Properties

Label 5225.2.a.k.1.2
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
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: no (minimal twist has level 1045)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.65636\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.65636 q^{2} -3.32622 q^{3} +0.743534 q^{4} +5.50942 q^{6} -2.81120 q^{7} +2.08116 q^{8} +8.06374 q^{9} +O(q^{10})\) \(q-1.65636 q^{2} -3.32622 q^{3} +0.743534 q^{4} +5.50942 q^{6} -2.81120 q^{7} +2.08116 q^{8} +8.06374 q^{9} -1.00000 q^{11} -2.47316 q^{12} +5.98258 q^{13} +4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -13.3565 q^{18} -1.00000 q^{19} +9.35067 q^{21} +1.65636 q^{22} -2.77149 q^{23} -6.92241 q^{24} -9.90932 q^{26} -16.8431 q^{27} -2.09022 q^{28} -1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} +3.32622 q^{33} -10.7589 q^{34} +5.99567 q^{36} -3.90101 q^{37} +1.65636 q^{38} -19.8994 q^{39} -4.63245 q^{41} -15.4881 q^{42} +2.40738 q^{43} -0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +16.4123 q^{48} +0.902838 q^{49} -21.6055 q^{51} +4.44825 q^{52} -11.5083 q^{53} +27.8983 q^{54} -5.85056 q^{56} +3.32622 q^{57} +2.95726 q^{58} +5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} -22.6688 q^{63} +3.22555 q^{64} -5.50942 q^{66} +13.2370 q^{67} +4.82964 q^{68} +9.21858 q^{69} -0.437851 q^{71} +16.7820 q^{72} -11.2457 q^{73} +6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +32.9606 q^{78} +4.44176 q^{79} +31.8327 q^{81} +7.67301 q^{82} -17.7115 q^{83} +6.95254 q^{84} -3.98750 q^{86} +5.93863 q^{87} -2.08116 q^{88} +17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} +10.5038 q^{93} -6.80106 q^{94} -13.3399 q^{96} -4.99386 q^{97} -1.49543 q^{98} -8.06374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) 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 \(-0.699146\pi\)
−0.585612 + 0.810591i \(0.699146\pi\)
\(3\) −3.32622 −1.92039 −0.960197 0.279323i \(-0.909890\pi\)
−0.960197 + 0.279323i \(0.909890\pi\)
\(4\) 0.743534 0.371767
\(5\) 0 0
\(6\) 5.50942 2.24921
\(7\) −2.81120 −1.06253 −0.531267 0.847205i \(-0.678284\pi\)
−0.531267 + 0.847205i \(0.678284\pi\)
\(8\) 2.08116 0.735802
\(9\) 8.06374 2.68791
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.47316 −0.713939
\(13\) 5.98258 1.65927 0.829635 0.558306i \(-0.188549\pi\)
0.829635 + 0.558306i \(0.188549\pi\)
\(14\) 4.65636 1.24446
\(15\) 0 0
\(16\) −4.93423 −1.23356
\(17\) 6.49551 1.57539 0.787697 0.616063i \(-0.211273\pi\)
0.787697 + 0.616063i \(0.211273\pi\)
\(18\) −13.3565 −3.14815
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 9.35067 2.04048
\(22\) 1.65636 0.353137
\(23\) −2.77149 −0.577895 −0.288948 0.957345i \(-0.593305\pi\)
−0.288948 + 0.957345i \(0.593305\pi\)
\(24\) −6.92241 −1.41303
\(25\) 0 0
\(26\) −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 \(0.553011\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 5.99567 0.999278
\(37\) −3.90101 −0.641322 −0.320661 0.947194i \(-0.603905\pi\)
−0.320661 + 0.947194i \(0.603905\pi\)
\(38\) 1.65636 0.268697
\(39\) −19.8994 −3.18645
\(40\) 0 0
\(41\) −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 \(-0.441237\pi\)
0.183561 + 0.983008i \(0.441237\pi\)
\(44\) −0.743534 −0.112092
\(45\) 0 0
\(46\) 4.59059 0.676845
\(47\) 4.10602 0.598925 0.299462 0.954108i \(-0.403193\pi\)
0.299462 + 0.954108i \(0.403193\pi\)
\(48\) 16.4123 2.36891
\(49\) 0.902838 0.128977
\(50\) 0 0
\(51\) −21.6055 −3.02538
\(52\) 4.44825 0.616862
\(53\) −11.5083 −1.58078 −0.790391 0.612603i \(-0.790123\pi\)
−0.790391 + 0.612603i \(0.790123\pi\)
\(54\) 27.8983 3.79648
