Properties

Label 9025.2.a.bt.1.4
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4227136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 7x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.11917\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.471884 q^{2} -1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} +1.17540 q^{7} -1.78246 q^{8} -1.91794 q^{9} +O(q^{10})\) \(q+0.471884 q^{2} -1.04022 q^{3} -1.77733 q^{4} -0.490864 q^{6} +1.17540 q^{7} -1.78246 q^{8} -1.91794 q^{9} +0.713538 q^{11} +1.84881 q^{12} -4.10315 q^{13} +0.554651 q^{14} +2.71354 q^{16} +2.55233 q^{17} -0.905045 q^{18} -1.22267 q^{21} +0.336707 q^{22} -0.607061 q^{23} +1.85415 q^{24} -1.93621 q^{26} +5.11575 q^{27} -2.08907 q^{28} +0.859386 q^{29} -2.50914 q^{31} +4.84539 q^{32} -0.742237 q^{33} +1.20440 q^{34} +3.40880 q^{36} -9.38171 q^{37} +4.26819 q^{39} +4.12234 q^{41} -0.576960 q^{42} +10.1239 q^{43} -1.26819 q^{44} -0.286462 q^{46} +10.5184 q^{47} -2.82268 q^{48} -5.61844 q^{49} -2.65498 q^{51} +7.29264 q^{52} -4.98057 q^{53} +2.41404 q^{54} -2.09510 q^{56} +0.405530 q^{58} +6.24992 q^{59} -4.55465 q^{61} -1.18402 q^{62} -2.25434 q^{63} -3.14061 q^{64} -0.350250 q^{66} +5.18210 q^{67} -4.53632 q^{68} +0.631477 q^{69} +13.1679 q^{71} +3.41865 q^{72} -12.3783 q^{73} -4.42708 q^{74} +0.838691 q^{77} +2.01409 q^{78} +5.96345 q^{79} +0.432310 q^{81} +1.94527 q^{82} +13.8603 q^{83} +2.17309 q^{84} +4.77733 q^{86} -0.893952 q^{87} -1.27185 q^{88} -15.9635 q^{89} -4.82284 q^{91} +1.07894 q^{92} +2.61006 q^{93} +4.96345 q^{94} -5.04028 q^{96} -16.7020 q^{97} -2.65125 q^{98} -1.36852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} - 22 q^{14} + 14 q^{16} - 20 q^{21} + 2 q^{24} - 22 q^{26} + 12 q^{29} - 30 q^{31} - 10 q^{34} - 14 q^{36} - 2 q^{39} - 12 q^{41} + 20 q^{44} - 4 q^{46} + 2 q^{49} - 40 q^{51} - 4 q^{54} - 46 q^{56} - 20 q^{59} - 2 q^{61} - 12 q^{64} + 6 q^{66} - 18 q^{69} + 2 q^{71} - 22 q^{74} - 24 q^{79} + 14 q^{81} - 48 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} - 30 q^{94} + 26 q^{96} - 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.471884 0.333672 0.166836 0.985985i \(-0.446645\pi\)
0.166836 + 0.985985i \(0.446645\pi\)
\(3\) −1.04022 −0.600572 −0.300286 0.953849i \(-0.597082\pi\)
−0.300286 + 0.953849i \(0.597082\pi\)
\(4\) −1.77733 −0.888663
\(5\) 0 0
\(6\) −0.490864 −0.200394
\(7\) 1.17540 0.444259 0.222129 0.975017i \(-0.428699\pi\)
0.222129 + 0.975017i \(0.428699\pi\)
\(8\) −1.78246 −0.630194
\(9\) −1.91794 −0.639313
\(10\) 0 0
\(11\) 0.713538 0.215140 0.107570 0.994198i \(-0.465693\pi\)
0.107570 + 0.994198i \(0.465693\pi\)
\(12\) 1.84881 0.533706
\(13\) −4.10315 −1.13801 −0.569005 0.822334i \(-0.692672\pi\)
−0.569005 + 0.822334i \(0.692672\pi\)
\(14\) 0.554651 0.148237
\(15\) 0 0
\(16\) 2.71354 0.678384
\(17\) 2.55233 0.619030 0.309515 0.950895i \(-0.399833\pi\)
0.309515 + 0.950895i \(0.399833\pi\)
\(18\) −0.905045 −0.213321
\(19\) 0 0
\(20\) 0 0
\(21\) −1.22267 −0.266809
\(22\) 0.336707 0.0717862
\(23\) −0.607061 −0.126581 −0.0632904 0.997995i \(-0.520159\pi\)
−0.0632904 + 0.997995i \(0.520159\pi\)
\(24\) 1.85415 0.378477
\(25\) 0 0
\(26\) −1.93621 −0.379722
\(27\) 5.11575 0.984526
\(28\) −2.08907 −0.394796
\(29\) 0.859386 0.159584 0.0797920 0.996812i \(-0.474574\pi\)
0.0797920 + 0.996812i \(0.474574\pi\)
\(30\) 0 0
\(31\) −2.50914 −0.450654 −0.225327 0.974283i \(-0.572345\pi\)
−0.225327 + 0.974283i \(0.572345\pi\)
\(32\) 4.84539 0.856552
\(33\) −0.742237 −0.129207
\(34\) 1.20440 0.206553
\(35\) 0 0
\(36\) 3.40880 0.568134
\(37\) −9.38171 −1.54234 −0.771172 0.636627i \(-0.780329\pi\)
−0.771172 + 0.636627i \(0.780329\pi\)
\(38\) 0 0
\(39\) 4.26819 0.683457
\(40\) 0 0
\(41\) 4.12234 0.643802 0.321901 0.946773i \(-0.395678\pi\)
0.321901 + 0.946773i \(0.395678\pi\)
\(42\) −0.576960 −0.0890269
\(43\) 10.1239 1.54389 0.771944 0.635691i \(-0.219285\pi\)
0.771944 + 0.635691i \(0.219285\pi\)
\(44\) −1.26819 −0.191187
\(45\) 0 0
\(46\) −0.286462 −0.0422365
\(47\) 10.5184 1.53426 0.767132 0.641489i \(-0.221683\pi\)
0.767132 + 0.641489i \(0.221683\pi\)
\(48\) −2.82268 −0.407419
\(49\) −5.61844 −0.802634
\(50\) 0 0
\(51\) −2.65498 −0.371772
\(52\) 7.29264 1.01131
\(53\) −4.98057 −0.684134 −0.342067 0.939676i \(-0.611127\pi\)
−0.342067 + 0.939676i \(0.611127\pi\)
\(54\) 2.41404 0.328509
\(55\) 0 0
\(56\) −2.09510 −0.279969
\(57\) 0 0
\(58\) 0.405530 0.0532488
\(59\) 6.24992 0.813670 0.406835 0.913502i \(-0.366632\pi\)
0.406835 + 0.913502i \(0.366632\pi\)
\(60\) 0 0
\(61\) −4.55465 −0.583163 −0.291582 0.956546i \(-0.594182\pi\)
−0.291582 + 0.956546i \(0.594182\pi\)
\(62\) −1.18402 −0.150371
\(63\) −2.25434 −0.284020
\(64\) −3.14061 −0.392577
\(65\) 0 0
\(66\) −0.350250 −0.0431128
\(67\) 5.18210 0.633094 0.316547 0.948577i \(-0.397476\pi\)
0.316547 + 0.948577i \(0.397476\pi\)
\(68\) −4.53632 −0.550109
