Properties

Label 1045.2.a.g.1.5
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $6$
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.65636\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.65636 q^{2} +3.32622 q^{3} +0.743534 q^{4} +1.00000 q^{5} +5.50942 q^{6} +2.81120 q^{7} -2.08116 q^{8} +8.06374 q^{9} +O(q^{10})\) \(q+1.65636 q^{2} +3.32622 q^{3} +0.743534 q^{4} +1.00000 q^{5} +5.50942 q^{6} +2.81120 q^{7} -2.08116 q^{8} +8.06374 q^{9} +1.65636 q^{10} -1.00000 q^{11} +2.47316 q^{12} -5.98258 q^{13} +4.65636 q^{14} +3.32622 q^{15} -4.93423 q^{16} -6.49551 q^{17} +13.3565 q^{18} -1.00000 q^{19} +0.743534 q^{20} +9.35067 q^{21} -1.65636 q^{22} +2.77149 q^{23} -6.92241 q^{24} +1.00000 q^{25} -9.90932 q^{26} +16.8431 q^{27} +2.09022 q^{28} -1.78540 q^{29} +5.50942 q^{30} -3.15789 q^{31} -4.01054 q^{32} -3.32622 q^{33} -10.7589 q^{34} +2.81120 q^{35} +5.99567 q^{36} +3.90101 q^{37} -1.65636 q^{38} -19.8994 q^{39} -2.08116 q^{40} -4.63245 q^{41} +15.4881 q^{42} -2.40738 q^{43} -0.743534 q^{44} +8.06374 q^{45} +4.59059 q^{46} -4.10602 q^{47} -16.4123 q^{48} +0.902838 q^{49} +1.65636 q^{50} -21.6055 q^{51} -4.44825 q^{52} +11.5083 q^{53} +27.8983 q^{54} -1.00000 q^{55} -5.85056 q^{56} -3.32622 q^{57} -2.95726 q^{58} +5.33014 q^{59} +2.47316 q^{60} +7.18233 q^{61} -5.23060 q^{62} +22.6688 q^{63} +3.22555 q^{64} -5.98258 q^{65} -5.50942 q^{66} -13.2370 q^{67} -4.82964 q^{68} +9.21858 q^{69} +4.65636 q^{70} -0.437851 q^{71} -16.7820 q^{72} +11.2457 q^{73} +6.46148 q^{74} +3.32622 q^{75} -0.743534 q^{76} -2.81120 q^{77} -32.9606 q^{78} +4.44176 q^{79} -4.93423 q^{80} +31.8327 q^{81} -7.67301 q^{82} +17.7115 q^{83} +6.95254 q^{84} -6.49551 q^{85} -3.98750 q^{86} -5.93863 q^{87} +2.08116 q^{88} +17.5897 q^{89} +13.3565 q^{90} -16.8182 q^{91} +2.06070 q^{92} -10.5038 q^{93} -6.80106 q^{94} -1.00000 q^{95} -13.3399 q^{96} +4.99386 q^{97} +1.49543 q^{98} -8.06374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 8 q^{4} + 6 q^{5} + 2 q^{6} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 8 q^{4} + 6 q^{5} + 2 q^{6} + 5 q^{7} + 9 q^{9} - 6 q^{11} + 19 q^{12} - 9 q^{13} + 18 q^{14} + 3 q^{15} + 4 q^{16} - 5 q^{17} + 2 q^{18} - 6 q^{19} + 8 q^{20} + 3 q^{21} + 8 q^{23} - 7 q^{24} + 6 q^{25} - 22 q^{26} + 30 q^{27} + 10 q^{28} - 5 q^{29} + 2 q^{30} - q^{31} + 15 q^{32} - 3 q^{33} - 22 q^{34} + 5 q^{35} + 12 q^{36} + 9 q^{37} - 32 q^{39} + 25 q^{41} + 11 q^{42} + 15 q^{43} - 8 q^{44} + 9 q^{45} - 16 q^{46} + 24 q^{47} - 4 q^{48} + 13 q^{49} - 27 q^{52} + 5 q^{53} - 11 q^{54} - 6 q^{55} - 12 q^{56} - 3 q^{57} + 13 q^{58} + 39 q^{59} + 19 q^{60} - 11 q^{61} - 42 q^{62} + 38 q^{63} - 14 q^{64} - 9 q^{65} - 2 q^{66} + 24 q^{67} + 45 q^{68} + 14 q^{69} + 18 q^{70} - 24 q^{71} - 61 q^{72} - 26 q^{73} + q^{74} + 3 q^{75} - 8 q^{76} - 5 q^{77} - 29 q^{78} + 11 q^{79} + 4 q^{80} + 30 q^{81} + 8 q^{82} + 39 q^{83} + 25 q^{84} - 5 q^{85} + 18 q^{86} - 16 q^{87} + 22 q^{89} + 2 q^{90} - 26 q^{91} - 11 q^{92} - 6 q^{93} - 30 q^{94} - 6 q^{95} - 15 q^{96} + 22 q^{97} + 33 q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.65636 1.17122 0.585612 0.810591i \(-0.300854\pi\)
0.585612 + 0.810591i \(0.300854\pi\)
\(3\) 3.32622 1.92039 0.960197 0.279323i \(-0.0901100\pi\)
0.960197 + 0.279323i \(0.0901100\pi\)
\(4\) 0.743534 0.371767
\(5\) 1.00000 0.447214
\(6\) 5.50942 2.24921
\(7\) 2.81120 1.06253 0.531267 0.847205i \(-0.321716\pi\)
0.531267 + 0.847205i \(0.321716\pi\)
\(8\) −2.08116 −0.735802
\(9\) 8.06374 2.68791
\(10\) 1.65636 0.523788
\(11\) −1.00000 −0.301511
\(12\) 2.47316 0.713939
\(13\) −5.98258 −1.65927 −0.829635 0.558306i \(-0.811451\pi\)
−0.829635 + 0.558306i \(0.811451\pi\)
\(14\) 4.65636 1.24446
\(15\) 3.32622 0.858827
\(16\) −4.93423 −1.23356
\(17\) −6.49551 −1.57539 −0.787697 0.616063i \(-0.788727\pi\)
−0.787697 + 0.616063i \(0.788727\pi\)
\(18\) 13.3565 3.14815
\(19\) −1.00000 −0.229416
\(20\) 0.743534 0.166259
\(21\) 9.35067 2.04048
\(22\) −1.65636 −0.353137
\(23\) 2.77149 0.577895 0.288948 0.957345i \(-0.406695\pi\)
0.288948 + 0.957345i \(0.406695\pi\)
\(24\) −6.92241 −1.41303
\(25\) 1.00000 0.200000
\(26\) −9.90932 −1.94338
\(27\) 16.8431 3.24146
\(28\) 2.09022 0.395015
\(29\) −1.78540 −0.331540 −0.165770 0.986164i \(-0.553011\pi\)
−0.165770 + 0.986164i \(0.553011\pi\)
\(30\) 5.50942 1.00588
\(31\) −3.15789 −0.567173 −0.283587 0.958947i \(-0.591524\pi\)
−0.283587 + 0.958947i \(0.591524\pi\)
\(32\) −4.01054 −0.708969
\(33\) −3.32622 −0.579021
\(34\) −10.7589 −1.84514
\(35\) 2.81120 0.475179
\(36\) 5.99567 0.999278
\(37\) 3.90101 0.641322 0.320661 0.947194i \(-0.396095\pi\)
0.320661 + 0.947194i \(0.396095\pi\)
\(38\) −1.65636 −0.268697
\(39\) −19.8994 −3.18645
\(40\) −2.08116 −0.329061
\(41\) −4.63245 −0.723467 −0.361734 0.932281i \(-0.617815\pi\)
−0.361734 + 0.932281i \(0.617815\pi\)
\(42\) 15.4881 2.38986
\(43\) −2.40738 −0.367122 −0.183561 0.983008i \(-0.558763\pi\)
−0.183561 + 0.983008i \(0.558763\pi\)
\(44\) −0.743534 −0.112092
\(45\) 8.06374 1.20207
\(46\) 4.59059 0.676845
\(47\) −4.10602 −0.598925 −0.299462 0.954108i \(-0.596807\pi\)
−0.299462 + 0.954108i \(0.596807\pi\)
\(48\) −16.4123 −2.36891
\(49\) 0.902838 0.128977
