Properties

Label 9405.2.a.w.1.2
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.65636 q^{2} +0.743534 q^{4} -1.00000 q^{5} +2.81120 q^{7} +2.08116 q^{8} +O(q^{10})\) \(q-1.65636 q^{2} +0.743534 q^{4} -1.00000 q^{5} +2.81120 q^{7} +2.08116 q^{8} +1.65636 q^{10} +1.00000 q^{11} -5.98258 q^{13} -4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -1.00000 q^{19} -0.743534 q^{20} -1.65636 q^{22} -2.77149 q^{23} +1.00000 q^{25} +9.90932 q^{26} +2.09022 q^{28} +1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} -10.7589 q^{34} -2.81120 q^{35} +3.90101 q^{37} +1.65636 q^{38} -2.08116 q^{40} +4.63245 q^{41} -2.40738 q^{43} +0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +0.902838 q^{49} -1.65636 q^{50} -4.44825 q^{52} -11.5083 q^{53} -1.00000 q^{55} +5.85056 q^{56} -2.95726 q^{58} -5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} +3.22555 q^{64} +5.98258 q^{65} -13.2370 q^{67} +4.82964 q^{68} +4.65636 q^{70} +0.437851 q^{71} +11.2457 q^{73} -6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +4.44176 q^{79} +4.93423 q^{80} -7.67301 q^{82} -17.7115 q^{83} -6.49551 q^{85} +3.98750 q^{86} +2.08116 q^{88} -17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} -6.80106 q^{94} +1.00000 q^{95} +4.99386 q^{97} -1.49543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) 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 \(0.699146\pi\)
\(3\) 0 0
\(4\) 0.743534 0.371767
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.81120 1.06253 0.531267 0.847205i \(-0.321716\pi\)
0.531267 + 0.847205i \(0.321716\pi\)
\(8\) 2.08116 0.735802
\(9\) 0 0
\(10\) 1.65636 0.523788
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.98258 −1.65927 −0.829635 0.558306i \(-0.811451\pi\)
−0.829635 + 0.558306i \(0.811451\pi\)
\(14\) −4.65636 −1.24446
\(15\) 0 0
\(16\) −4.93423 −1.23356
\(17\) 6.49551 1.57539 0.787697 0.616063i \(-0.211273\pi\)
0.787697 + 0.616063i \(0.211273\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −0.743534 −0.166259
\(21\) 0 0
\(22\) −1.65636 −0.353137
\(23\) −2.77149 −0.577895 −0.288948 0.957345i \(-0.593305\pi\)
−0.288948 + 0.957345i \(0.593305\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 9.90932 1.94338
\(27\) 0 0
\(28\) 2.09022 0.395015
\(29\) 1.78540 0.331540 0.165770 0.986164i \(-0.446989\pi\)
0.165770 + 0.986164i \(0.446989\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −10.7589 −1.84514
\(35\) −2.81120 −0.475179
\(36\) 0 0
\(37\) 3.90101 0.641322 0.320661 0.947194i \(-0.396095\pi\)
0.320661 + 0.947194i \(0.396095\pi\)
\(38\) 1.65636 0.268697
\(39\) 0 0
\(40\) −2.08116 −0.329061
\(41\) 4.63245 0.723467 0.361734 0.932281i \(-0.382185\pi\)
0.361734 + 0.932281i \(0.382185\pi\)
\(42\) 0 0
\(43\) −2.40738 −0.367122 −0.183561 0.983008i \(-0.558763\pi\)
−0.183561 + 0.983008i \(0.558763\pi\)
\(44\) 0.743534 0.112092
\(45\) 0 0
\(46\) 4.59059 0.676845
\(47\) 4.10602 0.598925 0.299462 0.954108i \(-0.403193\pi\)
0.299462 + 0.954108i \(0.403193\pi\)
\(48\) 0 0
\(49\) 0.902838 0.128977
\(50\) −1.65636 −0.234245
\(51\) 0 0
\(52\) −4.44825 −0.616862
\(53\) −11.5083 −1.58078 −0.790391 0.612603i \(-0.790123\pi\)
−0.790391 + 0.612603i \(0.790123\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 5.85056 0.781814
\(57\) 0 0
\(58\) −2.95726 −0.388308
\(59\) −5.33014 −0.693925 −0.346963 0.937879i \(-0.612787\pi\)
−0.346963 + 0.937879i \(0.612787\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 3.22555 0.403194
\(65\) 5.98258 0.742048
\(66\) 0 0
\(67\) −13.2370 −1.61716 −0.808580 0.588387i \(-0.799763\pi\)
−0.808580 + 0.588387i \(0.799763\pi\)
\(68\) 4.82964 0.585679
\(69\) 0 0
\(70\) 4.65636 0.556542
\(71\) 0.437851 0.0519633 0.0259817 0.999662i \(-0.491729\pi\)
0.0259817 + 0.999662i \(0.491729\pi\)
\(72\) 0 0
\(73\) 11.2457 1.31621 0.658103 0.752928i \(-0.271359\pi\)
0.658103 + 0.752928i \(0.271359\pi\)
\(74\) −6.46148 −0.751132
\(75\) 0 0
\(76\) −0.743534 −0.0852892
\(77\) 2.81120 0.320366
\(78\) 0 0
\(79\) 4.44176 0.499737 0.249869 0.968280i \(-0.419613\pi\)