\(69\) 0.631477 0.0760209
\(70\) 0 0
\(71\) 13.1679 1.56274 0.781368 0.624070i \(-0.214522\pi\)
0.781368 + 0.624070i \(0.214522\pi\)
\(72\) 3.41865 0.402892
\(73\) −12.3783 −1.44877 −0.724384 0.689396i \(-0.757876\pi\)
−0.724384 + 0.689396i \(0.757876\pi\)
\(74\) −4.42708 −0.514637
\(75\) 0 0
\(76\) 0 0
\(77\) 0.838691 0.0955777
\(78\) 2.01409 0.228051
\(79\) 5.96345 0.670941 0.335471 0.942051i \(-0.391105\pi\)
0.335471 + 0.942051i \(0.391105\pi\)
\(80\) 0 0
\(81\) 0.432310 0.0480345
\(82\) 1.94527 0.214819
\(83\) 13.8603 1.52136 0.760682 0.649124i \(-0.224864\pi\)
0.760682 + 0.649124i \(0.224864\pi\)
\(84\) 2.17309 0.237104
\(85\) 0 0
\(86\) 4.77733 0.515152
\(87\) −0.893952 −0.0958417
\(88\) −1.27185 −0.135580
\(89\) −15.9635 −1.69212 −0.846061 0.533085i \(-0.821032\pi\)
−0.846061 + 0.533085i \(0.821032\pi\)
\(90\) 0 0
\(91\) −4.82284 −0.505571
\(92\) 1.07894 0.112488
\(93\) 2.61006 0.270650
\(94\) 4.96345 0.511941
\(95\) 0 0
\(96\) −5.04028 −0.514421
\(97\) −16.7020 −1.69583 −0.847915 0.530133i \(-0.822142\pi\)
−0.847915 + 0.530133i \(0.822142\pi\)
\(98\) −2.65125 −0.267817
\(99\) −1.36852 −0.137542
\(100\) 0 0
\(101\) 14.0272 1.39576 0.697881 0.716213i \(-0.254126\pi\)
0.697881 + 0.716213i \(0.254126\pi\)
\(102\) −1.25284 −0.124050
\(103\) −3.55382 −0.350169 −0.175084 0.984553i \(-0.556020\pi\)
−0.175084 + 0.984553i \(0.556020\pi\)
\(104\) 7.31370 0.717168
\(105\) 0 0
\(106\) −2.35025 −0.228276
\(107\) −3.63127 −0.351048 −0.175524 0.984475i \(-0.556162\pi\)
−0.175524 + 0.984475i \(0.556162\pi\)
\(108\) −9.09235 −0.874911
\(109\) 6.22267 0.596024 0.298012 0.954562i \(-0.403676\pi\)
0.298012 + 0.954562i \(0.403676\pi\)
\(110\) 0 0
\(111\) 9.75905 0.926288
\(112\) 3.18949 0.301378
\(113\) 12.2707 1.15433 0.577167 0.816626i \(-0.304158\pi\)
0.577167 + 0.816626i \(0.304158\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.52741 −0.141816
\(117\) 7.86960 0.727545
\(118\) 2.94923 0.271499
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.4909 −0.953715
\(122\) −2.14927 −0.194585
\(123\) −4.28815 −0.386649
\(124\) 4.45955 0.400480
\(125\) 0 0
\(126\) −1.06379 −0.0947697
\(127\) −2.15789 −0.191482 −0.0957408 0.995406i \(-0.530522\pi\)
−0.0957408 + 0.995406i \(0.530522\pi\)
\(128\) −11.1728 −0.987544
\(129\) −10.5311 −0.927216
\(130\) 0 0
\(131\) −17.5494 −1.53330 −0.766650 0.642065i \(-0.778078\pi\)
−0.766650 + 0.642065i \(0.778078\pi\)
\(132\) 1.31920 0.114821
\(133\) 0 0
\(134\) 2.44535 0.211246
\(135\) 0 0
\(136\) −4.54942 −0.390109
\(137\) 5.07455 0.433548 0.216774 0.976222i \(-0.430446\pi\)
0.216774 + 0.976222i \(0.430446\pi\)
\(138\) 0.297984 0.0253661
\(139\) 5.98173 0.507363 0.253682 0.967288i \(-0.418358\pi\)
0.253682 + 0.967288i \(0.418358\pi\)
\(140\) 0 0
\(141\) −10.9414 −0.921436
\(142\) 6.21370 0.521442
\(143\) −2.92776 −0.244831
\(144\) −5.20440 −0.433700
\(145\) 0 0
\(146\) −5.84111 −0.483414
\(147\) 5.84442 0.482040
\(148\) 16.6744 1.37062
\(149\) 9.60017 0.786476 0.393238 0.919437i \(-0.371355\pi\)
0.393238 + 0.919437i \(0.371355\pi\)
\(150\) 0 0
\(151\) −7.53638 −0.613302 −0.306651 0.951822i \(-0.599208\pi\)
−0.306651 + 0.951822i \(0.599208\pi\)
\(152\) 0 0
\(153\) −4.89521 −0.395754
\(154\) 0.395765 0.0318916
\(155\) 0 0
\(156\) −7.58596 −0.607363
\(157\) −9.85606 −0.786599 −0.393300 0.919410i \(-0.628667\pi\)
−0.393300 + 0.919410i \(0.628667\pi\)
\(158\) 2.81406 0.223874
\(159\) 5.18089 0.410872
\(160\) 0 0
\(161\) −0.713538 −0.0562347
\(162\) 0.204000 0.0160278
\(163\) −0.0688234 −0.00539067 −0.00269533 0.999996i \(-0.500858\pi\)
−0.00269533 + 0.999996i \(0.500858\pi\)
\(164\) −7.32674 −0.572122
\(165\) 0 0
\(166\) 6.54045 0.507637
\(167\) 16.6442 1.28797 0.643985 0.765038i \(-0.277280\pi\)
0.643985 + 0.765038i \(0.277280\pi\)
\(168\) 2.17937 0.168142
\(169\) 3.83588 0.295068
\(170\) 0 0
\(171\) 0 0
\(172\) −17.9935 −1.37200
\(173\) −12.1356 −0.922650 −0.461325 0.887231i \(-0.652626\pi\)
−0.461325 + 0.887231i \(0.652626\pi\)
\(174\) −0.421841 −0.0319797
\(175\) 0 0
\(176\) 1.93621 0.145947
\(177\) −6.50130 −0.488667
\(178\) −7.53290 −0.564614
\(179\) −11.7538 −0.878522 −0.439261 0.898360i \(-0.644760\pi\)
−0.439261 + 0.898360i \(0.644760\pi\)
\(180\) 0 0
\(181\) 8.83588 0.656766 0.328383 0.944545i \(-0.393496\pi\)
0.328383 + 0.944545i \(0.393496\pi\)
\(182\) −2.27582 −0.168695
\(183\) 4.73785 0.350232
\(184\) 1.08206 0.0797706
\(185\) 0 0
\(186\) 1.23164 0.0903085
\(187\) 1.82118 0.133178
\(188\) −18.6946 −1.36344
\(189\) 6.01304 0.437384
\(190\) 0 0
\(191\) 14.2447 1.03071 0.515355 0.856977i \(-0.327660\pi\)
0.515355 + 0.856977i \(0.327660\pi\)
\(192\) 3.26693 0.235771
\(193\) 19.5057 1.40405 0.702024 0.712153i \(-0.252280\pi\)
0.702024 + 0.712153i \(0.252280\pi\)
\(194\) −7.88139 −0.565851
\(195\) 0 0
\(196\) 9.98580 0.713271
\(197\) 4.33232 0.308665 0.154332 0.988019i \(-0.450677\pi\)
