Properties

Label 3850.2.a.bs.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $1$
Dimension $3$
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] = [3850,2,Mod(1,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, 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("3850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.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: \(30.7424047782\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.81361 q^{3} +1.00000 q^{4} +2.81361 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.91638 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.81361 q^{3} +1.00000 q^{4} +2.81361 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.91638 q^{9} +1.00000 q^{11} -2.81361 q^{12} +4.91638 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.81361 q^{17} -4.91638 q^{18} +1.28917 q^{19} -2.81361 q^{21} -1.00000 q^{22} -7.49472 q^{23} +2.81361 q^{24} -4.91638 q^{26} -5.39194 q^{27} +1.00000 q^{28} -1.86751 q^{29} +9.15165 q^{31} -1.00000 q^{32} -2.81361 q^{33} +4.81361 q^{34} +4.91638 q^{36} -6.75971 q^{37} -1.28917 q^{38} -13.8328 q^{39} -12.4111 q^{41} +2.81361 q^{42} +2.62721 q^{43} +1.00000 q^{44} +7.49472 q^{46} -11.0192 q^{47} -2.81361 q^{48} +1.00000 q^{49} +13.5436 q^{51} +4.91638 q^{52} +9.74914 q^{53} +5.39194 q^{54} -1.00000 q^{56} -3.62721 q^{57} +1.86751 q^{58} +1.39194 q^{59} +6.57834 q^{61} -9.15165 q^{62} +4.91638 q^{63} +1.00000 q^{64} +2.81361 q^{66} -3.62721 q^{67} -4.81361 q^{68} +21.0872 q^{69} -0.338044 q^{71} -4.91638 q^{72} +13.6952 q^{73} +6.75971 q^{74} +1.28917 q^{76} +1.00000 q^{77} +13.8328 q^{78} -6.28917 q^{79} +0.421663 q^{81} +12.4111 q^{82} -9.75971 q^{83} -2.81361 q^{84} -2.62721 q^{86} +5.25443 q^{87} -1.00000 q^{88} -6.71083 q^{89} +4.91638 q^{91} -7.49472 q^{92} -25.7491 q^{93} +11.0192 q^{94} +2.81361 q^{96} +0.338044 q^{97} -1.00000 q^{98} +4.91638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + q^{9} + 3 q^{11} - 2 q^{12} + q^{13} - 3 q^{14} + 3 q^{16} - 8 q^{17} - q^{18} + 3 q^{19} - 2 q^{21} - 3 q^{22} - 7 q^{23} + 2 q^{24} - q^{26} - 8 q^{27} + 3 q^{28} - 3 q^{29} + 9 q^{31} - 3 q^{32} - 2 q^{33} + 8 q^{34} + q^{36} - 10 q^{37} - 3 q^{38} - 14 q^{39} - 8 q^{41} + 2 q^{42} - 5 q^{43} + 3 q^{44} + 7 q^{46} - 12 q^{47} - 2 q^{48} + 3 q^{49} + 14 q^{51} + q^{52} - 12 q^{53} + 8 q^{54} - 3 q^{56} + 2 q^{57} + 3 q^{58} - 4 q^{59} + 18 q^{61} - 9 q^{62} + q^{63} + 3 q^{64} + 2 q^{66} + 2 q^{67} - 8 q^{68} + 10 q^{69} + 11 q^{71} - q^{72} - 4 q^{73} + 10 q^{74} + 3 q^{76} + 3 q^{77} + 14 q^{78} - 18 q^{79} + 3 q^{81} + 8 q^{82} - 19 q^{83} - 2 q^{84} + 5 q^{86} - 10 q^{87} - 3 q^{88} - 21 q^{89} + q^{91} - 7 q^{92} - 36 q^{93} + 12 q^{94} + 2 q^{96} - 11 q^{97} - 3 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.81361 −1.62444 −0.812218 0.583354i \(-0.801740\pi\)
