Properties

Label 3610.2.a.t.1.2
Level $3610$
Weight $2$
Character 3610.1
Self dual yes
Analytic conductor $28.826$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3610,2,Mod(1,3610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3610, 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("3610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.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: \(28.8259951297\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.56155 q^{6} -2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.56155 q^{6} -2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +1.00000 q^{10} +4.00000 q^{11} +2.56155 q^{12} -5.68466 q^{13} -2.56155 q^{14} +2.56155 q^{15} +1.00000 q^{16} +3.43845 q^{17} +3.56155 q^{18} +1.00000 q^{20} -6.56155 q^{21} +4.00000 q^{22} +7.68466 q^{23} +2.56155 q^{24} +1.00000 q^{25} -5.68466 q^{26} +1.43845 q^{27} -2.56155 q^{28} +5.68466 q^{29} +2.56155 q^{30} +5.12311 q^{31} +1.00000 q^{32} +10.2462 q^{33} +3.43845 q^{34} -2.56155 q^{35} +3.56155 q^{36} +6.00000 q^{37} -14.5616 q^{39} +1.00000 q^{40} -12.2462 q^{41} -6.56155 q^{42} -2.87689 q^{43} +4.00000 q^{44} +3.56155 q^{45} +7.68466 q^{46} +6.24621 q^{47} +2.56155 q^{48} -0.438447 q^{49} +1.00000 q^{50} +8.80776 q^{51} -5.68466 q^{52} +4.56155 q^{53} +1.43845 q^{54} +4.00000 q^{55} -2.56155 q^{56} +5.68466 q^{58} -2.56155 q^{59} +2.56155 q^{60} +11.1231 q^{61} +5.12311 q^{62} -9.12311 q^{63} +1.00000 q^{64} -5.68466 q^{65} +10.2462 q^{66} +2.56155 q^{67} +3.43845 q^{68} +19.6847 q^{69} -2.56155 q^{70} -10.2462 q^{71} +3.56155 q^{72} -1.68466 q^{73} +6.00000 q^{74} +2.56155 q^{75} -10.2462 q^{77} -14.5616 q^{78} +5.12311 q^{79} +1.00000 q^{80} -7.00000 q^{81} -12.2462 q^{82} +2.87689 q^{83} -6.56155 q^{84} +3.43845 q^{85} -2.87689 q^{86} +14.5616 q^{87} +4.00000 q^{88} -2.00000 q^{89} +3.56155 q^{90} +14.5616 q^{91} +7.68466 q^{92} +13.1231 q^{93} +6.24621 q^{94} +2.56155 q^{96} -6.00000 q^{97} -0.438447 q^{98} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - q^{7} + 2 q^{8} + 3 q^{9} + 2 q^{10} + 8 q^{11} + q^{12} + q^{13} - q^{14} + q^{15} + 2 q^{16} + 11 q^{17} + 3 q^{18} + 2 q^{20} - 9 q^{21} + 8 q^{22} + 3 q^{23} + q^{24} + 2 q^{25} + q^{26} + 7 q^{27} - q^{28} - q^{29} + q^{30} + 2 q^{31} + 2 q^{32} + 4 q^{33} + 11 q^{34} - q^{35} + 3 q^{36} + 12 q^{37} - 25 q^{39} + 2 q^{40} - 8 q^{41} - 9 q^{42} - 14 q^{43} + 8 q^{44} + 3 q^{45} + 3 q^{46} - 4 q^{47} + q^{48} - 5 q^{49} + 2 q^{50} - 3 q^{51} + q^{52} + 5 q^{53} + 7 q^{54} + 8 q^{55} - q^{56} - q^{58} - q^{59} + q^{60} + 14 q^{61} + 2 q^{62} - 10 q^{63} + 2 q^{64} + q^{65} + 4 q^{66} + q^{67} + 11 q^{68} + 27 q^{69} - q^{70} - 4 q^{71} + 3 q^{72} + 9 q^{73} + 12 q^{74} + q^{75} - 4 q^{77} - 25 q^{78} + 2 q^{79} + 2 q^{80} - 14 q^{81} - 8 q^{82} + 14 q^{83} - 9 q^{84} + 11 q^{85} - 14 q^{86} + 25 q^{87} + 8 q^{88} - 4 q^{89} + 3 q^{90} + 25 q^{91} + 3 q^{92} + 18 q^{93} - 4 q^{94} + q^{96} - 12 q^{97} - 5 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.56155 1.04575
\(7\) −2.56155 −0.968176 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.56155 1.18718
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 2.56155 0.739457
\(13\) −5.68466 −1.57664 −0.788320 0.615265i \(-0.789049\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) −2.56155 −0.684604
\(15\) 2.56155 0.661390
\(16\) 1.00000 0.250000
\(17\) 3.43845 0.833946 0.416973 0.908919i \(-0.363091\pi\)
0.416973 + 0.908919i \(0.363091\pi\)
\(18\) 3.56155 0.839466
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) −6.56155 −1.43185
\(22\) 4.00000 0.852803
\(23\) 7.68466 1.60236 0.801181 0.598422i \(-0.204205\pi\)
0.801181 + 0.598422i \(0.204205\pi\)
\(24\) 2.56155 0.522875
\(25\) 1.00000 0.200000
\(26\) −5.68466 −1.11485
\(27\) 1.43845 0.276829
\(28\) −2.56155 −0.484088
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 2.56155 0.467673
\(31\) 5.12311 0.920137 0.460068 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 1.00000 0.176777
\(33\) 10.2462 1.78364
\(34\) 3.43845 0.589689
\(35\) −2.56155 −0.432981
\(36\) 3.56155 0.593592
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −14.5616 −2.33171
\(40\) 1.00000 0.158114
\(41\) −12.2462 −1.91254 −0.956268 0.292490i \(-0.905516\pi\)
−0.956268 + 0.292490i \(0.905516\pi\)
\(42\) −6.56155 −1.01247
\(43\) −2.87689 −0.438722 −0.219361 0.975644i \(-0.570397\pi\)
−0.219361 + 0.975644i \(0.570397\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.56155 0.530925
\(46\) 7.68466 1.13304
\(47\) 6.24621 0.911104 0.455552 0.890209i \(-0.349442\pi\)
0.455552 + 0.890209i \(0.349442\pi\)
\(48\) 2.56155 0.369728
\(49\) −0.438447 −0.0626353
\(50\) 1.00000 0.141421
\(51\) 8.80776 1.23333
\(52\) −5.68466 −0.788320
\(53\) 4.56155 0.626577 0.313289 0.949658i \(-0.398569\pi\)
0.313289 + 0.949658i \(0.398569\pi\)
\(54\) 1.43845 0.195748
\(55\) 4.00000 0.539360
\(56\) −2.56155 −0.342302
\(57\) 0 0
\(58\) 5.68466 0.746432
\(59\) −2.56155 −0.333486 −0.166743 0.986000i \(-0.553325\pi\)
