.00865535 0.999963i \(-0.502755\pi\)
−0.00865535 + 0.999963i \(0.502755\pi\)
\(692\) −18.6450 −0.708775
\(693\) −1.78663 −0.0678686
\(694\) −40.5200 −1.53812
\(695\) 12.1235 0.459869
\(696\) 2.67379 0.101350
\(697\) −0.707612 −0.0268027
\(698\) 8.61078 0.325923
\(699\) −4.08268 −0.154421
\(700\) −2.56465 −0.0969348
\(701\) −11.7366 −0.443285 −0.221642 0.975128i \(-0.571142\pi\)
−0.221642 + 0.975128i \(0.571142\pi\)
\(702\) −9.17970 −0.346465
\(703\) 43.0619 1.62411
\(704\) −20.9028 −0.787803
\(705\) −10.7203 −0.403750
\(706\) −7.48011 −0.281518
\(707\) 8.27803 0.311327
\(708\) 7.93089 0.298061
\(709\) 43.9572 1.65085 0.825424 0.564513i \(-0.190936\pi\)
0.825424 + 0.564513i \(0.190936\pi\)
\(710\) −11.1967 −0.420203
\(711\) 11.7510 0.440698
\(712\) −6.16583 −0.231074
\(713\) −0.792668 −0.0296857
\(714\) −0.201603 −0.00754482
\(715\) 7.67644 0.287083
\(716\) 47.0600 1.75872
\(717\) −23.2202 −0.867173
\(718\) −16.4936 −0.615534
\(719\) −25.9590 −0.968106 −0.484053 0.875039i \(-0.660836\pi\)
−0.484053 + 0.875039i \(0.660836\pi\)
\(720\) −2.55186 −0.0951023
\(721\) 17.5937 0.655222
\(722\) −57.5587 −2.14211
\(723\) 27.0592 1.00634
\(724\) 54.2108 2.01473
\(725\) −2.21636 −0.0823137
\(726\) 16.6817 0.619116
\(727\) 33.0098 1.22427 0.612133 0.790755i \(-0.290312\pi\)
0.612133 + 0.790755i \(0.290312\pi\)
\(728\) 5.18334 0.192107
\(729\) 1.00000 0.0370370
\(730\) −24.6213 −0.911275
\(731\) 0.768195 0.0284127
\(732\) −14.9965 −0.554288
\(733\) −11.4160 −0.421659 −0.210830 0.977523i \(-0.567617\pi\)
−0.210830 + 0.977523i \(0.567617\pi\)
\(734\) 7.76581 0.286641
\(735\) 1.00000 0.0368856
\(736\) 7.86483 0.289902
\(737\) 24.2642 0.893783
\(738\) −16.0216 −0.589762
\(739\) −10.8797 −0.400215 −0.200107 0.979774i \(-0.564129\pi\)
−0.200107 + 0.979774i \(0.564129\pi\)
\(740\) 16.2939 0.598974
\(741\) 29.1221 1.06983
\(742\) −22.1733 −0.814008
\(743\) −33.1400 −1.21579 −0.607895 0.794017i \(-0.707986\pi\)
−0.607895 + 0.794017i \(0.707986\pi\)
\(744\) 0.956262 0.0350583
\(745\) −1.94223 −0.0711579
\(746\) −19.7125 −0.721725
\(747\) −9.22806 −0.337637
\(748\) −0.432373 −0.0158091
\(749\) 12.9655 0.473748
\(750\) −2.13650 −0.0780141
\(751\) −18.8474 −0.687752 −0.343876 0.939015i \(-0.611740\pi\)
−0.343876 + 0.939015i \(0.611740\pi\)
\(752\) 27.3567 0.997597
\(753\) 26.1630 0.953431
\(754\) 20.3456 0.740942
\(755\) 7.56553 0.275338
\(756\) −2.56465 −0.0932755
\(757\) 1.55270 0.0564338 0.0282169 0.999602i \(-0.491017\pi\)
0.0282169 + 0.999602i \(0.491017\pi\)
\(758\) −47.2245 −1.71527
\(759\) 1.78663 0.0648507
\(760\) −8.17680 −0.296604
\(761\) −23.2767 −0.843781 −0.421890 0.906647i \(-0.638633\pi\)
−0.421890 + 0.906647i \(0.638633\pi\)
\(762\) −25.5538 −0.925715
\(763\) −13.0849 −0.473705
\(764\) −12.2076 −0.441654
\(765\) −0.0943613 −0.00341164
\(766\) 48.5673 1.75481
\(767\) 13.2867 0.479756
\(768\) 3.60128 0.129950