\(55\) 0 0
\(56\) −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 \(0.387213\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −5.50942 −0.678163
\(67\) 13.2370 1.61716 0.808580 0.588387i \(-0.200237\pi\)
0.808580 + 0.588387i \(0.200237\pi\)
\(68\) 4.82964 0.585679
\(69\) 9.21858 1.10979
\(70\) 0 0
\(71\) −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 \(-0.728641\pi\)
−0.658103 + 0.752928i \(0.728641\pi\)
\(74\) 6.46148 0.751132
\(75\) 0 0
\(76\) −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 \(0.419613\pi\)
\(80\) 0 0
\(81\) 31.8327 3.53697
\(82\) 7.67301 0.847343
\(83\) −17.7115 −1.94409 −0.972046 0.234790i \(-0.924560\pi\)
−0.972046 + 0.234790i \(0.924560\pi\)
\(84\) 6.95254 0.758584
\(85\) 0 0
\(86\) −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 \(0.117840\pi\)
\(90\) 0 0
\(91\) −16.8182 −1.76303
\(92\) −2.06070 −0.214842
\(93\) 10.5038 1.08920
\(94\) −6.80106 −0.701475
\(95\) 0 0
\(96\) −13.3399 −1.36150
\(97\) −4.99386 −0.507049 −0.253525 0.967329i \(-0.581590\pi\)
−0.253525 + 0.967329i \(0.581590\pi\)
\(98\) −1.49543 −0.151061
\(99\) −8.06374 −0.810437
\(100\) 0 0
\(101\) −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 \(-0.570960\pi\)
−0.221085 + 0.975254i \(0.570960\pi\)
\(104\) 12.4507 1.22089
\(105\) 0 0
\(106\) 19.0618 1.85145
\(107\) −1.05672 −0.102157 −0.0510783 0.998695i \(-0.516266\pi\)
−0.0510783 + 0.998695i \(0.516266\pi\)
\(108\) −12.5234 −1.20507
\(109\) −3.48106 −0.333425 −0.166712 0.986006i \(-0.553315\pi\)
−0.166712 + 0.986006i \(0.553315\pi\)
\(110\) 0 0
\(111\) 12.9756 1.23159
\(112\) 13.8711 1.31069
\(113\) 8.94084 0.841084 0.420542 0.907273i \(-0.361840\pi\)
0.420542 + 0.907273i \(0.361840\pi\)
\(114\) −5.50942 −0.516005
\(115\) 0 0
\(116\) −1.32750 −0.123256
\(117\) 48.2420 4.45998
\(118\) −8.82864 −0.812742
\(119\) −18.2602 −1.67391
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.8965 −1.07706
\(123\) 15.4086 1.38934
\(124\) −2.34800 −0.210856
\(125\) 0 0
\(126\) 37.5477 3.34502
\(127\) −2.14707 −0.190522 −0.0952609 0.995452i \(-0.530369\pi\)
−0.0952609 + 0.995452i \(0.530369\pi\)
\(128\) −13.3638 −1.18120
\(129\) −8.00749 −0.705020
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 13.5182 1.15918
\(137\) 0.316167 0.0270120 0.0135060 0.999909i \(-0.495701\pi\)
0.0135060 + 0.999909i \(0.495701\pi\)
\(138\) −15.2693 −1.29981
\(139\) −4.07415 −0.345565 −0.172782 0.984960i \(-0.555276\pi\)
−0.172782 + 0.984960i \(0.555276\pi\)
\(140\) 0 0
\(141\) −13.6575 −1.15017
\(142\) 0.725239 0.0608607
\(143\) −5.98258 −0.500289
\(144\) −39.7883 −3.31569
\(145\) 0 0
\(146\) 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 \(0.436943\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) −14.7959 −1.18462
\(157\) 18.1161 1.44582 0.722910 0.690942i \(-0.242804\pi\)
0.722910 + 0.690942i \(0.242804\pi\)
\(158\) −7.35716 −0.585304
\(159\) 38.2790 3.03572
\(160\) 0 0
\(161\) 7.79120 0.614033
\(162\) −52.7265 −4.14259
\(163\) 23.9946 1.87940 0.939699 0.342002i \(-0.111105\pi\)
0.939699 + 0.342002i \(0.111105\pi\)
\(164\) −3.44438 −0.268961
\(165\) 0 0
\(166\) 29.3367 2.27697
\(167\) −3.10562 −0.240320 −0.120160 0.992755i \(-0.538341\pi\)
−0.120160 + 0.992755i \(0.538341\pi\)
\(168\) 19.4603 1.50139
\(169\) 22.7913 1.75318
\(170\) 0 0
\(171\) −8.06374 −0.616650
\(172\) 1.78997 0.136484
\(173\) −10.9411 −0.831839 −0.415919 0.909401i \(-0.636540\pi\)
−0.415919 + 0.909401i \(0.636540\pi\)
\(174\) −9.83652 −0.745704
\(175\) 0 0
\(176\) 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 \(0.477540\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −17.3981 −1.27569
