Properties

Label 507.2.a.g.1.1
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
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] = [507,2,Mod(1,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("507.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -3.56155 q^{5} -1.56155 q^{6} +0.561553 q^{7} +2.43845 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} -3.56155 q^{5} -1.56155 q^{6} +0.561553 q^{7} +2.43845 q^{8} +1.00000 q^{9} +5.56155 q^{10} -2.00000 q^{11} +0.438447 q^{12} -0.876894 q^{14} -3.56155 q^{15} -4.68466 q^{16} -1.56155 q^{17} -1.56155 q^{18} +7.12311 q^{19} -1.56155 q^{20} +0.561553 q^{21} +3.12311 q^{22} +2.00000 q^{23} +2.43845 q^{24} +7.68466 q^{25} +1.00000 q^{27} +0.246211 q^{28} +6.68466 q^{29} +5.56155 q^{30} +2.56155 q^{31} +2.43845 q^{32} -2.00000 q^{33} +2.43845 q^{34} -2.00000 q^{35} +0.438447 q^{36} +7.56155 q^{37} -11.1231 q^{38} -8.68466 q^{40} -1.56155 q^{41} -0.876894 q^{42} +4.56155 q^{43} -0.876894 q^{44} -3.56155 q^{45} -3.12311 q^{46} +8.24621 q^{47} -4.68466 q^{48} -6.68466 q^{49} -12.0000 q^{50} -1.56155 q^{51} -0.684658 q^{53} -1.56155 q^{54} +7.12311 q^{55} +1.36932 q^{56} +7.12311 q^{57} -10.4384 q^{58} -2.87689 q^{59} -1.56155 q^{60} +3.87689 q^{61} -4.00000 q^{62} +0.561553 q^{63} +5.56155 q^{64} +3.12311 q^{66} +4.56155 q^{67} -0.684658 q^{68} +2.00000 q^{69} +3.12311 q^{70} +14.0000 q^{71} +2.43845 q^{72} -10.1231 q^{73} -11.8078 q^{74} +7.68466 q^{75} +3.12311 q^{76} -1.12311 q^{77} +5.43845 q^{79} +16.6847 q^{80} +1.00000 q^{81} +2.43845 q^{82} -0.876894 q^{83} +0.246211 q^{84} +5.56155 q^{85} -7.12311 q^{86} +6.68466 q^{87} -4.87689 q^{88} +4.87689 q^{89} +5.56155 q^{90} +0.876894 q^{92} +2.56155 q^{93} -12.8769 q^{94} -25.3693 q^{95} +2.43845 q^{96} -8.56155 q^{97} +10.4384 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9} + 7 q^{10} - 4 q^{11} + 5 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + q^{18} + 6 q^{19} + q^{20} - 3 q^{21} - 2 q^{22} + 4 q^{23} + 9 q^{24} + 3 q^{25} + 2 q^{27} - 16 q^{28} + q^{29} + 7 q^{30} + q^{31} + 9 q^{32} - 4 q^{33} + 9 q^{34} - 4 q^{35} + 5 q^{36} + 11 q^{37} - 14 q^{38} - 5 q^{40} + q^{41} - 10 q^{42} + 5 q^{43} - 10 q^{44} - 3 q^{45} + 2 q^{46} + 3 q^{48} - q^{49} - 24 q^{50} + q^{51} + 11 q^{53} + q^{54} + 6 q^{55} - 22 q^{56} + 6 q^{57} - 25 q^{58} - 14 q^{59} + q^{60} + 16 q^{61} - 8 q^{62} - 3 q^{63} + 7 q^{64} - 2 q^{66} + 5 q^{67} + 11 q^{68} + 4 q^{69} - 2 q^{70} + 28 q^{71} + 9 q^{72} - 12 q^{73} - 3 q^{74} + 3 q^{75} - 2 q^{76} + 6 q^{77} + 15 q^{79} + 21 q^{80} + 2 q^{81} + 9 q^{82} - 10 q^{83} - 16 q^{84} + 7 q^{85} - 6 q^{86} + q^{87} - 18 q^{88} + 18 q^{89} + 7 q^{90} + 10 q^{92} + q^{93} - 34 q^{94} - 26 q^{95} + 9 q^{96} - 13 q^{97} + 25 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.438447 0.219224
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) −1.56155 −0.637501
\(7\) 0.561553 0.212247 0.106124 0.994353i \(-0.466156\pi\)
0.106124 + 0.994353i \(0.466156\pi\)
\(8\) 2.43845 0.862121
\(9\) 1.00000 0.333333
\(10\) 5.56155 1.75872
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0.438447 0.126569
\(13\) 0 0
\(14\) −0.876894 −0.234360
\(15\) −3.56155 −0.919589
\(16\) −4.68466 −1.17116
\(17\) −1.56155 −0.378732 −0.189366 0.981907i \(-0.560643\pi\)
−0.189366 + 0.981907i \(0.560643\pi\)
\(18\) −1.56155 −0.368062
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) −1.56155 −0.349174
\(21\) 0.561553 0.122541
\(22\) 3.12311 0.665848
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 2.43845 0.497746
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.246211 0.0465296
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 5.56155 1.01540
\(31\) 2.56155 0.460068 0.230034 0.973183i \(-0.426116\pi\)
0.230034 + 0.973183i \(0.426116\pi\)
\(32\) 2.43845 0.431061
\(33\) −2.00000 −0.348155
\(34\) 2.43845 0.418190
\(35\) −2.00000 −0.338062
\(36\) 0.438447 0.0730745
\(37\) 7.56155 1.24311 0.621556 0.783370i \(-0.286501\pi\)
0.621556 + 0.783370i \(0.286501\pi\)
\(38\) −11.1231 −1.80441
\(39\) 0 0
\(40\) −8.68466 −1.37317
\(41\) −1.56155 −0.243874 −0.121937 0.992538i \(-0.538911\pi\)
−0.121937 + 0.992538i \(0.538911\pi\)
\(42\) −0.876894 −0.135308
\(43\) 4.56155 0.695630 0.347815 0.937563i \(-0.386924\pi\)
0.347815 + 0.937563i \(0.386924\pi\)
\(44\) −0.876894 −0.132197
\(45\) −3.56155 −0.530925
\(46\) −3.12311 −0.460477
\(47\) 8.24621 1.20283 0.601417 0.798935i \(-0.294603\pi\)
0.601417 + 0.798935i \(0.294603\pi\)
\(48\) −4.68466 −0.676172
\(49\) −6.68466 −0.954951
\(50\) −12.0000 −1.69706
\(51\) −1.56155 −0.218661
\(52\) 0 0
\(53\) −0.684658 −0.0940451 −0.0470225 0.998894i \(-0.514973\pi\)
−0.0470225 + 0.998894i \(0.514973\pi\)
\(54\) −1.56155 −0.212500
\(55\) 7.12311 0.960479
\(56\) 1.36932 0.182983
\(57\) 7.12311 0.943478
\(58\) −10.4384 −1.37064
\(59\) −2.87689 −0.374540 −0.187270 0.982309i \(-0.559964\pi\)
−0.187270 + 0.982309i \(0.559964\pi\)
\(60\) −1.56155 −0.201596
\(61\) 3.87689 0.496385 0.248193 0.968711i \(-0.420163\pi\)
0.248193 + 0.968711i \(0.420163\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0.561553 0.0707490
\(64\) 5.56155 0.695194
\(65\) 0 0