0.249869 + 0.968280i \(0.419613\pi\)
\(80\) 4.93423 0.551663
\(81\) 0 0
\(82\) −7.67301 −0.847343
\(83\) −17.7115 −1.94409 −0.972046 0.234790i \(-0.924560\pi\)
−0.972046 + 0.234790i \(0.924560\pi\)
\(84\) 0 0
\(85\) −6.49551 −0.704537
\(86\) 3.98750 0.429983
\(87\) 0 0
\(88\) 2.08116 0.221853
\(89\) −17.5897 −1.86451 −0.932253 0.361807i \(-0.882160\pi\)
−0.932253 + 0.361807i \(0.882160\pi\)
\(90\) 0 0
\(91\) −16.8182 −1.76303
\(92\) −2.06070 −0.214842
\(93\) 0 0
\(94\) −6.80106 −0.701475
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 4.99386 0.507049 0.253525 0.967329i \(-0.418410\pi\)
0.253525 + 0.967329i \(0.418410\pi\)
\(98\) −1.49543 −0.151061
\(99\) 0 0
\(100\) 0.743534 0.0743534
\(101\) 16.7267 1.66437 0.832186 0.554497i \(-0.187089\pi\)
0.832186 + 0.554497i \(0.187089\pi\)
\(102\) 0 0
\(103\) 4.48754 0.442171 0.221085 0.975254i \(-0.429040\pi\)
0.221085 + 0.975254i \(0.429040\pi\)
\(104\) −12.4507 −1.22089
\(105\) 0 0
\(106\) 19.0618 1.85145
\(107\) −1.05672 −0.102157 −0.0510783 0.998695i \(-0.516266\pi\)
−0.0510783 + 0.998695i \(0.516266\pi\)
\(108\) 0 0
\(109\) −3.48106 −0.333425 −0.166712 0.986006i \(-0.553315\pi\)
−0.166712 + 0.986006i \(0.553315\pi\)
\(110\) 1.65636 0.157928
\(111\) 0 0
\(112\) −13.8711 −1.31069
\(113\) 8.94084 0.841084 0.420542 0.907273i \(-0.361840\pi\)
0.420542 + 0.907273i \(0.361840\pi\)
\(114\) 0 0
\(115\) 2.77149 0.258443
\(116\) 1.32750 0.123256
\(117\) 0 0
\(118\) 8.82864 0.812742
\(119\) 18.2602 1.67391
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.8965 −1.07706
\(123\) 0 0
\(124\) −2.34800 −0.210856
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.14707 0.190522 0.0952609 0.995452i \(-0.469631\pi\)
0.0952609 + 0.995452i \(0.469631\pi\)
\(128\) −13.3638 −1.18120
\(129\) 0 0
\(130\) −9.90932 −0.869105
\(131\) −13.1523 −1.14912 −0.574562 0.818461i \(-0.694828\pi\)
−0.574562 + 0.818461i \(0.694828\pi\)
\(132\) 0 0
\(133\) −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 \(0.495701\pi\)
\(138\) 0 0
\(139\) −4.07415 −0.345565 −0.172782 0.984960i \(-0.555276\pi\)
−0.172782 + 0.984960i \(0.555276\pi\)
\(140\) −2.09022 −0.176656
\(141\) 0 0
\(142\) −0.725239 −0.0608607
\(143\) −5.98258 −0.500289
\(144\) 0 0
\(145\) −1.78540 −0.148269
\(146\) −18.6269 −1.54157
\(147\) 0 0
\(148\) 2.90053 0.238422
\(149\) −4.80464 −0.393612 −0.196806 0.980442i \(-0.563057\pi\)
−0.196806 + 0.980442i \(0.563057\pi\)
\(150\) 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
\(153\) 0 0
\(154\) −4.65636 −0.375220
\(155\) 3.15789 0.253648
\(156\) 0 0
\(157\) −18.1161 −1.44582 −0.722910 0.690942i \(-0.757196\pi\)
−0.722910 + 0.690942i \(0.757196\pi\)
\(158\) −7.35716 −0.585304
\(159\) 0 0
\(160\) −4.01054 −0.317061
\(161\) −7.79120 −0.614033
\(162\) 0 0
\(163\) −23.9946 −1.87940 −0.939699 0.342002i \(-0.888895\pi\)
−0.939699 + 0.342002i \(0.888895\pi\)
\(164\) 3.44438 0.268961
\(165\) 0 0
\(166\) 29.3367 2.27697
\(167\) −3.10562 −0.240320 −0.120160 0.992755i \(-0.538341\pi\)
−0.120160 + 0.992755i \(0.538341\pi\)
\(168\) 0 0
\(169\) 22.7913 1.75318
\(170\) 10.7589 0.825172
\(171\) 0 0
\(172\) −1.78997 −0.136484
\(173\) −10.9411 −0.831839 −0.415919 0.909401i \(-0.636540\pi\)
−0.415919 + 0.909401i \(0.636540\pi\)
\(174\) 0 0
\(175\) 2.81120 0.212507
\(176\) −4.93423 −0.371931
\(177\) 0 0
\(178\) 29.1349 2.18376
\(179\) −1.88648 −0.141002 −0.0705011 0.997512i \(-0.522460\pi\)
−0.0705011 + 0.997512i \(0.522460\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −5.76792 −0.425216
\(185\) −3.90101 −0.286808
\(186\) 0 0
\(187\) 6.49551 0.474999
\(188\) 3.05297 0.222660
\(189\) 0 0
\(190\) −1.65636 −0.120165
\(191\) 23.2282 1.68073 0.840366 0.542019i \(-0.182340\pi\)
0.840366 + 0.542019i \(0.182340\pi\)
\(192\) 0 0
\(193\) 15.5701 1.12076 0.560380 0.828235i \(-0.310655\pi\)
0.560380 + 0.828235i \(0.310655\pi\)
\(194\) −8.27163 −0.593868
\(195\) 0 0
\(196\) 0.671291 0.0479493
\(197\) 22.8694 1.62938 0.814689 0.579898i \(-0.196908\pi\)