\(187\) −6.49551 −0.474999
\(188\) 3.05297 0.222660
\(189\) 47.3494 3.44416
\(190\) 0 0
\(191\) −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 \(-0.689345\pi\)
−0.560380 + 0.828235i \(0.689345\pi\)
\(194\) 8.27163 0.593868
\(195\) 0 0
\(196\) 0.671291 0.0479493
\(197\) 22.8694 1.62938 0.814689 0.579898i \(-0.196908\pi\)
0.814689 + 0.579898i \(0.196908\pi\)
\(198\) 13.3565 0.949204
\(199\) −23.7525 −1.68377 −0.841885 0.539657i \(-0.818554\pi\)
−0.841885 + 0.539657i \(0.818554\pi\)
\(200\) 0 0
\(201\) −44.0292 −3.10558
\(202\) 27.7055 1.94935
\(203\) 5.01911 0.352272
\(204\) −16.0644 −1.12474
\(205\) 0 0
\(206\) 7.43300 0.517881
\(207\) −22.3486 −1.55333
\(208\) −29.5194 −2.04680
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) −35.0533 −2.38507
\(217\) 8.87745 0.602640
\(218\) 5.76589 0.390515
\(219\) 37.4056 2.52763
\(220\) 0 0
\(221\) 38.8600 2.61400
\(222\) −21.4923 −1.44247
\(223\) 8.80258 0.589464 0.294732 0.955580i \(-0.404770\pi\)
0.294732 + 0.955580i \(0.404770\pi\)
\(224\) −11.2744 −0.753304
\(225\) 0 0
\(226\) −14.8093 −0.985098
\(227\) 12.5157 0.830695 0.415347 0.909663i \(-0.363660\pi\)
0.415347 + 0.909663i \(0.363660\pi\)
\(228\) 2.47316 0.163789
\(229\) −2.31366 −0.152891 −0.0764454 0.997074i \(-0.524357\pi\)
−0.0764454 + 0.997074i \(0.524357\pi\)
\(230\) 0 0
\(231\) −9.35067 −0.615229
\(232\) −3.71570 −0.243948
\(233\) 13.0851 0.857234 0.428617 0.903486i \(-0.359001\pi\)
0.428617 + 0.903486i \(0.359001\pi\)
\(234\) −79.9062 −5.22363
\(235\) 0 0
\(236\) 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 \(0.263057\pi\)
\(240\) 0 0
\(241\) −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 0
\(246\) −25.5221 −1.62723
\(247\) −5.98258 −0.380663
\(248\) −6.57207 −0.417327
\(249\) 58.9124 3.73342
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 15.6841 0.980257
\(257\) −7.27753 −0.453960 −0.226980 0.973899i \(-0.572885\pi\)
−0.226980 + 0.973899i \(0.572885\pi\)
\(258\) 13.2633 0.825737
\(259\) 10.9665 0.681426
\(260\) 0 0
\(261\) −14.3970 −0.891152
\(262\) −21.7850 −1.34588
\(263\) 16.6526 1.02684 0.513421 0.858137i \(-0.328378\pi\)
0.513421 + 0.858137i \(0.328378\pi\)
\(264\) 6.92241 0.426045
\(265\) 0 0
\(266\) −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 \(0.569582\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) 6.85433 0.412582
\(277\) 4.24890 0.255292 0.127646 0.991820i \(-0.459258\pi\)
0.127646 + 0.991820i \(0.459258\pi\)
\(278\) 6.74827 0.404734
\(279\) −25.4644 −1.52451
\(280\) 0 0
\(281\) 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 \(-0.617877\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(284\) −0.325557 −0.0193182
\(285\) 0 0
\(286\) 9.90932 0.585950
\(287\) 13.0227 0.768708
\(288\) 32.3399 1.90565
\(289\) 25.1917 1.48187
\(290\) 0 0
\(291\) 16.6107 0.973734
\(292\) −8.36153 −0.489322
\(293\) −24.1175 −1.40896 −0.704480 0.709724i \(-0.748820\pi\)
−0.704480 + 0.709724i \(0.748820\pi\)
\(294\) 4.97412 0.290096
\(295\) 0 0
\(296\) −8.11863 −0.471886
\(297\) 16.8431 0.977338
\(298\) −7.95822 −0.461008
\(299\) −16.5807 −0.958884
\(300\) 0 0
\(301\) −6.76763 −0.390080
\(302\) 30.2118 1.73849
\(303\) 55.6368 3.19625
\(304\) 4.93423 0.282997
\(305\) 0 0
\(306\) −86.7572 −4.95958
\(307\) −19.7231 −1.12566 −0.562829 0.826573i \(-0.690287\pi\)
−0.562829 + 0.826573i \(0.690287\pi\)
\(308\) 2.09022 0.119101
\(309\) 14.9266 0.849142
\(310\) 0 0
\(311\) −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 \(-0.524959\pi\)
−0.0783311 + 0.996927i \(0.524959\pi\)
\(314\) −30.0068 −1.69338
\(315\) 0 0
\(316\) 3.30260 0.185786