−0.166743 + 0.986000i \(0.553325\pi\)
\(60\) 2.56155 0.330695
\(61\) 11.1231 1.42417 0.712084 0.702094i \(-0.247752\pi\)
0.712084 + 0.702094i \(0.247752\pi\)
\(62\) 5.12311 0.650635
\(63\) −9.12311 −1.14940
\(64\) 1.00000 0.125000
\(65\) −5.68466 −0.705095
\(66\) 10.2462 1.26122
\(67\) 2.56155 0.312943 0.156472 0.987682i \(-0.449988\pi\)
0.156472 + 0.987682i \(0.449988\pi\)
\(68\) 3.43845 0.416973
\(69\) 19.6847 2.36975
\(70\) −2.56155 −0.306164
\(71\) −10.2462 −1.21600 −0.608001 0.793936i \(-0.708028\pi\)
−0.608001 + 0.793936i \(0.708028\pi\)
\(72\) 3.56155 0.419733
\(73\) −1.68466 −0.197174 −0.0985872 0.995128i \(-0.531432\pi\)
−0.0985872 + 0.995128i \(0.531432\pi\)
\(74\) 6.00000 0.697486
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) −10.2462 −1.16766
\(78\) −14.5616 −1.64877
\(79\) 5.12311 0.576394 0.288197 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.00000 −0.777778
\(82\) −12.2462 −1.35237
\(83\) 2.87689 0.315780 0.157890 0.987457i \(-0.449531\pi\)
0.157890 + 0.987457i \(0.449531\pi\)
\(84\) −6.56155 −0.715924
\(85\) 3.43845 0.372952
\(86\) −2.87689 −0.310223
\(87\) 14.5616 1.56116
\(88\) 4.00000 0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 3.56155 0.375421
\(91\) 14.5616 1.52647
\(92\) 7.68466 0.801181
\(93\) 13.1231 1.36080
\(94\) 6.24621 0.644247
\(95\) 0 0
\(96\) 2.56155 0.261437
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −0.438447 −0.0442899
\(99\) 14.2462 1.43180
\(100\) 1.00000 0.100000
\(101\) −17.3693 −1.72831 −0.864156 0.503224i \(-0.832147\pi\)
−0.864156 + 0.503224i \(0.832147\pi\)
\(102\) 8.80776 0.872099
\(103\) −2.24621 −0.221326 −0.110663 0.993858i \(-0.535297\pi\)
−0.110663 + 0.993858i \(0.535297\pi\)
\(104\) −5.68466 −0.557427
\(105\) −6.56155 −0.640342
\(106\) 4.56155 0.443057
\(107\) 5.43845 0.525755 0.262877 0.964829i \(-0.415329\pi\)
0.262877 + 0.964829i \(0.415329\pi\)
\(108\) 1.43845 0.138415
\(109\) 0.561553 0.0537870 0.0268935 0.999638i \(-0.491439\pi\)
0.0268935 + 0.999638i \(0.491439\pi\)
\(110\) 4.00000 0.381385
\(111\) 15.3693 1.45879
\(112\) −2.56155 −0.242044
\(113\) −8.87689 −0.835068 −0.417534 0.908661i \(-0.637106\pi\)
−0.417534 + 0.908661i \(0.637106\pi\)
\(114\) 0 0
\(115\) 7.68466 0.716598
\(116\) 5.68466 0.527807
\(117\) −20.2462 −1.87176
\(118\) −2.56155 −0.235810
\(119\) −8.80776 −0.807406
\(120\) 2.56155 0.233837
\(121\) 5.00000 0.454545
\(122\) 11.1231 1.00704
\(123\) −31.3693 −2.82848
\(124\) 5.12311 0.460068
\(125\) 1.00000 0.0894427
\(126\) −9.12311 −0.812751
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.36932 −0.648832
\(130\) −5.68466 −0.498578
\(131\) −16.4924 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(132\) 10.2462 0.891818
\(133\) 0 0
\(134\) 2.56155 0.221284
\(135\) 1.43845 0.123802
\(136\) 3.43845 0.294844
\(137\) −14.8078 −1.26511 −0.632556 0.774514i \(-0.717994\pi\)
−0.632556 + 0.774514i \(0.717994\pi\)
\(138\) 19.6847 1.67567
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) −2.56155 −0.216491
\(141\) 16.0000 1.34744
\(142\) −10.2462 −0.859843
\(143\) −22.7386 −1.90150
\(144\) 3.56155 0.296796
\(145\) 5.68466 0.472085
\(146\) −1.68466 −0.139423
\(147\) −1.12311 −0.0926322
\(148\) 6.00000 0.493197
\(149\) 13.3693 1.09526 0.547629 0.836722i \(-0.315531\pi\)
0.547629 + 0.836722i \(0.315531\pi\)
\(150\) 2.56155 0.209150
\(151\) −5.12311 −0.416912 −0.208456 0.978032i \(-0.566844\pi\)
−0.208456 + 0.978032i \(0.566844\pi\)
\(152\) 0 0
\(153\) 12.2462 0.990048
\(154\) −10.2462 −0.825663
\(155\) 5.12311 0.411498
\(156\) −14.5616 −1.16586
\(157\) −20.2462 −1.61582 −0.807912 0.589303i \(-0.799402\pi\)
−0.807912 + 0.589303i \(0.799402\pi\)
\(158\) 5.12311 0.407572
\(159\) 11.6847 0.926654
\(160\) 1.00000 0.0790569
\(161\) −19.6847 −1.55137
\(162\) −7.00000 −0.549972
\(163\) −15.3693 −1.20382 −0.601909 0.798565i \(-0.705593\pi\)
−0.601909 + 0.798565i \(0.705593\pi\)
\(164\) −12.2462 −0.956268
\(165\) 10.2462 0.797666
\(166\) 2.87689 0.223290
\(167\) 7.36932 0.570255 0.285127 0.958490i \(-0.407964\pi\)
0.285127 + 0.958490i \(0.407964\pi\)
\(168\) −6.56155 −0.506235
\(169\) 19.3153 1.48580
\(170\) 3.43845 0.263717
\(171\) 0 0
\(172\) −2.87689 −0.219361
\(173\) −20.2462 −1.53929 −0.769645 0.638471i \(-0.779567\pi\)
−0.769645 + 0.638471i \(0.779567\pi\)
\(174\) 14.5616 1.10391
\(175\) −2.56155 −0.193635
\(176\) 4.00000 0.301511
\(177\) −6.56155 −0.493197
\(178\) −2.00000 −0.149906
\(179\) 22.2462 1.66276 0.831380 0.555704i \(-0.187551\pi\)
0.831380 + 0.555704i \(0.187551\pi\)
\(180\) 3.56155 0.265462
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 14.5616 1.07937
\(183\) 28.4924 2.10622
\(184\) 7.68466 0.566521
\(185\) 6.00000 0.441129
\(186\) 13.1231 0.962233
\(187\) 13.7538 1.00578
\(188\) 6.24621 0.455552
\(189\) −3.68466 −0.268019
\(190\) 0 0
\(191\) −3.68466 −0.266613 −0.133306 0.991075i \(-0.542559\pi\)
−0.133306 + 0.991075i \(0.542559\pi\)
\(192\) 2.56155 0.184864
\(193\) 14.4924 1.04319 0.521594 0.853194i \(-0.325338\pi\)