\(66\) 3.12311 0.384428
\(67\) 4.56155 0.557282 0.278641 0.960395i \(-0.410116\pi\)
0.278641 + 0.960395i \(0.410116\pi\)
\(68\) −0.684658 −0.0830270
\(69\) 2.00000 0.240772
\(70\) 3.12311 0.373283
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 2.43845 0.287374
\(73\) −10.1231 −1.18482 −0.592410 0.805637i \(-0.701823\pi\)
−0.592410 + 0.805637i \(0.701823\pi\)
\(74\) −11.8078 −1.37262
\(75\) 7.68466 0.887348
\(76\) 3.12311 0.358245
\(77\) −1.12311 −0.127990
\(78\) 0 0
\(79\) 5.43845 0.611873 0.305937 0.952052i \(-0.401030\pi\)
0.305937 + 0.952052i \(0.401030\pi\)
\(80\) 16.6847 1.86540
\(81\) 1.00000 0.111111
\(82\) 2.43845 0.269281
\(83\) −0.876894 −0.0962517 −0.0481258 0.998841i \(-0.515325\pi\)
−0.0481258 + 0.998841i \(0.515325\pi\)
\(84\) 0.246211 0.0268638
\(85\) 5.56155 0.603235
\(86\) −7.12311 −0.768104
\(87\) 6.68466 0.716671
\(88\) −4.87689 −0.519879
\(89\) 4.87689 0.516950 0.258475 0.966018i \(-0.416780\pi\)
0.258475 + 0.966018i \(0.416780\pi\)
\(90\) 5.56155 0.586239
\(91\) 0 0
\(92\) 0.876894 0.0914226
\(93\) 2.56155 0.265621
\(94\) −12.8769 −1.32815
\(95\) −25.3693 −2.60284
\(96\) 2.43845 0.248873
\(97\) −8.56155 −0.869294 −0.434647 0.900601i \(-0.643127\pi\)
−0.434647 + 0.900601i \(0.643127\pi\)
\(98\) 10.4384 1.05444
\(99\) −2.00000 −0.201008
\(100\) 3.36932 0.336932
\(101\) −7.56155 −0.752403 −0.376201 0.926538i \(-0.622770\pi\)
−0.376201 + 0.926538i \(0.622770\pi\)
\(102\) 2.43845 0.241442
\(103\) −3.43845 −0.338800 −0.169400 0.985547i \(-0.554183\pi\)
−0.169400 + 0.985547i \(0.554183\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 1.06913 0.103843
\(107\) −8.24621 −0.797191 −0.398596 0.917127i \(-0.630502\pi\)
−0.398596 + 0.917127i \(0.630502\pi\)
\(108\) 0.438447 0.0421896
\(109\) −2.80776 −0.268935 −0.134468 0.990918i \(-0.542932\pi\)
−0.134468 + 0.990918i \(0.542932\pi\)
\(110\) −11.1231 −1.06055
\(111\) 7.56155 0.717711
\(112\) −2.63068 −0.248576
\(113\) 5.80776 0.546348 0.273174 0.961965i \(-0.411926\pi\)
0.273174 + 0.961965i \(0.411926\pi\)
\(114\) −11.1231 −1.04177
\(115\) −7.12311 −0.664233
\(116\) 2.93087 0.272124
\(117\) 0 0
\(118\) 4.49242 0.413561
\(119\) −0.876894 −0.0803848
\(120\) −8.68466 −0.792797
\(121\) −7.00000 −0.636364
\(122\) −6.05398 −0.548101
\(123\) −1.56155 −0.140800
\(124\) 1.12311 0.100858
\(125\) −9.56155 −0.855211
\(126\) −0.876894 −0.0781200
\(127\) 5.43845 0.482584 0.241292 0.970453i \(-0.422429\pi\)
0.241292 + 0.970453i \(0.422429\pi\)
\(128\) −13.5616 −1.19868
\(129\) 4.56155 0.401622
\(130\) 0 0
\(131\) 7.36932 0.643860 0.321930 0.946763i \(-0.395668\pi\)
0.321930 + 0.946763i \(0.395668\pi\)
\(132\) −0.876894 −0.0763239
\(133\) 4.00000 0.346844
\(134\) −7.12311 −0.615343
\(135\) −3.56155 −0.306530
\(136\) −3.80776 −0.326513
\(137\) −5.56155 −0.475156 −0.237578 0.971369i \(-0.576353\pi\)
−0.237578 + 0.971369i \(0.576353\pi\)
\(138\) −3.12311 −0.265856
\(139\) −17.9309 −1.52088 −0.760438 0.649410i \(-0.775016\pi\)
−0.760438 + 0.649410i \(0.775016\pi\)
\(140\) −0.876894 −0.0741111
\(141\) 8.24621 0.694456
\(142\) −21.8617 −1.83460
\(143\) 0 0
\(144\) −4.68466 −0.390388
\(145\) −23.8078 −1.97713
\(146\) 15.8078 1.30826
\(147\) −6.68466 −0.551341
\(148\) 3.31534 0.272519
\(149\) −2.43845 −0.199765 −0.0998827 0.994999i \(-0.531847\pi\)
−0.0998827 + 0.994999i \(0.531847\pi\)
\(150\) −12.0000 −0.979796
\(151\) −9.36932 −0.762464 −0.381232 0.924479i \(-0.624500\pi\)
−0.381232 + 0.924479i \(0.624500\pi\)
\(152\) 17.3693 1.40884
\(153\) −1.56155 −0.126244
\(154\) 1.75379 0.141324
\(155\) −9.12311 −0.732785
\(156\) 0 0
\(157\) 20.3693 1.62565 0.812824 0.582509i \(-0.197929\pi\)
0.812824 + 0.582509i \(0.197929\pi\)
\(158\) −8.49242 −0.675621
\(159\) −0.684658 −0.0542969
\(160\) −8.68466 −0.686583
\(161\) 1.12311 0.0885131
\(162\) −1.56155 −0.122687
\(163\) −4.80776 −0.376573 −0.188287 0.982114i \(-0.560293\pi\)
−0.188287 + 0.982114i \(0.560293\pi\)
\(164\) −0.684658 −0.0534628
\(165\) 7.12311 0.554533
\(166\) 1.36932 0.106280
\(167\) 10.2462 0.792876 0.396438 0.918062i \(-0.370246\pi\)
0.396438 + 0.918062i \(0.370246\pi\)
\(168\) 1.36932 0.105645
\(169\) 0 0
\(170\) −8.68466 −0.666083
\(171\) 7.12311 0.544718
\(172\) 2.00000 0.152499
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) −10.4384 −0.791337
\(175\) 4.31534 0.326209
\(176\) 9.36932 0.706239
\(177\) −2.87689 −0.216241
\(178\) −7.61553 −0.570808
\(179\) −4.87689 −0.364516 −0.182258 0.983251i \(-0.558341\pi\)
−0.182258 + 0.983251i \(0.558341\pi\)
\(180\) −1.56155 −0.116391
\(181\) −2.68466 −0.199549 −0.0997745 0.995010i \(-0.531812\pi\)
−0.0997745 + 0.995010i \(0.531812\pi\)
\(182\) 0 0
\(183\) 3.87689 0.286588
\(184\) 4.87689 0.359529
\(185\) −26.9309 −1.98000
\(186\) −4.00000 −0.293294
\(187\) 3.12311 0.228384
\(188\) 3.61553 0.263689
\(189\) 0.561553 0.0408470
\(190\) 39.6155 2.87401
\(191\) −9.12311 −0.660125 −0.330062 0.943959i \(-0.607070\pi\)
−0.330062 + 0.943959i \(0.607070\pi\)
\(192\) 5.56155 0.401371
\(193\) −13.4924 −0.971206 −0.485603 0.874180i \(-0.661400\pi\)
−0.485603 + 0.874180i \(0.661400\pi\)
\(194\) 13.3693 0.959861
\(195\) 0 0
\(196\) −2.93087 −0.209348
\(197\) 13.3693 0.952524 0.476262 0.879303i \(-0.341991\pi\)