\(769\) −27.4213 −0.988839 −0.494419 0.869224i \(-0.664619\pi\)
−0.494419 + 0.869224i \(0.664619\pi\)
\(770\) 3.81715 0.137561
\(771\) 23.5375 0.847681
\(772\) 17.1920 0.618754
\(773\) −49.0281 −1.76342 −0.881708 0.471795i \(-0.843606\pi\)
−0.881708 + 0.471795i \(0.843606\pi\)
\(774\) 17.3933 0.625189
\(775\) −0.792668 −0.0284735
\(776\) −6.46630 −0.232127
\(777\) −6.35324 −0.227921
\(778\) −75.6516 −2.71224
\(779\) 50.8276 1.82109
\(780\) 11.0193 0.394553
\(781\) 9.36310 0.335038
\(782\) 0.201603 0.00720933
\(783\) −2.21636 −0.0792064
\(784\) −2.55186 −0.0911379
\(785\) −21.3972 −0.763698
\(786\) 22.1182 0.788929
\(787\) −16.1608 −0.576069 −0.288034 0.957620i \(-0.593002\pi\)
−0.288034 + 0.957620i \(0.593002\pi\)
\(788\) 54.2330 1.93197
\(789\) −18.4727 −0.657644
\(790\) −25.1061 −0.893236
\(791\) −3.26165 −0.115971
\(792\) −2.15536 −0.0765876
\(793\) −25.1239 −0.892176
\(794\) 76.8977 2.72900
\(795\) −10.3783 −0.368081
\(796\) 61.7368 2.18820
\(797\) 20.1299 0.713038 0.356519 0.934288i \(-0.383963\pi\)
0.356519 + 0.934288i \(0.383963\pi\)
\(798\) 14.4811 0.512626
\(799\) 1.01158 0.0357872
\(800\) 7.86483 0.278064
\(801\) 5.11100 0.180588
\(802\) −35.8963 −1.26754
\(803\) 20.5893 0.726582
\(804\) 34.8305 1.22838
\(805\) −1.00000 −0.0352454
\(806\) 7.27646 0.256302
\(807\) 14.3774 0.506109
\(808\) 9.98648 0.351323
\(809\) −10.0935 −0.354868 −0.177434 0.984133i \(-0.556780\pi\)
−0.177434 + 0.984133i \(0.556780\pi\)
\(810\) −2.13650 −0.0750691
\(811\) 7.27148 0.255336 0.127668 0.991817i \(-0.459251\pi\)
0.127668 + 0.991817i \(0.459251\pi\)
\(812\) 5.68421 0.199477
\(813\) −18.0079 −0.631564
\(814\) −24.2513 −0.850007
\(815\) 0.103163 0.00361363
\(816\) 0.240797 0.00842959
\(817\) −55.1793 −1.93048
\(818\) 32.5827 1.13923
\(819\) −4.29660 −0.150135
\(820\) 19.2322 0.671619
\(821\) −55.1620 −1.92517 −0.962584 0.270984i \(-0.912651\pi\)
−0.962584 + 0.270984i \(0.912651\pi\)
\(822\) 15.1524 0.528501
\(823\) 22.5494 0.786024 0.393012 0.919533i \(-0.371433\pi\)
0.393012 + 0.919533i \(0.371433\pi\)
\(824\) 21.2247 0.739398
\(825\) 1.78663 0.0622026
\(826\) 6.60689 0.229883
\(827\) −12.2478 −0.425898 −0.212949 0.977063i \(-0.568307\pi\)
−0.212949 + 0.977063i \(0.568307\pi\)
\(828\) 2.56465 0.0891278
\(829\) −17.4614 −0.606460 −0.303230 0.952917i \(-0.598065\pi\)
−0.303230 + 0.952917i \(0.598065\pi\)
\(830\) 19.7158 0.684345
\(831\) 5.94604 0.206266
\(832\) −50.2681 −1.74273
\(833\) −0.0943613 −0.00326943
\(834\) −25.9018 −0.896907
\(835\) −15.7873 −0.546343
\(836\) 31.0572 1.07414
\(837\) −0.792668 −0.0273986
\(838\) −44.2650 −1.52911
\(839\) 34.6725 1.19703 0.598514 0.801113i \(-0.295758\pi\)
0.598514 + 0.801113i \(0.295758\pi\)
\(840\) 1.20638 0.0416242
\(841\) −24.0877 −0.830611
\(842\) 5.01828 0.172941
\(843\) −17.9816 −0.619320
\(844\) −2.18352 −0.0751598
\(845\) 5.46074 0.187855
\(846\) 22.9040 0.787455
\(847\) 7.80794 0.268284
\(848\) 26.4840 0.909464