\(50\) 1.65636 0.234245
\(51\) −21.6055 −3.02538
\(52\) −4.44825 −0.616862
\(53\) 11.5083 1.58078 0.790391 0.612603i \(-0.209877\pi\)
0.790391 + 0.612603i \(0.209877\pi\)
\(54\) 27.8983 3.79648
\(55\) −1.00000 −0.134840
\(56\) −5.85056 −0.781814
\(57\) −3.32622 −0.440569
\(58\) −2.95726 −0.388308
\(59\) 5.33014 0.693925 0.346963 0.937879i \(-0.387213\pi\)
0.346963 + 0.937879i \(0.387213\pi\)
\(60\) 2.47316 0.319283
\(61\) 7.18233 0.919603 0.459802 0.888022i \(-0.347920\pi\)
0.459802 + 0.888022i \(0.347920\pi\)
\(62\) −5.23060 −0.664287
\(63\) 22.6688 2.85600
\(64\) 3.22555 0.403194
\(65\) −5.98258 −0.742048
\(66\) −5.50942 −0.678163
\(67\) −13.2370 −1.61716 −0.808580 0.588387i \(-0.799763\pi\)
−0.808580 + 0.588387i \(0.799763\pi\)
\(68\) −4.82964 −0.585679
\(69\) 9.21858 1.10979
\(70\) 4.65636 0.556542
\(71\) −0.437851 −0.0519633 −0.0259817 0.999662i \(-0.508271\pi\)
−0.0259817 + 0.999662i \(0.508271\pi\)
\(72\) −16.7820 −1.97777
\(73\) 11.2457 1.31621 0.658103 0.752928i \(-0.271359\pi\)
0.658103 + 0.752928i \(0.271359\pi\)
\(74\) 6.46148 0.751132
\(75\) 3.32622 0.384079
\(76\) −0.743534 −0.0852892
\(77\) −2.81120 −0.320366
\(78\) −32.9606 −3.73205
\(79\) 4.44176 0.499737 0.249869 0.968280i \(-0.419613\pi\)
0.249869 + 0.968280i \(0.419613\pi\)
\(80\) −4.93423 −0.551663
\(81\) 31.8327 3.53697
\(82\) −7.67301 −0.847343
\(83\) 17.7115 1.94409 0.972046 0.234790i \(-0.0754402\pi\)
0.972046 + 0.234790i \(0.0754402\pi\)
\(84\) 6.95254 0.758584
\(85\) −6.49551 −0.704537
\(86\) −3.98750 −0.429983
\(87\) −5.93863 −0.636688
\(88\) 2.08116 0.221853
\(89\) 17.5897 1.86451 0.932253 0.361807i \(-0.117840\pi\)
0.932253 + 0.361807i \(0.117840\pi\)
\(90\) 13.3565 1.40790
\(91\) −16.8182 −1.76303
\(92\) 2.06070 0.214842
\(93\) −10.5038 −1.08920
\(94\) −6.80106 −0.701475
\(95\) −1.00000 −0.102598
\(96\) −13.3399 −1.36150
\(97\) 4.99386 0.507049 0.253525 0.967329i \(-0.418410\pi\)
0.253525 + 0.967329i \(0.418410\pi\)
\(98\) 1.49543 0.151061
\(99\) −8.06374 −0.810437
\(100\) 0.743534 0.0743534
\(101\) −16.7267 −1.66437 −0.832186 0.554497i \(-0.812911\pi\)
−0.832186 + 0.554497i \(0.812911\pi\)
\(102\) −35.7865 −3.54340
\(103\) 4.48754 0.442171 0.221085 0.975254i \(-0.429040\pi\)
0.221085 + 0.975254i \(0.429040\pi\)
\(104\) 12.4507 1.22089
\(105\) 9.35067 0.912532
\(106\) 19.0618 1.85145
\(107\) 1.05672 0.102157 0.0510783 0.998695i \(-0.483734\pi\)
0.0510783 + 0.998695i \(0.483734\pi\)
\(108\) 12.5234 1.20507
\(109\) −3.48106 −0.333425 −0.166712 0.986006i \(-0.553315\pi\)
−0.166712 + 0.986006i \(0.553315\pi\)
\(110\) −1.65636 −0.157928
\(111\) 12.9756 1.23159
\(112\) −13.8711 −1.31069
\(113\) −8.94084 −0.841084 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(114\) −5.50942 −0.516005
\(115\) 2.77149 0.258443
\(116\) −1.32750 −0.123256
\(117\) −48.2420 −4.45998
\(118\) 8.82864 0.812742
\(119\) −18.2602 −1.67391
\(120\) −6.92241 −0.631926
\(121\) 1.00000 0.0909091
\(122\) 11.8965 1.07706
\(123\) −15.4086 −1.38934
\(124\) −2.34800 −0.210856
\(125\) 1.00000 0.0894427
\(126\) 37.5477 3.34502
\(127\) 2.14707 0.190522 0.0952609 0.995452i \(-0.469631\pi\)
0.0952609 + 0.995452i \(0.469631\pi\)
\(128\) 13.3638 1.18120
\(129\) −8.00749 −0.705020
\(130\) −9.90932 −0.869105
\(131\) 13.1523 1.14912 0.574562 0.818461i \(-0.305172\pi\)
0.574562 + 0.818461i \(0.305172\pi\)
\(132\) −2.47316 −0.215261
\(133\) −2.81120 −0.243762
\(134\) −21.9253 −1.89406
\(135\) 16.8431 1.44963
\(136\) 13.5182 1.15918
\(137\) −0.316167 −0.0270120 −0.0135060 0.999909i \(-0.504299\pi\)
−0.0135060 + 0.999909i \(0.504299\pi\)
\(138\) 15.2693 1.29981
\(139\) −4.07415 −0.345565 −0.172782 0.984960i \(-0.555276\pi\)
−0.172782 + 0.984960i \(0.555276\pi\)
\(140\) 2.09022 0.176656
\(141\) −13.6575 −1.15017
\(142\) −0.725239 −0.0608607
\(143\) 5.98258 0.500289
\(144\) −39.7883 −3.31569
\(145\) −1.78540 −0.148269
\(146\) 18.6269 1.54157
\(147\) 3.00304 0.247686
\(148\) 2.90053 0.238422
\(149\) 4.80464 0.393612 0.196806 0.980442i \(-0.436943\pi\)
0.196806 + 0.980442i \(0.436943\pi\)
\(150\) 5.50942 0.449843
\(151\) −18.2399 −1.48434 −0.742169 0.670213i \(-0.766203\pi\)
−0.742169 + 0.670213i \(0.766203\pi\)
\(152\) 2.08116 0.168805
\(153\) −52.3782 −4.23452
\(154\) −4.65636 −0.375220
\(155\) −3.15789 −0.253648
\(156\) −14.7959 −1.18462
\(157\) −18.1161 −1.44582 −0.722910 0.690942i \(-0.757196\pi\)
−0.722910 + 0.690942i \(0.757196\pi\)
\(158\) 7.35716 0.585304
\(159\) 38.2790 3.03572
\(160\) −4.01054 −0.317061
\(161\) 7.79120 0.614033
\(162\) 52.7265 4.14259
\(163\) −23.9946 −1.87940 −0.939699 0.342002i \(-0.888895\pi\)
−0.939699 + 0.342002i \(0.888895\pi\)
\(164\) −3.44438 −0.268961
\(165\) −3.32622 −0.258946
\(166\) 29.3367 2.27697
\(167\) 3.10562 0.240320 0.120160 0.992755i \(-0.461659\pi\)
0.120160 + 0.992755i \(0.461659\pi\)
\(168\) −19.4603 −1.50139
\(169\) 22.7913 1.75318
\(170\) −10.7589 −0.825172
\(171\) −8.06374 −0.616650
\(172\) −1.78997 −0.136484
\(173\) 10.9411 0.831839 0.415919 0.909401i \(-0.363460\pi\)
0.415919 + 0.909401i \(0.363460\pi\)
\(174\) −9.83652 −0.745704
\(175\) 2.81120 0.212507
\(176\) 4.93423 0.371931
\(177\) 17.7292 1.33261
\(178\) 29.1349 2.18376
\(179\) 1.88648 0.141002 0.0705011 0.997512i \(-0.477540\pi\)