0.476262 + 0.879303i \(0.341991\pi\)
\(198\) 3.12311 0.221949
\(199\) 22.1771 1.57209 0.786046 0.618168i \(-0.212125\pi\)
0.786046 + 0.618168i \(0.212125\pi\)
\(200\) 18.7386 1.32502
\(201\) 4.56155 0.321747
\(202\) 11.8078 0.830791
\(203\) 3.75379 0.263464
\(204\) −0.684658 −0.0479357
\(205\) 5.56155 0.388436
\(206\) 5.36932 0.374098
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −14.2462 −0.985431
\(210\) 3.12311 0.215515
\(211\) 19.6847 1.35515 0.677574 0.735455i \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(212\) −0.300187 −0.0206169
\(213\) 14.0000 0.959264
\(214\) 12.8769 0.880246
\(215\) −16.2462 −1.10798
\(216\) 2.43845 0.165915
\(217\) 1.43845 0.0976482
\(218\) 4.38447 0.296954
\(219\) −10.1231 −0.684056
\(220\) 3.12311 0.210560
\(221\) 0 0
\(222\) −11.8078 −0.792485
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.36932 0.0914913
\(225\) 7.68466 0.512311
\(226\) −9.06913 −0.603270
\(227\) 7.12311 0.472777 0.236389 0.971659i \(-0.424036\pi\)
0.236389 + 0.971659i \(0.424036\pi\)
\(228\) 3.12311 0.206833
\(229\) −16.2462 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(230\) 11.1231 0.733436
\(231\) −1.12311 −0.0738949
\(232\) 16.3002 1.07016
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −29.3693 −1.91584
\(236\) −1.26137 −0.0821079
\(237\) 5.43845 0.353265
\(238\) 1.36932 0.0887596
\(239\) −25.3693 −1.64100 −0.820502 0.571643i \(-0.806306\pi\)
−0.820502 + 0.571643i \(0.806306\pi\)
\(240\) 16.6847 1.07699
\(241\) −17.8078 −1.14710 −0.573549 0.819171i \(-0.694434\pi\)
−0.573549 + 0.819171i \(0.694434\pi\)
\(242\) 10.9309 0.702663
\(243\) 1.00000 0.0641500
\(244\) 1.69981 0.108819
\(245\) 23.8078 1.52102
\(246\) 2.43845 0.155470
\(247\) 0 0
\(248\) 6.24621 0.396635
\(249\) −0.876894 −0.0555709
\(250\) 14.9309 0.944311
\(251\) 18.7386 1.18277 0.591386 0.806389i \(-0.298581\pi\)
0.591386 + 0.806389i \(0.298581\pi\)
\(252\) 0.246211 0.0155099
\(253\) −4.00000 −0.251478
\(254\) −8.49242 −0.532862
\(255\) 5.56155 0.348278
\(256\) 10.0540 0.628373
\(257\) 29.1771 1.82002 0.910008 0.414590i \(-0.136075\pi\)
0.910008 + 0.414590i \(0.136075\pi\)
\(258\) −7.12311 −0.443465
\(259\) 4.24621 0.263847
\(260\) 0 0
\(261\) 6.68466 0.413770
\(262\) −11.5076 −0.710941
\(263\) 9.36932 0.577737 0.288868 0.957369i \(-0.406721\pi\)
0.288868 + 0.957369i \(0.406721\pi\)
\(264\) −4.87689 −0.300152
\(265\) 2.43845 0.149793
\(266\) −6.24621 −0.382980
\(267\) 4.87689 0.298461
\(268\) 2.00000 0.122169
\(269\) −21.3693 −1.30291 −0.651455 0.758687i \(-0.725841\pi\)
−0.651455 + 0.758687i \(0.725841\pi\)
\(270\) 5.56155 0.338465
\(271\) −29.9309 −1.81817 −0.909085 0.416610i \(-0.863218\pi\)
−0.909085 + 0.416610i \(0.863218\pi\)
\(272\) 7.31534 0.443558
\(273\) 0 0
\(274\) 8.68466 0.524659
\(275\) −15.3693 −0.926805
\(276\) 0.876894 0.0527828
\(277\) 5.31534 0.319368 0.159684 0.987168i \(-0.448952\pi\)
0.159684 + 0.987168i \(0.448952\pi\)
\(278\) 28.0000 1.67933
\(279\) 2.56155 0.153356
\(280\) −4.87689 −0.291450
\(281\) 17.8078 1.06232 0.531161 0.847271i \(-0.321756\pi\)
0.531161 + 0.847271i \(0.321756\pi\)
\(282\) −12.8769 −0.766808
\(283\) −13.6847 −0.813469 −0.406734 0.913547i \(-0.633333\pi\)
−0.406734 + 0.913547i \(0.633333\pi\)
\(284\) 6.13826 0.364239
\(285\) −25.3693 −1.50275
\(286\) 0 0
\(287\) −0.876894 −0.0517614
\(288\) 2.43845 0.143687
\(289\) −14.5616 −0.856562
\(290\) 37.1771 2.18311
\(291\) −8.56155 −0.501887
\(292\) −4.43845 −0.259740
\(293\) 20.4384 1.19403 0.597013 0.802231i \(-0.296354\pi\)
0.597013 + 0.802231i \(0.296354\pi\)
\(294\) 10.4384 0.608783
\(295\) 10.2462 0.596557
\(296\) 18.4384 1.07171
\(297\) −2.00000 −0.116052
\(298\) 3.80776 0.220578
\(299\) 0 0
\(300\) 3.36932 0.194528
\(301\) 2.56155 0.147645
\(302\) 14.6307 0.841901
\(303\) −7.56155 −0.434400
\(304\) −33.3693 −1.91386
\(305\) −13.8078 −0.790630
\(306\) 2.43845 0.139397
\(307\) 30.8078 1.75829 0.879146 0.476553i \(-0.158114\pi\)
0.879146 + 0.476553i \(0.158114\pi\)
\(308\) −0.492423 −0.0280584
\(309\) −3.43845 −0.195606
\(310\) 14.2462 0.809130
\(311\) −19.1231 −1.08437 −0.542186 0.840259i \(-0.682403\pi\)
−0.542186 + 0.840259i \(0.682403\pi\)
\(312\) 0 0
\(313\) −13.6847 −0.773503 −0.386751 0.922184i \(-0.626403\pi\)
−0.386751 + 0.922184i \(0.626403\pi\)
\(314\) −31.8078 −1.79502
\(315\) −2.00000 −0.112687
\(316\) 2.38447 0.134137
\(317\) 14.0540 0.789350 0.394675 0.918821i \(-0.370857\pi\)
0.394675 + 0.918821i \(0.370857\pi\)
\(318\) 1.06913 0.0599539
\(319\) −13.3693 −0.748538
\(320\) −19.8078 −1.10729
\(321\) −8.24621 −0.460259
\(322\) −1.75379 −0.0977348
\(323\) −11.1231 −0.618906
\(324\) 0.438447 0.0243582
\(325\) 0 0
\(326\) 7.50758 0.415806
\(327\) −2.80776 −0.155270
\(328\) −3.80776 −0.210249
\(329\) 4.63068 0.255298
\(330\) −11.1231 −0.612307
\(331\) −3.19224 −0.175461 −0.0877306 0.996144i \(-0.527961\pi\)
−0.0877306 + 0.996144i \(0.527961\pi\)
\(332\) −0.384472 −0.0211006
\(333\) 7.56155 0.414371
\(334\) −16.0000 −0.875481
\(335\) −16.2462 −0.887625
\(336\) −2.63068 −0.143516
\(337\) −6.12311 −0.333547 −0.166773 0.985995i \(-0.553335\pi\)
−0.166773 + 0.985995i \(0.553335\pi\)
\(338\) 0 0