0.154332 + 0.988019i \(0.450677\pi\)
\(198\) −0.645784 −0.0458938
\(199\) −12.6315 −0.895422 −0.447711 0.894178i \(-0.647761\pi\)
−0.447711 + 0.894178i \(0.647761\pi\)
\(200\) 0 0
\(201\) −5.39053 −0.380219
\(202\) 6.61923 0.465727
\(203\) 1.01012 0.0708966
\(204\) 4.71877 0.330380
\(205\) 0 0
\(206\) −1.67699 −0.116842
\(207\) 1.16431 0.0809248
\(208\) −11.1341 −0.772009
\(209\) 0 0
\(210\) 0 0
\(211\) −22.2447 −1.53139 −0.765694 0.643206i \(-0.777604\pi\)
−0.765694 + 0.643206i \(0.777604\pi\)
\(212\) 8.85209 0.607964
\(213\) −13.6975 −0.938536
\(214\) −1.71354 −0.117135
\(215\) 0 0
\(216\) −9.11861 −0.620443
\(217\) −2.94923 −0.200207
\(218\) 2.93638 0.198877
\(219\) 12.8762 0.870090
\(220\) 0 0
\(221\) −10.4726 −0.704463
\(222\) 4.60514 0.309077
\(223\) 13.6225 0.912231 0.456115 0.889921i \(-0.349240\pi\)
0.456115 + 0.889921i \(0.349240\pi\)
\(224\) 5.69527 0.380531
\(225\) 0 0
\(226\) 5.79036 0.385169
\(227\) 6.58913 0.437336 0.218668 0.975799i \(-0.429829\pi\)
0.218668 + 0.975799i \(0.429829\pi\)
\(228\) 0 0
\(229\) −0.585962 −0.0387215 −0.0193607 0.999813i \(-0.506163\pi\)
−0.0193607 + 0.999813i \(0.506163\pi\)
\(230\) 0 0
\(231\) −0.872425 −0.0574013
\(232\) −1.53182 −0.100569
\(233\) −7.15747 −0.468901 −0.234451 0.972128i \(-0.575329\pi\)
−0.234451 + 0.972128i \(0.575329\pi\)
\(234\) 3.71354 0.242762
\(235\) 0 0
\(236\) −11.1081 −0.723078
\(237\) −6.20331 −0.402948
\(238\) 1.41565 0.0917631
\(239\) 3.65498 0.236421 0.118211 0.992989i \(-0.462284\pi\)
0.118211 + 0.992989i \(0.462284\pi\)
\(240\) 0 0
\(241\) −17.2630 −1.11200 −0.556002 0.831181i \(-0.687665\pi\)
−0.556002 + 0.831181i \(0.687665\pi\)
\(242\) −4.95047 −0.318228
\(243\) −15.7969 −1.01337
\(244\) 8.09510 0.518236
\(245\) 0 0
\(246\) −2.02351 −0.129014
\(247\) 0 0
\(248\) 4.47243 0.284000
\(249\) −14.4178 −0.913689
\(250\) 0 0
\(251\) −19.4140 −1.22540 −0.612702 0.790314i \(-0.709917\pi\)
−0.612702 + 0.790314i \(0.709917\pi\)
\(252\) 4.00670 0.252398
\(253\) −0.433161 −0.0272326
\(254\) −1.01827 −0.0638921
\(255\) 0 0
\(256\) 1.00897 0.0630606
\(257\) −13.8929 −0.866613 −0.433306 0.901247i \(-0.642653\pi\)
−0.433306 + 0.901247i \(0.642653\pi\)
\(258\) −4.96948 −0.309386
\(259\) −11.0272 −0.685199
\(260\) 0 0
\(261\) −1.64825 −0.102024
\(262\) −8.28129 −0.511620
\(263\) 14.9883 0.924221 0.462110 0.886822i \(-0.347092\pi\)
0.462110 + 0.886822i \(0.347092\pi\)
\(264\) 1.32301 0.0814255
\(265\) 0 0
\(266\) 0 0
\(267\) 16.6055 1.01624
\(268\) −9.21028 −0.562607
\(269\) −19.3723 −1.18115 −0.590574 0.806984i \(-0.701098\pi\)
−0.590574 + 0.806984i \(0.701098\pi\)
\(270\) 0 0
\(271\) 11.9179 0.723963 0.361982 0.932185i \(-0.382100\pi\)
0.361982 + 0.932185i \(0.382100\pi\)
\(272\) 6.92583 0.419940
\(273\) 5.01682 0.303632
\(274\) 2.39460 0.144663
\(275\) 0 0
\(276\) −1.12234 −0.0675570
\(277\) −10.9824 −0.659866 −0.329933 0.944004i \(-0.607026\pi\)
−0.329933 + 0.944004i \(0.607026\pi\)
\(278\) 2.82268 0.169293
\(279\) 4.81237 0.288109
\(280\) 0 0
\(281\) −14.6498 −0.873931 −0.436965 0.899478i \(-0.643947\pi\)
−0.436965 + 0.899478i \(0.643947\pi\)
\(282\) −5.16309 −0.307458
\(283\) 18.9287 1.12519 0.562597 0.826731i \(-0.309802\pi\)
0.562597 + 0.826731i \(0.309802\pi\)
\(284\) −23.4036 −1.38875
\(285\) 0 0
\(286\) −1.38156 −0.0816934
\(287\) 4.84539 0.286014
\(288\) −9.29317 −0.547605
\(289\) −10.4856 −0.616802
\(290\) 0 0
\(291\) 17.3738 1.01847
\(292\) 22.0002 1.28747
\(293\) 5.59625 0.326937 0.163468 0.986549i \(-0.447732\pi\)
0.163468 + 0.986549i \(0.447732\pi\)
\(294\) 2.75789 0.160843
\(295\) 0 0
\(296\) 16.7225 0.971976
\(297\) 3.65028 0.211811
\(298\) 4.53016 0.262425
\(299\) 2.49086 0.144050
\(300\) 0 0
\(301\) 11.8997 0.685885
\(302\) −3.55629 −0.204642
\(303\) −14.5914 −0.838256
\(304\) 0 0
\(305\) 0 0
\(306\) −2.30997 −0.132052
\(307\) −20.6319 −1.17753 −0.588764 0.808305i \(-0.700385\pi\)
−0.588764 + 0.808305i \(0.700385\pi\)
\(308\) −1.49063 −0.0849364
\(309\) 3.69676 0.210302
\(310\) 0 0
\(311\) −24.6587 −1.39827 −0.699134 0.714991i \(-0.746431\pi\)
−0.699134 + 0.714991i \(0.746431\pi\)
\(312\) −7.60787 −0.430711
\(313\) 3.49856 0.197751 0.0988753 0.995100i \(-0.468476\pi\)
0.0988753 + 0.995100i \(0.468476\pi\)
\(314\) −4.65092 −0.262466
\(315\) 0 0
\(316\) −10.5990 −0.596240
\(317\) −31.3709 −1.76196 −0.880982 0.473150i \(-0.843117\pi\)
−0.880982 + 0.473150i \(0.843117\pi\)
\(318\) 2.44478 0.137096
\(319\) 0.613205 0.0343329
\(320\) 0 0
\(321\) 3.77733 0.210830
\(322\) −0.336707 −0.0187639
\(323\) 0 0
\(324\) −0.768356 −0.0426865
\(325\) 0 0
\(326\) −0.0324767 −0.00179872
\(327\) −6.47296 −0.357955
\(328\) −7.34790 −0.405720
\(329\) 12.3633 0.681610
\(330\) 0 0
\(331\) −35.7497 −1.96498 −0.982492 0.186305i \(-0.940349\pi\)
−0.982492 + 0.186305i \(0.940349\pi\)
\(332\) −24.6343 −1.35198
\(333\) 17.9935 0.986040
\(334\) 7.85415 0.429760
\(335\) 0 0