−0.812218 + 0.583354i \(0.801740\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.81361 1.14865
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.91638 1.63879
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.81361 −0.812218
\(13\) 4.91638 1.36356 0.681779 0.731558i \(-0.261206\pi\)
0.681779 + 0.731558i \(0.261206\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.81361 −1.16747 −0.583736 0.811944i \(-0.698410\pi\)
−0.583736 + 0.811944i \(0.698410\pi\)
\(18\) −4.91638 −1.15880
\(19\) 1.28917 0.295756 0.147878 0.989006i \(-0.452756\pi\)
0.147878 + 0.989006i \(0.452756\pi\)
\(20\) 0 0
\(21\) −2.81361 −0.613979
\(22\) −1.00000 −0.213201
\(23\) −7.49472 −1.56276 −0.781378 0.624057i \(-0.785483\pi\)
−0.781378 + 0.624057i \(0.785483\pi\)
\(24\) 2.81361 0.574325
\(25\) 0 0
\(26\) −4.91638 −0.964182
\(27\) −5.39194 −1.03768
\(28\) 1.00000 0.188982
\(29\) −1.86751 −0.346787 −0.173394 0.984853i \(-0.555473\pi\)
−0.173394 + 0.984853i \(0.555473\pi\)
\(30\) 0 0
\(31\) 9.15165 1.64369 0.821843 0.569715i \(-0.192946\pi\)
0.821843 + 0.569715i \(0.192946\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.81361 −0.489786
\(34\) 4.81361 0.825527
\(35\) 0 0
\(36\) 4.91638 0.819397
\(37\) −6.75971 −1.11129 −0.555645 0.831420i \(-0.687528\pi\)
−0.555645 + 0.831420i \(0.687528\pi\)
\(38\) −1.28917 −0.209131
\(39\) −13.8328 −2.21501
\(40\) 0 0
\(41\) −12.4111 −1.93829 −0.969144 0.246495i \(-0.920721\pi\)
−0.969144 + 0.246495i \(0.920721\pi\)
\(42\) 2.81361 0.434149
\(43\) 2.62721 0.400646 0.200323 0.979730i \(-0.435801\pi\)
0.200323 + 0.979730i \(0.435801\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 7.49472 1.10504
\(47\) −11.0192 −1.60731 −0.803655 0.595096i \(-0.797114\pi\)
−0.803655 + 0.595096i \(0.797114\pi\)
\(48\) −2.81361 −0.406109
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.5436 1.89648
\(52\) 4.91638 0.681779
\(53\) 9.74914 1.33915 0.669574 0.742745i \(-0.266477\pi\)
0.669574 + 0.742745i \(0.266477\pi\)
\(54\) 5.39194 0.733751
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −3.62721 −0.480436
\(58\) 1.86751 0.245216
\(59\) 1.39194 0.181216 0.0906078 0.995887i \(-0.471119\pi\)
0.0906078 + 0.995887i \(0.471119\pi\)
\(60\) 0 0
\(61\) 6.57834 0.842270 0.421135 0.906998i \(-0.361632\pi\)
0.421135 + 0.906998i \(0.361632\pi\)
\(62\) −9.15165 −1.16226
\(63\) 4.91638 0.619406
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.81361 0.346331
\(67\) −3.62721 −0.443135 −0.221567 0.975145i \(-0.571117\pi\)
−0.221567 + 0.975145i \(0.571117\pi\)
\(68\) −4.81361 −0.583736
\(69\) 21.0872 2.53860
\(70\) 0 0
\(71\) −0.338044 −0.0401185 −0.0200592 0.999799i \(-0.506385\pi\)
−0.0200592 + 0.999799i \(0.506385\pi\)