0.521594 + 0.853194i \(0.325338\pi\)
\(194\) −6.00000 −0.430775
\(195\) −14.5616 −1.04277
\(196\) −0.438447 −0.0313177
\(197\) −20.2462 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(198\) 14.2462 1.01243
\(199\) 16.8078 1.19147 0.595735 0.803181i \(-0.296861\pi\)
0.595735 + 0.803181i \(0.296861\pi\)
\(200\) 1.00000 0.0707107
\(201\) 6.56155 0.462816
\(202\) −17.3693 −1.22210
\(203\) −14.5616 −1.02202
\(204\) 8.80776 0.616667
\(205\) −12.2462 −0.855312
\(206\) −2.24621 −0.156501
\(207\) 27.3693 1.90230
\(208\) −5.68466 −0.394160
\(209\) 0 0
\(210\) −6.56155 −0.452790
\(211\) −8.31534 −0.572452 −0.286226 0.958162i \(-0.592401\pi\)
−0.286226 + 0.958162i \(0.592401\pi\)
\(212\) 4.56155 0.313289
\(213\) −26.2462 −1.79836
\(214\) 5.43845 0.371765
\(215\) −2.87689 −0.196203
\(216\) 1.43845 0.0978739
\(217\) −13.1231 −0.890854
\(218\) 0.561553 0.0380332
\(219\) −4.31534 −0.291604
\(220\) 4.00000 0.269680
\(221\) −19.5464 −1.31483
\(222\) 15.3693 1.03152
\(223\) 23.3693 1.56493 0.782463 0.622698i \(-0.213963\pi\)
0.782463 + 0.622698i \(0.213963\pi\)
\(224\) −2.56155 −0.171151
\(225\) 3.56155 0.237437
\(226\) −8.87689 −0.590482
\(227\) −25.9309 −1.72109 −0.860546 0.509373i \(-0.829878\pi\)
−0.860546 + 0.509373i \(0.829878\pi\)
\(228\) 0 0
\(229\) −14.4924 −0.957686 −0.478843 0.877900i \(-0.658944\pi\)
−0.478843 + 0.877900i \(0.658944\pi\)
\(230\) 7.68466 0.506711
\(231\) −26.2462 −1.72687
\(232\) 5.68466 0.373216
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −20.2462 −1.32354
\(235\) 6.24621 0.407458
\(236\) −2.56155 −0.166743
\(237\) 13.1231 0.852437
\(238\) −8.80776 −0.570923
\(239\) −1.43845 −0.0930454 −0.0465227 0.998917i \(-0.514814\pi\)
−0.0465227 + 0.998917i \(0.514814\pi\)
\(240\) 2.56155 0.165348
\(241\) −23.1231 −1.48949 −0.744745 0.667349i \(-0.767429\pi\)
−0.744745 + 0.667349i \(0.767429\pi\)
\(242\) 5.00000 0.321412
\(243\) −22.2462 −1.42710
\(244\) 11.1231 0.712084
\(245\) −0.438447 −0.0280114
\(246\) −31.3693 −2.00003
\(247\) 0 0
\(248\) 5.12311 0.325318
\(249\) 7.36932 0.467011
\(250\) 1.00000 0.0632456
\(251\) 6.24621 0.394257 0.197129 0.980378i \(-0.436838\pi\)
0.197129 + 0.980378i \(0.436838\pi\)
\(252\) −9.12311 −0.574702
\(253\) 30.7386 1.93252
\(254\) −13.1231 −0.823417
\(255\) 8.80776 0.551564
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −7.36932 −0.458794
\(259\) −15.3693 −0.955003
\(260\) −5.68466 −0.352548
\(261\) 20.2462 1.25321
\(262\) −16.4924 −1.01891
\(263\) −22.2462 −1.37176 −0.685880 0.727715i \(-0.740583\pi\)
−0.685880 + 0.727715i \(0.740583\pi\)
\(264\) 10.2462 0.630611
\(265\) 4.56155 0.280214
\(266\) 0 0
\(267\) −5.12311 −0.313529
\(268\) 2.56155 0.156472
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 1.43845 0.0875411
\(271\) −21.9309 −1.33221 −0.666103 0.745860i \(-0.732039\pi\)
−0.666103 + 0.745860i \(0.732039\pi\)
\(272\) 3.43845 0.208486
\(273\) 37.3002 2.25751
\(274\) −14.8078 −0.894570
\(275\) 4.00000 0.241209
\(276\) 19.6847 1.18488
\(277\) 0.876894 0.0526875 0.0263437 0.999653i \(-0.491614\pi\)
0.0263437 + 0.999653i \(0.491614\pi\)
\(278\) 16.4924 0.989150
\(279\) 18.2462 1.09237
\(280\) −2.56155 −0.153082
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 16.0000 0.952786
\(283\) −21.1231 −1.25564 −0.627819 0.778359i \(-0.716052\pi\)
−0.627819 + 0.778359i \(0.716052\pi\)
\(284\) −10.2462 −0.608001
\(285\) 0 0
\(286\) −22.7386 −1.34456
\(287\) 31.3693 1.85167
\(288\) 3.56155 0.209867
\(289\) −5.17708 −0.304534
\(290\) 5.68466 0.333815
\(291\) −15.3693 −0.900965
\(292\) −1.68466 −0.0985872
\(293\) 22.1771 1.29560 0.647799 0.761811i \(-0.275689\pi\)
0.647799 + 0.761811i \(0.275689\pi\)
\(294\) −1.12311 −0.0655009
\(295\) −2.56155 −0.149139
\(296\) 6.00000 0.348743
\(297\) 5.75379 0.333869
\(298\) 13.3693 0.774464
\(299\) −43.6847 −2.52635
\(300\) 2.56155 0.147891
\(301\) 7.36932 0.424760
\(302\) −5.12311 −0.294802
\(303\) −44.4924 −2.55602
\(304\) 0 0
\(305\) 11.1231 0.636907
\(306\) 12.2462 0.700069
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) −10.2462 −0.583832
\(309\) −5.75379 −0.327322
\(310\) 5.12311 0.290973
\(311\) −3.68466 −0.208938 −0.104469 0.994528i \(-0.533314\pi\)
−0.104469 + 0.994528i \(0.533314\pi\)
\(312\) −14.5616 −0.824386
\(313\) 5.05398 0.285668 0.142834 0.989747i \(-0.454379\pi\)
0.142834 + 0.989747i \(0.454379\pi\)
\(314\) −20.2462 −1.14256
\(315\) −9.12311 −0.514029
\(316\) 5.12311 0.288197
\(317\) −13.0540 −0.733184 −0.366592 0.930382i \(-0.619476\pi\)
−0.366592 + 0.930382i \(0.619476\pi\)
\(318\) 11.6847 0.655243
\(319\) 22.7386 1.27312
\(320\) 1.00000 0.0559017
\(321\) 13.9309 0.777545
\(322\) −19.6847 −1.09698
\(323\) 0 0
\(324\) −7.00000 −0.388889
\(325\) −5.68466 −0.315328
\(326\) −15.3693 −0.851228
\(327\) 1.43845 0.0795463
\(328\) −12.2462 −0.676184
\(329\) −16.0000 −0.882109
\(330\) 10.2462 0.564035
\(331\) 2.56155 0.140796 0.0703978 0.997519i \(-0.477573\pi\)
0.0703978 + 0.997519i \(0.477573\pi\)
\(332\) 2.87689 0.157890
\(333\) 21.3693 1.17103