\(336\) −3.31777 −0.180999
\(337\) −26.4892 −1.44296 −0.721480 0.692435i \(-0.756538\pi\)
−0.721480 + 0.692435i \(0.756538\pi\)
\(338\) 1.81009 0.0984559
\(339\) −12.7643 −0.693261
\(340\) 0 0
\(341\) −1.79036 −0.0969536
\(342\) 0 0
\(343\) −14.8317 −0.800836
\(344\) −18.0455 −0.972949
\(345\) 0 0
\(346\) −5.72658 −0.307863
\(347\) 34.3398 1.84346 0.921729 0.387835i \(-0.126777\pi\)
0.921729 + 0.387835i \(0.126777\pi\)
\(348\) 1.58884 0.0851710
\(349\) −21.4308 −1.14717 −0.573583 0.819148i \(-0.694447\pi\)
−0.573583 + 0.819148i \(0.694447\pi\)
\(350\) 0 0
\(351\) −20.9907 −1.12040
\(352\) 3.45737 0.184279
\(353\) 19.5659 1.04139 0.520693 0.853744i \(-0.325674\pi\)
0.520693 + 0.853744i \(0.325674\pi\)
\(354\) −3.06786 −0.163055
\(355\) 0 0
\(356\) 28.3723 1.50373
\(357\) −3.12066 −0.165163
\(358\) −5.54644 −0.293138
\(359\) −16.4726 −0.869390 −0.434695 0.900578i \(-0.643144\pi\)
−0.434695 + 0.900578i \(0.643144\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 4.16951 0.219144
\(363\) 10.9128 0.572775
\(364\) 8.57176 0.449282
\(365\) 0 0
\(366\) 2.23571 0.116863
\(367\) 12.7347 0.664746 0.332373 0.943148i \(-0.392151\pi\)
0.332373 + 0.943148i \(0.392151\pi\)
\(368\) −1.64728 −0.0858705
\(369\) −7.90640 −0.411591
\(370\) 0 0
\(371\) −5.85415 −0.303932
\(372\) −4.63892 −0.240517
\(373\) 19.3342 1.00109 0.500544 0.865711i \(-0.333133\pi\)
0.500544 + 0.865711i \(0.333133\pi\)
\(374\) 0.859386 0.0444378
\(375\) 0 0
\(376\) −18.7486 −0.966884
\(377\) −3.52619 −0.181608
\(378\) 2.83746 0.145943
\(379\) −9.85789 −0.506366 −0.253183 0.967418i \(-0.581477\pi\)
−0.253183 + 0.967418i \(0.581477\pi\)
\(380\) 0 0
\(381\) 2.24468 0.114999
\(382\) 6.72183 0.343919
\(383\) 20.3333 1.03898 0.519490 0.854476i \(-0.326122\pi\)
0.519490 + 0.854476i \(0.326122\pi\)
\(384\) 11.6222 0.593092
\(385\) 0 0
\(386\) 9.20440 0.468492
\(387\) −19.4171 −0.987027
\(388\) 29.6849 1.50702
\(389\) −1.00523 −0.0509674 −0.0254837 0.999675i \(-0.508113\pi\)
−0.0254837 + 0.999675i \(0.508113\pi\)
\(390\) 0 0
\(391\) −1.54942 −0.0783574
\(392\) 10.0146 0.505816
\(393\) 18.2553 0.920857
\(394\) 2.04435 0.102993
\(395\) 0 0
\(396\) 2.43231 0.122228
\(397\) −9.86221 −0.494970 −0.247485 0.968892i \(-0.579604\pi\)
−0.247485 + 0.968892i \(0.579604\pi\)
\(398\) −5.96059 −0.298777
\(399\) 0 0
\(400\) 0 0
\(401\) −29.8318 −1.48973 −0.744865 0.667215i \(-0.767486\pi\)
−0.744865 + 0.667215i \(0.767486\pi\)
\(402\) −2.54370 −0.126868
\(403\) 10.2954 0.512849
\(404\) −24.9310 −1.24036
\(405\) 0 0
\(406\) 0.476660 0.0236562
\(407\) −6.69420 −0.331819
\(408\) 4.73240 0.234289
\(409\) −27.6442 −1.36692 −0.683458 0.729989i \(-0.739525\pi\)
−0.683458 + 0.729989i \(0.739525\pi\)
\(410\) 0 0
\(411\) −5.27866 −0.260377
\(412\) 6.31630 0.311182
\(413\) 7.34614 0.361480
\(414\) 0.549417 0.0270024
\(415\) 0 0
\(416\) −19.8814 −0.974766
\(417\) −6.22232 −0.304708
\(418\) 0 0
\(419\) −11.2902 −0.551562 −0.275781 0.961220i \(-0.588936\pi\)
−0.275781 + 0.961220i \(0.588936\pi\)
\(420\) 0 0
\(421\) −4.54942 −0.221725 −0.110863 0.993836i \(-0.535361\pi\)
−0.110863 + 0.993836i \(0.535361\pi\)
\(422\) −10.4969 −0.510981
\(423\) −20.1736 −0.980875
\(424\) 8.87766 0.431137
\(425\) 0 0
\(426\) −6.46362 −0.313163
\(427\) −5.35353 −0.259075
\(428\) 6.45395 0.311964
\(429\) 3.04551 0.147039
\(430\) 0 0
\(431\) −32.9034 −1.58490 −0.792451 0.609936i \(-0.791195\pi\)
−0.792451 + 0.609936i \(0.791195\pi\)
\(432\) 13.8818 0.667887
\(433\) 22.3093 1.07212 0.536059 0.844181i \(-0.319912\pi\)
0.536059 + 0.844181i \(0.319912\pi\)
\(434\) −1.39170 −0.0668035
\(435\) 0 0
\(436\) −11.0597 −0.529664
\(437\) 0 0
\(438\) 6.07605 0.290325
\(439\) −5.70573 −0.272320 −0.136160 0.990687i \(-0.543476\pi\)
−0.136160 + 0.990687i \(0.543476\pi\)
\(440\) 0 0
\(441\) 10.7758 0.513135
\(442\) −4.94185 −0.235060
\(443\) −11.5776 −0.550069 −0.275034 0.961434i \(-0.588689\pi\)
−0.275034 + 0.961434i \(0.588689\pi\)
\(444\) −17.3450 −0.823158
\(445\) 0 0
\(446\) 6.42824 0.304386
\(447\) −9.98630 −0.472336
\(448\) −3.69147 −0.174406
\(449\) 25.4726 1.20213 0.601063 0.799202i \(-0.294744\pi\)
0.601063 + 0.799202i \(0.294744\pi\)
\(450\) 0 0
\(451\) 2.94145 0.138507
\(452\) −21.8091 −1.02581
\(453\) 7.83950 0.368332
\(454\) 3.10930 0.145927
\(455\) 0 0
\(456\) 0 0
\(457\) 6.92830 0.324092 0.162046 0.986783i \(-0.448191\pi\)
0.162046 + 0.986783i \(0.448191\pi\)
\(458\) −0.276506 −0.0129203
\(459\) 13.0571 0.609451
\(460\) 0 0
\(461\) 32.4946 1.51342 0.756712 0.653748i \(-0.226804\pi\)
0.756712 + 0.653748i \(0.226804\pi\)
\(462\) −0.411683 −0.0191532
\(463\) −21.5062 −0.999477 −0.499738 0.866176i \(-0.666571\pi\)
−0.499738 + 0.866176i \(0.666571\pi\)
\(464\) 2.33198 0.108259
\(465\) 0 0
\(466\) −3.37749 −0.156459
\(467\) 16.9509 0.784392 0.392196 0.919882i \(-0.371715\pi\)
0.392196 + 0.919882i \(0.371715\pi\)
\(468\) −13.9868 −0.646542