\(72\) −4.91638 −0.579401
\(73\) 13.6952 1.60291 0.801454 0.598057i \(-0.204060\pi\)
0.801454 + 0.598057i \(0.204060\pi\)
\(74\) 6.75971 0.785800
\(75\) 0 0
\(76\) 1.28917 0.147878
\(77\) 1.00000 0.113961
\(78\) 13.8328 1.56625
\(79\) −6.28917 −0.707587 −0.353793 0.935324i \(-0.615108\pi\)
−0.353793 + 0.935324i \(0.615108\pi\)
\(80\) 0 0
\(81\) 0.421663 0.0468514
\(82\) 12.4111 1.37058
\(83\) −9.75971 −1.07127 −0.535634 0.844451i \(-0.679927\pi\)
−0.535634 + 0.844451i \(0.679927\pi\)
\(84\) −2.81361 −0.306990
\(85\) 0 0
\(86\) −2.62721 −0.283300
\(87\) 5.25443 0.563334
\(88\) −1.00000 −0.106600
\(89\) −6.71083 −0.711347 −0.355673 0.934610i \(-0.615748\pi\)
−0.355673 + 0.934610i \(0.615748\pi\)
\(90\) 0 0
\(91\) 4.91638 0.515377
\(92\) −7.49472 −0.781378
\(93\) −25.7491 −2.67006
\(94\) 11.0192 1.13654
\(95\) 0 0
\(96\) 2.81361 0.287163
\(97\) 0.338044 0.0343232 0.0171616 0.999853i \(-0.494537\pi\)
0.0171616 + 0.999853i \(0.494537\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.91638 0.494115
\(100\) 0 0
\(101\) 16.5139 1.64319 0.821596 0.570070i \(-0.193084\pi\)
0.821596 + 0.570070i \(0.193084\pi\)
\(102\) −13.5436 −1.34102
\(103\) −5.83779 −0.575214 −0.287607 0.957748i \(-0.592860\pi\)
−0.287607 + 0.957748i \(0.592860\pi\)
\(104\) −4.91638 −0.482091
\(105\) 0 0
\(106\) −9.74914 −0.946921
\(107\) 0.0488759 0.00472501 0.00236251 0.999997i \(-0.499248\pi\)
0.00236251 + 0.999997i \(0.499248\pi\)
\(108\) −5.39194 −0.518840
\(109\) 0.240293 0.0230159 0.0115079 0.999934i \(-0.496337\pi\)
0.0115079 + 0.999934i \(0.496337\pi\)
\(110\) 0 0
\(111\) 19.0192 1.80522
\(112\) 1.00000 0.0944911
\(113\) 21.0036 1.97585 0.987925 0.154934i \(-0.0495164\pi\)
0.987925 + 0.154934i \(0.0495164\pi\)
\(114\) 3.62721 0.339720
\(115\) 0 0
\(116\) −1.86751 −0.173394
\(117\) 24.1708 2.23459
\(118\) −1.39194 −0.128139
\(119\) −4.81361 −0.441263
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.57834 −0.595575
\(123\) 34.9200 3.14863
\(124\) 9.15165 0.821843
\(125\) 0 0
\(126\) −4.91638 −0.437986
\(127\) −4.96526 −0.440595 −0.220298 0.975433i \(-0.570703\pi\)
−0.220298 + 0.975433i \(0.570703\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.39194 −0.650824
\(130\) 0 0
\(131\) 12.1325 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(132\) −2.81361 −0.244893
\(133\) 1.28917 0.111785
\(134\) 3.62721 0.313343
\(135\) 0 0
\(136\) 4.81361 0.412763
\(137\) 16.7491 1.43098 0.715488 0.698625i \(-0.246204\pi\)
0.715488 + 0.698625i \(0.246204\pi\)
\(138\) −21.0872 −1.79506
\(139\) −22.5819 −1.91537 −0.957686 0.287814i \(-0.907071\pi\)
−0.957686 + 0.287814i \(0.907071\pi\)
\(140\) 0 0
\(141\) 31.0036 2.61097
\(142\) 0.338044 0.0283681
\(143\) 4.91638 0.411128
\(144\) 4.91638 0.409698
\(145\) 0 0
\(146\) −13.6952 −1.13343