\(334\) 7.36932 0.403231
\(335\) 2.56155 0.139953
\(336\) −6.56155 −0.357962
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 19.3153 1.05062
\(339\) −22.7386 −1.23499
\(340\) 3.43845 0.186476
\(341\) 20.4924 1.10973
\(342\) 0 0
\(343\) 19.0540 1.02882
\(344\) −2.87689 −0.155112
\(345\) 19.6847 1.05979
\(346\) −20.2462 −1.08844
\(347\) −8.63068 −0.463319 −0.231660 0.972797i \(-0.574416\pi\)
−0.231660 + 0.972797i \(0.574416\pi\)
\(348\) 14.5616 0.780581
\(349\) 3.75379 0.200936 0.100468 0.994940i \(-0.467966\pi\)
0.100468 + 0.994940i \(0.467966\pi\)
\(350\) −2.56155 −0.136921
\(351\) −8.17708 −0.436460
\(352\) 4.00000 0.213201
\(353\) −3.93087 −0.209219 −0.104610 0.994513i \(-0.533359\pi\)
−0.104610 + 0.994513i \(0.533359\pi\)
\(354\) −6.56155 −0.348743
\(355\) −10.2462 −0.543812
\(356\) −2.00000 −0.106000
\(357\) −22.5616 −1.19408
\(358\) 22.2462 1.17575
\(359\) 1.43845 0.0759183 0.0379592 0.999279i \(-0.487914\pi\)
0.0379592 + 0.999279i \(0.487914\pi\)
\(360\) 3.56155 0.187710
\(361\) 0 0
\(362\) 18.0000 0.946059
\(363\) 12.8078 0.672233
\(364\) 14.5616 0.763233
\(365\) −1.68466 −0.0881791
\(366\) 28.4924 1.48932
\(367\) 6.24621 0.326050 0.163025 0.986622i \(-0.447875\pi\)
0.163025 + 0.986622i \(0.447875\pi\)
\(368\) 7.68466 0.400591
\(369\) −43.6155 −2.27053
\(370\) 6.00000 0.311925
\(371\) −11.6847 −0.606637
\(372\) 13.1231 0.680401
\(373\) 23.4384 1.21360 0.606798 0.794856i \(-0.292454\pi\)
0.606798 + 0.794856i \(0.292454\pi\)
\(374\) 13.7538 0.711191
\(375\) 2.56155 0.132278
\(376\) 6.24621 0.322124
\(377\) −32.3153 −1.66432
\(378\) −3.68466 −0.189518
\(379\) −10.5616 −0.542511 −0.271255 0.962507i \(-0.587439\pi\)
−0.271255 + 0.962507i \(0.587439\pi\)
\(380\) 0 0
\(381\) −33.6155 −1.72218
\(382\) −3.68466 −0.188524
\(383\) −13.7538 −0.702786 −0.351393 0.936228i \(-0.614292\pi\)
−0.351393 + 0.936228i \(0.614292\pi\)
\(384\) 2.56155 0.130719
\(385\) −10.2462 −0.522195
\(386\) 14.4924 0.737645
\(387\) −10.2462 −0.520844
\(388\) −6.00000 −0.304604
\(389\) −7.12311 −0.361156 −0.180578 0.983561i \(-0.557797\pi\)
−0.180578 + 0.983561i \(0.557797\pi\)
\(390\) −14.5616 −0.737353
\(391\) 26.4233 1.33628
\(392\) −0.438447 −0.0221449
\(393\) −42.2462 −2.13104
\(394\) −20.2462 −1.01999
\(395\) 5.12311 0.257771
\(396\) 14.2462 0.715899
\(397\) −7.12311 −0.357498 −0.178749 0.983895i \(-0.557205\pi\)
−0.178749 + 0.983895i \(0.557205\pi\)
\(398\) 16.8078 0.842497
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 3.75379 0.187455 0.0937276 0.995598i \(-0.470122\pi\)
0.0937276 + 0.995598i \(0.470122\pi\)
\(402\) 6.56155 0.327261
\(403\) −29.1231 −1.45073
\(404\) −17.3693 −0.864156
\(405\) −7.00000 −0.347833
\(406\) −14.5616 −0.722678
\(407\) 24.0000 1.18964
\(408\) 8.80776 0.436049
\(409\) −24.7386 −1.22325 −0.611623 0.791149i \(-0.709483\pi\)
−0.611623 + 0.791149i \(0.709483\pi\)
\(410\) −12.2462 −0.604797
\(411\) −37.9309 −1.87099
\(412\) −2.24621 −0.110663
\(413\) 6.56155 0.322873
\(414\) 27.3693 1.34513
\(415\) 2.87689 0.141221
\(416\) −5.68466 −0.278713
\(417\) 42.2462 2.06881
\(418\) 0 0
\(419\) 23.8617 1.16572 0.582861 0.812572i \(-0.301933\pi\)
0.582861 + 0.812572i \(0.301933\pi\)
\(420\) −6.56155 −0.320171
\(421\) 23.9309 1.16632 0.583160 0.812358i \(-0.301816\pi\)
0.583160 + 0.812358i \(0.301816\pi\)
\(422\) −8.31534 −0.404784
\(423\) 22.2462 1.08165
\(424\) 4.56155 0.221529
\(425\) 3.43845 0.166789
\(426\) −26.2462 −1.27163
\(427\) −28.4924 −1.37884
\(428\) 5.43845 0.262877
\(429\) −58.2462 −2.81215
\(430\) −2.87689 −0.138736
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.43845 0.0692073
\(433\) −14.6307 −0.703106 −0.351553 0.936168i \(-0.614346\pi\)
−0.351553 + 0.936168i \(0.614346\pi\)
\(434\) −13.1231 −0.629929
\(435\) 14.5616 0.698173
\(436\) 0.561553 0.0268935
\(437\) 0 0
\(438\) −4.31534 −0.206195
\(439\) 13.1231 0.626332 0.313166 0.949698i \(-0.398610\pi\)
0.313166 + 0.949698i \(0.398610\pi\)
\(440\) 4.00000 0.190693
\(441\) −1.56155 −0.0743597
\(442\) −19.5464 −0.929727
\(443\) 2.24621 0.106721 0.0533604 0.998575i \(-0.483007\pi\)
0.0533604 + 0.998575i \(0.483007\pi\)
\(444\) 15.3693 0.729396
\(445\) −2.00000 −0.0948091
\(446\) 23.3693 1.10657
\(447\) 34.2462 1.61979
\(448\) −2.56155 −0.121022
\(449\) 28.7386 1.35626 0.678130 0.734942i \(-0.262791\pi\)
0.678130 + 0.734942i \(0.262791\pi\)
\(450\) 3.56155 0.167893
\(451\) −48.9848 −2.30661
\(452\) −8.87689 −0.417534
\(453\) −13.1231 −0.616577
\(454\) −25.9309 −1.21700
\(455\) 14.5616 0.682656
\(456\) 0 0
\(457\) 6.31534 0.295419 0.147710 0.989031i \(-0.452810\pi\)
0.147710 + 0.989031i \(0.452810\pi\)
\(458\) −14.4924 −0.677186
\(459\) 4.94602 0.230861
\(460\) 7.68466 0.358299
\(461\) 3.75379 0.174831 0.0874157 0.996172i \(-0.472139\pi\)
0.0874157 + 0.996172i \(0.472139\pi\)
\(462\) −26.2462 −1.22108
\(463\) −30.2462 −1.40566 −0.702830 0.711358i \(-0.748081\pi\)
−0.702830 + 0.711358i \(0.748081\pi\)
\(464\) 5.68466 0.263904
\(465\) 13.1231 0.608569
\(466\) 10.0000 0.463241