\(849\) −10.0103 −0.343552
\(850\) 0.201603 0.00691494
\(851\) 6.35324 0.217786
\(852\) 13.4404 0.460461
\(853\) 42.7465 1.46361 0.731807 0.681512i \(-0.238677\pi\)
0.731807 + 0.681512i \(0.238677\pi\)
\(854\) −12.4930 −0.427502
\(855\) 6.77795 0.231801
\(856\) 15.6413 0.534609
\(857\) 46.8484 1.60031 0.800155 0.599793i \(-0.204750\pi\)
0.800155 + 0.599793i \(0.204750\pi\)
\(858\) −16.4008 −0.559913
\(859\) 4.28101 0.146066 0.0730331 0.997330i \(-0.476732\pi\)
0.0730331 + 0.997330i \(0.476732\pi\)
\(860\) −20.8788 −0.711962
\(861\) −7.49896 −0.255564
\(862\) 35.2522 1.20069
\(863\) 15.1862 0.516943 0.258472 0.966019i \(-0.416781\pi\)
0.258472 + 0.966019i \(0.416781\pi\)
\(864\) 7.86483 0.267567
\(865\) −7.26997 −0.247187
\(866\) 45.9759 1.56233
\(867\) −16.9911 −0.577048
\(868\) 2.03292 0.0690018
\(869\) 20.9948 0.712199
\(870\) 4.73527 0.160541
\(871\) 58.3519 1.97718
\(872\) −15.7854 −0.534561
\(873\) 5.36007 0.181411
\(874\) −14.4811 −0.489831
\(875\) −1.00000 −0.0338062
\(876\) 29.5553 0.998581
\(877\) −22.9270 −0.774190 −0.387095 0.922040i \(-0.626521\pi\)
−0.387095 + 0.922040i \(0.626521\pi\)
\(878\) 38.9343 1.31397
\(879\) 28.8258 0.972271
\(880\) −4.55924 −0.153692
\(881\) −27.7091 −0.933542 −0.466771 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(882\) −2.13650 −0.0719399
\(883\) 45.3010 1.52450 0.762249 0.647283i \(-0.224095\pi\)
0.762249 + 0.647283i \(0.224095\pi\)
\(884\) −1.03979 −0.0349721
\(885\) 3.09238 0.103949
\(886\) −39.1718 −1.31600
\(887\) 42.9145 1.44093 0.720464 0.693492i \(-0.243929\pi\)
0.720464 + 0.693492i \(0.243929\pi\)
\(888\) −7.66444 −0.257202
\(889\) −11.9605 −0.401144
\(890\) −10.9197 −0.366029
\(891\) 1.78663 0.0598545
\(892\) 16.6535 0.557601
\(893\) −72.6616 −2.43153
\(894\) 4.14959 0.138783
\(895\) 18.3495 0.613355
\(896\) −9.26643 −0.309570
\(897\) 4.29660 0.143459
\(898\) 54.9142 1.83251
\(899\) 1.75684 0.0585940
\(900\) 2.56465 0.0854884
\(901\) 0.979311 0.0326256
\(902\) −28.6247 −0.953097
\(903\) 8.14100 0.270915
\(904\) −3.93480 −0.130870
\(905\) 21.1377 0.702641
\(906\) −16.1638 −0.537006
\(907\) −1.29413 −0.0429707 −0.0214854 0.999769i \(-0.506840\pi\)
−0.0214854 + 0.999769i \(0.506840\pi\)
\(908\) 37.7991 1.25441
\(909\) −8.27803 −0.274565
\(910\) 9.17970 0.304304
\(911\) 8.05207 0.266777 0.133388 0.991064i \(-0.457414\pi\)
0.133388 + 0.991064i \(0.457414\pi\)
\(912\) −17.2964 −0.572741
\(913\) −16.4872 −0.545645
\(914\) 66.2464 2.19123
\(915\) −5.84740 −0.193309
\(916\) −43.6773 −1.44314
\(917\) 10.3525 0.341870
\(918\) 0.201603 0.00665391
\(919\) −48.3126 −1.59369 −0.796843 0.604186i \(-0.793498\pi\)
−0.796843 + 0.604186i \(0.793498\pi\)
\(920\) −1.20638 −0.0397733
\(921\) −21.9082 −0.721900
\(922\) 8.58184 0.282628
\(923\) 22.5169 0.741153
\(924\) −4.58209 −0.150740
\(925\) 6.35324 0.208893
\(926\) 6.20956 0.204059
\(927\) −17.5937 −0.577852
\(928\) −17.4313 −0.572212