\(147\) −2.81361 −0.232062
\(148\) −6.75971 −0.555645
\(149\) −13.4217 −1.09955 −0.549773 0.835314i \(-0.685286\pi\)
−0.549773 + 0.835314i \(0.685286\pi\)
\(150\) 0 0
\(151\) −15.2544 −1.24139 −0.620694 0.784053i \(-0.713149\pi\)
−0.620694 + 0.784053i \(0.713149\pi\)
\(152\) −1.28917 −0.104565
\(153\) −23.6655 −1.91324
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −13.8328 −1.10751
\(157\) 8.50885 0.679080 0.339540 0.940592i \(-0.389729\pi\)
0.339540 + 0.940592i \(0.389729\pi\)
\(158\) 6.28917 0.500339
\(159\) −27.4303 −2.17536
\(160\) 0 0
\(161\) −7.49472 −0.590667
\(162\) −0.421663 −0.0331290
\(163\) −15.7250 −1.23167 −0.615837 0.787873i \(-0.711182\pi\)
−0.615837 + 0.787873i \(0.711182\pi\)
\(164\) −12.4111 −0.969144
\(165\) 0 0
\(166\) 9.75971 0.757500
\(167\) 14.6167 1.13107 0.565535 0.824724i \(-0.308670\pi\)
0.565535 + 0.824724i \(0.308670\pi\)
\(168\) 2.81361 0.217074
\(169\) 11.1708 0.859293
\(170\) 0 0
\(171\) 6.33804 0.484682
\(172\) 2.62721 0.200323
\(173\) −3.08362 −0.234443 −0.117222 0.993106i \(-0.537399\pi\)
−0.117222 + 0.993106i \(0.537399\pi\)
\(174\) −5.25443 −0.398337
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −3.91638 −0.294373
\(178\) 6.71083 0.502998
\(179\) −17.0489 −1.27429 −0.637146 0.770743i \(-0.719885\pi\)
−0.637146 + 0.770743i \(0.719885\pi\)
\(180\) 0 0
\(181\) −11.0872 −0.824104 −0.412052 0.911160i \(-0.635188\pi\)
−0.412052 + 0.911160i \(0.635188\pi\)
\(182\) −4.91638 −0.364426
\(183\) −18.5089 −1.36821
\(184\) 7.49472 0.552518
\(185\) 0 0
\(186\) 25.7491 1.88802
\(187\) −4.81361 −0.352006
\(188\) −11.0192 −0.803655
\(189\) −5.39194 −0.392206
\(190\) 0 0
\(191\) −1.55416 −0.112455 −0.0562274 0.998418i \(-0.517907\pi\)
−0.0562274 + 0.998418i \(0.517907\pi\)
\(192\) −2.81361 −0.203055
\(193\) −12.9894 −0.935000 −0.467500 0.883993i \(-0.654845\pi\)
−0.467500 + 0.883993i \(0.654845\pi\)
\(194\) −0.338044 −0.0242702
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 0.710831 0.0506446 0.0253223 0.999679i \(-0.491939\pi\)
0.0253223 + 0.999679i \(0.491939\pi\)
\(198\) −4.91638 −0.349392
\(199\) 15.2197 1.07889 0.539447 0.842019i \(-0.318633\pi\)
0.539447 + 0.842019i \(0.318633\pi\)
\(200\) 0 0
\(201\) 10.2056 0.719844
\(202\) −16.5139 −1.16191
\(203\) −1.86751 −0.131073
\(204\) 13.5436 0.948241
\(205\) 0 0
\(206\) 5.83779 0.406738
\(207\) −36.8469 −2.56104
\(208\) 4.91638 0.340890
\(209\) 1.28917 0.0891737
\(210\) 0 0
\(211\) −12.0242 −0.827779 −0.413889 0.910327i \(-0.635830\pi\)
−0.413889 + 0.910327i \(0.635830\pi\)
\(212\) 9.74914 0.669574
\(213\) 0.951124 0.0651699
\(214\) −0.0488759 −0.00334109
\(215\) 0 0
\(216\) 5.39194 0.366875
\(217\) 9.15165 0.621255
\(218\) −0.240293 −0.0162747
\(219\) −38.5330 −2.60382
\(220\) 0 0
\(221\) −23.6655 −1.59192