\(317\) −4.00082 −0.224709 −0.112354 0.993668i \(-0.535839\pi\)
−0.112354 + 0.993668i \(0.535839\pi\)
\(318\) −63.4039 −3.55552
\(319\) 1.78540 0.0999631
\(320\) 0 0
\(321\) 3.51487 0.196181
\(322\) −12.9051 −0.719170
\(323\) −6.49551 −0.361420
\(324\) 23.6687 1.31493
\(325\) 0 0
\(326\) −39.7437 −2.20120
\(327\) 11.5788 0.640307
\(328\) −9.64088 −0.532329
\(329\) −11.5428 −0.636377
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) −46.1383 −2.51705
\(337\) −32.7681 −1.78499 −0.892497 0.451054i \(-0.851048\pi\)
−0.892497 + 0.451054i \(0.851048\pi\)
\(338\) −37.7506 −2.05336
\(339\) −29.7392 −1.61521
\(340\) 0 0
\(341\) 3.15789 0.171009
\(342\) 13.3565 0.722236
\(343\) 17.1403 0.925491
\(344\) 5.01015 0.270129
\(345\) 0 0
\(346\) 18.1225 0.974270
\(347\) 19.6495 1.05484 0.527421 0.849604i \(-0.323159\pi\)
0.527421 + 0.849604i \(0.323159\pi\)
\(348\) 4.41557 0.236699
\(349\) 19.9610 1.06849 0.534243 0.845331i \(-0.320597\pi\)
0.534243 + 0.845331i \(0.320597\pi\)
\(350\) 0 0
\(351\) −100.765 −5.37846
\(352\) −4.01054 −0.213762
\(353\) 23.2030 1.23497 0.617486 0.786582i \(-0.288151\pi\)
0.617486 + 0.786582i \(0.288151\pi\)
\(354\) 29.3660 1.56079
\(355\) 0 0
\(356\) 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 \(0.583561\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0833 −0.740201
\(363\) −3.32622 −0.174581
\(364\) −12.5049 −0.655436
\(365\) 0 0
\(366\) 39.5705 2.06838
\(367\) −1.73835 −0.0907411 −0.0453705 0.998970i \(-0.514447\pi\)
−0.0453705 + 0.998970i \(0.514447\pi\)
\(368\) 13.6751 0.712866
\(369\) −37.3549 −1.94462
\(370\) 0 0
\(371\) 32.3520 1.67963
\(372\) 7.80995 0.404927
\(373\) 6.58428 0.340921 0.170460 0.985365i \(-0.445474\pi\)
0.170460 + 0.985365i \(0.445474\pi\)
\(374\) 10.7589 0.556331
\(375\) 0 0
\(376\) 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 0
\(381\) 7.14163 0.365877
\(382\) 38.4743 1.96852
\(383\) −27.5711 −1.40882 −0.704409 0.709795i \(-0.748788\pi\)
−0.704409 + 0.709795i \(0.748788\pi\)
\(384\) 44.4508 2.26837
\(385\) 0 0
\(386\) 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 \(0.643224\pi\)
\(390\) 0 0
\(391\) −18.0022 −0.910413
\(392\) 1.87895 0.0949014
\(393\) −43.7475 −2.20677
\(394\) −37.8800 −1.90837
\(395\) 0 0
\(396\) −5.99567 −0.301294
\(397\) 14.1506 0.710201 0.355100 0.934828i \(-0.384447\pi\)
0.355100 + 0.934828i \(0.384447\pi\)
\(398\) 39.3427 1.97207
\(399\) −9.35067 −0.468119
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.467587\pi\)
\(410\) 0 0
\(411\) −1.05164 −0.0518737
\(412\) −3.33664 −0.164385
\(413\) −14.9841 −0.737319
\(414\) 37.0173 1.81930
\(415\) 0 0
\(416\) 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 \(0.368384\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −2.41230 −0.116877
\(427\) −20.1910 −0.977109
\(428\) −0.785704 −0.0379784
\(429\) 19.8994 0.960752
\(430\) 0 0
\(431\) −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 \(-0.684872\pi\)
−0.548686 + 0.836028i \(0.684872\pi\)
\(434\) −14.7043 −0.705827
\(435\) 0 0
\(436\) −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 \(0.688404\pi\)
\(440\) 0 0
\(441\) 7.28025 0.346679
\(442\) −64.3661 −3.06158
\(443\) −14.6539 −0.696228 −0.348114 0.937452i \(-0.613178\pi\)
−0.348114 + 0.937452i \(0.613178\pi\)
\(444\) 9.64781 0.457865
\(445\) 0 0
\(446\) −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 \(0.347011\pi\)
\(450\) 0 0
\(451\) 4.63245 0.218134
\(452\) 6.64782 0.312687
\(453\) 60.6698 2.85052
\(454\) −20.7305 −0.972930
\(455\) 0 0
\(456\) 6.92241 0.324171
\(457\) 32.4549 1.51817 0.759087 0.650989i \(-0.225646\pi\)