\(467\) 18.2462 0.844334 0.422167 0.906518i \(-0.361270\pi\)
0.422167 + 0.906518i \(0.361270\pi\)
\(468\) −20.2462 −0.935881
\(469\) −6.56155 −0.302984
\(470\) 6.24621 0.288116
\(471\) −51.8617 −2.38966
\(472\) −2.56155 −0.117905
\(473\) −11.5076 −0.529119
\(474\) 13.1231 0.602764
\(475\) 0 0
\(476\) −8.80776 −0.403703
\(477\) 16.2462 0.743863
\(478\) −1.43845 −0.0657930
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 2.56155 0.116918
\(481\) −34.1080 −1.55519
\(482\) −23.1231 −1.05323
\(483\) −50.4233 −2.29434
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) −22.2462 −1.00911
\(487\) 17.6155 0.798236 0.399118 0.916900i \(-0.369316\pi\)
0.399118 + 0.916900i \(0.369316\pi\)
\(488\) 11.1231 0.503519
\(489\) −39.3693 −1.78034
\(490\) −0.438447 −0.0198070
\(491\) −1.12311 −0.0506850 −0.0253425 0.999679i \(-0.508068\pi\)
−0.0253425 + 0.999679i \(0.508068\pi\)
\(492\) −31.3693 −1.41424
\(493\) 19.5464 0.880325
\(494\) 0 0
\(495\) 14.2462 0.640320
\(496\) 5.12311 0.230034
\(497\) 26.2462 1.17730
\(498\) 7.36932 0.330227
\(499\) 42.1080 1.88501 0.942505 0.334191i \(-0.108463\pi\)
0.942505 + 0.334191i \(0.108463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 18.8769 0.843357
\(502\) 6.24621 0.278782
\(503\) −7.05398 −0.314521 −0.157261 0.987557i \(-0.550266\pi\)
−0.157261 + 0.987557i \(0.550266\pi\)
\(504\) −9.12311 −0.406375
\(505\) −17.3693 −0.772924
\(506\) 30.7386 1.36650
\(507\) 49.4773 2.19736
\(508\) −13.1231 −0.582244
\(509\) −2.49242 −0.110475 −0.0552373 0.998473i \(-0.517592\pi\)
−0.0552373 + 0.998473i \(0.517592\pi\)
\(510\) 8.80776 0.390014
\(511\) 4.31534 0.190899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −2.24621 −0.0989799
\(516\) −7.36932 −0.324416
\(517\) 24.9848 1.09883
\(518\) −15.3693 −0.675289
\(519\) −51.8617 −2.27648
\(520\) −5.68466 −0.249289
\(521\) 3.12311 0.136826 0.0684129 0.997657i \(-0.478206\pi\)
0.0684129 + 0.997657i \(0.478206\pi\)
\(522\) 20.2462 0.886153
\(523\) 31.6847 1.38547 0.692737 0.721191i \(-0.256405\pi\)
0.692737 + 0.721191i \(0.256405\pi\)
\(524\) −16.4924 −0.720475
\(525\) −6.56155 −0.286370
\(526\) −22.2462 −0.969981
\(527\) 17.6155 0.767344
\(528\) 10.2462 0.445909
\(529\) 36.0540 1.56756
\(530\) 4.56155 0.198141
\(531\) −9.12311 −0.395909
\(532\) 0 0
\(533\) 69.6155 3.01538
\(534\) −5.12311 −0.221698
\(535\) 5.43845 0.235125
\(536\) 2.56155 0.110642
\(537\) 56.9848 2.45908
\(538\) 26.0000 1.12094
\(539\) −1.75379 −0.0755410
\(540\) 1.43845 0.0619009
\(541\) −0.384472 −0.0165297 −0.00826487 0.999966i \(-0.502631\pi\)
−0.00826487 + 0.999966i \(0.502631\pi\)
\(542\) −21.9309 −0.942012
\(543\) 46.1080 1.97868
\(544\) 3.43845 0.147422
\(545\) 0.561553 0.0240543
\(546\) 37.3002 1.59630
\(547\) −16.4924 −0.705165 −0.352583 0.935781i \(-0.614696\pi\)
−0.352583 + 0.935781i \(0.614696\pi\)
\(548\) −14.8078 −0.632556
\(549\) 39.6155 1.69075
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) 19.6847 0.837835
\(553\) −13.1231 −0.558051
\(554\) 0.876894 0.0372557
\(555\) 15.3693 0.652391
\(556\) 16.4924 0.699435
\(557\) 39.6155 1.67856 0.839282 0.543697i \(-0.182976\pi\)
0.839282 + 0.543697i \(0.182976\pi\)
\(558\) 18.2462 0.772424
\(559\) 16.3542 0.691707
\(560\) −2.56155 −0.108245
\(561\) 35.2311 1.48746
\(562\) −2.00000 −0.0843649
\(563\) 8.49242 0.357913 0.178956 0.983857i \(-0.442728\pi\)
0.178956 + 0.983857i \(0.442728\pi\)
\(564\) 16.0000 0.673722
\(565\) −8.87689 −0.373454
\(566\) −21.1231 −0.887870
\(567\) 17.9309 0.753026
\(568\) −10.2462 −0.429921
\(569\) 3.12311 0.130927 0.0654637 0.997855i \(-0.479147\pi\)
0.0654637 + 0.997855i \(0.479147\pi\)
\(570\) 0 0
\(571\) 21.6155 0.904582 0.452291 0.891870i \(-0.350607\pi\)
0.452291 + 0.891870i \(0.350607\pi\)
\(572\) −22.7386 −0.950750
\(573\) −9.43845 −0.394297
\(574\) 31.3693 1.30933
\(575\) 7.68466 0.320472
\(576\) 3.56155 0.148398
\(577\) −10.3153 −0.429433 −0.214717 0.976676i \(-0.568883\pi\)
−0.214717 + 0.976676i \(0.568883\pi\)
\(578\) −5.17708 −0.215338
\(579\) 37.1231 1.54278
\(580\) 5.68466 0.236043
\(581\) −7.36932 −0.305731
\(582\) −15.3693 −0.637079
\(583\) 18.2462 0.755681
\(584\) −1.68466 −0.0697117
\(585\) −20.2462 −0.837078
\(586\) 22.1771 0.916127
\(587\) −7.36932 −0.304164 −0.152082 0.988368i \(-0.548598\pi\)
−0.152082 + 0.988368i \(0.548598\pi\)
\(588\) −1.12311 −0.0463161
\(589\) 0 0
\(590\) −2.56155 −0.105457
\(591\) −51.8617 −2.13331
\(592\) 6.00000 0.246598
\(593\) 7.75379 0.318410 0.159205 0.987246i \(-0.449107\pi\)
0.159205 + 0.987246i \(0.449107\pi\)
\(594\) 5.75379 0.236081
\(595\) −8.80776 −0.361083
\(596\) 13.3693 0.547629
\(597\) 43.0540 1.76208
\(598\) −43.6847 −1.78640
\(599\) 11.8617 0.484658 0.242329 0.970194i \(-0.422089\pi\)
0.242329 + 0.970194i \(0.422089\pi\)
\(600\) 2.56155 0.104575
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 7.36932 0.300351
\(603\) 9.12311 0.371522
\(604\) −5.12311 −0.208456
\(605\) 5.00000 0.203279
\(606\) −44.4924 −1.80738
\(607\) 21.1231 0.857360 0.428680 0.903456i \(-0.358979\pi\)