0.0705011 + 0.997512i \(0.477540\pi\)
\(180\) 5.99567 0.446891
\(181\) 8.50254 0.631989 0.315995 0.948761i \(-0.397662\pi\)
0.315995 + 0.948761i \(0.397662\pi\)
\(182\) −27.8571 −2.06490
\(183\) 23.8900 1.76600
\(184\) −5.76792 −0.425216
\(185\) 3.90101 0.286808
\(186\) −17.3981 −1.27569
\(187\) 6.49551 0.474999
\(188\) −3.05297 −0.222660
\(189\) 47.3494 3.44416
\(190\) −1.65636 −0.120165
\(191\) −23.2282 −1.68073 −0.840366 0.542019i \(-0.817660\pi\)
−0.840366 + 0.542019i \(0.817660\pi\)
\(192\) 10.7289 0.774291
\(193\) 15.5701 1.12076 0.560380 0.828235i \(-0.310655\pi\)
0.560380 + 0.828235i \(0.310655\pi\)
\(194\) 8.27163 0.593868
\(195\) −19.8994 −1.42502
\(196\) 0.671291 0.0479493
\(197\) −22.8694 −1.62938 −0.814689 0.579898i \(-0.803092\pi\)
−0.814689 + 0.579898i \(0.803092\pi\)
\(198\) −13.3565 −0.949204
\(199\) −23.7525 −1.68377 −0.841885 0.539657i \(-0.818554\pi\)
−0.841885 + 0.539657i \(0.818554\pi\)
\(200\) −2.08116 −0.147160
\(201\) −44.0292 −3.10558
\(202\) −27.7055 −1.94935
\(203\) −5.01911 −0.352272
\(204\) −16.0644 −1.12474
\(205\) −4.63245 −0.323544
\(206\) 7.43300 0.517881
\(207\) 22.3486 1.55333
\(208\) 29.5194 2.04680
\(209\) 1.00000 0.0691714
\(210\) 15.4881 1.06878
\(211\) 7.48728 0.515446 0.257723 0.966219i \(-0.417028\pi\)
0.257723 + 0.966219i \(0.417028\pi\)
\(212\) 8.55679 0.587682
\(213\) −1.45639 −0.0997900
\(214\) 1.75030 0.119648
\(215\) −2.40738 −0.164182
\(216\) −35.0533 −2.38507
\(217\) −8.87745 −0.602640
\(218\) −5.76589 −0.390515
\(219\) 37.4056 2.52763
\(220\) −0.743534 −0.0501290
\(221\) 38.8600 2.61400
\(222\) 21.4923 1.44247
\(223\) −8.80258 −0.589464 −0.294732 0.955580i \(-0.595230\pi\)
−0.294732 + 0.955580i \(0.595230\pi\)
\(224\) −11.2744 −0.753304
\(225\) 8.06374 0.537583
\(226\) −14.8093 −0.985098
\(227\) −12.5157 −0.830695 −0.415347 0.909663i \(-0.636340\pi\)
−0.415347 + 0.909663i \(0.636340\pi\)
\(228\) −2.47316 −0.163789
\(229\) −2.31366 −0.152891 −0.0764454 0.997074i \(-0.524357\pi\)
−0.0764454 + 0.997074i \(0.524357\pi\)
\(230\) 4.59059 0.302694
\(231\) −9.35067 −0.615229
\(232\) 3.71570 0.243948
\(233\) −13.0851 −0.857234 −0.428617 0.903486i \(-0.640999\pi\)
−0.428617 + 0.903486i \(0.640999\pi\)
\(234\) −79.9062 −5.22363
\(235\) −4.10602 −0.267847
\(236\) 3.96314 0.257978
\(237\) 14.7743 0.959692
\(238\) −30.2455 −1.96052
\(239\) 20.9482 1.35503 0.677514 0.735510i \(-0.263057\pi\)
0.677514 + 0.735510i \(0.263057\pi\)
\(240\) −16.4123 −1.05941
\(241\) −14.2896 −0.920477 −0.460238 0.887795i \(-0.652236\pi\)
−0.460238 + 0.887795i \(0.652236\pi\)
\(242\) 1.65636 0.106475
\(243\) 55.3533 3.55092
\(244\) 5.34031 0.341878
\(245\) 0.902838 0.0576802
\(246\) −25.5221 −1.62723
\(247\) 5.98258 0.380663
\(248\) 6.57207 0.417327
\(249\) 58.9124 3.73342
\(250\) 1.65636 0.104758
\(251\) 11.1296 0.702495 0.351247 0.936283i \(-0.385758\pi\)
0.351247 + 0.936283i \(0.385758\pi\)
\(252\) 16.8550 1.06177
\(253\) −2.77149 −0.174242
\(254\) 3.55633 0.223144
\(255\) −21.6055 −1.35299
\(256\) 15.6841 0.980257
\(257\) 7.27753 0.453960 0.226980 0.973899i \(-0.427115\pi\)
0.226980 + 0.973899i \(0.427115\pi\)
\(258\) −13.2633 −0.825737
\(259\) 10.9665 0.681426
\(260\) −4.44825 −0.275869
\(261\) −14.3970 −0.891152
\(262\) 21.7850 1.34588
\(263\) −16.6526 −1.02684 −0.513421 0.858137i \(-0.671622\pi\)
−0.513421 + 0.858137i \(0.671622\pi\)
\(264\) 6.92241 0.426045
\(265\) 11.5083 0.706947
\(266\) −4.65636 −0.285500
\(267\) 58.5073 3.58059
\(268\) −9.84217 −0.601206
\(269\) −7.11358 −0.433723 −0.216861 0.976202i \(-0.569582\pi\)
−0.216861 + 0.976202i \(0.569582\pi\)
\(270\) 27.8983 1.69784
\(271\) 11.2711 0.684669 0.342334 0.939578i \(-0.388782\pi\)
0.342334 + 0.939578i \(0.388782\pi\)
\(272\) 32.0503 1.94334
\(273\) −55.9411 −3.38571
\(274\) −0.523687 −0.0316371
\(275\) −1.00000 −0.0603023
\(276\) 6.85433 0.412582
\(277\) −4.24890 −0.255292 −0.127646 0.991820i \(-0.540742\pi\)
−0.127646 + 0.991820i \(0.540742\pi\)
\(278\) −6.74827 −0.404734
\(279\) −25.4644 −1.52451
\(280\) −5.85056 −0.349638
\(281\) 0.222832 0.0132931 0.00664653 0.999978i \(-0.497884\pi\)
0.00664653 + 0.999978i \(0.497884\pi\)
\(282\) −22.6218 −1.34711
\(283\) 12.1767 0.723828 0.361914 0.932211i \(-0.382123\pi\)
0.361914 + 0.932211i \(0.382123\pi\)
\(284\) −0.325557 −0.0193182
\(285\) −3.32622 −0.197028
\(286\) 9.90932 0.585950
\(287\) −13.0227 −0.768708
\(288\) −32.3399 −1.90565
\(289\) 25.1917 1.48187
\(290\) −2.95726 −0.173657
\(291\) 16.6107 0.973734
\(292\) 8.36153 0.489322
\(293\) 24.1175 1.40896 0.704480 0.709724i \(-0.251180\pi\)
0.704480 + 0.709724i \(0.251180\pi\)
\(294\) 4.97412 0.290096
\(295\) 5.33014 0.310333
\(296\) −8.11863 −0.471886
\(297\) −16.8431 −0.977338
\(298\) 7.95822 0.461008
\(299\) −16.5807 −0.958884
\(300\) 2.47316 0.142788
\(301\) −6.76763 −0.390080
\(302\) −30.2118 −1.73849
\(303\) −55.6368 −3.19625
\(304\) 4.93423 0.282997
\(305\) 7.18233 0.411259
\(306\) −86.7572 −4.95958
\(307\) 19.7231 1.12566 0.562829 0.826573i \(-0.309713\pi\)
0.562829 + 0.826573i \(0.309713\pi\)
\(308\) −2.09022 −0.119101
\(309\) 14.9266 0.849142
\(310\) −5.23060 −0.297078
\(311\) −26.2571 −1.48890 −0.744451 0.667677i \(-0.767289\pi\)