0.814689 + 0.579898i \(0.196908\pi\)
\(198\) 0 0
\(199\) −23.7525 −1.68377 −0.841885 0.539657i \(-0.818554\pi\)
−0.841885 + 0.539657i \(0.818554\pi\)
\(200\) 2.08116 0.147160
\(201\) 0 0
\(202\) −27.7055 −1.94935
\(203\) 5.01911 0.352272
\(204\) 0 0
\(205\) −4.63245 −0.323544
\(206\) −7.43300 −0.517881
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 1.75030 0.119648
\(215\) 2.40738 0.164182
\(216\) 0 0
\(217\) −8.87745 −0.602640
\(218\) 5.76589 0.390515
\(219\) 0 0
\(220\) −0.743534 −0.0501290
\(221\) −38.8600 −2.61400
\(222\) 0 0
\(223\) −8.80258 −0.589464 −0.294732 0.955580i \(-0.595230\pi\)
−0.294732 + 0.955580i \(0.595230\pi\)
\(224\) 11.2744 0.753304
\(225\) 0 0
\(226\) −14.8093 −0.985098
\(227\) 12.5157 0.830695 0.415347 0.909663i \(-0.363660\pi\)
0.415347 + 0.909663i \(0.363660\pi\)
\(228\) 0 0
\(229\) −2.31366 −0.152891 −0.0764454 0.997074i \(-0.524357\pi\)
−0.0764454 + 0.997074i \(0.524357\pi\)
\(230\) −4.59059 −0.302694
\(231\) 0 0
\(232\) 3.71570 0.243948
\(233\) 13.0851 0.857234 0.428617 0.903486i \(-0.359001\pi\)
0.428617 + 0.903486i \(0.359001\pi\)
\(234\) 0 0
\(235\) −4.10602 −0.267847
\(236\) −3.96314 −0.257978
\(237\) 0 0
\(238\) −30.2455 −1.96052
\(239\) −20.9482 −1.35503 −0.677514 0.735510i \(-0.736943\pi\)
−0.677514 + 0.735510i \(0.736943\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 5.34031 0.341878
\(245\) −0.902838 −0.0576802
\(246\) 0 0
\(247\) 5.98258 0.380663
\(248\) −6.57207 −0.417327
\(249\) 0 0
\(250\) 1.65636 0.104758
\(251\) −11.1296 −0.702495 −0.351247 0.936283i \(-0.614242\pi\)
−0.351247 + 0.936283i \(0.614242\pi\)
\(252\) 0 0
\(253\) −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 \(0.572885\pi\)
\(258\) 0 0
\(259\) 10.9665 0.681426
\(260\) 4.44825 0.275869
\(261\) 0 0
\(262\) 21.7850 1.34588
\(263\) 16.6526 1.02684 0.513421 0.858137i \(-0.328378\pi\)
0.513421 + 0.858137i \(0.328378\pi\)
\(264\) 0 0
\(265\) 11.5083 0.706947
\(266\) 4.65636 0.285500
\(267\) 0 0
\(268\) −9.84217 −0.601206
\(269\) 7.11358 0.433723 0.216861 0.976202i \(-0.430418\pi\)
0.216861 + 0.976202i \(0.430418\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −0.523687 −0.0316371
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −4.24890 −0.255292 −0.127646 0.991820i \(-0.540742\pi\)
−0.127646 + 0.991820i \(0.540742\pi\)
\(278\) 6.74827 0.404734
\(279\) 0 0
\(280\) −5.85056 −0.349638
\(281\) −0.222832 −0.0132931 −0.00664653 0.999978i \(-0.502116\pi\)
−0.00664653 + 0.999978i \(0.502116\pi\)
\(282\) 0 0
\(283\) 12.1767 0.723828 0.361914 0.932211i \(-0.382123\pi\)
0.361914 + 0.932211i \(0.382123\pi\)
\(284\) 0.325557 0.0193182
\(285\) 0 0
\(286\) 9.90932 0.585950
\(287\) 13.0227 0.768708
\(288\) 0 0
\(289\) 25.1917 1.48187
\(290\) 2.95726 0.173657
\(291\) 0 0
\(292\) 8.36153 0.489322
\(293\) −24.1175 −1.40896 −0.704480 0.709724i \(-0.748820\pi\)
−0.704480 + 0.709724i \(0.748820\pi\)
\(294\) 0 0
\(295\) 5.33014 0.310333
\(296\) 8.11863 0.471886
\(297\) 0 0
\(298\) 7.95822 0.461008
\(299\) 16.5807 0.958884
\(300\) 0 0
\(301\) −6.76763 −0.390080
\(302\) 30.2118 1.73849
\(303\) 0 0
\(304\) 4.93423 0.282997
\(305\) −7.18233 −0.411259
\(306\) 0 0
\(307\) 19.7231 1.12566 0.562829 0.826573i \(-0.309713\pi\)
0.562829 + 0.826573i \(0.309713\pi\)
\(308\) 2.09022 0.119101
\(309\) 0 0
\(310\) −5.23060 −0.297078
\(311\) 26.2571 1.48890 0.744451 0.667677i \(-0.232711\pi\)
0.744451 + 0.667677i \(0.232711\pi\)
\(312\) 0 0
\(313\) 2.77164 0.156662 0.0783311 0.996927i \(-0.475041\pi\)
0.0783311 + 0.996927i \(0.475041\pi\)
\(314\) 30.0068 1.69338
\(315\) 0 0
\(316\) 3.30260 0.185786
\(317\) −4.00082 −0.224709 −0.112354 0.993668i \(-0.535839\pi\)
−0.112354 + 0.993668i \(0.535839\pi\)
\(318\) 0 0
\(319\) 1.78540 0.0999631
\(320\) −3.22555 −0.180314
\(321\) 0 0
\(322\) 12.9051 0.719170
\(323\) −6.49551 −0.361420
\(324\) 0 0
\(325\) −5.98258 −0.331854
\(326\) 39.7437 2.20120
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 5.14403 0.281469
\(335\) 13.2370 0.723216
\(336\) 0 0