0.428680 + 0.903456i \(0.358979\pi\)
\(608\) 0 0
\(609\) −37.3002 −1.51148
\(610\) 11.1231 0.450361
\(611\) −35.5076 −1.43648
\(612\) 12.2462 0.495024
\(613\) 5.36932 0.216865 0.108432 0.994104i \(-0.465417\pi\)
0.108432 + 0.994104i \(0.465417\pi\)
\(614\) −32.4924 −1.31129
\(615\) −31.3693 −1.26493
\(616\) −10.2462 −0.412832
\(617\) 12.2462 0.493014 0.246507 0.969141i \(-0.420717\pi\)
0.246507 + 0.969141i \(0.420717\pi\)
\(618\) −5.75379 −0.231451
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 5.12311 0.205749
\(621\) 11.0540 0.443581
\(622\) −3.68466 −0.147741
\(623\) 5.12311 0.205253
\(624\) −14.5616 −0.582929
\(625\) 1.00000 0.0400000
\(626\) 5.05398 0.201997
\(627\) 0 0
\(628\) −20.2462 −0.807912
\(629\) 20.6307 0.822599
\(630\) −9.12311 −0.363473
\(631\) −20.4924 −0.815790 −0.407895 0.913029i \(-0.633737\pi\)
−0.407895 + 0.913029i \(0.633737\pi\)
\(632\) 5.12311 0.203786
\(633\) −21.3002 −0.846606
\(634\) −13.0540 −0.518440
\(635\) −13.1231 −0.520775
\(636\) 11.6847 0.463327
\(637\) 2.49242 0.0987534
\(638\) 22.7386 0.900231
\(639\) −36.4924 −1.44362
\(640\) 1.00000 0.0395285
\(641\) −21.8617 −0.863487 −0.431743 0.901996i \(-0.642101\pi\)
−0.431743 + 0.901996i \(0.642101\pi\)
\(642\) 13.9309 0.549808
\(643\) −33.6155 −1.32567 −0.662834 0.748767i \(-0.730646\pi\)
−0.662834 + 0.748767i \(0.730646\pi\)
\(644\) −19.6847 −0.775684
\(645\) −7.36932 −0.290167
\(646\) 0 0
\(647\) 5.43845 0.213807 0.106904 0.994269i \(-0.465906\pi\)
0.106904 + 0.994269i \(0.465906\pi\)
\(648\) −7.00000 −0.274986
\(649\) −10.2462 −0.402199
\(650\) −5.68466 −0.222971
\(651\) −33.6155 −1.31750
\(652\) −15.3693 −0.601909
\(653\) −7.12311 −0.278749 −0.139374 0.990240i \(-0.544509\pi\)
−0.139374 + 0.990240i \(0.544509\pi\)
\(654\) 1.43845 0.0562477
\(655\) −16.4924 −0.644412
\(656\) −12.2462 −0.478134
\(657\) −6.00000 −0.234082
\(658\) −16.0000 −0.623745
\(659\) 14.0691 0.548056 0.274028 0.961722i \(-0.411644\pi\)
0.274028 + 0.961722i \(0.411644\pi\)
\(660\) 10.2462 0.398833
\(661\) 10.8078 0.420373 0.210187 0.977661i \(-0.432593\pi\)
0.210187 + 0.977661i \(0.432593\pi\)
\(662\) 2.56155 0.0995576
\(663\) −50.0691 −1.94452
\(664\) 2.87689 0.111645
\(665\) 0 0
\(666\) 21.3693 0.828044
\(667\) 43.6847 1.69148
\(668\) 7.36932 0.285127
\(669\) 59.8617 2.31439
\(670\) 2.56155 0.0989614
\(671\) 44.4924 1.71761
\(672\) −6.56155 −0.253117
\(673\) −21.3693 −0.823727 −0.411863 0.911246i \(-0.635122\pi\)
−0.411863 + 0.911246i \(0.635122\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.43845 0.0553659
\(676\) 19.3153 0.742898
\(677\) −40.5616 −1.55891 −0.779454 0.626460i \(-0.784503\pi\)
−0.779454 + 0.626460i \(0.784503\pi\)
\(678\) −22.7386 −0.873272
\(679\) 15.3693 0.589820
\(680\) 3.43845 0.131858
\(681\) −66.4233 −2.54535
\(682\) 20.4924 0.784695
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −14.8078 −0.565776
\(686\) 19.0540 0.727484
\(687\) −37.1231 −1.41633
\(688\) −2.87689 −0.109681
\(689\) −25.9309 −0.987887
\(690\) 19.6847 0.749382
\(691\) −17.1231 −0.651394 −0.325697 0.945474i \(-0.605599\pi\)
−0.325697 + 0.945474i \(0.605599\pi\)
\(692\) −20.2462 −0.769645
\(693\) −36.4924 −1.38623
\(694\) −8.63068 −0.327616
\(695\) 16.4924 0.625593
\(696\) 14.5616 0.551954
\(697\) −42.1080 −1.59495
\(698\) 3.75379 0.142083
\(699\) 25.6155 0.968868
\(700\) −2.56155 −0.0968176
\(701\) −20.8769 −0.788509 −0.394255 0.919001i \(-0.628997\pi\)
−0.394255 + 0.919001i \(0.628997\pi\)
\(702\) −8.17708 −0.308624
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 16.0000 0.602595
\(706\) −3.93087 −0.147940
\(707\) 44.4924 1.67331
\(708\) −6.56155 −0.246598
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −10.2462 −0.384533
\(711\) 18.2462 0.684286
\(712\) −2.00000 −0.0749532
\(713\) 39.3693 1.47439
\(714\) −22.5616 −0.844345
\(715\) −22.7386 −0.850377
\(716\) 22.2462 0.831380
\(717\) −3.68466 −0.137606
\(718\) 1.43845 0.0536824
\(719\) −25.4384 −0.948694 −0.474347 0.880338i \(-0.657316\pi\)
−0.474347 + 0.880338i \(0.657316\pi\)
\(720\) 3.56155 0.132731
\(721\) 5.75379 0.214282
\(722\) 0 0
\(723\) −59.2311 −2.20283
\(724\) 18.0000 0.668965
\(725\) 5.68466 0.211123
\(726\) 12.8078 0.475341
\(727\) −24.3153 −0.901806 −0.450903 0.892573i \(-0.648898\pi\)
−0.450903 + 0.892573i \(0.648898\pi\)
\(728\) 14.5616 0.539687
\(729\) −35.9848 −1.33277
\(730\) −1.68466 −0.0623520
\(731\) −9.89205 −0.365871
\(732\) 28.4924 1.05311
\(733\) −36.8769 −1.36208 −0.681040 0.732247i \(-0.738472\pi\)
−0.681040 + 0.732247i \(0.738472\pi\)
\(734\) 6.24621 0.230552
\(735\) −1.12311 −0.0414264
\(736\) 7.68466 0.283260
\(737\) 10.2462 0.377424
\(738\) −43.6155 −1.60551
\(739\) 25.1231 0.924168 0.462084 0.886836i \(-0.347102\pi\)
0.462084 + 0.886836i \(0.347102\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) −11.6847 −0.428957
\(743\) −18.8769 −0.692526 −0.346263 0.938137i \(-0.612550\pi\)
−0.346263 + 0.938137i \(0.612550\pi\)
\(744\) 13.1231 0.481116
\(745\) 13.3693 0.489814