−0.744451 + 0.667677i \(0.767289\pi\)
\(312\) 41.4139 2.34460
\(313\) 2.77164 0.156662 0.0783311 0.996927i \(-0.475041\pi\)
0.0783311 + 0.996927i \(0.475041\pi\)
\(314\) −30.0068 −1.69338
\(315\) 22.6688 1.27724
\(316\) 3.30260 0.185786
\(317\) 4.00082 0.224709 0.112354 0.993668i \(-0.464161\pi\)
0.112354 + 0.993668i \(0.464161\pi\)
\(318\) 63.4039 3.55552
\(319\) 1.78540 0.0999631
\(320\) 3.22555 0.180314
\(321\) 3.51487 0.196181
\(322\) 12.9051 0.719170
\(323\) 6.49551 0.361420
\(324\) 23.6687 1.31493
\(325\) −5.98258 −0.331854
\(326\) −39.7437 −2.20120
\(327\) −11.5788 −0.640307
\(328\) 9.64088 0.532329
\(329\) −11.5428 −0.636377
\(330\) −5.50942 −0.303284
\(331\) −17.5839 −0.966500 −0.483250 0.875483i \(-0.660544\pi\)
−0.483250 + 0.875483i \(0.660544\pi\)
\(332\) 13.1691 0.722749
\(333\) 31.4567 1.72382
\(334\) 5.14403 0.281469
\(335\) −13.2370 −0.723216
\(336\) −46.1383 −2.51705
\(337\) 32.7681 1.78499 0.892497 0.451054i \(-0.148952\pi\)
0.892497 + 0.451054i \(0.148952\pi\)
\(338\) 37.7506 2.05336
\(339\) −29.7392 −1.61521
\(340\) −4.82964 −0.261924
\(341\) 3.15789 0.171009
\(342\) −13.3565 −0.722236
\(343\) −17.1403 −0.925491
\(344\) 5.01015 0.270129
\(345\) 9.21858 0.496312
\(346\) 18.1225 0.974270
\(347\) −19.6495 −1.05484 −0.527421 0.849604i \(-0.676841\pi\)
−0.527421 + 0.849604i \(0.676841\pi\)
\(348\) −4.41557 −0.236699
\(349\) 19.9610 1.06849 0.534243 0.845331i \(-0.320597\pi\)
0.534243 + 0.845331i \(0.320597\pi\)
\(350\) 4.65636 0.248893
\(351\) −100.765 −5.37846
\(352\) 4.01054 0.213762
\(353\) −23.2030 −1.23497 −0.617486 0.786582i \(-0.711849\pi\)
−0.617486 + 0.786582i \(0.711849\pi\)
\(354\) 29.3660 1.56079
\(355\) −0.437851 −0.0232387
\(356\) 13.0786 0.693162
\(357\) −60.7374 −3.21456
\(358\) 3.12469 0.165145
\(359\) −9.83399 −0.519018 −0.259509 0.965741i \(-0.583561\pi\)
−0.259509 + 0.965741i \(0.583561\pi\)
\(360\) −16.7820 −0.884487
\(361\) 1.00000 0.0526316
\(362\) 14.0833 0.740201
\(363\) 3.32622 0.174581
\(364\) −12.5049 −0.655436
\(365\) 11.2457 0.588625
\(366\) 39.5705 2.06838
\(367\) 1.73835 0.0907411 0.0453705 0.998970i \(-0.485553\pi\)
0.0453705 + 0.998970i \(0.485553\pi\)
\(368\) −13.6751 −0.712866
\(369\) −37.3549 −1.94462
\(370\) 6.46148 0.335916
\(371\) 32.3520 1.67963
\(372\) −7.80995 −0.404927
\(373\) −6.58428 −0.340921 −0.170460 0.985365i \(-0.554526\pi\)
−0.170460 + 0.985365i \(0.554526\pi\)
\(374\) 10.7589 0.556331
\(375\) 3.32622 0.171765
\(376\) 8.54530 0.440690
\(377\) 10.6813 0.550114
\(378\) 78.4277 4.03389
\(379\) −3.74050 −0.192137 −0.0960684 0.995375i \(-0.530627\pi\)
−0.0960684 + 0.995375i \(0.530627\pi\)
\(380\) −0.743534 −0.0381425
\(381\) 7.14163 0.365877
\(382\) −38.4743 −1.96852
\(383\) 27.5711 1.40882 0.704409 0.709795i \(-0.251212\pi\)
0.704409 + 0.709795i \(0.251212\pi\)
\(384\) 44.4508 2.26837
\(385\) −2.81120 −0.143272
\(386\) 25.7897 1.31266
\(387\) −19.4125 −0.986794
\(388\) 3.71310 0.188504
\(389\) −17.1560 −0.869844 −0.434922 0.900468i \(-0.643224\pi\)
−0.434922 + 0.900468i \(0.643224\pi\)
\(390\) −32.9606 −1.66902
\(391\) −18.0022 −0.910413
\(392\) −1.87895 −0.0949014
\(393\) 43.7475 2.20677
\(394\) −37.8800 −1.90837
\(395\) 4.44176 0.223489
\(396\) −5.99567 −0.301294
\(397\) −14.1506 −0.710201 −0.355100 0.934828i \(-0.615553\pi\)
−0.355100 + 0.934828i \(0.615553\pi\)
\(398\) −39.3427 −1.97207
\(399\) −9.35067 −0.468119
\(400\) −4.93423 −0.246711
\(401\) 10.3891 0.518807 0.259403 0.965769i \(-0.416474\pi\)
0.259403 + 0.965769i \(0.416474\pi\)
\(402\) −72.9284 −3.63734
\(403\) 18.8923 0.941093
\(404\) −12.4369 −0.618758
\(405\) 31.8327 1.58178
\(406\) −8.31346 −0.412590
\(407\) −3.90101 −0.193366
\(408\) 44.9646 2.22608
\(409\) 4.11163 0.203307 0.101653 0.994820i \(-0.467587\pi\)
0.101653 + 0.994820i \(0.467587\pi\)
\(410\) −7.67301 −0.378943
\(411\) −1.05164 −0.0518737
\(412\) 3.33664 0.164385
\(413\) 14.9841 0.737319
\(414\) 37.0173 1.81930
\(415\) 17.7115 0.869424
\(416\) 23.9934 1.17637
\(417\) −13.5515 −0.663621
\(418\) 1.65636 0.0810153
\(419\) 16.4494 0.803604 0.401802 0.915726i \(-0.368384\pi\)
0.401802 + 0.915726i \(0.368384\pi\)
\(420\) 6.95254 0.339249
\(421\) −38.6187 −1.88216 −0.941079 0.338187i \(-0.890186\pi\)
−0.941079 + 0.338187i \(0.890186\pi\)
\(422\) 12.4016 0.603703
\(423\) −33.1099 −1.60986
\(424\) −23.9506 −1.16314
\(425\) −6.49551 −0.315079
\(426\) −2.41230 −0.116877
\(427\) 20.1910 0.977109
\(428\) 0.785704 0.0379784
\(429\) 19.8994 0.960752
\(430\) −3.98750 −0.192294
\(431\) −24.2965 −1.17032 −0.585161 0.810917i \(-0.698969\pi\)
−0.585161 + 0.810917i \(0.698969\pi\)
\(432\) −83.1078 −3.99853
\(433\) 22.8348 1.09737 0.548686 0.836028i \(-0.315128\pi\)
0.548686 + 0.836028i \(0.315128\pi\)
\(434\) −14.7043 −0.705827
\(435\) −5.93863 −0.284735
\(436\) −2.58828 −0.123956
\(437\) −2.77149 −0.132578
\(438\) 61.9571 2.96043
\(439\) −23.3798 −1.11586 −0.557929 0.829889i \(-0.688404\pi\)
−0.557929 + 0.829889i \(0.688404\pi\)
\(440\) 2.08116 0.0992155
\(441\) 7.28025 0.346679
\(442\) 64.3661 3.06158
\(443\) 14.6539 0.696228 0.348114 0.937452i \(-0.386822\pi\)
0.348114 + 0.937452i \(0.386822\pi\)
\(444\) 9.64781 0.457865
\(445\) 17.5897 0.833833
\(446\) −14.5803 −0.690395