0.759087 + 0.650989i \(0.225646\pi\)
\(458\) 3.83225 0.179069
\(459\) −109.405 −5.10658
\(460\) 0 0
\(461\) 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 \(-0.574626\pi\)
−0.232302 + 0.972644i \(0.574626\pi\)
\(464\) 8.80956 0.408973
\(465\) 0 0
\(466\) −21.6737 −1.00401
\(467\) −26.3637 −1.21997 −0.609984 0.792414i \(-0.708824\pi\)
−0.609984 + 0.792414i \(0.708824\pi\)
\(468\) 35.8696 1.65807
\(469\) −37.2119 −1.71829
\(470\) 0 0
\(471\) −60.2581 −2.77655
\(472\) 11.0929 0.510592
\(473\) −2.40738 −0.110692
\(474\) 24.4715 1.12402
\(475\) 0 0
\(476\) −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 \(0.354188\pi\)
\(480\) 0 0
\(481\) −23.3381 −1.06413
\(482\) 23.6688 1.07809
\(483\) −25.9153 −1.17919
\(484\) 0.743534 0.0337970
\(485\) 0 0
\(486\) 91.6851 4.15892
\(487\) −15.4579 −0.700462 −0.350231 0.936663i \(-0.613897\pi\)
−0.350231 + 0.936663i \(0.613897\pi\)
\(488\) 14.9476 0.676646
\(489\) −79.8112 −3.60919
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) 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 0
\(501\) 10.3300 0.461509
\(502\) −18.4347 −0.822779
\(503\) −8.51652 −0.379733 −0.189866 0.981810i \(-0.560805\pi\)
−0.189866 + 0.981810i \(0.560805\pi\)
\(504\) −47.1774 −2.10145
\(505\) 0 0
\(506\) −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 \(0.355094\pi\)
\(510\) 0 0
\(511\) 31.6138 1.39851
\(512\) 0.748955 0.0330994
\(513\) 16.8431 0.743642
\(514\) 12.0542 0.531689
\(515\) 0 0
\(516\) −5.95384 −0.262103
\(517\) −4.10602 −0.180583
\(518\) −18.1645 −0.798102
\(519\) 36.3926 1.59746
\(520\) 0 0
\(521\) 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 \(-0.286348\pi\)
0.621931 + 0.783072i \(0.286348\pi\)
\(524\) 9.77920 0.427206
\(525\) 0 0
\(526\) −27.5827 −1.20266
\(527\) −20.5121 −0.893521
\(528\) −16.4123 −0.714255
\(529\) −15.3189 −0.666037
\(530\) 0 0
\(531\) 42.9809 1.86521
\(532\) 2.09022 0.0906226
\(533\) −27.7140 −1.20043
\(534\) 96.9092 4.19367
\(535\) 0 0
\(536\) 27.5484 1.18991
\(537\) −6.27485 −0.270780
\(538\) 11.7827 0.507987
\(539\) −0.902838 −0.0388880
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) −92.6588 −3.96543
\(547\) −22.8994 −0.979109 −0.489555 0.871973i \(-0.662841\pi\)
−0.489555 + 0.871973i \(0.662841\pi\)
\(548\) 0.235081 0.0100422
\(549\) 57.9165 2.47182
\(550\) 0 0
\(551\) 1.78540 0.0760605
\(552\) 19.1854 0.816583
\(553\) −12.4867 −0.530987
\(554\) −7.03772 −0.299004
\(555\) 0 0
\(556\) −3.02927 −0.128470
\(557\) −13.6065 −0.576524 −0.288262 0.957552i \(-0.593077\pi\)
−0.288262 + 0.957552i \(0.593077\pi\)
\(558\) 42.1782 1.78555
\(559\) 14.4024 0.609155
\(560\) 0 0
\(561\) 21.6055 0.912186
\(562\) −0.369091 −0.0155692
\(563\) 38.8581 1.63767 0.818836 0.574028i \(-0.194620\pi\)
0.818836 + 0.574028i \(0.194620\pi\)
\(564\) −10.1548 −0.427596
\(565\) 0 0
\(566\) 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 \(0.698947\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 26.0100 1.08375
\(577\) −12.3632 −0.514685 −0.257343 0.966320i \(-0.582847\pi\)
−0.257343 + 0.966320i \(0.582847\pi\)
\(578\) −41.7266 −1.73560
\(579\) 51.7896 2.15230
\(580\) 0 0
\(581\) 49.7906 2.06566
\(582\) −27.5133 −1.14046
\(583\) 11.5083 0.476624
\(584\) −23.4041 −0.968467
\(585\) 0 0
\(586\) 39.9473 1.65021
\(587\) −6.03747 −0.249193 −0.124597 0.992207i \(-0.539764\pi\)
−0.124597 + 0.992207i \(0.539764\pi\)
\(588\) −2.23286 −0.0920816
\(589\) 3.15789 0.130118
\(590\) 0 0
\(591\) −76.0687 −3.12905
\(592\) 19.2485 0.791106
\(593\) 1.44341 0.0592739 0.0296369 0.999561i \(-0.490565\pi\)
0.0296369 + 0.999561i \(0.490565\pi\)
\(594\) −27.8983 −1.14468