\(929\) 16.2587 0.533432 0.266716 0.963775i \(-0.414061\pi\)
0.266716 + 0.963775i \(0.414061\pi\)
\(930\) 1.69354 0.0555333
\(931\) 6.77795 0.222138
\(932\) −10.4707 −0.342978
\(933\) 31.9210 1.04505
\(934\) 29.5735 0.967675
\(935\) −0.168589 −0.00551345
\(936\) −5.18334 −0.169423
\(937\) 0.0791510 0.00258575 0.00129287 0.999999i \(-0.499588\pi\)
0.00129287 + 0.999999i \(0.499588\pi\)
\(938\) 29.0158 0.947399
\(939\) −0.0950302 −0.00310119
\(940\) −27.4938 −0.896750
\(941\) −30.3413 −0.989098 −0.494549 0.869150i \(-0.664667\pi\)
−0.494549 + 0.869150i \(0.664667\pi\)
\(942\) 45.7152 1.48948
\(943\) 7.49896 0.244200
\(944\) −7.89133 −0.256841
\(945\) −1.00000 −0.0325300
\(946\) 31.0754 1.01035
\(947\) −26.3732 −0.857015 −0.428507 0.903538i \(-0.640960\pi\)
−0.428507 + 0.903538i \(0.640960\pi\)
\(948\) 30.1373 0.978814
\(949\) 49.5144 1.60731
\(950\) −14.4811 −0.469830
\(951\) 9.58935 0.310956
\(952\) −0.113836 −0.00368944
\(953\) 8.93301 0.289368 0.144684 0.989478i \(-0.453783\pi\)
0.144684 + 0.989478i \(0.453783\pi\)
\(954\) 22.1733 0.717887
\(955\) −4.75992 −0.154028
\(956\) −59.5517 −1.92604
\(957\) −3.95983 −0.128003
\(958\) 52.4484 1.69453
\(959\) 7.09215 0.229018
\(960\) −11.6995 −0.377601
\(961\) −30.3717 −0.979732
\(962\) −58.3208 −1.88034
\(963\) −12.9655 −0.417806
\(964\) 69.3976 2.23514
\(965\) 6.70344 0.215791
\(966\) 2.13650 0.0687409
\(967\) −2.58674 −0.0831840 −0.0415920 0.999135i \(-0.513243\pi\)
−0.0415920 + 0.999135i \(0.513243\pi\)
\(968\) 9.41937 0.302750
\(969\) −0.639576 −0.0205462
\(970\) −11.4518 −0.367696
\(971\) −16.5154 −0.530005 −0.265003 0.964248i \(-0.585373\pi\)
−0.265003 + 0.964248i \(0.585373\pi\)
\(972\) 2.56465 0.0822613
\(973\) −12.1235 −0.388660
\(974\) 53.9198 1.72770
\(975\) 4.29660 0.137601
\(976\) 14.9218 0.477634
\(977\) 35.6320 1.13997 0.569984 0.821656i \(-0.306949\pi\)
0.569984 + 0.821656i \(0.306949\pi\)
\(978\) −0.220408 −0.00704786
\(979\) 9.13149 0.291844
\(980\) 2.56465 0.0819248
\(981\) 13.0849 0.417768
\(982\) 43.5333 1.38920
\(983\) −23.3567 −0.744964 −0.372482 0.928039i \(-0.621493\pi\)
−0.372482 + 0.928039i \(0.621493\pi\)
\(984\) −9.04662 −0.288396
\(985\) 21.1463 0.673778
\(986\) −0.446827 −0.0142299
\(987\) 10.7203 0.341231
\(988\) 74.6881 2.37615
\(989\) −8.14100 −0.258869
\(990\) −3.81715 −0.121317
\(991\) −4.68461 −0.148812 −0.0744058 0.997228i \(-0.523706\pi\)
−0.0744058 + 0.997228i \(0.523706\pi\)
\(992\) −6.23420 −0.197936
\(993\) 4.28108 0.135856
\(994\) 11.1967 0.355136
\(995\) 24.0722 0.763140
\(996\) −23.6668 −0.749910
\(997\) 15.2149 0.481862 0.240931 0.970542i \(-0.422547\pi\)
0.240931 + 0.970542i \(0.422547\pi\)
\(998\) −4.07141 −0.128878
\(999\) 6.35324 0.201008
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 2415.2.a.v.1.2 10
3.2 odd 2 7245.2.a.bu.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.v.1.2 10 1.1 even 1 trivial
7245.2.a.bu.1.9 10 3.2 odd 2