\(222\) −19.0192 −1.27648
\(223\) −6.03831 −0.404355 −0.202178 0.979349i \(-0.564802\pi\)
−0.202178 + 0.979349i \(0.564802\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −21.0036 −1.39714
\(227\) −4.60303 −0.305514 −0.152757 0.988264i \(-0.548815\pi\)
−0.152757 + 0.988264i \(0.548815\pi\)
\(228\) −3.62721 −0.240218
\(229\) −5.42166 −0.358274 −0.179137 0.983824i \(-0.557330\pi\)
−0.179137 + 0.983824i \(0.557330\pi\)
\(230\) 0 0
\(231\) −2.81361 −0.185122
\(232\) 1.86751 0.122608
\(233\) 2.57834 0.168912 0.0844562 0.996427i \(-0.473085\pi\)
0.0844562 + 0.996427i \(0.473085\pi\)
\(234\) −24.1708 −1.58010
\(235\) 0 0
\(236\) 1.39194 0.0906078
\(237\) 17.6952 1.14943
\(238\) 4.81361 0.312020
\(239\) −21.0630 −1.36245 −0.681226 0.732073i \(-0.738553\pi\)
−0.681226 + 0.732073i \(0.738553\pi\)
\(240\) 0 0
\(241\) 21.0192 1.35396 0.676981 0.736000i \(-0.263288\pi\)
0.676981 + 0.736000i \(0.263288\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 14.9894 0.961573
\(244\) 6.57834 0.421135
\(245\) 0 0
\(246\) −34.9200 −2.22641
\(247\) 6.33804 0.403280
\(248\) −9.15165 −0.581130
\(249\) 27.4600 1.74021
\(250\) 0 0
\(251\) −27.4897 −1.73513 −0.867567 0.497320i \(-0.834317\pi\)
−0.867567 + 0.497320i \(0.834317\pi\)
\(252\) 4.91638 0.309703
\(253\) −7.49472 −0.471189
\(254\) 4.96526 0.311548
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.23025 0.513389 0.256694 0.966493i \(-0.417367\pi\)
0.256694 + 0.966493i \(0.417367\pi\)
\(258\) 7.39194 0.460202
\(259\) −6.75971 −0.420028
\(260\) 0 0
\(261\) −9.18137 −0.568313
\(262\) −12.1325 −0.749548
\(263\) −8.96526 −0.552821 −0.276411 0.961040i \(-0.589145\pi\)
−0.276411 + 0.961040i \(0.589145\pi\)
\(264\) 2.81361 0.173166
\(265\) 0 0
\(266\) −1.28917 −0.0790440
\(267\) 18.8816 1.15554
\(268\) −3.62721 −0.221567
\(269\) 7.56777 0.461415 0.230708 0.973023i \(-0.425896\pi\)
0.230708 + 0.973023i \(0.425896\pi\)
\(270\) 0 0
\(271\) 20.6167 1.25237 0.626186 0.779674i \(-0.284615\pi\)
0.626186 + 0.779674i \(0.284615\pi\)
\(272\) −4.81361 −0.291868
\(273\) −13.8328 −0.837197
\(274\) −16.7491 −1.01185
\(275\) 0 0
\(276\) 21.0872 1.26930
\(277\) −22.3517 −1.34298 −0.671491 0.741013i \(-0.734346\pi\)
−0.671491 + 0.741013i \(0.734346\pi\)
\(278\) 22.5819 1.35437
\(279\) 44.9930 2.69366
\(280\) 0 0
\(281\) −6.20555 −0.370192 −0.185096 0.982720i \(-0.559260\pi\)
−0.185096 + 0.982720i \(0.559260\pi\)
\(282\) −31.0036 −1.84624
\(283\) 10.6761 0.634628 0.317314 0.948321i \(-0.397219\pi\)
0.317314 + 0.948321i \(0.397219\pi\)
\(284\) −0.338044 −0.0200592
\(285\) 0 0
\(286\) −4.91638 −0.290712
\(287\) −12.4111 −0.732604
\(288\) −4.91638 −0.289701
\(289\) 6.17081 0.362989
\(290\) 0 0
\(291\) −0.951124 −0.0557559
\(292\) 13.6952 0.801454
\(293\) 12.2736 0.717030 0.358515 0.933524i \(-0.383283\pi\)