\(595\) 0 0
\(596\) 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 \(0.363099\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) −92.1546 −3.74353
\(607\) −7.79823 −0.316520 −0.158260 0.987397i \(-0.550588\pi\)
−0.158260 + 0.987397i \(0.550588\pi\)
\(608\) −4.01054 −0.162649
\(609\) −16.6947 −0.676502
\(610\) 0 0
\(611\) 24.5646 0.993778
\(612\) 38.9449 1.57426
\(613\) −28.4672 −1.14978 −0.574890 0.818231i \(-0.694955\pi\)
−0.574890 + 0.818231i \(0.694955\pi\)
\(614\) 32.6686 1.31840
\(615\) 0 0
\(616\) 5.85056 0.235726
\(617\) −1.26620 −0.0509751 −0.0254876 0.999675i \(-0.508114\pi\)
−0.0254876 + 0.999675i \(0.508114\pi\)
\(618\) −24.7238 −0.994536
\(619\) 2.57985 0.103693 0.0518464 0.998655i \(-0.483489\pi\)
0.0518464 + 0.998655i \(0.483489\pi\)
\(620\) 0 0
\(621\) 46.6805 1.87323
\(622\) 43.4912 1.74384
\(623\) −49.4482 −1.98110
\(624\) 98.1881 3.93067
\(625\) 0 0
\(626\) 4.59083 0.183487
\(627\) −3.32622 −0.132836
\(628\) 13.4699 0.537508
\(629\) −25.3391 −1.01033
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 28.4618 1.12858
\(637\) 5.40130 0.214007
\(638\) −2.95726 −0.117079
\(639\) −3.53072 −0.139673
\(640\) 0 0
\(641\) 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 \(-0.559151\pi\)
−0.184761 + 0.982784i \(0.559151\pi\)
\(644\) 5.79302 0.228277
\(645\) 0 0
\(646\) 10.7589 0.423304
\(647\) 15.3143 0.602069 0.301034 0.953613i \(-0.402668\pi\)
0.301034 + 0.953613i \(0.402668\pi\)
\(648\) 66.2491 2.60251
\(649\) −5.33014 −0.209226
\(650\) 0 0
\(651\) −29.5283 −1.15731
\(652\) 17.8408 0.698698
\(653\) −7.61281 −0.297912 −0.148956 0.988844i \(-0.547591\pi\)
−0.148956 + 0.988844i \(0.547591\pi\)
\(654\) −19.1786 −0.749944
\(655\) 0 0
\(656\) 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 \(0.317322\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) 52.1037 2.01898
\(667\) 4.94821 0.191595
\(668\) −2.30913 −0.0893431
\(669\) −29.2793 −1.13200
\(670\) 0 0
\(671\) −7.18233 −0.277271
\(672\) 37.5012 1.44664
\(673\) 41.4748 1.59874 0.799369 0.600841i \(-0.205168\pi\)
0.799369 + 0.600841i \(0.205168\pi\)
\(674\) 54.2759 2.09063
\(675\) 0 0
\(676\) 16.9461 0.651773
\(677\) 0.266167 0.0102296 0.00511482 0.999987i \(-0.498372\pi\)
0.00511482 + 0.999987i \(0.498372\pi\)
\(678\) 49.2589 1.89178
\(679\) 14.0387 0.538757
\(680\) 0 0
\(681\) −41.6299 −1.59526
\(682\) −5.23060 −0.200290
\(683\) −29.3614 −1.12348 −0.561742 0.827312i \(-0.689869\pi\)
−0.561742 + 0.827312i \(0.689869\pi\)
\(684\) −5.99567 −0.229250
\(685\) 0 0
\(686\) −28.3906 −1.08396
\(687\) 7.69574 0.293611
\(688\) −11.8786 −0.452866
\(689\) −68.8491 −2.62294
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 12.3592 0.468476
\(697\) −30.0902 −1.13975
\(698\) −33.0626 −1.25144
\(699\) −43.5240 −1.64623
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 0
\(711\) 35.8172 1.34325
\(712\) 36.6071 1.37191
\(713\) 8.75204 0.327767
\(714\) −100.603 −3.76498
\(715\) 0 0
\(716\) 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 \(0.522888\pi\)
\(720\) 0 0
\(721\) 12.6154 0.469821
\(722\) −1.65636 −0.0616434
\(723\) 47.5305 1.76768
\(724\) 6.32193 0.234953
\(725\) 0 0
\(726\) 5.50942 0.204474
\(727\) 35.5951 1.32015 0.660074 0.751200i \(-0.270525\pi\)
0.660074 + 0.751200i \(0.270525\pi\)
\(728\) −35.0015 −1.29724
\(729\) 88.6192 3.28219
\(730\) 0 0
\(731\) 15.6372 0.578362
\(732\) −17.7630 −0.656541
\(733\) 22.8053 0.842332 0.421166 0.906984i \(-0.361621\pi\)
0.421166 + 0.906984i \(0.361621\pi\)
\(734\) 2.87933 0.106278
\(735\) 0 0
\(736\) −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 \(0.740563\pi\)