\(469\) 6.09103 0.281258
\(470\) 0 0
\(471\) 10.2525 0.472410
\(472\) −11.1402 −0.512770
\(473\) 7.22382 0.332152
\(474\) −2.92724 −0.134453
\(475\) 0 0
\(476\) −5.33198 −0.244391
\(477\) 9.55243 0.437376
\(478\) 1.72473 0.0788872
\(479\) 8.85789 0.404727 0.202364 0.979310i \(-0.435138\pi\)
0.202364 + 0.979310i \(0.435138\pi\)
\(480\) 0 0
\(481\) 38.4946 1.75520
\(482\) −8.14611 −0.371045
\(483\) 0.742237 0.0337730
\(484\) 18.6457 0.847531
\(485\) 0 0
\(486\) −7.45432 −0.338135
\(487\) −30.0628 −1.36227 −0.681137 0.732156i \(-0.738514\pi\)
−0.681137 + 0.732156i \(0.738514\pi\)
\(488\) 8.11848 0.367506
\(489\) 0.0715916 0.00323748
\(490\) 0 0
\(491\) 3.10033 0.139916 0.0699580 0.997550i \(-0.477713\pi\)
0.0699580 + 0.997550i \(0.477713\pi\)
\(492\) 7.62143 0.343601
\(493\) 2.19343 0.0987873
\(494\) 0 0
\(495\) 0 0
\(496\) −6.80864 −0.305717
\(497\) 15.4775 0.694260
\(498\) −6.80351 −0.304873
\(499\) 32.1391 1.43874 0.719372 0.694625i \(-0.244430\pi\)
0.719372 + 0.694625i \(0.244430\pi\)
\(500\) 0 0
\(501\) −17.3137 −0.773519
\(502\) −9.16117 −0.408883
\(503\) 8.41822 0.375350 0.187675 0.982231i \(-0.439905\pi\)
0.187675 + 0.982231i \(0.439905\pi\)
\(504\) 4.01827 0.178988
\(505\) 0 0
\(506\) −0.204402 −0.00908676
\(507\) −3.99016 −0.177209
\(508\) 3.83527 0.170163
\(509\) −20.3450 −0.901777 −0.450888 0.892580i \(-0.648893\pi\)
−0.450888 + 0.892580i \(0.648893\pi\)
\(510\) 0 0
\(511\) −14.5494 −0.643628
\(512\) 22.8217 1.00859
\(513\) 0 0
\(514\) −6.55582 −0.289165
\(515\) 0 0
\(516\) 18.7173 0.823982
\(517\) 7.50527 0.330081
\(518\) −5.20358 −0.228632
\(519\) 12.6237 0.554118
\(520\) 0 0
\(521\) −15.3502 −0.672507 −0.336253 0.941772i \(-0.609160\pi\)
−0.336253 + 0.941772i \(0.609160\pi\)
\(522\) −0.777783 −0.0340426
\(523\) −13.2367 −0.578800 −0.289400 0.957208i \(-0.593456\pi\)
−0.289400 + 0.957208i \(0.593456\pi\)
\(524\) 31.1910 1.36259
\(525\) 0 0
\(526\) 7.07276 0.308387
\(527\) −6.40414 −0.278969
\(528\) −2.01409 −0.0876520
\(529\) −22.6315 −0.983977
\(530\) 0 0
\(531\) −11.9870 −0.520190
\(532\) 0 0
\(533\) −16.9146 −0.732653
\(534\) 7.83588 0.339092
\(535\) 0 0
\(536\) −9.23688 −0.398972
\(537\) 12.2266 0.527616
\(538\) −9.14145 −0.394116
\(539\) −4.00897 −0.172679
\(540\) 0 0
\(541\) −27.7408 −1.19267 −0.596335 0.802736i \(-0.703377\pi\)
−0.596335 + 0.802736i \(0.703377\pi\)
\(542\) 5.62388 0.241566
\(543\) −9.19127 −0.394435
\(544\) 12.3670 0.530232
\(545\) 0 0
\(546\) 2.36736 0.101314
\(547\) −17.0602 −0.729440 −0.364720 0.931117i \(-0.618835\pi\)
−0.364720 + 0.931117i \(0.618835\pi\)
\(548\) −9.01913 −0.385278
\(549\) 8.73555 0.372824
\(550\) 0 0
\(551\) 0 0
\(552\) −1.12558 −0.0479080
\(553\) 7.00943 0.298071
\(554\) −5.18239 −0.220179
\(555\) 0 0
\(556\) −10.6315 −0.450875
\(557\) −26.5415 −1.12460 −0.562300 0.826933i \(-0.690083\pi\)
−0.562300 + 0.826933i \(0.690083\pi\)
\(558\) 2.27088 0.0961340
\(559\) −41.5401 −1.75696
\(560\) 0 0
\(561\) −1.89443 −0.0799830
\(562\) −6.91298 −0.291606
\(563\) −35.5594 −1.49865 −0.749325 0.662203i \(-0.769622\pi\)
−0.749325 + 0.662203i \(0.769622\pi\)
\(564\) 19.4465 0.818846
\(565\) 0 0
\(566\) 8.93214 0.375446
\(567\) 0.508137 0.0213397
\(568\) −23.4712 −0.984828
\(569\) 14.9713 0.627628 0.313814 0.949485i \(-0.398393\pi\)
0.313814 + 0.949485i \(0.398393\pi\)
\(570\) 0 0
\(571\) 5.92724 0.248047 0.124024 0.992279i \(-0.460420\pi\)
0.124024 + 0.992279i \(0.460420\pi\)
\(572\) 5.20358 0.217572
\(573\) −14.8176 −0.619015
\(574\) 2.28646 0.0954351
\(575\) 0 0
\(576\) 6.02351 0.250979
\(577\) 14.5559 0.605970 0.302985 0.952995i \(-0.402017\pi\)
0.302985 + 0.952995i \(0.402017\pi\)
\(578\) −4.94800 −0.205810
\(579\) −20.2902 −0.843232
\(580\) 0 0
\(581\) 16.2914 0.675880
\(582\) 8.19839 0.339834
\(583\) −3.55382 −0.147184
\(584\) 22.0638 0.913006
\(585\) 0 0
\(586\) 2.64078 0.109090
\(587\) −20.5839 −0.849588 −0.424794 0.905290i \(-0.639653\pi\)
−0.424794 + 0.905290i \(0.639653\pi\)
\(588\) −10.3874 −0.428371
\(589\) 0 0
\(590\) 0 0
\(591\) −4.50657 −0.185375
\(592\) −25.4576 −1.04630
\(593\) −14.3322 −0.588552 −0.294276 0.955720i \(-0.595078\pi\)
−0.294276 + 0.955720i \(0.595078\pi\)
\(594\) 1.72251 0.0706753
\(595\) 0 0
\(596\) −17.0626 −0.698912
\(597\) 13.1395 0.537765
\(598\) 1.17540 0.0480656
\(599\) −2.50017 −0.102154 −0.0510770 0.998695i \(-0.516265\pi\)
−0.0510770 + 0.998695i \(0.516265\pi\)
\(600\) 0 0
\(601\) −14.5259 −0.592524 −0.296262 0.955107i \(-0.595740\pi\)
−0.296262 + 0.955107i \(0.595740\pi\)
\(602\) 5.61526 0.228861
\(603\) −9.93895 −0.404745
\(604\) 13.3946 0.545019
\(605\) 0 0
\(606\) −6.88546 −0.279703
\(607\) 31.8704 1.29358 0.646789 0.762669i \(-0.276111\pi\)
0.646789 + 0.762669i \(0.276111\pi\)
\(608\) 0 0
\(609\) −1.05075 −0.0425785
\(610\) 0 0
\(611\) −43.1586 −1.74601
\(612\) 8.70038 0.351692
\(613\) 20.0488 0.809765 0.404882 0.914369i \(-0.367312\pi\)