\(337\) 32.7681 1.78499 0.892497 0.451054i \(-0.148952\pi\)
0.892497 + 0.451054i \(0.148952\pi\)
\(338\) −37.7506 −2.05336
\(339\) 0 0
\(340\) −4.82964 −0.261924
\(341\) −3.15789 −0.171009
\(342\) 0 0
\(343\) −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 \(0.323159\pi\)
\(348\) 0 0
\(349\) 19.9610 1.06849 0.534243 0.845331i \(-0.320597\pi\)
0.534243 + 0.845331i \(0.320597\pi\)
\(350\) −4.65636 −0.248893
\(351\) 0 0
\(352\) 4.01054 0.213762
\(353\) 23.2030 1.23497 0.617486 0.786582i \(-0.288151\pi\)
0.617486 + 0.786582i \(0.288151\pi\)
\(354\) 0 0
\(355\) −0.437851 −0.0232387
\(356\) −13.0786 −0.693162
\(357\) 0 0
\(358\) 3.12469 0.165145
\(359\) 9.83399 0.519018 0.259509 0.965741i \(-0.416439\pi\)
0.259509 + 0.965741i \(0.416439\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0833 −0.740201
\(363\) 0 0
\(364\) −12.5049 −0.655436
\(365\) −11.2457 −0.588625
\(366\) 0 0
\(367\) 1.73835 0.0907411 0.0453705 0.998970i \(-0.485553\pi\)
0.0453705 + 0.998970i \(0.485553\pi\)
\(368\) 13.6751 0.712866
\(369\) 0 0
\(370\) 6.46148 0.335916
\(371\) −32.3520 −1.67963
\(372\) 0 0
\(373\) −6.58428 −0.340921 −0.170460 0.985365i \(-0.554526\pi\)
−0.170460 + 0.985365i \(0.554526\pi\)
\(374\) −10.7589 −0.556331
\(375\) 0 0
\(376\) 8.54530 0.440690
\(377\) −10.6813 −0.550114
\(378\) 0 0
\(379\) −3.74050 −0.192137 −0.0960684 0.995375i \(-0.530627\pi\)
−0.0960684 + 0.995375i \(0.530627\pi\)
\(380\) 0.743534 0.0381425
\(381\) 0 0
\(382\) −38.4743 −1.96852
\(383\) −27.5711 −1.40882 −0.704409 0.709795i \(-0.748788\pi\)
−0.704409 + 0.709795i \(0.748788\pi\)
\(384\) 0 0
\(385\) −2.81120 −0.143272
\(386\) −25.7897 −1.31266
\(387\) 0 0
\(388\) 3.71310 0.188504
\(389\) 17.1560 0.869844 0.434922 0.900468i \(-0.356776\pi\)
0.434922 + 0.900468i \(0.356776\pi\)
\(390\) 0 0
\(391\) −18.0022 −0.910413
\(392\) 1.87895 0.0949014
\(393\) 0 0
\(394\) −37.8800 −1.90837
\(395\) −4.44176 −0.223489
\(396\) 0 0
\(397\) −14.1506 −0.710201 −0.355100 0.934828i \(-0.615553\pi\)
−0.355100 + 0.934828i \(0.615553\pi\)
\(398\) 39.3427 1.97207
\(399\) 0 0
\(400\) −4.93423 −0.246711
\(401\) −10.3891 −0.518807 −0.259403 0.965769i \(-0.583526\pi\)
−0.259403 + 0.965769i \(0.583526\pi\)
\(402\) 0 0
\(403\) 18.8923 0.941093
\(404\) 12.4369 0.618758
\(405\) 0 0
\(406\) −8.31346 −0.412590
\(407\) 3.90101 0.193366
\(408\) 0 0
\(409\) 4.11163 0.203307 0.101653 0.994820i \(-0.467587\pi\)
0.101653 + 0.994820i \(0.467587\pi\)
\(410\) 7.67301 0.378943
\(411\) 0 0
\(412\) 3.33664 0.164385
\(413\) −14.9841 −0.737319
\(414\) 0 0
\(415\) 17.7115 0.869424
\(416\) −23.9934 −1.17637
\(417\) 0 0
\(418\) 1.65636 0.0810153
\(419\) −16.4494 −0.803604 −0.401802 0.915726i \(-0.631616\pi\)
−0.401802 + 0.915726i \(0.631616\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −23.9506 −1.16314
\(425\) 6.49551 0.315079
\(426\) 0 0
\(427\) 20.1910 0.977109
\(428\) −0.785704 −0.0379784
\(429\) 0 0
\(430\) −3.98750 −0.192294
\(431\) 24.2965 1.17032 0.585161 0.810917i \(-0.301031\pi\)
0.585161 + 0.810917i \(0.301031\pi\)
\(432\) 0 0
\(433\) 22.8348 1.09737 0.548686 0.836028i \(-0.315128\pi\)
0.548686 + 0.836028i \(0.315128\pi\)
\(434\) 14.7043 0.705827
\(435\) 0 0
\(436\) −2.58828 −0.123956
\(437\) 2.77149 0.132578
\(438\) 0 0
\(439\) −23.3798 −1.11586 −0.557929 0.829889i \(-0.688404\pi\)
−0.557929 + 0.829889i \(0.688404\pi\)
\(440\) −2.08116 −0.0992155
\(441\) 0 0
\(442\) 64.3661 3.06158
\(443\) −14.6539 −0.696228 −0.348114 0.937452i \(-0.613178\pi\)
−0.348114 + 0.937452i \(0.613178\pi\)
\(444\) 0 0
\(445\) 17.5897 0.833833
\(446\) 14.5803 0.690395
\(447\) 0 0
\(448\) 9.06766 0.428407
\(449\) −19.5934 −0.924672 −0.462336 0.886705i \(-0.652989\pi\)
−0.462336 + 0.886705i \(0.652989\pi\)
\(450\) 0 0
\(451\) 4.63245 0.218134
\(452\) 6.64782 0.312687
\(453\) 0 0
\(454\) −20.7305 −0.972930
\(455\) 16.8182 0.788451
\(456\) 0 0
\(457\) −32.4549 −1.51817 −0.759087 0.650989i \(-0.774354\pi\)
−0.759087 + 0.650989i \(0.774354\pi\)
\(458\) 3.83225 0.179069
\(459\) 0 0
\(460\) 2.06070 0.0960804