\(339\) 5.80776 0.315434
\(340\) 2.43845 0.132243
\(341\) −5.12311 −0.277432
\(342\) −11.1231 −0.601469
\(343\) −7.68466 −0.414933
\(344\) 11.1231 0.599718
\(345\) −7.12311 −0.383495
\(346\) −31.6155 −1.69966
\(347\) 27.6155 1.48248 0.741240 0.671241i \(-0.234238\pi\)
0.741240 + 0.671241i \(0.234238\pi\)
\(348\) 2.93087 0.157111
\(349\) 6.80776 0.364411 0.182206 0.983260i \(-0.441676\pi\)
0.182206 + 0.983260i \(0.441676\pi\)
\(350\) −6.73863 −0.360195
\(351\) 0 0
\(352\) −4.87689 −0.259939
\(353\) 5.31534 0.282907 0.141454 0.989945i \(-0.454822\pi\)
0.141454 + 0.989945i \(0.454822\pi\)
\(354\) 4.49242 0.238770
\(355\) −49.8617 −2.64639
\(356\) 2.13826 0.113328
\(357\) −0.876894 −0.0464102
\(358\) 7.61553 0.402493
\(359\) −9.36932 −0.494494 −0.247247 0.968953i \(-0.579526\pi\)
−0.247247 + 0.968953i \(0.579526\pi\)
\(360\) −8.68466 −0.457722
\(361\) 31.7386 1.67045
\(362\) 4.19224 0.220339
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 36.0540 1.88715
\(366\) −6.05398 −0.316446
\(367\) −17.0540 −0.890210 −0.445105 0.895478i \(-0.646834\pi\)
−0.445105 + 0.895478i \(0.646834\pi\)
\(368\) −9.36932 −0.488409
\(369\) −1.56155 −0.0812912
\(370\) 42.0540 2.18628
\(371\) −0.384472 −0.0199608
\(372\) 1.12311 0.0582303
\(373\) 28.3693 1.46891 0.734454 0.678659i \(-0.237438\pi\)
0.734454 + 0.678659i \(0.237438\pi\)
\(374\) −4.87689 −0.252178
\(375\) −9.56155 −0.493756
\(376\) 20.1080 1.03699
\(377\) 0 0
\(378\) −0.876894 −0.0451026
\(379\) 23.6847 1.21660 0.608300 0.793708i \(-0.291852\pi\)
0.608300 + 0.793708i \(0.291852\pi\)
\(380\) −11.1231 −0.570603
\(381\) 5.43845 0.278620
\(382\) 14.2462 0.728900
\(383\) −22.7386 −1.16189 −0.580945 0.813943i \(-0.697317\pi\)
−0.580945 + 0.813943i \(0.697317\pi\)
\(384\) −13.5616 −0.692060
\(385\) 4.00000 0.203859
\(386\) 21.0691 1.07239
\(387\) 4.56155 0.231877
\(388\) −3.75379 −0.190570
\(389\) 34.0540 1.72661 0.863303 0.504687i \(-0.168392\pi\)
0.863303 + 0.504687i \(0.168392\pi\)
\(390\) 0 0
\(391\) −3.12311 −0.157942
\(392\) −16.3002 −0.823284
\(393\) 7.36932 0.371733
\(394\) −20.8769 −1.05176
\(395\) −19.3693 −0.974576
\(396\) −0.876894 −0.0440656
\(397\) 25.0540 1.25742 0.628711 0.777639i \(-0.283583\pi\)
0.628711 + 0.777639i \(0.283583\pi\)
\(398\) −34.6307 −1.73588
\(399\) 4.00000 0.200250
\(400\) −36.0000 −1.80000
\(401\) 14.4384 0.721022 0.360511 0.932755i \(-0.382602\pi\)
0.360511 + 0.932755i \(0.382602\pi\)
\(402\) −7.12311 −0.355268
\(403\) 0 0
\(404\) −3.31534 −0.164944
\(405\) −3.56155 −0.176975
\(406\) −5.86174 −0.290913
\(407\) −15.1231 −0.749625
\(408\) −3.80776 −0.188512
\(409\) 6.36932 0.314942 0.157471 0.987524i \(-0.449666\pi\)
0.157471 + 0.987524i \(0.449666\pi\)
\(410\) −8.68466 −0.428905
\(411\) −5.56155 −0.274331
\(412\) −1.50758 −0.0742730
\(413\) −1.61553 −0.0794949
\(414\) −3.12311 −0.153492
\(415\) 3.12311 0.153307
\(416\) 0 0
\(417\) −17.9309 −0.878078
\(418\) 22.2462 1.08810
\(419\) −34.2462 −1.67304 −0.836518 0.547939i \(-0.815413\pi\)
−0.836518 + 0.547939i \(0.815413\pi\)
\(420\) −0.876894 −0.0427881
\(421\) 31.2462 1.52285 0.761424 0.648255i \(-0.224501\pi\)
0.761424 + 0.648255i \(0.224501\pi\)
\(422\) −30.7386 −1.49633
\(423\) 8.24621 0.400945
\(424\) −1.66950 −0.0810783
\(425\) −12.0000 −0.582086
\(426\) −21.8617 −1.05920
\(427\) 2.17708 0.105356
\(428\) −3.61553 −0.174763
\(429\) 0 0
\(430\) 25.3693 1.22342
\(431\) −11.1231 −0.535781 −0.267891 0.963449i \(-0.586327\pi\)
−0.267891 + 0.963449i \(0.586327\pi\)
\(432\) −4.68466 −0.225391
\(433\) 8.75379 0.420680 0.210340 0.977628i \(-0.432543\pi\)
0.210340 + 0.977628i \(0.432543\pi\)
\(434\) −2.24621 −0.107822
\(435\) −23.8078 −1.14149
\(436\) −1.23106 −0.0589569
\(437\) 14.2462 0.681489
\(438\) 15.8078 0.755324
\(439\) −13.6847 −0.653133 −0.326567 0.945174i \(-0.605892\pi\)
−0.326567 + 0.945174i \(0.605892\pi\)
\(440\) 17.3693 0.828050
\(441\) −6.68466 −0.318317
\(442\) 0 0
\(443\) −34.7386 −1.65048 −0.825241 0.564781i \(-0.808961\pi\)
−0.825241 + 0.564781i \(0.808961\pi\)
\(444\) 3.31534 0.157339
\(445\) −17.3693 −0.823385
\(446\) −12.4924 −0.591533
\(447\) −2.43845 −0.115335
\(448\) 3.12311 0.147553
\(449\) −8.24621 −0.389163 −0.194581 0.980886i \(-0.562335\pi\)
−0.194581 + 0.980886i \(0.562335\pi\)
\(450\) −12.0000 −0.565685
\(451\) 3.12311 0.147061
\(452\) 2.54640 0.119772
\(453\) −9.36932 −0.440209
\(454\) −11.1231 −0.522033
\(455\) 0 0
\(456\) 17.3693 0.813393
\(457\) 12.6155 0.590130 0.295065 0.955477i \(-0.404659\pi\)
0.295065 + 0.955477i \(0.404659\pi\)
\(458\) 25.3693 1.18543
\(459\) −1.56155 −0.0728870
\(460\) −3.12311 −0.145616
\(461\) −16.1922 −0.754148 −0.377074 0.926183i \(-0.623070\pi\)
−0.377074 + 0.926183i \(0.623070\pi\)
\(462\) 1.75379 0.0815936
\(463\) −14.3153 −0.665290 −0.332645 0.943052i \(-0.607941\pi\)
−0.332645 + 0.943052i \(0.607941\pi\)
\(464\) −31.3153 −1.45378
\(465\) −9.12311 −0.423074
\(466\) −40.6004 −1.88078
\(467\) −26.0000 −1.20314 −0.601568 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) 0 0
\(469\) 2.56155 0.118282
\(470\) 45.8617 2.11544
\(471\) 20.3693 0.938569
\(472\) −7.01515 −0.322899
\(473\) −9.12311 −0.419481
\(474\) −8.49242 −0.390070