0.358515 + 0.933524i \(0.383283\pi\)
\(294\) 2.81361 0.164093
\(295\) 0 0
\(296\) 6.75971 0.392900
\(297\) −5.39194 −0.312872
\(298\) 13.4217 0.777496
\(299\) −36.8469 −2.13091
\(300\) 0 0
\(301\) 2.62721 0.151430
\(302\) 15.2544 0.877794
\(303\) −46.4635 −2.66926
\(304\) 1.28917 0.0739389
\(305\) 0 0
\(306\) 23.6655 1.35287
\(307\) −19.2544 −1.09891 −0.549454 0.835524i \(-0.685164\pi\)
−0.549454 + 0.835524i \(0.685164\pi\)
\(308\) 1.00000 0.0569803
\(309\) 16.4252 0.934399
\(310\) 0 0
\(311\) −1.36725 −0.0775295 −0.0387647 0.999248i \(-0.512342\pi\)
−0.0387647 + 0.999248i \(0.512342\pi\)
\(312\) 13.8328 0.783126
\(313\) 32.2822 1.82470 0.912348 0.409415i \(-0.134267\pi\)
0.912348 + 0.409415i \(0.134267\pi\)
\(314\) −8.50885 −0.480182
\(315\) 0 0
\(316\) −6.28917 −0.353793
\(317\) −11.3380 −0.636808 −0.318404 0.947955i \(-0.603147\pi\)
−0.318404 + 0.947955i \(0.603147\pi\)
\(318\) 27.4303 1.53821
\(319\) −1.86751 −0.104560
\(320\) 0 0
\(321\) −0.137518 −0.00767548
\(322\) 7.49472 0.417664
\(323\) −6.20555 −0.345286
\(324\) 0.421663 0.0234257
\(325\) 0 0
\(326\) 15.7250 0.870925
\(327\) −0.676089 −0.0373878
\(328\) 12.4111 0.685288
\(329\) −11.0192 −0.607506
\(330\) 0 0
\(331\) −32.7144 −1.79815 −0.899073 0.437799i \(-0.855758\pi\)
−0.899073 + 0.437799i \(0.855758\pi\)
\(332\) −9.75971 −0.535634
\(333\) −33.2333 −1.82117
\(334\) −14.6167 −0.799788
\(335\) 0 0
\(336\) −2.81361 −0.153495
\(337\) −18.5472 −1.01033 −0.505164 0.863023i \(-0.668568\pi\)
−0.505164 + 0.863023i \(0.668568\pi\)
\(338\) −11.1708 −0.607612
\(339\) −59.0958 −3.20964
\(340\) 0 0
\(341\) 9.15165 0.495590
\(342\) −6.33804 −0.342722
\(343\) 1.00000 0.0539949
\(344\) −2.62721 −0.141650
\(345\) 0 0
\(346\) 3.08362 0.165776
\(347\) −6.28917 −0.337620 −0.168810 0.985649i \(-0.553992\pi\)
−0.168810 + 0.985649i \(0.553992\pi\)
\(348\) 5.25443 0.281667
\(349\) −23.2283 −1.24338 −0.621691 0.783263i \(-0.713554\pi\)
−0.621691 + 0.783263i \(0.713554\pi\)
\(350\) 0 0
\(351\) −26.5089 −1.41494
\(352\) −1.00000 −0.0533002
\(353\) 9.21968 0.490714 0.245357 0.969433i \(-0.421095\pi\)
0.245357 + 0.969433i \(0.421095\pi\)
\(354\) 3.91638 0.208153
\(355\) 0 0
\(356\) −6.71083 −0.355673
\(357\) 13.5436 0.716803
\(358\) 17.0489 0.901061
\(359\) 28.0766 1.48183 0.740914 0.671600i \(-0.234393\pi\)
0.740914 + 0.671600i \(0.234393\pi\)
\(360\) 0 0
\(361\) −17.3380 −0.912529
\(362\) 11.0872 0.582730
\(363\) −2.81361 −0.147676
\(364\) 4.91638 0.257688
\(365\) 0 0
\(366\) 18.5089 0.967473
\(367\) −6.87807 −0.359032 −0.179516 0.983755i \(-0.557453\pi\)
−0.179516 + 0.983755i \(0.557453\pi\)
\(368\) −7.49472 −0.390689
\(369\) −61.0177 −3.17645
\(370\) 0 0
\(371\) 9.74914 0.506150
\(372\) −25.7491 −1.33503
\(373\) −10.3627 −0.536562 −0.268281 0.963341i \(-0.586456\pi\)