\(461\) −9.88505 −0.460393 −0.230196 0.973144i \(-0.573937\pi\)
−0.230196 + 0.973144i \(0.573937\pi\)
\(462\) 0 0
\(463\) 9.99709 0.464604 0.232302 0.972644i \(-0.425374\pi\)
0.232302 + 0.972644i \(0.425374\pi\)
\(464\) −8.80956 −0.408973
\(465\) 0 0
\(466\) −21.6737 −1.00401
\(467\) −26.3637 −1.21997 −0.609984 0.792414i \(-0.708824\pi\)
−0.609984 + 0.792414i \(0.708824\pi\)
\(468\) 0 0
\(469\) −37.2119 −1.71829
\(470\) 6.80106 0.313709
\(471\) 0 0
\(472\) −11.0929 −0.510592
\(473\) −2.40738 −0.110692
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 13.5771 0.622304
\(477\) 0 0
\(478\) 34.6978 1.58704
\(479\) −19.3573 −0.884459 −0.442230 0.896902i \(-0.645812\pi\)
−0.442230 + 0.896902i \(0.645812\pi\)
\(480\) 0 0
\(481\) −23.3381 −1.06413
\(482\) 23.6688 1.07809
\(483\) 0 0
\(484\) 0.743534 0.0337970
\(485\) −4.99386 −0.226759
\(486\) 0 0
\(487\) 15.4579 0.700462 0.350231 0.936663i \(-0.386103\pi\)
0.350231 + 0.936663i \(0.386103\pi\)
\(488\) 14.9476 0.676646
\(489\) 0 0
\(490\) 1.49543 0.0675565
\(491\) −33.7098 −1.52130 −0.760650 0.649162i \(-0.775120\pi\)
−0.760650 + 0.649162i \(0.775120\pi\)
\(492\) 0 0
\(493\) 11.5971 0.522306
\(494\) −9.90932 −0.445841
\(495\) 0 0
\(496\) 15.5817 0.699640
\(497\) 1.23089 0.0552127
\(498\) 0 0
\(499\) 32.9388 1.47454 0.737272 0.675596i \(-0.236114\pi\)
0.737272 + 0.675596i \(0.236114\pi\)
\(500\) −0.743534 −0.0332518
\(501\) 0 0
\(502\) 18.4347 0.822779
\(503\) −8.51652 −0.379733 −0.189866 0.981810i \(-0.560805\pi\)
−0.189866 + 0.981810i \(0.560805\pi\)
\(504\) 0 0
\(505\) −16.7267 −0.744329
\(506\) 4.59059 0.204076
\(507\) 0 0
\(508\) 1.59642 0.0708297
\(509\) −19.8390 −0.879350 −0.439675 0.898157i \(-0.644906\pi\)
−0.439675 + 0.898157i \(0.644906\pi\)
\(510\) 0 0
\(511\) 31.6138 1.39851
\(512\) 0.748955 0.0330994
\(513\) 0 0
\(514\) 12.0542 0.531689
\(515\) −4.48754 −0.197745
\(516\) 0 0
\(517\) 4.10602 0.180583
\(518\) −18.1645 −0.798102
\(519\) 0 0
\(520\) 12.4507 0.546000
\(521\) −42.2657 −1.85169 −0.925846 0.377900i \(-0.876646\pi\)
−0.925846 + 0.377900i \(0.876646\pi\)
\(522\) 0 0
\(523\) −28.4461 −1.24386 −0.621931 0.783072i \(-0.713652\pi\)
−0.621931 + 0.783072i \(0.713652\pi\)
\(524\) −9.77920 −0.427206
\(525\) 0 0
\(526\) −27.5827 −1.20266
\(527\) −20.5121 −0.893521
\(528\) 0 0
\(529\) −15.3189 −0.666037
\(530\) −19.0618 −0.827994
\(531\) 0 0
\(532\) −2.09022 −0.0906226
\(533\) −27.7140 −1.20043
\(534\) 0 0
\(535\) 1.05672 0.0456858
\(536\) −27.5484 −1.18991
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) 26.0505 1.11691
\(545\) 3.48106 0.149112
\(546\) 0 0
\(547\) 22.8994 0.979109 0.489555 0.871973i \(-0.337159\pi\)
0.489555 + 0.871973i \(0.337159\pi\)
\(548\) 0.235081 0.0100422
\(549\) 0 0
\(550\) −1.65636 −0.0706275
\(551\) −1.78540 −0.0760605
\(552\) 0 0
\(553\) 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 \(0.593077\pi\)
\(558\) 0 0
\(559\) 14.4024 0.609155
\(560\) 13.8711 0.586160
\(561\) 0 0
\(562\) 0.369091 0.0155692
\(563\) 38.8581 1.63767 0.818836 0.574028i \(-0.194620\pi\)
0.818836 + 0.574028i \(0.194620\pi\)
\(564\) 0 0
\(565\) −8.94084 −0.376144
\(566\) −20.1690 −0.847765
\(567\) 0 0
\(568\) 0.911238 0.0382347
\(569\) 27.9139 1.17021 0.585107 0.810956i \(-0.301053\pi\)
0.585107 + 0.810956i \(0.301053\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −21.5704 −0.900330
\(575\) −2.77149 −0.115579
\(576\) 0 0
\(577\) 12.3632 0.514685 0.257343 0.966320i \(-0.417153\pi\)
0.257343 + 0.966320i \(0.417153\pi\)
\(578\) −41.7266 −1.73560
\(579\) 0 0
\(580\) −1.32750 −0.0551216
\(581\) −49.7906 −2.06566
\(582\) 0 0
\(583\) −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 \(0.539764\pi\)
\(588\) 0 0
\(589\) 3.15789 0.130118
\(590\) −8.82864 −0.363469
\(591\) 0 0
\(592\) −19.2485 −0.791106
\(593\) 1.44341 0.0592739 0.0296369 0.999561i \(-0.490565\pi\)
0.0296369 + 0.999561i \(0.490565\pi\)
\(594\) 0 0
\(595\) −18.2602 −0.748594
\(596\) −3.57241 −0.146332
\(597\) 0 0
\(598\) −27.4636 −1.12307