\(475\) 54.7386 2.51158
\(476\) −0.384472 −0.0176222
\(477\) −0.684658 −0.0313484
\(478\) 39.6155 1.81197
\(479\) 10.2462 0.468161 0.234081 0.972217i \(-0.424792\pi\)
0.234081 + 0.972217i \(0.424792\pi\)
\(480\) −8.68466 −0.396399
\(481\) 0 0
\(482\) 27.8078 1.26661
\(483\) 1.12311 0.0511031
\(484\) −3.06913 −0.139506
\(485\) 30.4924 1.38459
\(486\) −1.56155 −0.0708335
\(487\) 7.12311 0.322779 0.161389 0.986891i \(-0.448403\pi\)
0.161389 + 0.986891i \(0.448403\pi\)
\(488\) 9.45360 0.427944
\(489\) −4.80776 −0.217415
\(490\) −37.1771 −1.67949
\(491\) −36.2462 −1.63577 −0.817884 0.575383i \(-0.804853\pi\)
−0.817884 + 0.575383i \(0.804853\pi\)
\(492\) −0.684658 −0.0308668
\(493\) −10.4384 −0.470124
\(494\) 0 0
\(495\) 7.12311 0.320160
\(496\) −12.0000 −0.538816
\(497\) 7.86174 0.352647
\(498\) 1.36932 0.0613606
\(499\) 4.49242 0.201108 0.100554 0.994932i \(-0.467938\pi\)
0.100554 + 0.994932i \(0.467938\pi\)
\(500\) −4.19224 −0.187482
\(501\) 10.2462 0.457767
\(502\) −29.2614 −1.30600
\(503\) −28.2462 −1.25944 −0.629718 0.776824i \(-0.716830\pi\)
−0.629718 + 0.776824i \(0.716830\pi\)
\(504\) 1.36932 0.0609942
\(505\) 26.9309 1.19841
\(506\) 6.24621 0.277678
\(507\) 0 0
\(508\) 2.38447 0.105794
\(509\) −13.8078 −0.612018 −0.306009 0.952029i \(-0.598994\pi\)
−0.306009 + 0.952029i \(0.598994\pi\)
\(510\) −8.68466 −0.384563
\(511\) −5.68466 −0.251474
\(512\) 11.4233 0.504843
\(513\) 7.12311 0.314493
\(514\) −45.5616 −2.00963
\(515\) 12.2462 0.539633
\(516\) 2.00000 0.0880451
\(517\) −16.4924 −0.725336
\(518\) −6.63068 −0.291335
\(519\) 20.2462 0.888710
\(520\) 0 0
\(521\) −9.06913 −0.397326 −0.198663 0.980068i \(-0.563660\pi\)
−0.198663 + 0.980068i \(0.563660\pi\)
\(522\) −10.4384 −0.456878
\(523\) 33.8617 1.48067 0.740335 0.672238i \(-0.234667\pi\)
0.740335 + 0.672238i \(0.234667\pi\)
\(524\) 3.23106 0.141149
\(525\) 4.31534 0.188337
\(526\) −14.6307 −0.637928
\(527\) −4.00000 −0.174243
\(528\) 9.36932 0.407747
\(529\) −19.0000 −0.826087
\(530\) −3.80776 −0.165399
\(531\) −2.87689 −0.124847
\(532\) 1.75379 0.0760364
\(533\) 0 0
\(534\) −7.61553 −0.329556
\(535\) 29.3693 1.26975
\(536\) 11.1231 0.480445
\(537\) −4.87689 −0.210454
\(538\) 33.3693 1.43865
\(539\) 13.3693 0.575857
\(540\) −1.56155 −0.0671985
\(541\) 19.7386 0.848630 0.424315 0.905515i \(-0.360515\pi\)
0.424315 + 0.905515i \(0.360515\pi\)
\(542\) 46.7386 2.00760
\(543\) −2.68466 −0.115210
\(544\) −3.80776 −0.163257
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 3.93087 0.168072 0.0840359 0.996463i \(-0.473219\pi\)
0.0840359 + 0.996463i \(0.473219\pi\)
\(548\) −2.43845 −0.104165
\(549\) 3.87689 0.165462
\(550\) 24.0000 1.02336
\(551\) 47.6155 2.02849
\(552\) 4.87689 0.207574
\(553\) 3.05398 0.129868
\(554\) −8.30019 −0.352641
\(555\) −26.9309 −1.14315
\(556\) −7.86174 −0.333412
\(557\) −42.9309 −1.81904 −0.909520 0.415661i \(-0.863550\pi\)
−0.909520 + 0.415661i \(0.863550\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 9.36932 0.395926
\(561\) 3.12311 0.131858
\(562\) −27.8078 −1.17300
\(563\) −23.3693 −0.984899 −0.492450 0.870341i \(-0.663898\pi\)
−0.492450 + 0.870341i \(0.663898\pi\)
\(564\) 3.61553 0.152241
\(565\) −20.6847 −0.870210
\(566\) 21.3693 0.898219
\(567\) 0.561553 0.0235830
\(568\) 34.1383 1.43241
\(569\) −8.73863 −0.366343 −0.183171 0.983081i \(-0.558636\pi\)
−0.183171 + 0.983081i \(0.558636\pi\)
\(570\) 39.6155 1.65931
\(571\) −5.36932 −0.224699 −0.112349 0.993669i \(-0.535838\pi\)
−0.112349 + 0.993669i \(0.535838\pi\)
\(572\) 0 0
\(573\) −9.12311 −0.381123
\(574\) 1.36932 0.0571542
\(575\) 15.3693 0.640945
\(576\) 5.56155 0.231731
\(577\) −17.3153 −0.720847 −0.360424 0.932789i \(-0.617368\pi\)
−0.360424 + 0.932789i \(0.617368\pi\)
\(578\) 22.7386 0.945802
\(579\) −13.4924 −0.560726
\(580\) −10.4384 −0.433433
\(581\) −0.492423 −0.0204291
\(582\) 13.3693 0.554176
\(583\) 1.36932 0.0567113
\(584\) −24.6847 −1.02146
\(585\) 0 0
\(586\) −31.9157 −1.31843
\(587\) 39.3693 1.62495 0.812473 0.582999i \(-0.198121\pi\)
0.812473 + 0.582999i \(0.198121\pi\)
\(588\) −2.93087 −0.120867
\(589\) 18.2462 0.751822
\(590\) −16.0000 −0.658710
\(591\) 13.3693 0.549940
\(592\) −35.4233 −1.45589
\(593\) −17.4233 −0.715489 −0.357744 0.933820i \(-0.616454\pi\)
−0.357744 + 0.933820i \(0.616454\pi\)
\(594\) 3.12311 0.128143
\(595\) 3.12311 0.128035
\(596\) −1.06913 −0.0437933
\(597\) 22.1771 0.907648
\(598\) 0 0
\(599\) −41.6155 −1.70036 −0.850182 0.526489i \(-0.823508\pi\)
−0.850182 + 0.526489i \(0.823508\pi\)
\(600\) 18.7386 0.765002
\(601\) −7.06913 −0.288356 −0.144178 0.989552i \(-0.546054\pi\)
−0.144178 + 0.989552i \(0.546054\pi\)
\(602\) −4.00000 −0.163028
\(603\) 4.56155 0.185761
\(604\) −4.10795 −0.167150
\(605\) 24.9309 1.01358
\(606\) 11.8078 0.479658
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 17.3693 0.704419
\(609\) 3.75379 0.152111
\(610\) 21.5616 0.873002
\(611\) 0 0
\(612\) −0.684658 −0.0276757
\(613\) −34.8617 −1.40805 −0.704026 0.710174i \(-0.748616\pi\)
−0.704026 + 0.710174i \(0.748616\pi\)
\(614\) −48.1080 −1.94148
\(615\) 5.56155 0.224263
\(616\) −2.73863 −0.110343
\(617\) 9.80776 0.394846 0.197423 0.980318i \(-0.436743\pi\)