\(447\) 15.9813 0.755890
\(448\) 9.06766 0.428407
\(449\) 19.5934 0.924672 0.462336 0.886705i \(-0.347011\pi\)
0.462336 + 0.886705i \(0.347011\pi\)
\(450\) 13.3565 0.629630
\(451\) 4.63245 0.218134
\(452\) −6.64782 −0.312687
\(453\) −60.6698 −2.85052
\(454\) −20.7305 −0.972930
\(455\) −16.8182 −0.788451
\(456\) 6.92241 0.324171
\(457\) −32.4549 −1.51817 −0.759087 0.650989i \(-0.774354\pi\)
−0.759087 + 0.650989i \(0.774354\pi\)
\(458\) −3.83225 −0.179069
\(459\) −109.405 −5.10658
\(460\) 2.06070 0.0960804
\(461\) 9.88505 0.460393 0.230196 0.973144i \(-0.426063\pi\)
0.230196 + 0.973144i \(0.426063\pi\)
\(462\) −15.4881 −0.720571
\(463\) 9.99709 0.464604 0.232302 0.972644i \(-0.425374\pi\)
0.232302 + 0.972644i \(0.425374\pi\)
\(464\) 8.80956 0.408973
\(465\) −10.5038 −0.487103
\(466\) −21.6737 −1.00401
\(467\) 26.3637 1.21997 0.609984 0.792414i \(-0.291176\pi\)
0.609984 + 0.792414i \(0.291176\pi\)
\(468\) −35.8696 −1.65807
\(469\) −37.2119 −1.71829
\(470\) −6.80106 −0.313709
\(471\) −60.2581 −2.77655
\(472\) −11.0929 −0.510592
\(473\) 2.40738 0.110692
\(474\) 24.4715 1.12402
\(475\) −1.00000 −0.0458831
\(476\) −13.5771 −0.622304
\(477\) 92.7997 4.24901
\(478\) 34.6978 1.58704
\(479\) 19.3573 0.884459 0.442230 0.896902i \(-0.354188\pi\)
0.442230 + 0.896902i \(0.354188\pi\)
\(480\) −13.3399 −0.608882
\(481\) −23.3381 −1.06413
\(482\) −23.6688 −1.07809
\(483\) 25.9153 1.17919
\(484\) 0.743534 0.0337970
\(485\) 4.99386 0.226759
\(486\) 91.6851 4.15892
\(487\) 15.4579 0.700462 0.350231 0.936663i \(-0.386103\pi\)
0.350231 + 0.936663i \(0.386103\pi\)
\(488\) −14.9476 −0.676646
\(489\) −79.8112 −3.60919
\(490\) 1.49543 0.0675565
\(491\) 33.7098 1.52130 0.760650 0.649162i \(-0.224880\pi\)
0.760650 + 0.649162i \(0.224880\pi\)
\(492\) −11.4568 −0.516512
\(493\) 11.5971 0.522306
\(494\) 9.90932 0.445841
\(495\) −8.06374 −0.362438
\(496\) 15.5817 0.699640
\(497\) −1.23089 −0.0552127
\(498\) 97.5803 4.37268
\(499\) 32.9388 1.47454 0.737272 0.675596i \(-0.236114\pi\)
0.737272 + 0.675596i \(0.236114\pi\)
\(500\) 0.743534 0.0332518
\(501\) 10.3300 0.461509
\(502\) 18.4347 0.822779
\(503\) 8.51652 0.379733 0.189866 0.981810i \(-0.439195\pi\)
0.189866 + 0.981810i \(0.439195\pi\)
\(504\) −47.1774 −2.10145
\(505\) −16.7267 −0.744329
\(506\) −4.59059 −0.204076
\(507\) 75.8089 3.36679
\(508\) 1.59642 0.0708297
\(509\) 19.8390 0.879350 0.439675 0.898157i \(-0.355094\pi\)
0.439675 + 0.898157i \(0.355094\pi\)
\(510\) −35.7865 −1.58465
\(511\) 31.6138 1.39851
\(512\) −0.748955 −0.0330994
\(513\) −16.8431 −0.743642
\(514\) 12.0542 0.531689
\(515\) 4.48754 0.197745
\(516\) −5.95384 −0.262103
\(517\) 4.10602 0.180583
\(518\) 18.1645 0.798102
\(519\) 36.3926 1.59746
\(520\) 12.4507 0.546000
\(521\) 42.2657 1.85169 0.925846 0.377900i \(-0.123354\pi\)
0.925846 + 0.377900i \(0.123354\pi\)
\(522\) −23.8466 −1.04374
\(523\) −28.4461 −1.24386 −0.621931 0.783072i \(-0.713652\pi\)
−0.621931 + 0.783072i \(0.713652\pi\)
\(524\) 9.77920 0.427206
\(525\) 9.35067 0.408097
\(526\) −27.5827 −1.20266
\(527\) 20.5121 0.893521
\(528\) 16.4123 0.714255
\(529\) −15.3189 −0.666037
\(530\) 19.0618 0.827994
\(531\) 42.9809 1.86521
\(532\) −2.09022 −0.0906226
\(533\) 27.7140 1.20043
\(534\) 96.9092 4.19367
\(535\) 1.05672 0.0456858
\(536\) 27.5484 1.18991
\(537\) 6.27485 0.270780
\(538\) −11.7827 −0.507987
\(539\) −0.902838 −0.0388880
\(540\) 12.5234 0.538923
\(541\) 18.7126 0.804515 0.402258 0.915526i \(-0.368226\pi\)
0.402258 + 0.915526i \(0.368226\pi\)
\(542\) 18.6690 0.801901
\(543\) 28.2813 1.21367
\(544\) 26.0505 1.11691
\(545\) −3.48106 −0.149112
\(546\) −92.6588 −3.96543
\(547\) 22.8994 0.979109 0.489555 0.871973i \(-0.337159\pi\)
0.489555 + 0.871973i \(0.337159\pi\)
\(548\) −0.235081 −0.0100422
\(549\) 57.9165 2.47182
\(550\) −1.65636 −0.0706275
\(551\) 1.78540 0.0760605
\(552\) −19.1854 −0.816583
\(553\) 12.4867 0.530987
\(554\) −7.03772 −0.299004
\(555\) 12.9756 0.550784
\(556\) −3.02927 −0.128470
\(557\) 13.6065 0.576524 0.288262 0.957552i \(-0.406923\pi\)
0.288262 + 0.957552i \(0.406923\pi\)
\(558\) −42.1782 −1.78555
\(559\) 14.4024 0.609155
\(560\) −13.8711 −0.586160
\(561\) 21.6055 0.912186
\(562\) 0.369091 0.0155692
\(563\) −38.8581 −1.63767 −0.818836 0.574028i \(-0.805380\pi\)
−0.818836 + 0.574028i \(0.805380\pi\)
\(564\) −10.1548 −0.427596
\(565\) −8.94084 −0.376144
\(566\) 20.1690 0.847765
\(567\) 89.4882 3.75815
\(568\) 0.911238 0.0382347
\(569\) −27.9139 −1.17021 −0.585107 0.810956i \(-0.698947\pi\)
−0.585107 + 0.810956i \(0.698947\pi\)
\(570\) −5.50942 −0.230764
\(571\) −39.5183 −1.65379 −0.826895 0.562357i \(-0.809895\pi\)
−0.826895 + 0.562357i \(0.809895\pi\)
\(572\) 4.44825 0.185991
\(573\) −77.2621 −3.22767
\(574\) −21.5704 −0.900330
\(575\) 2.77149 0.115579
\(576\) 26.0100 1.08375
\(577\) 12.3632 0.514685 0.257343 0.966320i \(-0.417153\pi\)
0.257343 + 0.966320i \(0.417153\pi\)
\(578\) 41.7266 1.73560
\(579\) 51.7896 2.15230
\(580\) −1.32750 −0.0551216
\(581\) 49.7906 2.06566
\(582\) 27.5133 1.14046
\(583\) −11.5083 −0.476624
\(584\) −23.4041 −0.968467
\(585\) −48.2420 −1.99456
\(586\) 39.9473 1.65021
\(587\) 6.03747 0.249193 0.124597 0.992207i \(-0.460236\pi\)