\(599\) −20.4093 −0.833902 −0.416951 0.908929i \(-0.636901\pi\)
−0.416951 + 0.908929i \(0.636901\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −13.5620 −0.551828
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 7.79823 0.316520 0.158260 0.987397i \(-0.449412\pi\)
0.158260 + 0.987397i \(0.449412\pi\)
\(608\) −4.01054 −0.162649
\(609\) 0 0
\(610\) 11.8965 0.481677
\(611\) −24.5646 −0.993778
\(612\) 0 0
\(613\) 28.4672 1.14978 0.574890 0.818231i \(-0.305045\pi\)
0.574890 + 0.818231i \(0.305045\pi\)
\(614\) −32.6686 −1.31840
\(615\) 0 0
\(616\) 5.85056 0.235726
\(617\) −1.26620 −0.0509751 −0.0254876 0.999675i \(-0.508114\pi\)
−0.0254876 + 0.999675i \(0.508114\pi\)
\(618\) 0 0
\(619\) 2.57985 0.103693 0.0518464 0.998655i \(-0.483489\pi\)
0.0518464 + 0.998655i \(0.483489\pi\)
\(620\) 2.34800 0.0942978
\(621\) 0 0
\(622\) −43.4912 −1.74384
\(623\) −49.4482 −1.98110
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.59083 −0.183487
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 6.62681 0.263184
\(635\) −2.14707 −0.0852039
\(636\) 0 0
\(637\) −5.40130 −0.214007
\(638\) −2.95726 −0.117079
\(639\) 0 0
\(640\) 13.3638 0.528249
\(641\) −20.9430 −0.827199 −0.413600 0.910459i \(-0.635729\pi\)
−0.413600 + 0.910459i \(0.635729\pi\)
\(642\) 0 0
\(643\) 9.37012 0.369522 0.184761 0.982784i \(-0.440849\pi\)
0.184761 + 0.982784i \(0.440849\pi\)
\(644\) −5.79302 −0.228277
\(645\) 0 0
\(646\) 10.7589 0.423304
\(647\) 15.3143 0.602069 0.301034 0.953613i \(-0.402668\pi\)
0.301034 + 0.953613i \(0.402668\pi\)
\(648\) 0 0
\(649\) −5.33014 −0.209226
\(650\) 9.90932 0.388676
\(651\) 0 0
\(652\) −17.8408 −0.698698
\(653\) −7.61281 −0.297912 −0.148956 0.988844i \(-0.547591\pi\)
−0.148956 + 0.988844i \(0.547591\pi\)
\(654\) 0 0
\(655\) 13.1523 0.513904
\(656\) −22.8576 −0.892438
\(657\) 0 0
\(658\) −19.1191 −0.745341
\(659\) −27.8742 −1.08582 −0.542912 0.839789i \(-0.682678\pi\)
−0.542912 + 0.839789i \(0.682678\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −36.8606 −1.43047
\(665\) 2.81120 0.109014
\(666\) 0 0
\(667\) −4.94821 −0.191595
\(668\) −2.30913 −0.0893431
\(669\) 0 0
\(670\) −21.9253 −0.847048
\(671\) 7.18233 0.277271
\(672\) 0 0
\(673\) −41.4748 −1.59874 −0.799369 0.600841i \(-0.794832\pi\)
−0.799369 + 0.600841i \(0.794832\pi\)
\(674\) −54.2759 −2.09063
\(675\) 0 0
\(676\) 16.9461 0.651773
\(677\) 0.266167 0.0102296 0.00511482 0.999987i \(-0.498372\pi\)
0.00511482 + 0.999987i \(0.498372\pi\)
\(678\) 0 0
\(679\) 14.0387 0.538757
\(680\) −13.5182 −0.518400
\(681\) 0 0
\(682\) 5.23060 0.200290
\(683\) −29.3614 −1.12348 −0.561742 0.827312i \(-0.689869\pi\)
−0.561742 + 0.827312i \(0.689869\pi\)
\(684\) 0 0
\(685\) −0.316167 −0.0120801
\(686\) 28.3906 1.08396
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) −32.5468 −1.23546
\(695\) 4.07415 0.154541
\(696\) 0 0
\(697\) 30.0902 1.13975
\(698\) −33.0626 −1.25144
\(699\) 0 0
\(700\) 2.09022 0.0790029
\(701\) 13.6101 0.514046 0.257023 0.966405i \(-0.417258\pi\)
0.257023 + 0.966405i \(0.417258\pi\)
\(702\) 0 0
\(703\) −3.90101 −0.147129
\(704\) 3.22555 0.121568
\(705\) 0 0
\(706\) −38.4326 −1.44643
\(707\) 47.0221 1.76845
\(708\) 0 0
\(709\) −32.9904 −1.23898 −0.619491 0.785004i \(-0.712661\pi\)
−0.619491 + 0.785004i \(0.712661\pi\)
\(710\) 0.725239 0.0272177
\(711\) 0 0
\(712\) −36.6071 −1.37191
\(713\) 8.75204 0.327767
\(714\) 0 0
\(715\) 5.98258 0.223736
\(716\) −1.40266 −0.0524199
\(717\) 0 0
\(718\) −16.2886 −0.607886
\(719\) 3.85276 0.143684 0.0718418 0.997416i \(-0.477112\pi\)
0.0718418 + 0.997416i \(0.477112\pi\)
\(720\) 0 0
\(721\) 12.6154 0.469821
\(722\) −1.65636 −0.0616434
\(723\) 0 0
\(724\) 6.32193 0.234953
\(725\) 1.78540 0.0663080
\(726\) 0 0
\(727\) −35.5951 −1.32015 −0.660074 0.751200i \(-0.729475\pi\)
−0.660074 + 0.751200i \(0.729475\pi\)
\(728\) −35.0015 −1.29724
\(729\) 0 0
\(730\) 18.6269 0.689412
\(731\) −15.6372 −0.578362
\(732\) 0 0
\(733\) −22.8053 −0.842332 −0.421166 0.906984i \(-0.638379\pi\)
−0.421166 + 0.906984i \(0.638379\pi\)