\(740\) 0 0
\(741\) 19.8994 0.731022
\(742\) −53.5866 −1.96723
\(743\) −24.9220 −0.914301 −0.457150 0.889389i \(-0.651130\pi\)
−0.457150 + 0.889389i \(0.651130\pi\)
\(744\) 21.8602 0.801433
\(745\) 0 0
\(746\) −10.9059 −0.399295
\(747\) −142.821 −5.22555
\(748\) −4.82964 −0.176589
\(749\) 2.97064 0.108545
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 35.2059 1.28043
\(757\) −4.66986 −0.169729 −0.0848645 0.996393i \(-0.527046\pi\)
−0.0848645 + 0.996393i \(0.527046\pi\)
\(758\) 6.19562 0.225035
\(759\) −9.21858 −0.334613
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) 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 \(0.520524\pi\)
\(770\) 0 0
\(771\) 24.2067 0.871782
\(772\) −11.5769 −0.416662
\(773\) −34.6561 −1.24649 −0.623246 0.782026i \(-0.714186\pi\)
−0.623246 + 0.782026i \(0.714186\pi\)
\(774\) −32.1542 −1.15576
\(775\) 0 0
\(776\) −10.3930 −0.373088
\(777\) −36.4770 −1.30861
\(778\) 28.4166 1.01878
\(779\) 4.63245 0.165975
\(780\) 0 0
\(781\) 0.437851 0.0156675
\(782\) 29.8182 1.06630
\(783\) 30.0717 1.07467
\(784\) −4.45481 −0.159100
\(785\) 0 0
\(786\) 72.4617 2.58462
\(787\) −30.3693 −1.08255 −0.541274 0.840846i \(-0.682058\pi\)
−0.541274 + 0.840846i \(0.682058\pi\)
\(788\) 17.0042 0.605749
\(789\) −55.3902 −1.97194
\(790\) 0 0
\(791\) −25.1345 −0.893679
\(792\) −16.7820 −0.596321
\(793\) 42.9689 1.52587
\(794\) −23.4386 −0.831804
\(795\) 0 0
\(796\) −17.6608 −0.625970
\(797\) 34.4392 1.21990 0.609950 0.792440i \(-0.291190\pi\)
0.609950 + 0.792440i \(0.291190\pi\)
\(798\) 15.4881 0.548272
\(799\) 26.6707 0.943542
\(800\) 0 0
\(801\) 141.839 5.01163
\(802\) −17.2081 −0.607639
\(803\) 11.2457 0.396851
\(804\) −32.7372 −1.15455
\(805\) 0 0
\(806\) 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 \(0.546230\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 106.606 3.73197
\(817\) −2.40738 −0.0842237
\(818\) −6.81034 −0.238118
\(819\) −135.618 −4.73887
\(820\) 0 0
\(821\) 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 \(-0.454320\pi\)
0.143015 + 0.989720i \(0.454320\pi\)
\(824\) −9.33931 −0.325350
\(825\) 0 0
\(826\) 24.8191 0.863566
\(827\) 15.5275 0.539943 0.269972 0.962868i \(-0.412986\pi\)
0.269972 + 0.962868i \(0.412986\pi\)
\(828\) −16.6169 −0.577478
\(829\) 3.31730 0.115215 0.0576074 0.998339i \(-0.481653\pi\)
0.0576074 + 0.998339i \(0.481653\pi\)
\(830\) 0 0
\(831\) −14.1328 −0.490261
\(832\) 19.2971 0.669007
\(833\) 5.86440 0.203189
\(834\) −22.4462 −0.777249
\(835\) 0 0
\(836\) 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 \(0.461330\pi\)
\(840\) 0 0
\(841\) −25.8124 −0.890081
\(842\) 63.9665 2.20443
\(843\) −0.741189 −0.0255279
\(844\) 5.56705 0.191626
\(845\) 0 0
\(846\) −54.8420 −1.88551
\(847\) −2.81120 −0.0965939
\(848\) 56.7844 1.94998
\(849\) 40.5023 1.39004
\(850\) 0 0
\(851\) 10.8116 0.370617
\(852\) 1.08287 0.0370986
\(853\) −3.62821 −0.124227 −0.0621137 0.998069i \(-0.519784\pi\)
−0.0621137 + 0.998069i \(0.519784\pi\)
\(854\) 33.4435 1.14441
\(855\) 0 0
\(856\) −2.19920 −0.0751670
\(857\) −16.1864 −0.552918 −0.276459 0.961026i \(-0.589161\pi\)
−0.276459 + 0.961026i \(0.589161\pi\)
\(858\) −32.9606 −1.12526
\(859\) 25.5004 0.870063 0.435032 0.900415i \(-0.356737\pi\)
0.435032 + 0.900415i \(0.356737\pi\)
\(860\) 0 0
\(861\) −43.3165 −1.47622
\(862\) 40.2438 1.37071
\(863\) 39.9012 1.35825 0.679126 0.734022i \(-0.262359\pi\)
0.679126 + 0.734022i \(0.262359\pi\)
\(864\) −67.5500 −2.29810
\(865\) 0 0
\(866\) 37.8228 1.28527
\(867\) −83.7932 −2.84577
\(868\) 6.60068 0.224042
\(869\) −4.44176 −0.150676
\(870\) 0 0
\(871\) 79.1916 2.68330