0.124597 + 0.992207i \(0.460236\pi\)
\(588\) 2.23286 0.0920816
\(589\) 3.15789 0.130118
\(590\) 8.82864 0.363469
\(591\) −76.0687 −3.12905
\(592\) −19.2485 −0.791106
\(593\) −1.44341 −0.0592739 −0.0296369 0.999561i \(-0.509435\pi\)
−0.0296369 + 0.999561i \(0.509435\pi\)
\(594\) −27.8983 −1.14468
\(595\) −18.2602 −0.748594
\(596\) 3.57241 0.146332
\(597\) −79.0061 −3.23350
\(598\) −27.4636 −1.12307
\(599\) 20.4093 0.833902 0.416951 0.908929i \(-0.363099\pi\)
0.416951 + 0.908929i \(0.363099\pi\)
\(600\) −6.92241 −0.282606
\(601\) −33.1258 −1.35123 −0.675616 0.737254i \(-0.736122\pi\)
−0.675616 + 0.737254i \(0.736122\pi\)
\(602\) −11.2096 −0.456871
\(603\) −106.740 −4.34679
\(604\) −13.5620 −0.551828
\(605\) 1.00000 0.0406558
\(606\) −92.1546 −3.74353
\(607\) 7.79823 0.316520 0.158260 0.987397i \(-0.449412\pi\)
0.158260 + 0.987397i \(0.449412\pi\)
\(608\) 4.01054 0.162649
\(609\) −16.6947 −0.676502
\(610\) 11.8965 0.481677
\(611\) 24.5646 0.993778
\(612\) −38.9449 −1.57426
\(613\) 28.4672 1.14978 0.574890 0.818231i \(-0.305045\pi\)
0.574890 + 0.818231i \(0.305045\pi\)
\(614\) 32.6686 1.31840
\(615\) −15.4086 −0.621333
\(616\) 5.85056 0.235726
\(617\) 1.26620 0.0509751 0.0254876 0.999675i \(-0.491886\pi\)
0.0254876 + 0.999675i \(0.491886\pi\)
\(618\) 24.7238 0.994536
\(619\) 2.57985 0.103693 0.0518464 0.998655i \(-0.483489\pi\)
0.0518464 + 0.998655i \(0.483489\pi\)
\(620\) −2.34800 −0.0942978
\(621\) 46.6805 1.87323
\(622\) −43.4912 −1.74384
\(623\) 49.4482 1.98110
\(624\) 98.1881 3.93067
\(625\) 1.00000 0.0400000
\(626\) 4.59083 0.183487
\(627\) 3.32622 0.132836
\(628\) −13.4699 −0.537508
\(629\) −25.3391 −1.01033
\(630\) 37.5477 1.49594
\(631\) −18.3838 −0.731848 −0.365924 0.930645i \(-0.619247\pi\)
−0.365924 + 0.930645i \(0.619247\pi\)
\(632\) −9.24402 −0.367707
\(633\) 24.9043 0.989859
\(634\) 6.62681 0.263184
\(635\) 2.14707 0.0852039
\(636\) 28.4618 1.12858
\(637\) −5.40130 −0.214007
\(638\) 2.95726 0.117079
\(639\) −3.53072 −0.139673
\(640\) 13.3638 0.528249
\(641\) 20.9430 0.827199 0.413600 0.910459i \(-0.364271\pi\)
0.413600 + 0.910459i \(0.364271\pi\)
\(642\) 5.82189 0.229772
\(643\) 9.37012 0.369522 0.184761 0.982784i \(-0.440849\pi\)
0.184761 + 0.982784i \(0.440849\pi\)
\(644\) 5.79302 0.228277
\(645\) −8.00749 −0.315295
\(646\) 10.7589 0.423304
\(647\) −15.3143 −0.602069 −0.301034 0.953613i \(-0.597332\pi\)
−0.301034 + 0.953613i \(0.597332\pi\)
\(648\) −66.2491 −2.60251
\(649\) −5.33014 −0.209226
\(650\) −9.90932 −0.388676
\(651\) −29.5283 −1.15731
\(652\) −17.8408 −0.698698
\(653\) 7.61281 0.297912 0.148956 0.988844i \(-0.452409\pi\)
0.148956 + 0.988844i \(0.452409\pi\)
\(654\) −19.1786 −0.749944
\(655\) 13.1523 0.513904
\(656\) 22.8576 0.892438
\(657\) 90.6822 3.53785
\(658\) −19.1191 −0.745341
\(659\) 27.8742 1.08582 0.542912 0.839789i \(-0.317322\pi\)
0.542912 + 0.839789i \(0.317322\pi\)
\(660\) −2.47316 −0.0962675
\(661\) −40.0185 −1.55654 −0.778269 0.627931i \(-0.783902\pi\)
−0.778269 + 0.627931i \(0.783902\pi\)
\(662\) −29.1253 −1.13199
\(663\) 129.257 5.01992
\(664\) −36.8606 −1.43047
\(665\) −2.81120 −0.109014
\(666\) 52.1037 2.01898
\(667\) −4.94821 −0.191595
\(668\) 2.30913 0.0893431
\(669\) −29.2793 −1.13200
\(670\) −21.9253 −0.847048
\(671\) −7.18233 −0.277271
\(672\) −37.5012 −1.44664
\(673\) −41.4748 −1.59874 −0.799369 0.600841i \(-0.794832\pi\)
−0.799369 + 0.600841i \(0.794832\pi\)
\(674\) 54.2759 2.09063
\(675\) 16.8431 0.648292
\(676\) 16.9461 0.651773
\(677\) −0.266167 −0.0102296 −0.00511482 0.999987i \(-0.501628\pi\)
−0.00511482 + 0.999987i \(0.501628\pi\)
\(678\) −49.2589 −1.89178
\(679\) 14.0387 0.538757
\(680\) 13.5182 0.518400
\(681\) −41.6299 −1.59526
\(682\) 5.23060 0.200290
\(683\) 29.3614 1.12348 0.561742 0.827312i \(-0.310131\pi\)
0.561742 + 0.827312i \(0.310131\pi\)
\(684\) −5.99567 −0.229250
\(685\) −0.316167 −0.0120801
\(686\) −28.3906 −1.08396
\(687\) −7.69574 −0.293611
\(688\) 11.8786 0.452866
\(689\) −68.8491 −2.62294
\(690\) 15.2693 0.581293
\(691\) −22.7603 −0.865844 −0.432922 0.901431i \(-0.642517\pi\)
−0.432922 + 0.901431i \(0.642517\pi\)
\(692\) 8.13510 0.309250
\(693\) −22.6688 −0.861116
\(694\) −32.5468 −1.23546
\(695\) −4.07415 −0.154541
\(696\) 12.3592 0.468476
\(697\) 30.0902 1.13975
\(698\) 33.0626 1.25144
\(699\) −43.5240 −1.64623
\(700\) 2.09022 0.0790029
\(701\) −13.6101 −0.514046 −0.257023 0.966405i \(-0.582742\pi\)
−0.257023 + 0.966405i \(0.582742\pi\)
\(702\) −166.904 −6.29939
\(703\) −3.90101 −0.147129
\(704\) −3.22555 −0.121568
\(705\) −13.6575 −0.514372
\(706\) −38.4326 −1.44643
\(707\) −47.0221 −1.76845
\(708\) 13.1823 0.495420
\(709\) −32.9904 −1.23898 −0.619491 0.785004i \(-0.712661\pi\)
−0.619491 + 0.785004i \(0.712661\pi\)
\(710\) −0.725239 −0.0272177
\(711\) 35.8172 1.34325
\(712\) −36.6071 −1.37191
\(713\) −8.75204 −0.327767
\(714\) −100.603 −3.76498
\(715\) 5.98258 0.223736
\(716\) 1.40266 0.0524199
\(717\) 69.6784 2.60219
\(718\) −16.2886 −0.607886
\(719\) −3.85276 −0.143684 −0.0718418 0.997416i \(-0.522888\pi\)
−0.0718418 + 0.997416i \(0.522888\pi\)
\(720\) −39.7883 −1.48282
\(721\) 12.6154 0.469821
\(722\) 1.65636 0.0616434
\(723\) −47.5305 −1.76768
\(724\) 6.32193 0.234953