\(734\) −2.87933 −0.106278
\(735\) 0 0
\(736\) −11.1152 −0.409710
\(737\) −13.2370 −0.487592
\(738\) 0 0
\(739\) −37.2882 −1.37167 −0.685835 0.727757i \(-0.740563\pi\)
−0.685835 + 0.727757i \(0.740563\pi\)
\(740\) −2.90053 −0.106626
\(741\) 0 0
\(742\) 53.5866 1.96723
\(743\) −24.9220 −0.914301 −0.457150 0.889389i \(-0.651130\pi\)
−0.457150 + 0.889389i \(0.651130\pi\)
\(744\) 0 0
\(745\) 4.80464 0.176028
\(746\) 10.9059 0.399295
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 17.6921 0.644308
\(755\) 18.2399 0.663816
\(756\) 0 0
\(757\) 4.66986 0.169729 0.0848645 0.996393i \(-0.472954\pi\)
0.0848645 + 0.996393i \(0.472954\pi\)
\(758\) 6.19562 0.225035
\(759\) 0 0
\(760\) 2.08116 0.0754917
\(761\) −18.4084 −0.667303 −0.333652 0.942696i \(-0.608281\pi\)
−0.333652 + 0.942696i \(0.608281\pi\)
\(762\) 0 0
\(763\) −9.78595 −0.354275
\(764\) 17.2709 0.624841
\(765\) 0 0
\(766\) 45.6677 1.65004
\(767\) 31.8880 1.15141
\(768\) 0 0
\(769\) −3.57354 −0.128865 −0.0644326 0.997922i \(-0.520524\pi\)
−0.0644326 + 0.997922i \(0.520524\pi\)
\(770\) 4.65636 0.167804
\(771\) 0 0
\(772\) 11.5769 0.416662
\(773\) −34.6561 −1.24649 −0.623246 0.782026i \(-0.714186\pi\)
−0.623246 + 0.782026i \(0.714186\pi\)
\(774\) 0 0
\(775\) −3.15789 −0.113435
\(776\) 10.3930 0.373088
\(777\) 0 0
\(778\) −28.4166 −1.01878
\(779\) −4.63245 −0.165975
\(780\) 0 0
\(781\) 0.437851 0.0156675
\(782\) 29.8182 1.06630
\(783\) 0 0
\(784\) −4.45481 −0.159100
\(785\) 18.1161 0.646591
\(786\) 0 0
\(787\) 30.3693 1.08255 0.541274 0.840846i \(-0.317942\pi\)
0.541274 + 0.840846i \(0.317942\pi\)
\(788\) 17.0042 0.605749
\(789\) 0 0
\(790\) 7.35716 0.261756
\(791\) 25.1345 0.893679
\(792\) 0 0
\(793\) −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 \(0.291190\pi\)
\(798\) 0 0
\(799\) 26.6707 0.943542
\(800\) 4.01054 0.141794
\(801\) 0 0
\(802\) 17.2081 0.607639
\(803\) 11.2457 0.396851
\(804\) 0 0
\(805\) 7.79120 0.274604
\(806\) −31.2925 −1.10223
\(807\) 0 0
\(808\) 34.8110 1.22465
\(809\) 8.23292 0.289454 0.144727 0.989472i \(-0.453770\pi\)
0.144727 + 0.989472i \(0.453770\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −6.46148 −0.226475
\(815\) 23.9946 0.840492
\(816\) 0 0
\(817\) 2.40738 0.0842237
\(818\) −6.81034 −0.238118
\(819\) 0 0
\(820\) −3.44438 −0.120283
\(821\) −31.2285 −1.08988 −0.544941 0.838474i \(-0.683448\pi\)
−0.544941 + 0.838474i \(0.683448\pi\)
\(822\) 0 0
\(823\) −8.20565 −0.286031 −0.143015 0.989720i \(-0.545680\pi\)
−0.143015 + 0.989720i \(0.545680\pi\)
\(824\) 9.33931 0.325350
\(825\) 0 0
\(826\) 24.8191 0.863566
\(827\) 15.5275 0.539943 0.269972 0.962868i \(-0.412986\pi\)
0.269972 + 0.962868i \(0.412986\pi\)
\(828\) 0 0
\(829\) 3.31730 0.115215 0.0576074 0.998339i \(-0.481653\pi\)
0.0576074 + 0.998339i \(0.481653\pi\)
\(830\) −29.3367 −1.01829
\(831\) 0 0
\(832\) −19.2971 −0.669007
\(833\) 5.86440 0.203189
\(834\) 0 0
\(835\) 3.10562 0.107474
\(836\) −0.743534 −0.0257157
\(837\) 0 0
\(838\) 27.2461 0.941201
\(839\) −7.02051 −0.242375 −0.121188 0.992630i \(-0.538670\pi\)
−0.121188 + 0.992630i \(0.538670\pi\)
\(840\) 0 0
\(841\) −25.8124 −0.890081
\(842\) 63.9665 2.20443
\(843\) 0 0
\(844\) 5.56705 0.191626
\(845\) −22.7913 −0.784044
\(846\) 0 0
\(847\) 2.81120 0.0965939
\(848\) 56.7844 1.94998
\(849\) 0 0
\(850\) −10.7589 −0.369028
\(851\) −10.8116 −0.370617
\(852\) 0 0
\(853\) 3.62821 0.124227 0.0621137 0.998069i \(-0.480216\pi\)
0.0621137 + 0.998069i \(0.480216\pi\)
\(854\) −33.4435 −1.14441
\(855\) 0 0
\(856\) −2.19920 −0.0751670
\(857\) −16.1864 −0.552918 −0.276459 0.961026i \(-0.589161\pi\)
−0.276459 + 0.961026i \(0.589161\pi\)
\(858\) 0 0
\(859\) 25.5004 0.870063 0.435032 0.900415i \(-0.356737\pi\)
0.435032 + 0.900415i \(0.356737\pi\)
\(860\) 1.78997 0.0610375
\(861\) 0 0
\(862\) −40.2438 −1.37071
\(863\) 39.9012 1.35825 0.679126 0.734022i \(-0.262359\pi\)
0.679126 + 0.734022i \(0.262359\pi\)
\(864\) 0 0
\(865\) 10.9411 0.372010
\(866\) −37.8228 −1.28527
\(867\) 0 0
\(868\) −6.60068 −0.224042