0.404882 + 0.914369i \(0.367312\pi\)
\(614\) −9.73588 −0.392908
\(615\) 0 0
\(616\) −1.49493 −0.0602325
\(617\) −20.0819 −0.808467 −0.404234 0.914656i \(-0.632462\pi\)
−0.404234 + 0.914656i \(0.632462\pi\)
\(618\) 1.74444 0.0701718
\(619\) 5.62217 0.225974 0.112987 0.993596i \(-0.463958\pi\)
0.112987 + 0.993596i \(0.463958\pi\)
\(620\) 0 0
\(621\) −3.10557 −0.124622
\(622\) −11.6361 −0.466563
\(623\) −18.7634 −0.751740
\(624\) 11.5819 0.463647
\(625\) 0 0
\(626\) 1.65092 0.0659839
\(627\) 0 0
\(628\) 17.5174 0.699022
\(629\) −23.9452 −0.954757
\(630\) 0 0
\(631\) −4.84368 −0.192824 −0.0964120 0.995342i \(-0.530737\pi\)
−0.0964120 + 0.995342i \(0.530737\pi\)
\(632\) −10.6296 −0.422823
\(633\) 23.1394 0.919708
\(634\) −14.8034 −0.587918
\(635\) 0 0
\(636\) −9.20814 −0.365126
\(637\) 23.0533 0.913406
\(638\) 0.289361 0.0114559
\(639\) −25.2552 −0.999078
\(640\) 0 0
\(641\) 25.3007 0.999316 0.499658 0.866223i \(-0.333459\pi\)
0.499658 + 0.866223i \(0.333459\pi\)
\(642\) 1.78246 0.0703480
\(643\) 13.9457 0.549963 0.274981 0.961450i \(-0.411328\pi\)
0.274981 + 0.961450i \(0.411328\pi\)
\(644\) 1.26819 0.0499737
\(645\) 0 0
\(646\) 0 0
\(647\) −2.51360 −0.0988200 −0.0494100 0.998779i \(-0.515734\pi\)
−0.0494100 + 0.998779i \(0.515734\pi\)
\(648\) −0.770575 −0.0302711
\(649\) 4.45955 0.175053
\(650\) 0 0
\(651\) 3.06786 0.120239
\(652\) 0.122322 0.00479049
\(653\) 1.92873 0.0754770 0.0377385 0.999288i \(-0.487985\pi\)
0.0377385 + 0.999288i \(0.487985\pi\)
\(654\) −3.05448 −0.119440
\(655\) 0 0
\(656\) 11.1861 0.436745
\(657\) 23.7408 0.926217
\(658\) 5.83404 0.227434
\(659\) −15.8008 −0.615513 −0.307757 0.951465i \(-0.599578\pi\)
−0.307757 + 0.951465i \(0.599578\pi\)
\(660\) 0 0
\(661\) −16.4036 −0.638025 −0.319012 0.947751i \(-0.603351\pi\)
−0.319012 + 0.947751i \(0.603351\pi\)
\(662\) −16.8697 −0.655661
\(663\) 10.8938 0.423081
\(664\) −24.7054 −0.958755
\(665\) 0 0
\(666\) 8.49086 0.329014
\(667\) −0.521700 −0.0202003
\(668\) −29.5823 −1.14457
\(669\) −14.1704 −0.547860
\(670\) 0 0
\(671\) −3.24992 −0.125462
\(672\) −5.92434 −0.228536
\(673\) −17.0878 −0.658686 −0.329343 0.944210i \(-0.606827\pi\)
−0.329343 + 0.944210i \(0.606827\pi\)
\(674\) −12.4998 −0.481476
\(675\) 0 0
\(676\) −6.81761 −0.262216
\(677\) 4.57680 0.175901 0.0879504 0.996125i \(-0.471968\pi\)
0.0879504 + 0.996125i \(0.471968\pi\)
\(678\) −6.02326 −0.231322
\(679\) −19.6315 −0.753387
\(680\) 0 0
\(681\) −6.85415 −0.262652
\(682\) −0.844844 −0.0323507
\(683\) 29.0692 1.11230 0.556151 0.831082i \(-0.312278\pi\)
0.556151 + 0.831082i \(0.312278\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −6.99883 −0.267217
\(687\) 0.609531 0.0232550
\(688\) 27.4717 1.04735
\(689\) 20.4360 0.778551
\(690\) 0 0
\(691\) −6.07833 −0.231230 −0.115615 0.993294i \(-0.536884\pi\)
−0.115615 + 0.993294i \(0.536884\pi\)
\(692\) 21.5689 0.819925
\(693\) −1.60856 −0.0611041
\(694\) 16.2044 0.615111
\(695\) 0 0
\(696\) 1.59343 0.0603989
\(697\) 10.5216 0.398533
\(698\) −10.1129 −0.382777
\(699\) 7.44535 0.281609
\(700\) 0 0
\(701\) 10.4114 0.393232 0.196616 0.980481i \(-0.437005\pi\)
0.196616 + 0.980481i \(0.437005\pi\)
\(702\) −9.90517 −0.373847
\(703\) 0 0
\(704\) −2.24095 −0.0844589
\(705\) 0 0
\(706\) 9.23281 0.347481
\(707\) 16.4876 0.620080
\(708\) 11.5549 0.434261
\(709\) −0.531144 −0.0199475 −0.00997377 0.999950i \(-0.503175\pi\)
−0.00997377 + 0.999950i \(0.503175\pi\)
\(710\) 0 0
\(711\) −11.4375 −0.428941
\(712\) 28.4542 1.06637
\(713\) 1.52320 0.0570442
\(714\) −1.47259 −0.0551103
\(715\) 0 0
\(716\) 20.8904 0.780710
\(717\) −3.80199 −0.141988
\(718\) −7.77315 −0.290091
\(719\) 22.4271 0.836389 0.418194 0.908358i \(-0.362663\pi\)
0.418194 + 0.908358i \(0.362663\pi\)
\(720\) 0 0
\(721\) −4.17716 −0.155566
\(722\) 0 0
\(723\) 17.9573 0.667839
\(724\) −15.7042 −0.583643
\(725\) 0 0
\(726\) 5.14958 0.191119
\(727\) −45.9734 −1.70506 −0.852529 0.522679i \(-0.824933\pi\)
−0.852529 + 0.522679i \(0.824933\pi\)
\(728\) 8.59651 0.318608
\(729\) 15.1354 0.560570
\(730\) 0 0
\(731\) 25.8396 0.955713
\(732\) −8.42069 −0.311238
\(733\) 42.5178 1.57043 0.785215 0.619223i \(-0.212552\pi\)
0.785215 + 0.619223i \(0.212552\pi\)
\(734\) 6.00930 0.221807
\(735\) 0 0
\(736\) −2.94145 −0.108423
\(737\) 3.69762 0.136204
\(738\) −3.73090 −0.137336
\(739\) −3.41777 −0.125725 −0.0628624 0.998022i \(-0.520023\pi\)
−0.0628624 + 0.998022i \(0.520023\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.76248 −0.101414
\(743\) 9.72088 0.356625 0.178312 0.983974i \(-0.442936\pi\)
0.178312 + 0.983974i \(0.442936\pi\)
\(744\) −4.65232 −0.170562
\(745\) 0 0
\(746\) 9.12351 0.334035
\(747\) −26.5832 −0.972629
\(748\) −3.23683 −0.118350
\(749\) −4.26819 −0.155956
\(750\) 0 0
\(751\) 11.3853 0.415455 0.207728 0.978187i \(-0.433393\pi\)
0.207728 + 0.978187i \(0.433393\pi\)
\(752\) 28.5420 1.04082