\(725\) −1.78540 −0.0663080
\(726\) 5.50942 0.204474
\(727\) −35.5951 −1.32015 −0.660074 0.751200i \(-0.729475\pi\)
−0.660074 + 0.751200i \(0.729475\pi\)
\(728\) 35.0015 1.29724
\(729\) 88.6192 3.28219
\(730\) 18.6269 0.689412
\(731\) 15.6372 0.578362
\(732\) 17.7630 0.656541
\(733\) −22.8053 −0.842332 −0.421166 0.906984i \(-0.638379\pi\)
−0.421166 + 0.906984i \(0.638379\pi\)
\(734\) 2.87933 0.106278
\(735\) 3.00304 0.110769
\(736\) −11.1152 −0.409710
\(737\) 13.2370 0.487592
\(738\) −61.8732 −2.27758
\(739\) −37.2882 −1.37167 −0.685835 0.727757i \(-0.740563\pi\)
−0.685835 + 0.727757i \(0.740563\pi\)
\(740\) 2.90053 0.106626
\(741\) 19.8994 0.731022
\(742\) 53.5866 1.96723
\(743\) 24.9220 0.914301 0.457150 0.889389i \(-0.348870\pi\)
0.457150 + 0.889389i \(0.348870\pi\)
\(744\) 21.8602 0.801433
\(745\) 4.80464 0.176028
\(746\) −10.9059 −0.399295
\(747\) 142.821 5.22555
\(748\) 4.82964 0.176589
\(749\) 2.97064 0.108545
\(750\) 5.50942 0.201176
\(751\) −1.46203 −0.0533501 −0.0266751 0.999644i \(-0.508492\pi\)
−0.0266751 + 0.999644i \(0.508492\pi\)
\(752\) 20.2600 0.738807
\(753\) 37.0195 1.34907
\(754\) 17.6921 0.644308
\(755\) −18.2399 −0.663816
\(756\) 35.2059 1.28043
\(757\) 4.66986 0.169729 0.0848645 0.996393i \(-0.472954\pi\)
0.0848645 + 0.996393i \(0.472954\pi\)
\(758\) −6.19562 −0.225035
\(759\) −9.21858 −0.334613
\(760\) 2.08116 0.0754917
\(761\) 18.4084 0.667303 0.333652 0.942696i \(-0.391719\pi\)
0.333652 + 0.942696i \(0.391719\pi\)
\(762\) 11.8291 0.428524
\(763\) −9.78595 −0.354275
\(764\) −17.2709 −0.624841
\(765\) −52.3782 −1.89374
\(766\) 45.6677 1.65004
\(767\) −31.8880 −1.15141
\(768\) 52.1688 1.88248
\(769\) −3.57354 −0.128865 −0.0644326 0.997922i \(-0.520524\pi\)
−0.0644326 + 0.997922i \(0.520524\pi\)
\(770\) −4.65636 −0.167804
\(771\) 24.2067 0.871782
\(772\) 11.5769 0.416662
\(773\) 34.6561 1.24649 0.623246 0.782026i \(-0.285814\pi\)
0.623246 + 0.782026i \(0.285814\pi\)
\(774\) −32.1542 −1.15576
\(775\) −3.15789 −0.113435
\(776\) −10.3930 −0.373088
\(777\) 36.4770 1.30861
\(778\) −28.4166 −1.01878
\(779\) 4.63245 0.165975
\(780\) −14.7959 −0.529777
\(781\) 0.437851 0.0156675
\(782\) −29.8182 −1.06630
\(783\) −30.0717 −1.07467
\(784\) −4.45481 −0.159100
\(785\) −18.1161 −0.646591
\(786\) 72.4617 2.58462
\(787\) 30.3693 1.08255 0.541274 0.840846i \(-0.317942\pi\)
0.541274 + 0.840846i \(0.317942\pi\)
\(788\) −17.0042 −0.605749
\(789\) −55.3902 −1.97194
\(790\) 7.35716 0.261756
\(791\) −25.1345 −0.893679
\(792\) 16.7820 0.596321
\(793\) −42.9689 −1.52587
\(794\) −23.4386 −0.831804
\(795\) 38.2790 1.35762
\(796\) −17.6608 −0.625970
\(797\) −34.4392 −1.21990 −0.609950 0.792440i \(-0.708810\pi\)
−0.609950 + 0.792440i \(0.708810\pi\)
\(798\) −15.4881 −0.548272
\(799\) 26.6707 0.943542
\(800\) −4.01054 −0.141794
\(801\) 141.839 5.01163
\(802\) 17.2081 0.607639
\(803\) −11.2457 −0.396851
\(804\) −32.7372 −1.15455
\(805\) 7.79120 0.274604
\(806\) 31.2925 1.10223
\(807\) −23.6613 −0.832918
\(808\) 34.8110 1.22465
\(809\) −8.23292 −0.289454 −0.144727 0.989472i \(-0.546230\pi\)
−0.144727 + 0.989472i \(0.546230\pi\)
\(810\) 52.7265 1.85262
\(811\) 35.9943 1.26393 0.631966 0.774996i \(-0.282248\pi\)
0.631966 + 0.774996i \(0.282248\pi\)
\(812\) −3.73188 −0.130963
\(813\) 37.4901 1.31483
\(814\) −6.46148 −0.226475
\(815\) −23.9946 −0.840492
\(816\) 106.606 3.73197
\(817\) 2.40738 0.0842237
\(818\) 6.81034 0.238118
\(819\) −135.618 −4.73887
\(820\) −3.44438 −0.120283
\(821\) 31.2285 1.08988 0.544941 0.838474i \(-0.316552\pi\)
0.544941 + 0.838474i \(0.316552\pi\)
\(822\) −1.74190 −0.0607557
\(823\) −8.20565 −0.286031 −0.143015 0.989720i \(-0.545680\pi\)
−0.143015 + 0.989720i \(0.545680\pi\)
\(824\) −9.33931 −0.325350
\(825\) −3.32622 −0.115804
\(826\) 24.8191 0.863566
\(827\) −15.5275 −0.539943 −0.269972 0.962868i \(-0.587014\pi\)
−0.269972 + 0.962868i \(0.587014\pi\)
\(828\) 16.6169 0.577478
\(829\) 3.31730 0.115215 0.0576074 0.998339i \(-0.481653\pi\)
0.0576074 + 0.998339i \(0.481653\pi\)
\(830\) 29.3367 1.01829
\(831\) −14.1328 −0.490261
\(832\) −19.2971 −0.669007
\(833\) −5.86440 −0.203189
\(834\) −22.4462 −0.777249
\(835\) 3.10562 0.107474
\(836\) 0.743534 0.0257157
\(837\) −53.1887 −1.83847
\(838\) 27.2461 0.941201
\(839\) 7.02051 0.242375 0.121188 0.992630i \(-0.461330\pi\)
0.121188 + 0.992630i \(0.461330\pi\)
\(840\) −19.4603 −0.671443
\(841\) −25.8124 −0.890081
\(842\) −63.9665 −2.20443
\(843\) 0.741189 0.0255279
\(844\) 5.56705 0.191626
\(845\) 22.7913 0.784044
\(846\) −54.8420 −1.88551
\(847\) 2.81120 0.0965939
\(848\) −56.7844 −1.94998
\(849\) 40.5023 1.39004
\(850\) −10.7589 −0.369028
\(851\) 10.8116 0.370617
\(852\) −1.08287 −0.0370986
\(853\) 3.62821 0.124227 0.0621137 0.998069i \(-0.480216\pi\)
0.0621137 + 0.998069i \(0.480216\pi\)
\(854\) 33.4435 1.14441
\(855\) −8.06374 −0.275774
\(856\) −2.19920 −0.0751670
\(857\) 16.1864 0.552918 0.276459 0.961026i \(-0.410839\pi\)
0.276459 + 0.961026i \(0.410839\pi\)
\(858\) 32.9606 1.12526
\(859\) 25.5004 0.870063 0.435032 0.900415i \(-0.356737\pi\)
0.435032 + 0.900415i \(0.356737\pi\)
\(860\) −1.78997 −0.0610375
\(861\) −43.3165 −1.47622
\(862\) −40.2438 −1.37071