\(746\) 23.4384 0.858143
\(747\) 10.2462 0.374889
\(748\) 13.7538 0.502888
\(749\) −13.9309 −0.509023
\(750\) 2.56155 0.0935347
\(751\) 34.8769 1.27268 0.636338 0.771410i \(-0.280448\pi\)
0.636338 + 0.771410i \(0.280448\pi\)
\(752\) 6.24621 0.227776
\(753\) 16.0000 0.583072
\(754\) −32.3153 −1.17686
\(755\) −5.12311 −0.186449
\(756\) −3.68466 −0.134010
\(757\) 42.4924 1.54441 0.772207 0.635371i \(-0.219153\pi\)
0.772207 + 0.635371i \(0.219153\pi\)
\(758\) −10.5616 −0.383613
\(759\) 78.7386 2.85803
\(760\) 0 0
\(761\) 41.5464 1.50606 0.753028 0.657989i \(-0.228593\pi\)
0.753028 + 0.657989i \(0.228593\pi\)
\(762\) −33.6155 −1.21776
\(763\) −1.43845 −0.0520753
\(764\) −3.68466 −0.133306
\(765\) 12.2462 0.442763
\(766\) −13.7538 −0.496945
\(767\) 14.5616 0.525787
\(768\) 2.56155 0.0924321
\(769\) 27.4384 0.989456 0.494728 0.869048i \(-0.335268\pi\)
0.494728 + 0.869048i \(0.335268\pi\)
\(770\) −10.2462 −0.369248
\(771\) −35.8617 −1.29153
\(772\) 14.4924 0.521594
\(773\) −10.8078 −0.388728 −0.194364 0.980929i \(-0.562264\pi\)
−0.194364 + 0.980929i \(0.562264\pi\)
\(774\) −10.2462 −0.368292
\(775\) 5.12311 0.184027
\(776\) −6.00000 −0.215387
\(777\) −39.3693 −1.41237
\(778\) −7.12311 −0.255376
\(779\) 0 0
\(780\) −14.5616 −0.521387
\(781\) −40.9848 −1.46655
\(782\) 26.4233 0.944895
\(783\) 8.17708 0.292225
\(784\) −0.438447 −0.0156588
\(785\) −20.2462 −0.722618
\(786\) −42.2462 −1.50687
\(787\) −11.1922 −0.398960 −0.199480 0.979902i \(-0.563925\pi\)
−0.199480 + 0.979902i \(0.563925\pi\)
\(788\) −20.2462 −0.721241
\(789\) −56.9848 −2.02871
\(790\) 5.12311 0.182272
\(791\) 22.7386 0.808493
\(792\) 14.2462 0.506217
\(793\) −63.2311 −2.24540
\(794\) −7.12311 −0.252790
\(795\) 11.6847 0.414412
\(796\) 16.8078 0.595735
\(797\) 11.3002 0.400273 0.200137 0.979768i \(-0.435861\pi\)
0.200137 + 0.979768i \(0.435861\pi\)
\(798\) 0 0
\(799\) 21.4773 0.759811
\(800\) 1.00000 0.0353553
\(801\) −7.12311 −0.251683
\(802\) 3.75379 0.132551
\(803\) −6.73863 −0.237801
\(804\) 6.56155 0.231408
\(805\) −19.6847 −0.693793
\(806\) −29.1231 −1.02582
\(807\) 66.6004 2.34444
\(808\) −17.3693 −0.611050
\(809\) −12.5616 −0.441641 −0.220820 0.975315i \(-0.570873\pi\)
−0.220820 + 0.975315i \(0.570873\pi\)
\(810\) −7.00000 −0.245955
\(811\) 20.8078 0.730659 0.365330 0.930878i \(-0.380956\pi\)
0.365330 + 0.930878i \(0.380956\pi\)
\(812\) −14.5616 −0.511010
\(813\) −56.1771 −1.97022
\(814\) 24.0000 0.841200
\(815\) −15.3693 −0.538364
\(816\) 8.80776 0.308333
\(817\) 0 0
\(818\) −24.7386 −0.864966
\(819\) 51.8617 1.81220
\(820\) −12.2462 −0.427656
\(821\) −17.3693 −0.606193 −0.303097 0.952960i \(-0.598021\pi\)
−0.303097 + 0.952960i \(0.598021\pi\)
\(822\) −37.9309 −1.32299
\(823\) −29.4384 −1.02616 −0.513080 0.858341i \(-0.671496\pi\)
−0.513080 + 0.858341i \(0.671496\pi\)
\(824\) −2.24621 −0.0782505
\(825\) 10.2462 0.356727
\(826\) 6.56155 0.228306
\(827\) 10.5616 0.367261 0.183631 0.982995i \(-0.441215\pi\)
0.183631 + 0.982995i \(0.441215\pi\)
\(828\) 27.3693 0.951150
\(829\) 21.0540 0.731235 0.365617 0.930765i \(-0.380858\pi\)
0.365617 + 0.930765i \(0.380858\pi\)
\(830\) 2.87689 0.0998585
\(831\) 2.24621 0.0779202
\(832\) −5.68466 −0.197080
\(833\) −1.50758 −0.0522345
\(834\) 42.2462 1.46287
\(835\) 7.36932 0.255026
\(836\) 0 0
\(837\) 7.36932 0.254721
\(838\) 23.8617 0.824290
\(839\) −20.4924 −0.707477 −0.353738 0.935344i \(-0.615090\pi\)
−0.353738 + 0.935344i \(0.615090\pi\)
\(840\) −6.56155 −0.226395
\(841\) 3.31534 0.114322
\(842\) 23.9309 0.824712
\(843\) −5.12311 −0.176449
\(844\) −8.31534 −0.286226
\(845\) 19.3153 0.664468
\(846\) 22.2462 0.764840
\(847\) −12.8078 −0.440080
\(848\) 4.56155 0.156644
\(849\) −54.1080 −1.85698
\(850\) 3.43845 0.117938
\(851\) 46.1080 1.58056
\(852\) −26.2462 −0.899180
\(853\) −24.7386 −0.847035 −0.423517 0.905888i \(-0.639205\pi\)
−0.423517 + 0.905888i \(0.639205\pi\)
\(854\) −28.4924 −0.974991
\(855\) 0 0
\(856\) 5.43845 0.185882
\(857\) −14.6307 −0.499775 −0.249887 0.968275i \(-0.580394\pi\)
−0.249887 + 0.968275i \(0.580394\pi\)
\(858\) −58.2462 −1.98849
\(859\) −52.9848 −1.80782 −0.903910 0.427723i \(-0.859316\pi\)
−0.903910 + 0.427723i \(0.859316\pi\)
\(860\) −2.87689 −0.0981013
\(861\) 80.3542 2.73846
\(862\) 16.0000 0.544962
\(863\) −2.24621 −0.0764619 −0.0382310 0.999269i \(-0.512172\pi\)
−0.0382310 + 0.999269i \(0.512172\pi\)
\(864\) 1.43845 0.0489370
\(865\) −20.2462 −0.688392
\(866\) −14.6307 −0.497171
\(867\) −13.2614 −0.450380
\(868\) −13.1231 −0.445427
\(869\) 20.4924 0.695158
\(870\) 14.5616 0.493683
\(871\) −14.5616 −0.493399
\(872\) 0.561553 0.0190166
\(873\) −21.3693 −0.723242
\(874\) 0 0
\(875\) −2.56155 −0.0865963
\(876\) −4.31534 −0.145802
\(877\) 3.93087 0.132736 0.0663680 0.997795i \(-0.478859\pi\)
0.0663680 + 0.997795i \(0.478859\pi\)
\(878\) 13.1231 0.442883
\(879\) 56.8078 1.91608
\(880\) 4.00000 0.134840
\(881\) 42.9848 1.44820 0.724098 0.689697i \(-0.242256\pi\)
0.724098 + 0.689697i \(0.242256\pi\)
\(882\) −1.56155 −0.0525802