\(753\) 20.1949 0.735943
\(754\) −1.66395 −0.0605976
\(755\) 0 0
\(756\) −10.6871 −0.388687
\(757\) 22.7842 0.828105 0.414053 0.910253i \(-0.364113\pi\)
0.414053 + 0.910253i \(0.364113\pi\)
\(758\) −4.65178 −0.168960
\(759\) 0.450583 0.0163551
\(760\) 0 0
\(761\) −36.7665 −1.33279 −0.666393 0.745601i \(-0.732163\pi\)
−0.666393 + 0.745601i \(0.732163\pi\)
\(762\) 1.05923 0.0383718
\(763\) 7.31412 0.264789
\(764\) −25.3174 −0.915953
\(765\) 0 0
\(766\) 9.59493 0.346679
\(767\) −25.6444 −0.925965
\(768\) −1.04955 −0.0378724
\(769\) −20.8008 −0.750097 −0.375049 0.927005i \(-0.622374\pi\)
−0.375049 + 0.927005i \(0.622374\pi\)
\(770\) 0 0
\(771\) 14.4517 0.520464
\(772\) −34.6679 −1.24772
\(773\) −21.0091 −0.755646 −0.377823 0.925878i \(-0.623327\pi\)
−0.377823 + 0.925878i \(0.623327\pi\)
\(774\) −9.16262 −0.329344
\(775\) 0 0
\(776\) 29.7706 1.06870
\(777\) 11.4708 0.411512
\(778\) −0.474354 −0.0170064
\(779\) 0 0
\(780\) 0 0
\(781\) 9.39576 0.336207
\(782\) −0.731145 −0.0261457
\(783\) 4.39640 0.157115
\(784\) −15.2458 −0.544495
\(785\) 0 0
\(786\) 8.61437 0.307264
\(787\) 18.0606 0.643791 0.321896 0.946775i \(-0.395680\pi\)
0.321896 + 0.946775i \(0.395680\pi\)
\(788\) −7.69994 −0.274299
\(789\) −15.5912 −0.555061
\(790\) 0 0
\(791\) 14.4230 0.512823
\(792\) 2.43934 0.0866780
\(793\) 18.6884 0.663646
\(794\) −4.65382 −0.165158
\(795\) 0 0
\(796\) 22.4502 0.795728
\(797\) −24.2571 −0.859229 −0.429615 0.903012i \(-0.641351\pi\)
−0.429615 + 0.903012i \(0.641351\pi\)
\(798\) 0 0
\(799\) 26.8463 0.949756
\(800\) 0 0
\(801\) 30.6169 1.08180
\(802\) −14.0771 −0.497081
\(803\) −8.83238 −0.311688
\(804\) 9.58073 0.337886
\(805\) 0 0
\(806\) 4.85822 0.171124
\(807\) 20.1514 0.709364
\(808\) −25.0030 −0.879602
\(809\) −4.27192 −0.150193 −0.0750964 0.997176i \(-0.523926\pi\)
−0.0750964 + 0.997176i \(0.523926\pi\)
\(810\) 0 0
\(811\) −10.1824 −0.357552 −0.178776 0.983890i \(-0.557214\pi\)
−0.178776 + 0.983890i \(0.557214\pi\)
\(812\) −1.79531 −0.0630032
\(813\) −12.3973 −0.434792
\(814\) −3.15889 −0.110719
\(815\) 0 0
\(816\) −7.20440 −0.252205
\(817\) 0 0
\(818\) −13.0448 −0.456102
\(819\) 9.24992 0.323218
\(820\) 0 0
\(821\) −56.0217 −1.95517 −0.977585 0.210541i \(-0.932477\pi\)
−0.977585 + 0.210541i \(0.932477\pi\)
\(822\) −2.49091 −0.0868806
\(823\) −33.5416 −1.16919 −0.584593 0.811326i \(-0.698746\pi\)
−0.584593 + 0.811326i \(0.698746\pi\)
\(824\) 6.33455 0.220674
\(825\) 0 0
\(826\) 3.46652 0.120616
\(827\) −0.0362530 −0.00126064 −0.000630320 1.00000i \(-0.500201\pi\)
−0.000630320 1.00000i \(0.500201\pi\)
\(828\) −2.06935 −0.0719149
\(829\) −39.5662 −1.37419 −0.687095 0.726567i \(-0.741115\pi\)
−0.687095 + 0.726567i \(0.741115\pi\)
\(830\) 0 0
\(831\) 11.4241 0.396297
\(832\) 12.8864 0.446756
\(833\) −14.3401 −0.496855
\(834\) −2.93621 −0.101673
\(835\) 0 0
\(836\) 0 0
\(837\) −12.8361 −0.443681
\(838\) −5.32766 −0.184041
\(839\) −40.1623 −1.38656 −0.693278 0.720670i \(-0.743834\pi\)
−0.693278 + 0.720670i \(0.743834\pi\)
\(840\) 0 0
\(841\) −28.2615 −0.974533
\(842\) −2.14680 −0.0739835
\(843\) 15.2390 0.524858
\(844\) 39.5360 1.36089
\(845\) 0 0
\(846\) −9.51961 −0.327291
\(847\) −12.3309 −0.423696
\(848\) −13.5150 −0.464106
\(849\) −19.6900 −0.675760
\(850\) 0 0
\(851\) 5.69527 0.195231
\(852\) 24.3449 0.834042
\(853\) −16.4617 −0.563639 −0.281819 0.959467i \(-0.590938\pi\)
−0.281819 + 0.959467i \(0.590938\pi\)
\(854\) −2.52624 −0.0864463
\(855\) 0 0
\(856\) 6.47259 0.221229
\(857\) 5.60664 0.191519 0.0957595 0.995404i \(-0.469472\pi\)
0.0957595 + 0.995404i \(0.469472\pi\)
\(858\) 1.43713 0.0490628
\(859\) 25.0298 0.854006 0.427003 0.904250i \(-0.359569\pi\)
0.427003 + 0.904250i \(0.359569\pi\)
\(860\) 0 0
\(861\) −5.04028 −0.171772
\(862\) −15.5266 −0.528837
\(863\) −42.4307 −1.44436 −0.722178 0.691707i \(-0.756859\pi\)
−0.722178 + 0.691707i \(0.756859\pi\)
\(864\) 24.7878 0.843298
\(865\) 0 0
\(866\) 10.5274 0.357736
\(867\) 10.9074 0.370434
\(868\) 5.24175 0.177917
\(869\) 4.25515 0.144346
\(870\) 0 0
\(871\) −21.2630 −0.720468
\(872\) −11.0917 −0.375611
\(873\) 32.0334 1.08417
\(874\) 0 0
\(875\) 0 0
\(876\) −22.8851 −0.773217
\(877\) 0.557245 0.0188168 0.00940841 0.999956i \(-0.497005\pi\)
0.00940841 + 0.999956i \(0.497005\pi\)
\(878\) −2.69244 −0.0908656
\(879\) −5.82134 −0.196349
\(880\) 0 0
\(881\) 8.35432 0.281464 0.140732 0.990048i \(-0.455054\pi\)
0.140732 + 0.990048i \(0.455054\pi\)
\(882\) 5.08494 0.171219
\(883\) −19.4842 −0.655695 −0.327847 0.944731i \(-0.606323\pi\)
−0.327847 + 0.944731i \(0.606323\pi\)
\(884\) 18.6132 0.626030
\(885\) 0 0
\(886\) −5.46329 −0.183543
\(887\) 35.9373 1.20666 0.603328 0.797493i \(-0.293841\pi\)
0.603328 + 0.797493i \(0.293841\pi\)
\(888\) −17.3951 −0.583742
\(889\) −2.53638 −0.0850674
\(890\) 0 0
\(891\) 0.308470 0.0103341
\(892\) −24.2116 −0.810666
\(893\) 0 0
\(894\) −4.71237 −0.157605