\(863\) −39.9012 −1.35825 −0.679126 0.734022i \(-0.737641\pi\)
−0.679126 + 0.734022i \(0.737641\pi\)
\(864\) −67.5500 −2.29810
\(865\) 10.9411 0.372010
\(866\) 37.8228 1.28527
\(867\) 83.7932 2.84577
\(868\) −6.60068 −0.224042
\(869\) −4.44176 −0.150676
\(870\) −9.83652 −0.333489
\(871\) 79.1916 2.68330
\(872\) 7.24465 0.245335
\(873\) 40.2692 1.36291
\(874\) −4.59059 −0.155279
\(875\) 2.81120 0.0950359
\(876\) 27.8123 0.939691
\(877\) 11.1804 0.377537 0.188768 0.982022i \(-0.439550\pi\)
0.188768 + 0.982022i \(0.439550\pi\)
\(878\) −38.7254 −1.30692
\(879\) 80.2202 2.70576
\(880\) 4.93423 0.166333
\(881\) 2.82033 0.0950195 0.0475097 0.998871i \(-0.484871\pi\)
0.0475097 + 0.998871i \(0.484871\pi\)
\(882\) 12.0587 0.406039
\(883\) −17.0119 −0.572495 −0.286247 0.958156i \(-0.592408\pi\)
−0.286247 + 0.958156i \(0.592408\pi\)
\(884\) 28.8937 0.971800
\(885\) 17.7292 0.595961
\(886\) 24.2722 0.815439
\(887\) 10.8354 0.363817 0.181908 0.983316i \(-0.441773\pi\)
0.181908 + 0.983316i \(0.441773\pi\)
\(888\) −27.0044 −0.906207
\(889\) 6.03584 0.202436
\(890\) 29.1349 0.976605
\(891\) −31.8327 −1.06644
\(892\) −6.54501 −0.219143
\(893\) 4.10602 0.137403
\(894\) 26.4708 0.885316
\(895\) 1.88648 0.0630581
\(896\) 37.5682 1.25506
\(897\) −55.1509 −1.84144
\(898\) 32.4538 1.08300
\(899\) 5.63808 0.188041
\(900\) 5.99567 0.199856
\(901\) −74.7521 −2.49035
\(902\) 7.67301 0.255483
\(903\) −22.5106 −0.749107
\(904\) 18.6073 0.618871
\(905\) 8.50254 0.282634
\(906\) −100.491 −3.33859
\(907\) 38.8963 1.29153 0.645766 0.763536i \(-0.276538\pi\)
0.645766 + 0.763536i \(0.276538\pi\)
\(908\) −9.30583 −0.308825
\(909\) −134.880 −4.47369
\(910\) −27.8571 −0.923453
\(911\) 11.2431 0.372502 0.186251 0.982502i \(-0.440366\pi\)
0.186251 + 0.982502i \(0.440366\pi\)
\(912\) 16.4123 0.543466
\(913\) −17.7115 −0.586166
\(914\) −53.7570 −1.77812
\(915\) 23.8900 0.789780
\(916\) −1.72028 −0.0568398
\(917\) 36.9738 1.22098
\(918\) −181.214 −5.98095
\(919\) −9.48069 −0.312739 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(920\) −5.76792 −0.190163
\(921\) 65.6035 2.16171
\(922\) 16.3732 0.539223
\(923\) 2.61948 0.0862211
\(924\) −6.95254 −0.228722
\(925\) 3.90101 0.128264
\(926\) 16.5588 0.544156
\(927\) 36.1864 1.18852
\(928\) 7.16040 0.235052
\(929\) −39.8543 −1.30758 −0.653789 0.756677i \(-0.726822\pi\)
−0.653789 + 0.756677i \(0.726822\pi\)
\(930\) −17.3981 −0.570507
\(931\) −0.902838 −0.0295893
\(932\) −9.72922 −0.318691
\(933\) −87.3369 −2.85928
\(934\) 43.6679 1.42886
\(935\) 6.49551 0.212426
\(936\) 100.399 3.28166
\(937\) −56.3053 −1.83942 −0.919708 0.392604i \(-0.871574\pi\)
−0.919708 + 0.392604i \(0.871574\pi\)
\(938\) −61.6363 −2.01250
\(939\) 9.21908 0.300853
\(940\) −3.05297 −0.0995768
\(941\) −21.4276 −0.698519 −0.349259 0.937026i \(-0.613567\pi\)
−0.349259 + 0.937026i \(0.613567\pi\)
\(942\) −99.8092 −3.25196
\(943\) −12.8388 −0.418088
\(944\) −26.3001 −0.855996
\(945\) 47.3494 1.54028
\(946\) 3.98750 0.129645
\(947\) 16.2072 0.526662 0.263331 0.964706i \(-0.415179\pi\)
0.263331 + 0.964706i \(0.415179\pi\)
\(948\) 10.9852 0.356782
\(949\) −67.2781 −2.18394
\(950\) −1.65636 −0.0537395
\(951\) 13.3076 0.431529
\(952\) 38.0024 1.23166
\(953\) 58.8844 1.90745 0.953726 0.300676i \(-0.0972122\pi\)
0.953726 + 0.300676i \(0.0972122\pi\)
\(954\) 153.710 4.97654
\(955\) −23.2282 −0.751646
\(956\) 15.5757 0.503755
\(957\) 5.93863 0.191969
\(958\) 32.0628 1.03590
\(959\) −0.888809 −0.0287011
\(960\) 10.7289 0.346274
\(961\) −21.0278 −0.678315
\(962\) −38.6563 −1.24633
\(963\) 8.52108 0.274588
\(964\) −10.6248 −0.342203
\(965\) 15.5701 0.501220
\(966\) 42.9251 1.38109
\(967\) −24.4443 −0.786074 −0.393037 0.919523i \(-0.628576\pi\)
−0.393037 + 0.919523i \(0.628576\pi\)
\(968\) −2.08116 −0.0668911
\(969\) 21.6055 0.694069
\(970\) 8.27163 0.265586
\(971\) 59.3586 1.90491 0.952454 0.304683i \(-0.0985506\pi\)
0.952454 + 0.304683i \(0.0985506\pi\)
\(972\) 41.1571 1.32011
\(973\) −11.4532 −0.367174
\(974\) 25.6038 0.820399
\(975\) −19.8994 −0.637291
\(976\) −35.4392 −1.13438
\(977\) −5.33048 −0.170537 −0.0852686 0.996358i \(-0.527175\pi\)
−0.0852686 + 0.996358i \(0.527175\pi\)
\(978\) −132.196 −4.22717
\(979\) −17.5897 −0.562170
\(980\) 0.671291 0.0214436
\(981\) −28.0704 −0.896218
\(982\) 55.8356 1.78178
\(983\) −31.3817 −1.00092 −0.500460 0.865760i \(-0.666836\pi\)
−0.500460 + 0.865760i \(0.666836\pi\)
\(984\) 32.0677 1.02228
\(985\) −22.8694 −0.728680
\(986\) 19.2090 0.611738
\(987\) −38.3940 −1.22210
\(988\) 4.44825 0.141518
\(989\) −6.67203 −0.212158
\(990\) −13.3565 −0.424497
\(991\) −25.0162 −0.794667 −0.397334 0.917674i \(-0.630064\pi\)
−0.397334 + 0.917674i \(0.630064\pi\)
\(992\) 12.6648 0.402108
\(993\) −58.4880 −1.85606
\(994\) −2.03879 −0.0646665
\(995\) −23.7525 −0.753005
\(996\) 43.8034 1.38796
\(997\) −16.2641 −0.515091 −0.257545 0.966266i \(-0.582914\pi\)
−0.257545 + 0.966266i \(0.582914\pi\)
\(998\) 54.5586 1.72702
\(999\) 65.7052 2.07882
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 1045.2.a.g.1.5 6
3.2 odd 2 9405.2.a.w.1.2 6
5.4 even 2 5225.2.a.k.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.5 6 1.1 even 1 trivial
5225.2.a.k.1.2 6 5.4 even 2
9405.2.a.w.1.2 6 3.2 odd 2