0.197423 + 0.980318i \(0.436743\pi\)
\(618\) 5.36932 0.215986
\(619\) 29.3002 1.17767 0.588837 0.808252i \(-0.299586\pi\)
0.588837 + 0.808252i \(0.299586\pi\)
\(620\) −4.00000 −0.160644
\(621\) 2.00000 0.0802572
\(622\) 29.8617 1.19735
\(623\) 2.73863 0.109721
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 21.3693 0.854090
\(627\) −14.2462 −0.568939
\(628\) 8.93087 0.356380
\(629\) −11.8078 −0.470806
\(630\) 3.12311 0.124428
\(631\) −18.5616 −0.738924 −0.369462 0.929246i \(-0.620458\pi\)
−0.369462 + 0.929246i \(0.620458\pi\)
\(632\) 13.2614 0.527509
\(633\) 19.6847 0.782395
\(634\) −21.9460 −0.871588
\(635\) −19.3693 −0.768648
\(636\) −0.300187 −0.0119032
\(637\) 0 0
\(638\) 20.8769 0.826524
\(639\) 14.0000 0.553831
\(640\) 48.3002 1.90923
\(641\) −19.1771 −0.757449 −0.378725 0.925509i \(-0.623637\pi\)
−0.378725 + 0.925509i \(0.623637\pi\)
\(642\) 12.8769 0.508210
\(643\) −31.5464 −1.24407 −0.622034 0.782990i \(-0.713694\pi\)
−0.622034 + 0.782990i \(0.713694\pi\)
\(644\) 0.492423 0.0194042
\(645\) −16.2462 −0.639694
\(646\) 17.3693 0.683387
\(647\) −6.38447 −0.250999 −0.125500 0.992094i \(-0.540053\pi\)
−0.125500 + 0.992094i \(0.540053\pi\)
\(648\) 2.43845 0.0957913
\(649\) 5.75379 0.225856
\(650\) 0 0
\(651\) 1.43845 0.0563772
\(652\) −2.10795 −0.0825537
\(653\) 23.1231 0.904877 0.452439 0.891796i \(-0.350554\pi\)
0.452439 + 0.891796i \(0.350554\pi\)
\(654\) 4.38447 0.171446
\(655\) −26.2462 −1.02552
\(656\) 7.31534 0.285616
\(657\) −10.1231 −0.394940
\(658\) −7.23106 −0.281896
\(659\) −2.24621 −0.0875000 −0.0437500 0.999043i \(-0.513930\pi\)
−0.0437500 + 0.999043i \(0.513930\pi\)
\(660\) 3.12311 0.121567
\(661\) 5.63068 0.219008 0.109504 0.993986i \(-0.465074\pi\)
0.109504 + 0.993986i \(0.465074\pi\)
\(662\) 4.98485 0.193742
\(663\) 0 0
\(664\) −2.13826 −0.0829806
\(665\) −14.2462 −0.552444
\(666\) −11.8078 −0.457542
\(667\) 13.3693 0.517662
\(668\) 4.49242 0.173817
\(669\) 8.00000 0.309298
\(670\) 25.3693 0.980102
\(671\) −7.75379 −0.299332
\(672\) 1.36932 0.0528225
\(673\) −23.2462 −0.896076 −0.448038 0.894015i \(-0.647877\pi\)
−0.448038 + 0.894015i \(0.647877\pi\)
\(674\) 9.56155 0.368297
\(675\) 7.68466 0.295783
\(676\) 0 0
\(677\) −15.6155 −0.600153 −0.300077 0.953915i \(-0.597012\pi\)
−0.300077 + 0.953915i \(0.597012\pi\)
\(678\) −9.06913 −0.348298
\(679\) −4.80776 −0.184505
\(680\) 13.5616 0.520062
\(681\) 7.12311 0.272958
\(682\) 8.00000 0.306336
\(683\) −38.1080 −1.45816 −0.729080 0.684428i \(-0.760052\pi\)
−0.729080 + 0.684428i \(0.760052\pi\)
\(684\) 3.12311 0.119415
\(685\) 19.8078 0.756816
\(686\) 12.0000 0.458162
\(687\) −16.2462 −0.619832
\(688\) −21.3693 −0.814698
\(689\) 0 0
\(690\) 11.1231 0.423449
\(691\) −51.3002 −1.95155 −0.975776 0.218774i \(-0.929794\pi\)
−0.975776 + 0.218774i \(0.929794\pi\)
\(692\) 8.87689 0.337449
\(693\) −1.12311 −0.0426633
\(694\) −43.1231 −1.63693
\(695\) 63.8617 2.42241
\(696\) 16.3002 0.617857
\(697\) 2.43845 0.0923628
\(698\) −10.6307 −0.402377
\(699\) 26.0000 0.983410
\(700\) 1.89205 0.0715127
\(701\) −5.36932 −0.202796 −0.101398 0.994846i \(-0.532332\pi\)
−0.101398 + 0.994846i \(0.532332\pi\)
\(702\) 0 0
\(703\) 53.8617 2.03143
\(704\) −11.1231 −0.419218
\(705\) −29.3693 −1.10611
\(706\) −8.30019 −0.312382
\(707\) −4.24621 −0.159695
\(708\) −1.26137 −0.0474050
\(709\) −7.49242 −0.281384 −0.140692 0.990053i \(-0.544933\pi\)
−0.140692 + 0.990053i \(0.544933\pi\)
\(710\) 77.8617 2.92210
\(711\) 5.43845 0.203958
\(712\) 11.8920 0.445673
\(713\) 5.12311 0.191862
\(714\) 1.36932 0.0512454
\(715\) 0 0
\(716\) −2.13826 −0.0799106
\(717\) −25.3693 −0.947435
\(718\) 14.6307 0.546012
\(719\) −23.3693 −0.871528 −0.435764 0.900061i \(-0.643522\pi\)
−0.435764 + 0.900061i \(0.643522\pi\)
\(720\) 16.6847 0.621801
\(721\) −1.93087 −0.0719093
\(722\) −49.5616 −1.84449
\(723\) −17.8078 −0.662278
\(724\) −1.17708 −0.0437459
\(725\) 51.3693 1.90781
\(726\) 10.9309 0.405683
\(727\) −38.6695 −1.43417 −0.717086 0.696984i \(-0.754525\pi\)
−0.717086 + 0.696984i \(0.754525\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −56.3002 −2.08376
\(731\) −7.12311 −0.263458
\(732\) 1.69981 0.0628269
\(733\) 20.5076 0.757465 0.378732 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(734\) 26.6307 0.982956
\(735\) 23.8078 0.878163
\(736\) 4.87689 0.179765
\(737\) −9.12311 −0.336054
\(738\) 2.43845 0.0897605
\(739\) 10.2462 0.376913 0.188456 0.982082i \(-0.439652\pi\)
0.188456 + 0.982082i \(0.439652\pi\)
\(740\) −11.8078 −0.434062
\(741\) 0 0
\(742\) 0.600373 0.0220404
\(743\) −12.6307 −0.463375 −0.231687 0.972790i \(-0.574425\pi\)
−0.231687 + 0.972790i \(0.574425\pi\)
\(744\) 6.24621 0.228997
\(745\) 8.68466 0.318181
\(746\) −44.3002 −1.62195
\(747\) −0.876894 −0.0320839
\(748\) 1.36932 0.0500672
\(749\) −4.63068 −0.169201
\(750\) 14.9309 0.545198
\(751\) 44.1080 1.60952 0.804761 0.593599i \(-0.202293\pi\)
0.804761 + 0.593599i \(0.202293\pi\)
\(752\) −38.6307 −1.40872
\(753\) 18.7386 0.682874
\(754\) 0 0
\(755\) 33.3693 1.21443
\(756\) 0.246211 0.00895462
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −36.9848 −1.34335
\(759\) −4.00000 −0.145191
\(760\) −61.8617 −2.24396