\(883\) −6.38447 −0.214855 −0.107427 0.994213i \(-0.534261\pi\)
−0.107427 + 0.994213i \(0.534261\pi\)
\(884\) −19.5464 −0.657417
\(885\) −6.56155 −0.220564
\(886\) 2.24621 0.0754629
\(887\) −28.4924 −0.956682 −0.478341 0.878174i \(-0.658762\pi\)
−0.478341 + 0.878174i \(0.658762\pi\)
\(888\) 15.3693 0.515761
\(889\) 33.6155 1.12743
\(890\) −2.00000 −0.0670402
\(891\) −28.0000 −0.938035
\(892\) 23.3693 0.782463
\(893\) 0 0
\(894\) 34.2462 1.14536
\(895\) 22.2462 0.743609
\(896\) −2.56155 −0.0855755
\(897\) −111.901 −3.73625
\(898\) 28.7386 0.959021
\(899\) 29.1231 0.971310
\(900\) 3.56155 0.118718
\(901\) 15.6847 0.522532
\(902\) −48.9848 −1.63102
\(903\) 18.8769 0.628184
\(904\) −8.87689 −0.295241
\(905\) 18.0000 0.598340
\(906\) −13.1231 −0.435986
\(907\) 20.1771 0.669969 0.334984 0.942224i \(-0.391269\pi\)
0.334984 + 0.942224i \(0.391269\pi\)
\(908\) −25.9309 −0.860546
\(909\) −61.8617 −2.05182
\(910\) 14.5616 0.482711
\(911\) −4.49242 −0.148841 −0.0744203 0.997227i \(-0.523711\pi\)
−0.0744203 + 0.997227i \(0.523711\pi\)
\(912\) 0 0
\(913\) 11.5076 0.380845
\(914\) 6.31534 0.208893
\(915\) 28.4924 0.941930
\(916\) −14.4924 −0.478843
\(917\) 42.2462 1.39509
\(918\) 4.94602 0.163243
\(919\) −2.06913 −0.0682543 −0.0341272 0.999417i \(-0.510865\pi\)
−0.0341272 + 0.999417i \(0.510865\pi\)
\(920\) 7.68466 0.253356
\(921\) −83.2311 −2.74256
\(922\) 3.75379 0.123624
\(923\) 58.2462 1.91720
\(924\) −26.2462 −0.863437
\(925\) 6.00000 0.197279
\(926\) −30.2462 −0.993952
\(927\) −8.00000 −0.262754
\(928\) 5.68466 0.186608
\(929\) −19.3002 −0.633219 −0.316609 0.948556i \(-0.602544\pi\)
−0.316609 + 0.948556i \(0.602544\pi\)
\(930\) 13.1231 0.430324
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −9.43845 −0.309001
\(934\) 18.2462 0.597034
\(935\) 13.7538 0.449797
\(936\) −20.2462 −0.661768
\(937\) 40.5616 1.32509 0.662544 0.749023i \(-0.269477\pi\)
0.662544 + 0.749023i \(0.269477\pi\)
\(938\) −6.56155 −0.214242
\(939\) 12.9460 0.422478
\(940\) 6.24621 0.203729
\(941\) −54.8078 −1.78668 −0.893341 0.449379i \(-0.851645\pi\)
−0.893341 + 0.449379i \(0.851645\pi\)
\(942\) −51.8617 −1.68975
\(943\) −94.1080 −3.06458
\(944\) −2.56155 −0.0833714
\(945\) −3.68466 −0.119862
\(946\) −11.5076 −0.374144
\(947\) −34.2462 −1.11285 −0.556426 0.830897i \(-0.687828\pi\)
−0.556426 + 0.830897i \(0.687828\pi\)
\(948\) 13.1231 0.426219
\(949\) 9.57671 0.310873
\(950\) 0 0
\(951\) −33.4384 −1.08432
\(952\) −8.80776 −0.285461
\(953\) −44.1080 −1.42880 −0.714398 0.699739i \(-0.753300\pi\)
−0.714398 + 0.699739i \(0.753300\pi\)
\(954\) 16.2462 0.525991
\(955\) −3.68466 −0.119233
\(956\) −1.43845 −0.0465227
\(957\) 58.2462 1.88283
\(958\) 32.0000 1.03387
\(959\) 37.9309 1.22485
\(960\) 2.56155 0.0826738
\(961\) −4.75379 −0.153348
\(962\) −34.1080 −1.09968
\(963\) 19.3693 0.624168
\(964\) −23.1231 −0.744745
\(965\) 14.4924 0.466528
\(966\) −50.4233 −1.62234
\(967\) −15.5076 −0.498690 −0.249345 0.968415i \(-0.580215\pi\)
−0.249345 + 0.968415i \(0.580215\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) 56.4924 1.81293 0.906464 0.422283i \(-0.138771\pi\)
0.906464 + 0.422283i \(0.138771\pi\)
\(972\) −22.2462 −0.713548
\(973\) −42.2462 −1.35435
\(974\) 17.6155 0.564438
\(975\) −14.5616 −0.466343
\(976\) 11.1231 0.356042
\(977\) −28.7386 −0.919430 −0.459715 0.888066i \(-0.652048\pi\)
−0.459715 + 0.888066i \(0.652048\pi\)
\(978\) −39.3693 −1.25889
\(979\) −8.00000 −0.255681
\(980\) −0.438447 −0.0140057
\(981\) 2.00000 0.0638551
\(982\) −1.12311 −0.0358397
\(983\) −18.8769 −0.602079 −0.301040 0.953612i \(-0.597334\pi\)
−0.301040 + 0.953612i \(0.597334\pi\)
\(984\) −31.3693 −1.00002
\(985\) −20.2462 −0.645098
\(986\) 19.5464 0.622484
\(987\) −40.9848 −1.30456
\(988\) 0 0
\(989\) −22.1080 −0.702992
\(990\) 14.2462 0.452774
\(991\) −2.87689 −0.0913876 −0.0456938 0.998955i \(-0.514550\pi\)
−0.0456938 + 0.998955i \(0.514550\pi\)
\(992\) 5.12311 0.162659
\(993\) 6.56155 0.208225
\(994\) 26.2462 0.832479
\(995\) 16.8078 0.532842
\(996\) 7.36932 0.233506
\(997\) −16.7386 −0.530118 −0.265059 0.964232i \(-0.585391\pi\)
−0.265059 + 0.964232i \(0.585391\pi\)
\(998\) 42.1080 1.33290
\(999\) 8.63068 0.273063
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 3610.2.a.t.1.2 2
19.18 odd 2 190.2.a.d.1.1 2
57.56 even 2 1710.2.a.w.1.1 2
76.75 even 2 1520.2.a.n.1.2 2
95.18 even 4 950.2.b.f.799.3 4
95.37 even 4 950.2.b.f.799.2 4
95.94 odd 2 950.2.a.h.1.2 2
133.132 even 2 9310.2.a.bc.1.2 2
152.37 odd 2 6080.2.a.bh.1.2 2
152.75 even 2 6080.2.a.bb.1.1 2
285.284 even 2 8550.2.a.br.1.2 2
380.379 even 2 7600.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.d.1.1 2 19.18 odd 2
950.2.a.h.1.2 2 95.94 odd 2
950.2.b.f.799.2 4 95.37 even 4
950.2.b.f.799.3 4 95.18 even 4
1520.2.a.n.1.2 2 76.75 even 2
1710.2.a.w.1.1 2 57.56 even 2
3610.2.a.t.1.2 2 1.1 even 1 trivial
6080.2.a.bb.1.1 2 152.75 even 2
6080.2.a.bh.1.2 2 152.37 odd 2
7600.2.a.y.1.1 2 380.379 even 2
8550.2.a.br.1.2 2 285.284 even 2
9310.2.a.bc.1.2 2 133.132 even 2