\(869\) 4.44176 0.150676
\(870\) 0 0
\(871\) 79.1916 2.68330
\(872\) −7.24465 −0.245335
\(873\) 0 0
\(874\) −4.59059 −0.155279
\(875\) −2.81120 −0.0950359
\(876\) 0 0
\(877\) 11.1804 0.377537 0.188768 0.982022i \(-0.439550\pi\)
0.188768 + 0.982022i \(0.439550\pi\)
\(878\) 38.7254 1.30692
\(879\) 0 0
\(880\) 4.93423 0.166333
\(881\) −2.82033 −0.0950195 −0.0475097 0.998871i \(-0.515129\pi\)
−0.0475097 + 0.998871i \(0.515129\pi\)
\(882\) 0 0
\(883\) −17.0119 −0.572495 −0.286247 0.958156i \(-0.592408\pi\)
−0.286247 + 0.958156i \(0.592408\pi\)
\(884\) −28.8937 −0.971800
\(885\) 0 0
\(886\) 24.2722 0.815439
\(887\) −10.8354 −0.363817 −0.181908 0.983316i \(-0.558227\pi\)
−0.181908 + 0.983316i \(0.558227\pi\)
\(888\) 0 0
\(889\) 6.03584 0.202436
\(890\) −29.1349 −0.976605
\(891\) 0 0
\(892\) −6.54501 −0.219143
\(893\) −4.10602 −0.137403
\(894\) 0 0
\(895\) 1.88648 0.0630581
\(896\) −37.5682 −1.25506
\(897\) 0 0
\(898\) 32.4538 1.08300
\(899\) −5.63808 −0.188041
\(900\) 0 0
\(901\) −74.7521 −2.49035
\(902\) −7.67301 −0.255483
\(903\) 0 0
\(904\) 18.6073 0.618871
\(905\) −8.50254 −0.282634
\(906\) 0 0
\(907\) 38.8963 1.29153 0.645766 0.763536i \(-0.276538\pi\)
0.645766 + 0.763536i \(0.276538\pi\)
\(908\) 9.30583 0.308825
\(909\) 0 0
\(910\) −27.8571 −0.923453
\(911\) −11.2431 −0.372502 −0.186251 0.982502i \(-0.559634\pi\)
−0.186251 + 0.982502i \(0.559634\pi\)
\(912\) 0 0
\(913\) −17.7115 −0.586166
\(914\) 53.7570 1.77812
\(915\) 0 0
\(916\) −1.72028 −0.0568398
\(917\) −36.9738 −1.22098
\(918\) 0 0
\(919\) −9.48069 −0.312739 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(920\) 5.76792 0.190163
\(921\) 0 0
\(922\) 16.3732 0.539223
\(923\) −2.61948 −0.0862211
\(924\) 0 0
\(925\) 3.90101 0.128264
\(926\) −16.5588 −0.544156
\(927\) 0 0
\(928\) 7.16040 0.235052
\(929\) 39.8543 1.30758 0.653789 0.756677i \(-0.273178\pi\)
0.653789 + 0.756677i \(0.273178\pi\)
\(930\) 0 0
\(931\) −0.902838 −0.0295893
\(932\) 9.72922 0.318691
\(933\) 0 0
\(934\) 43.6679 1.42886
\(935\) −6.49551 −0.212426
\(936\) 0 0
\(937\) −56.3053 −1.83942 −0.919708 0.392604i \(-0.871574\pi\)
−0.919708 + 0.392604i \(0.871574\pi\)
\(938\) 61.6363 2.01250
\(939\) 0 0
\(940\) −3.05297 −0.0995768
\(941\) 21.4276 0.698519 0.349259 0.937026i \(-0.386433\pi\)
0.349259 + 0.937026i \(0.386433\pi\)
\(942\) 0 0
\(943\) −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 \(0.584821\pi\)
\(948\) 0 0
\(949\) −67.2781 −2.18394
\(950\) 1.65636 0.0537395
\(951\) 0 0
\(952\) 38.0024 1.23166
\(953\) −58.8844 −1.90745 −0.953726 0.300676i \(-0.902788\pi\)
−0.953726 + 0.300676i \(0.902788\pi\)
\(954\) 0 0
\(955\) −23.2282 −0.751646
\(956\) −15.5757 −0.503755
\(957\) 0 0
\(958\) 32.0628 1.03590
\(959\) 0.888809 0.0287011
\(960\) 0 0
\(961\) −21.0278 −0.678315
\(962\) 38.6563 1.24633
\(963\) 0 0
\(964\) −10.6248 −0.342203
\(965\) −15.5701 −0.501220
\(966\) 0 0
\(967\) −24.4443 −0.786074 −0.393037 0.919523i \(-0.628576\pi\)
−0.393037 + 0.919523i \(0.628576\pi\)
\(968\) 2.08116 0.0668911
\(969\) 0 0
\(970\) 8.27163 0.265586
\(971\) −59.3586 −1.90491 −0.952454 0.304683i \(-0.901449\pi\)
−0.952454 + 0.304683i \(0.901449\pi\)
\(972\) 0 0
\(973\) −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 \(0.472825\pi\)
\(978\) 0 0
\(979\) −17.5897 −0.562170
\(980\) −0.671291 −0.0214436
\(981\) 0 0
\(982\) 55.8356 1.78178
\(983\) 31.3817 1.00092 0.500460 0.865760i \(-0.333164\pi\)
0.500460 + 0.865760i \(0.333164\pi\)
\(984\) 0 0
\(985\) −22.8694 −0.728680
\(986\) −19.2090 −0.611738
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −2.03879 −0.0646665
\(995\) 23.7525 0.753005
\(996\) 0 0
\(997\) −16.2641 −0.515091 −0.257545 0.966266i \(-0.582914\pi\)
−0.257545 + 0.966266i \(0.582914\pi\)
\(998\) −54.5586 −1.72702
\(999\) 0 0
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 9405.2.a.w.1.2 6
3.2 odd 2 1045.2.a.g.1.5 6
15.14 odd 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 3.2 odd 2
5225.2.a.k.1.2 6 15.14 odd 2
9405.2.a.w.1.2 6 1.1 even 1 trivial