\(761\) 9.36932 0.339637 0.169819 0.985475i \(-0.445682\pi\)
0.169819 + 0.985475i \(0.445682\pi\)
\(762\) −8.49242 −0.307648
\(763\) −1.57671 −0.0570807
\(764\) −4.00000 −0.144715
\(765\) 5.56155 0.201078
\(766\) 35.5076 1.28294
\(767\) 0 0
\(768\) 10.0540 0.362792
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) −6.24621 −0.225098
\(771\) 29.1771 1.05079
\(772\) −5.91571 −0.212911
\(773\) −24.2462 −0.872076 −0.436038 0.899928i \(-0.643619\pi\)
−0.436038 + 0.899928i \(0.643619\pi\)
\(774\) −7.12311 −0.256035
\(775\) 19.6847 0.707094
\(776\) −20.8769 −0.749437
\(777\) 4.24621 0.152332
\(778\) −53.1771 −1.90649
\(779\) −11.1231 −0.398527
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 4.87689 0.174397
\(783\) 6.68466 0.238890
\(784\) 31.3153 1.11841
\(785\) −72.5464 −2.58929
\(786\) −11.5076 −0.410462
\(787\) 44.1771 1.57474 0.787371 0.616479i \(-0.211441\pi\)
0.787371 + 0.616479i \(0.211441\pi\)
\(788\) 5.86174 0.208816
\(789\) 9.36932 0.333557
\(790\) 30.2462 1.07611
\(791\) 3.26137 0.115961
\(792\) −4.87689 −0.173293
\(793\) 0 0
\(794\) −39.1231 −1.38843
\(795\) 2.43845 0.0864828
\(796\) 9.72348 0.344640
\(797\) −0.384472 −0.0136187 −0.00680935 0.999977i \(-0.502167\pi\)
−0.00680935 + 0.999977i \(0.502167\pi\)
\(798\) −6.24621 −0.221113
\(799\) −12.8769 −0.455552
\(800\) 18.7386 0.662511
\(801\) 4.87689 0.172317
\(802\) −22.5464 −0.796141
\(803\) 20.2462 0.714473
\(804\) 2.00000 0.0705346
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −21.3693 −0.752236
\(808\) −18.4384 −0.648662
\(809\) −16.3002 −0.573084 −0.286542 0.958068i \(-0.592506\pi\)
−0.286542 + 0.958068i \(0.592506\pi\)
\(810\) 5.56155 0.195413
\(811\) 2.56155 0.0899483 0.0449741 0.998988i \(-0.485679\pi\)
0.0449741 + 0.998988i \(0.485679\pi\)
\(812\) 1.64584 0.0577576
\(813\) −29.9309 −1.04972
\(814\) 23.6155 0.827724
\(815\) 17.1231 0.599796
\(816\) 7.31534 0.256088
\(817\) 32.4924 1.13677
\(818\) −9.94602 −0.347755
\(819\) 0 0
\(820\) 2.43845 0.0851543
\(821\) 6.49242 0.226587 0.113294 0.993562i \(-0.463860\pi\)
0.113294 + 0.993562i \(0.463860\pi\)
\(822\) 8.68466 0.302912
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −8.38447 −0.292087
\(825\) −15.3693 −0.535091
\(826\) 2.52273 0.0877771
\(827\) −14.7386 −0.512513 −0.256256 0.966609i \(-0.582489\pi\)
−0.256256 + 0.966609i \(0.582489\pi\)
\(828\) 0.876894 0.0304742
\(829\) 13.4924 0.468611 0.234306 0.972163i \(-0.424718\pi\)
0.234306 + 0.972163i \(0.424718\pi\)
\(830\) −4.87689 −0.169279
\(831\) 5.31534 0.184387
\(832\) 0 0
\(833\) 10.4384 0.361671
\(834\) 28.0000 0.969561
\(835\) −36.4924 −1.26287
\(836\) −6.24621 −0.216030
\(837\) 2.56155 0.0885402
\(838\) 53.4773 1.84734
\(839\) 21.6155 0.746251 0.373125 0.927781i \(-0.378286\pi\)
0.373125 + 0.927781i \(0.378286\pi\)
\(840\) −4.87689 −0.168269
\(841\) 15.6847 0.540850
\(842\) −48.7926 −1.68150
\(843\) 17.8078 0.613332
\(844\) 8.63068 0.297080
\(845\) 0 0
\(846\) −12.8769 −0.442717
\(847\) −3.93087 −0.135066
\(848\) 3.20739 0.110142
\(849\) −13.6847 −0.469656
\(850\) 18.7386 0.642730
\(851\) 15.1231 0.518413
\(852\) 6.13826 0.210293
\(853\) 2.12311 0.0726938 0.0363469 0.999339i \(-0.488428\pi\)
0.0363469 + 0.999339i \(0.488428\pi\)
\(854\) −3.39963 −0.116333
\(855\) −25.3693 −0.867612
\(856\) −20.1080 −0.687276
\(857\) 35.5616 1.21476 0.607380 0.794412i \(-0.292221\pi\)
0.607380 + 0.794412i \(0.292221\pi\)
\(858\) 0 0
\(859\) 24.5616 0.838029 0.419015 0.907979i \(-0.362376\pi\)
0.419015 + 0.907979i \(0.362376\pi\)
\(860\) −7.12311 −0.242896
\(861\) −0.876894 −0.0298845
\(862\) 17.3693 0.591601
\(863\) 30.4924 1.03797 0.518987 0.854782i \(-0.326309\pi\)
0.518987 + 0.854782i \(0.326309\pi\)
\(864\) 2.43845 0.0829577
\(865\) −72.1080 −2.45174
\(866\) −13.6695 −0.464509
\(867\) −14.5616 −0.494536
\(868\) 0.630683 0.0214068
\(869\) −10.8769 −0.368973
\(870\) 37.1771 1.26042
\(871\) 0 0
\(872\) −6.84658 −0.231855
\(873\) −8.56155 −0.289765
\(874\) −22.2462 −0.752489
\(875\) −5.36932 −0.181516
\(876\) −4.43845 −0.149961
\(877\) 23.5616 0.795617 0.397809 0.917468i \(-0.369771\pi\)
0.397809 + 0.917468i \(0.369771\pi\)
\(878\) 21.3693 0.721180
\(879\) 20.4384 0.689372
\(880\) −33.3693 −1.12488
\(881\) −9.06913 −0.305547 −0.152773 0.988261i \(-0.548820\pi\)
−0.152773 + 0.988261i \(0.548820\pi\)
\(882\) 10.4384 0.351481
\(883\) 8.80776 0.296405 0.148202 0.988957i \(-0.452651\pi\)
0.148202 + 0.988957i \(0.452651\pi\)
\(884\) 0 0
\(885\) 10.2462 0.344423
\(886\) 54.2462 1.82244
\(887\) 24.6307 0.827017 0.413509 0.910500i \(-0.364303\pi\)
0.413509 + 0.910500i \(0.364303\pi\)
\(888\) 18.4384 0.618754
\(889\) 3.05398 0.102427
\(890\) 27.1231 0.909169
\(891\) −2.00000 −0.0670025
\(892\) 3.50758 0.117442
\(893\) 58.7386 1.96561
\(894\) 3.80776 0.127351
\(895\) 17.3693 0.580592
\(896\) −7.61553 −0.254417
\(897\) 0 0
\(898\) 12.8769 0.429708
\(899\) 17.1231 0.571088
\(900\) 3.36932 0.112311
\(901\) 1.06913 0.0356179
\(902\) −4.87689 −0.162383
\(903\) 2.56155 0.0852431
\(904\) 14.1619 0.471019
\(905\) 9.56155 0.317837
\(906\) 14.6307 0.486072
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 3.12311 0.103644
\(909\) −7.56155 −0.250801
\(910\) 0 0