\(872\) −7.24465 −0.245335
\(873\) −40.2692 −1.36291
\(874\) −4.59059 −0.155279
\(875\) 0 0
\(876\) 27.8123 0.939691
\(877\) −11.1804 −0.377537 −0.188768 0.982022i \(-0.560450\pi\)
−0.188768 + 0.982022i \(0.560450\pi\)
\(878\) 38.7254 1.30692
\(879\) 80.2202 2.70576
\(880\) 0 0
\(881\) 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 \(-0.407592\pi\)
0.286247 + 0.958156i \(0.407592\pi\)
\(884\) 28.8937 0.971800
\(885\) 0 0
\(886\) 24.2722 0.815439
\(887\) −10.8354 −0.363817 −0.181908 0.983316i \(-0.558227\pi\)
−0.181908 + 0.983316i \(0.558227\pi\)
\(888\) 27.0044 0.906207
\(889\) 6.03584 0.202436
\(890\) 0 0
\(891\) −31.8327 −1.06644
\(892\) 6.54501 0.219143
\(893\) −4.10602 −0.137403
\(894\) 26.4708 0.885316
\(895\) 0 0
\(896\) 37.5682 1.25506
\(897\) 55.1509 1.84144
\(898\) −32.4538 −1.08300
\(899\) 5.63808 0.188041
\(900\) 0 0
\(901\) −74.7521 −2.49035
\(902\) −7.67301 −0.255483
\(903\) 22.5106 0.749107
\(904\) 18.6073 0.618871
\(905\) 0 0
\(906\) −100.491 −3.33859
\(907\) −38.8963 −1.29153 −0.645766 0.763536i \(-0.723462\pi\)
−0.645766 + 0.763536i \(0.723462\pi\)
\(908\) 9.30583 0.308825
\(909\) −134.880 −4.47369
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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 \(0.549979\pi\)
\(920\) 0 0
\(921\) 65.6035 2.16171
\(922\) −16.3732 −0.539223
\(923\) −2.61948 −0.0862211
\(924\) −6.95254 −0.228722
\(925\) 0 0
\(926\) 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 \(0.726822\pi\)
\(930\) 0 0
\(931\) −0.902838 −0.0295893
\(932\) 9.72922 0.318691
\(933\) 87.3369 2.85928
\(934\) 43.6679 1.42886
\(935\) 0 0
\(936\) 100.399 3.28166
\(937\) 56.3053 1.83942 0.919708 0.392604i \(-0.128426\pi\)
0.919708 + 0.392604i \(0.128426\pi\)
\(938\) 61.6363 2.01250
\(939\) 9.21908 0.300853
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 3.98750 0.129645
\(947\) −16.2072 −0.526662 −0.263331 0.964706i \(-0.584821\pi\)
−0.263331 + 0.964706i \(0.584821\pi\)
\(948\) −10.9852 −0.356782
\(949\) −67.2781 −2.18394
\(950\) 0 0
\(951\) 13.3076 0.431529
\(952\) −38.0024 −1.23166
\(953\) −58.8844 −1.90745 −0.953726 0.300676i \(-0.902788\pi\)
−0.953726 + 0.300676i \(0.902788\pi\)
\(954\) 153.710 4.97654
\(955\) 0 0
\(956\) 15.5757 0.503755
\(957\) −5.93863 −0.191969
\(958\) −32.0628 −1.03590
\(959\) −0.888809 −0.0287011
\(960\) 0 0
\(961\) −21.0278 −0.678315
\(962\) 38.6563 1.24633
\(963\) −8.52108 −0.274588
\(964\) −10.6248 −0.342203
\(965\) 0 0
\(966\) 42.9251 1.38109
\(967\) 24.4443 0.786074 0.393037 0.919523i \(-0.371424\pi\)
0.393037 + 0.919523i \(0.371424\pi\)
\(968\) 2.08116 0.0668911
\(969\) 21.6055 0.694069
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −35.4392 −1.13438
\(977\) 5.33048 0.170537 0.0852686 0.996358i \(-0.472825\pi\)
0.0852686 + 0.996358i \(0.472825\pi\)
\(978\) 132.196 4.22717
\(979\) −17.5897 −0.562170
\(980\) 0 0
\(981\) −28.0704 −0.896218
\(982\) −55.8356 −1.78178
\(983\) 31.3817 1.00092 0.500460 0.865760i \(-0.333164\pi\)
0.500460 + 0.865760i \(0.333164\pi\)
\(984\) 32.0677 1.02228
\(985\) 0 0
\(986\) 19.2090 0.611738
\(987\) 38.3940 1.22210
\(988\) −4.44825 −0.141518
\(989\) −6.67203 −0.212158
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) 43.8034 1.38796
\(997\) 16.2641 0.515091 0.257545 0.966266i \(-0.417086\pi\)
0.257545 + 0.966266i \(0.417086\pi\)
\(998\) −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 5225.2.a.k.1.2 6
5.4 even 2 1045.2.a.g.1.5 6
15.14 odd 2 9405.2.a.w.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.5 6 5.4 even 2
5225.2.a.k.1.2 6 1.1 even 1 trivial
9405.2.a.w.1.2 6 15.14 odd 2