\(895\) 0 0
\(896\) −13.1325 −0.438725
\(897\) −2.59105 −0.0865126
\(898\) 12.0201 0.401116
\(899\) −2.15632 −0.0719172
\(900\) 0 0
\(901\) −12.7120 −0.423499
\(902\) 1.38802 0.0462160
\(903\) −12.3783 −0.411924
\(904\) −21.8721 −0.727455
\(905\) 0 0
\(906\) 3.69933 0.122902
\(907\) 48.9503 1.62537 0.812683 0.582706i \(-0.198006\pi\)
0.812683 + 0.582706i \(0.198006\pi\)
\(908\) −11.7110 −0.388644
\(909\) −26.9034 −0.892330
\(910\) 0 0
\(911\) 19.7811 0.655376 0.327688 0.944786i \(-0.393731\pi\)
0.327688 + 0.944786i \(0.393731\pi\)
\(912\) 0 0
\(913\) 9.88984 0.327306
\(914\) 3.26935 0.108141
\(915\) 0 0
\(916\) 1.04145 0.0344103
\(917\) −20.6276 −0.681182
\(918\) 6.16141 0.203357
\(919\) 26.0582 0.859581 0.429791 0.902929i \(-0.358587\pi\)
0.429791 + 0.902929i \(0.358587\pi\)
\(920\) 0 0
\(921\) 21.4618 0.707190
\(922\) 15.3337 0.504988
\(923\) −54.0298 −1.77841
\(924\) 1.55058 0.0510104
\(925\) 0 0
\(926\) −10.1484 −0.333498
\(927\) 6.81602 0.223868
\(928\) 4.16406 0.136692
\(929\) −7.97799 −0.261749 −0.130875 0.991399i \(-0.541779\pi\)
−0.130875 + 0.991399i \(0.541779\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.7211 0.416695
\(933\) 25.6505 0.839761
\(934\) 7.99883 0.261730
\(935\) 0 0
\(936\) −14.0272 −0.458495
\(937\) 46.6542 1.52413 0.762063 0.647502i \(-0.224186\pi\)
0.762063 + 0.647502i \(0.224186\pi\)
\(938\) 2.87426 0.0938479
\(939\) −3.63928 −0.118763
\(940\) 0 0
\(941\) 1.40100 0.0456713 0.0228356 0.999739i \(-0.492731\pi\)
0.0228356 + 0.999739i \(0.492731\pi\)
\(942\) 4.83798 0.157630
\(943\) −2.50251 −0.0814930
\(944\) 16.9594 0.551981
\(945\) 0 0
\(946\) 3.40880 0.110830
\(947\) 38.2501 1.24296 0.621480 0.783430i \(-0.286532\pi\)
0.621480 + 0.783430i \(0.286532\pi\)
\(948\) 11.0253 0.358085
\(949\) 50.7900 1.64871
\(950\) 0 0
\(951\) 32.6326 1.05819
\(952\) −5.34738 −0.173309
\(953\) −54.4390 −1.76345 −0.881726 0.471763i \(-0.843618\pi\)
−0.881726 + 0.471763i \(0.843618\pi\)
\(954\) 4.50764 0.145940
\(955\) 0 0
\(956\) −6.49610 −0.210099
\(957\) −0.637869 −0.0206194
\(958\) 4.17989 0.135046
\(959\) 5.96462 0.192608
\(960\) 0 0
\(961\) −24.7042 −0.796911
\(962\) 18.1650 0.585662
\(963\) 6.96456 0.224430
\(964\) 30.6819 0.988197
\(965\) 0 0
\(966\) 0.350250 0.0112691
\(967\) 4.98919 0.160442 0.0802208 0.996777i \(-0.474437\pi\)
0.0802208 + 0.996777i \(0.474437\pi\)
\(968\) 18.6995 0.601026
\(969\) 0 0
\(970\) 0 0
\(971\) 14.5401 0.466614 0.233307 0.972403i \(-0.425045\pi\)
0.233307 + 0.972403i \(0.425045\pi\)
\(972\) 28.0763 0.900548
\(973\) 7.03091 0.225401
\(974\) −14.1861 −0.454553
\(975\) 0 0
\(976\) −12.3592 −0.395609
\(977\) −26.1353 −0.836141 −0.418071 0.908415i \(-0.637294\pi\)
−0.418071 + 0.908415i \(0.637294\pi\)
\(978\) 0.0337829 0.00108026
\(979\) −11.3905 −0.364043
\(980\) 0 0
\(981\) −11.9347 −0.381046
\(982\) 1.46300 0.0466861
\(983\) −46.1160 −1.47087 −0.735436 0.677595i \(-0.763022\pi\)
−0.735436 + 0.677595i \(0.763022\pi\)
\(984\) 7.64345 0.243664
\(985\) 0 0
\(986\) 1.03505 0.0329626
\(987\) −12.8606 −0.409356
\(988\) 0 0
\(989\) −6.14585 −0.195427
\(990\) 0 0
\(991\) 31.3279 0.995164 0.497582 0.867417i \(-0.334221\pi\)
0.497582 + 0.867417i \(0.334221\pi\)
\(992\) −12.1577 −0.386009
\(993\) 37.1877 1.18011
\(994\) 7.30357 0.231655
\(995\) 0 0
\(996\) 25.6251 0.811962
\(997\) −21.6144 −0.684536 −0.342268 0.939602i \(-0.611195\pi\)
−0.342268 + 0.939602i \(0.611195\pi\)
\(998\) 15.1659 0.480069
\(999\) −47.9944 −1.51848
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 9025.2.a.bt.1.4 6
5.2 odd 4 1805.2.b.g.1084.4 6
5.3 odd 4 1805.2.b.g.1084.3 6
5.4 even 2 inner 9025.2.a.bt.1.3 6
19.8 odd 6 475.2.e.g.26.4 12
19.12 odd 6 475.2.e.g.201.4 12
19.18 odd 2 9025.2.a.bu.1.3 6
95.8 even 12 95.2.i.b.64.4 yes 12
95.12 even 12 95.2.i.b.49.4 yes 12
95.18 even 4 1805.2.b.f.1084.4 6
95.27 even 12 95.2.i.b.64.3 yes 12
95.37 even 4 1805.2.b.f.1084.3 6
95.69 odd 6 475.2.e.g.201.3 12
95.84 odd 6 475.2.e.g.26.3 12
95.88 even 12 95.2.i.b.49.3 12
95.94 odd 2 9025.2.a.bu.1.4 6
285.8 odd 12 855.2.be.d.64.3 12
285.107 odd 12 855.2.be.d.334.3 12
285.122 odd 12 855.2.be.d.64.4 12
285.278 odd 12 855.2.be.d.334.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.b.49.3 12 95.88 even 12
95.2.i.b.49.4 yes 12 95.12 even 12
95.2.i.b.64.3 yes 12 95.27 even 12
95.2.i.b.64.4 yes 12 95.8 even 12
475.2.e.g.26.3 12 95.84 odd 6
475.2.e.g.26.4 12 19.8 odd 6
475.2.e.g.201.3 12 95.69 odd 6
475.2.e.g.201.4 12 19.12 odd 6
855.2.be.d.64.3 12 285.8 odd 12
855.2.be.d.64.4 12 285.122 odd 12
855.2.be.d.334.3 12 285.107 odd 12
855.2.be.d.334.4 12 285.278 odd 12
1805.2.b.f.1084.3 6 95.37 even 4
1805.2.b.f.1084.4 6 95.18 even 4
1805.2.b.g.1084.3 6 5.3 odd 4
1805.2.b.g.1084.4 6 5.2 odd 4
9025.2.a.bt.1.3 6 5.4 even 2 inner
9025.2.a.bt.1.4 6 1.1 even 1 trivial
9025.2.a.bu.1.3 6 19.18 odd 2
9025.2.a.bu.1.4 6 95.94 odd 2