\(911\) 38.7386 1.28347 0.641734 0.766927i \(-0.278215\pi\)
0.641734 + 0.766927i \(0.278215\pi\)
\(912\) −33.3693 −1.10497
\(913\) 1.75379 0.0580419
\(914\) −19.6998 −0.651612
\(915\) −13.8078 −0.456471
\(916\) −7.12311 −0.235354
\(917\) 4.13826 0.136657
\(918\) 2.43845 0.0804807
\(919\) 11.5076 0.379600 0.189800 0.981823i \(-0.439216\pi\)
0.189800 + 0.981823i \(0.439216\pi\)
\(920\) −17.3693 −0.572649
\(921\) 30.8078 1.01515
\(922\) 25.2850 0.832718
\(923\) 0 0
\(924\) −0.492423 −0.0161995
\(925\) 58.1080 1.91058
\(926\) 22.3542 0.734603
\(927\) −3.43845 −0.112933
\(928\) 16.3002 0.535080
\(929\) −7.80776 −0.256164 −0.128082 0.991764i \(-0.540882\pi\)
−0.128082 + 0.991764i \(0.540882\pi\)
\(930\) 14.2462 0.467152
\(931\) −47.6155 −1.56054
\(932\) 11.3996 0.373407
\(933\) −19.1231 −0.626062
\(934\) 40.6004 1.32848
\(935\) −11.1231 −0.363764
\(936\) 0 0
\(937\) −7.56155 −0.247025 −0.123513 0.992343i \(-0.539416\pi\)
−0.123513 + 0.992343i \(0.539416\pi\)
\(938\) −4.00000 −0.130605
\(939\) −13.6847 −0.446582
\(940\) −12.8769 −0.419998
\(941\) 30.4924 0.994025 0.497012 0.867744i \(-0.334430\pi\)
0.497012 + 0.867744i \(0.334430\pi\)
\(942\) −31.8078 −1.03635
\(943\) −3.12311 −0.101702
\(944\) 13.4773 0.438648
\(945\) −2.00000 −0.0650600
\(946\) 14.2462 0.463184
\(947\) −38.7386 −1.25884 −0.629418 0.777067i \(-0.716707\pi\)
−0.629418 + 0.777067i \(0.716707\pi\)
\(948\) 2.38447 0.0774440
\(949\) 0 0
\(950\) −85.4773 −2.77325
\(951\) 14.0540 0.455731
\(952\) −2.13826 −0.0693014
\(953\) 30.9848 1.00370 0.501849 0.864955i \(-0.332653\pi\)
0.501849 + 0.864955i \(0.332653\pi\)
\(954\) 1.06913 0.0346144
\(955\) 32.4924 1.05143
\(956\) −11.1231 −0.359747
\(957\) −13.3693 −0.432169
\(958\) −16.0000 −0.516937
\(959\) −3.12311 −0.100850
\(960\) −19.8078 −0.639293
\(961\) −24.4384 −0.788337
\(962\) 0 0
\(963\) −8.24621 −0.265730
\(964\) −7.80776 −0.251471
\(965\) 48.0540 1.54691
\(966\) −1.75379 −0.0564272
\(967\) −0.876894 −0.0281990 −0.0140995 0.999901i \(-0.504488\pi\)
−0.0140995 + 0.999901i \(0.504488\pi\)
\(968\) −17.0691 −0.548623
\(969\) −11.1231 −0.357326
\(970\) −47.6155 −1.52884
\(971\) −12.9848 −0.416704 −0.208352 0.978054i \(-0.566810\pi\)
−0.208352 + 0.978054i \(0.566810\pi\)
\(972\) 0.438447 0.0140632
\(973\) −10.0691 −0.322801
\(974\) −11.1231 −0.356407
\(975\) 0 0
\(976\) −18.1619 −0.581349
\(977\) 61.1771 1.95723 0.978614 0.205705i \(-0.0659486\pi\)
0.978614 + 0.205705i \(0.0659486\pi\)
\(978\) 7.50758 0.240066
\(979\) −9.75379 −0.311732
\(980\) 10.4384 0.333444
\(981\) −2.80776 −0.0896450
\(982\) 56.6004 1.80619
\(983\) 13.6155 0.434268 0.217134 0.976142i \(-0.430329\pi\)
0.217134 + 0.976142i \(0.430329\pi\)
\(984\) −3.80776 −0.121387
\(985\) −47.6155 −1.51716
\(986\) 16.3002 0.519104
\(987\) 4.63068 0.147396
\(988\) 0 0
\(989\) 9.12311 0.290098
\(990\) −11.1231 −0.353516
\(991\) −50.3542 −1.59955 −0.799776 0.600298i \(-0.795049\pi\)
−0.799776 + 0.600298i \(0.795049\pi\)
\(992\) 6.24621 0.198317
\(993\) −3.19224 −0.101303
\(994\) −12.2765 −0.389388
\(995\) −78.9848 −2.50399
\(996\) −0.384472 −0.0121825
\(997\) −20.6155 −0.652900 −0.326450 0.945214i \(-0.605853\pi\)
−0.326450 + 0.945214i \(0.605853\pi\)
\(998\) −7.01515 −0.222061
\(999\) 7.56155 0.239237
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 507.2.a.g.1.1 2
3.2 odd 2 1521.2.a.g.1.2 2
4.3 odd 2 8112.2.a.bk.1.1 2
13.2 odd 12 507.2.j.g.316.2 8
13.3 even 3 39.2.e.b.22.2 yes 4
13.4 even 6 507.2.e.g.484.1 4
13.5 odd 4 507.2.b.d.337.3 4
13.6 odd 12 507.2.j.g.361.3 8
13.7 odd 12 507.2.j.g.361.2 8
13.8 odd 4 507.2.b.d.337.2 4
13.9 even 3 39.2.e.b.16.2 4
13.10 even 6 507.2.e.g.22.1 4
13.11 odd 12 507.2.j.g.316.3 8
13.12 even 2 507.2.a.d.1.2 2
39.5 even 4 1521.2.b.h.1351.2 4
39.8 even 4 1521.2.b.h.1351.3 4
39.29 odd 6 117.2.g.c.100.1 4
39.35 odd 6 117.2.g.c.55.1 4
39.38 odd 2 1521.2.a.m.1.1 2
52.3 odd 6 624.2.q.h.529.1 4
52.35 odd 6 624.2.q.h.289.1 4
52.51 odd 2 8112.2.a.bo.1.2 2
65.3 odd 12 975.2.bb.i.724.2 8
65.9 even 6 975.2.i.k.601.1 4
65.22 odd 12 975.2.bb.i.874.2 8
65.29 even 6 975.2.i.k.451.1 4
65.42 odd 12 975.2.bb.i.724.3 8
65.48 odd 12 975.2.bb.i.874.3 8
156.35 even 6 1872.2.t.r.289.2 4
156.107 even 6 1872.2.t.r.1153.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.2 4 13.9 even 3
39.2.e.b.22.2 yes 4 13.3 even 3
117.2.g.c.55.1 4 39.35 odd 6
117.2.g.c.100.1 4 39.29 odd 6
507.2.a.d.1.2 2 13.12 even 2
507.2.a.g.1.1 2 1.1 even 1 trivial
507.2.b.d.337.2 4 13.8 odd 4
507.2.b.d.337.3 4 13.5 odd 4
507.2.e.g.22.1 4 13.10 even 6
507.2.e.g.484.1 4 13.4 even 6
507.2.j.g.316.2 8 13.2 odd 12
507.2.j.g.316.3 8 13.11 odd 12
507.2.j.g.361.2 8 13.7 odd 12
507.2.j.g.361.3 8 13.6 odd 12
624.2.q.h.289.1 4 52.35 odd 6
624.2.q.h.529.1 4 52.3 odd 6
975.2.i.k.451.1 4 65.29 even 6
975.2.i.k.601.1 4 65.9 even 6
975.2.bb.i.724.2 8 65.3 odd 12
975.2.bb.i.724.3 8 65.42 odd 12
975.2.bb.i.874.2 8 65.22 odd 12
975.2.bb.i.874.3 8 65.48 odd 12
1521.2.a.g.1.2 2 3.2 odd 2
1521.2.a.m.1.1 2 39.38 odd 2
1521.2.b.h.1351.2 4 39.5 even 4
1521.2.b.h.1351.3 4 39.8 even 4
1872.2.t.r.289.2 4 156.35 even 6
1872.2.t.r.1153.2 4 156.107 even 6
8112.2.a.bk.1.1 2 4.3 odd 2
8112.2.a.bo.1.2 2 52.51 odd 2