−0.268281 + 0.963341i \(0.586456\pi\)
\(374\) 4.81361 0.248906
\(375\) 0 0
\(376\) 11.0192 0.568270
\(377\) −9.18137 −0.472865
\(378\) 5.39194 0.277332
\(379\) −32.4494 −1.66681 −0.833407 0.552659i \(-0.813613\pi\)
−0.833407 + 0.552659i \(0.813613\pi\)
\(380\) 0 0
\(381\) 13.9703 0.715719
\(382\) 1.55416 0.0795176
\(383\) −30.7194 −1.56969 −0.784845 0.619693i \(-0.787257\pi\)
−0.784845 + 0.619693i \(0.787257\pi\)
\(384\) 2.81361 0.143581
\(385\) 0 0
\(386\) 12.9894 0.661145
\(387\) 12.9164 0.656577
\(388\) 0.338044 0.0171616
\(389\) −29.3169 −1.48643 −0.743213 0.669054i \(-0.766699\pi\)
−0.743213 + 0.669054i \(0.766699\pi\)
\(390\) 0 0
\(391\) 36.0766 1.82447
\(392\) −1.00000 −0.0505076
\(393\) −34.1361 −1.72194
\(394\) −0.710831 −0.0358112
\(395\) 0 0
\(396\) 4.91638 0.247057
\(397\) −22.1955 −1.11396 −0.556980 0.830526i \(-0.688040\pi\)
−0.556980 + 0.830526i \(0.688040\pi\)
\(398\) −15.2197 −0.762894
\(399\) −3.62721 −0.181588
\(400\) 0 0
\(401\) −20.2544 −1.01146 −0.505729 0.862692i \(-0.668776\pi\)
−0.505729 + 0.862692i \(0.668776\pi\)
\(402\) −10.2056 −0.509007
\(403\) 44.9930 2.24126
\(404\) 16.5139 0.821596
\(405\) 0 0
\(406\) 1.86751 0.0926827
\(407\) −6.75971 −0.335066
\(408\) −13.5436 −0.670508
\(409\) −22.1658 −1.09603 −0.548014 0.836469i \(-0.684616\pi\)
−0.548014 + 0.836469i \(0.684616\pi\)
\(410\) 0 0
\(411\) −47.1255 −2.32453
\(412\) −5.83779 −0.287607
\(413\) 1.39194 0.0684931
\(414\) 36.8469 1.81093
\(415\) 0 0
\(416\) −4.91638 −0.241045
\(417\) 63.5366 3.11140
\(418\) −1.28917 −0.0630553
\(419\) 36.5769 1.78690 0.893449 0.449165i \(-0.148278\pi\)
0.893449 + 0.449165i \(0.148278\pi\)
\(420\) 0 0
\(421\) −2.94108 −0.143339 −0.0716697 0.997428i \(-0.522833\pi\)
−0.0716697 + 0.997428i \(0.522833\pi\)
\(422\) 12.0242 0.585328
\(423\) −54.1744 −2.63405
\(424\) −9.74914 −0.473460
\(425\) 0 0
\(426\) −0.951124 −0.0460821
\(427\) 6.57834 0.318348
\(428\) 0.0488759 0.00236251
\(429\) −13.8328 −0.667852
\(430\) 0 0
\(431\) 6.53303 0.314685 0.157343 0.987544i \(-0.449707\pi\)
0.157343 + 0.987544i \(0.449707\pi\)
\(432\) −5.39194 −0.259420
\(433\) 16.0347 0.770581 0.385290 0.922795i \(-0.374101\pi\)
0.385290 + 0.922795i \(0.374101\pi\)
\(434\) −9.15165 −0.439293
\(435\) 0 0
\(436\) 0.240293 0.0115079
\(437\) −9.66196 −0.462194
\(438\) 38.5330 1.84118
\(439\) −14.2650 −0.680831 −0.340415 0.940275i \(-0.610568\pi\)
−0.340415 + 0.940275i \(0.610568\pi\)
\(440\) 0 0
\(441\) 4.91638 0.234113
\(442\) 23.6655 1.12565
\(443\) −13.9022 −0.660516 −0.330258 0.943891i \(-0.607136\pi\)
−0.330258 + 0.943891i \(0.607136\pi\)
\(444\) 19.0192 0.902609
\(445\) 0 0
\(446\) 6.03831 0.285922
\(447\) 37.7633 1.78614
\(448\) 1.00000 0.0472456
\(449\) 11.5089 0.543136 0.271568 0.962419i \(-0.412458\pi\)
0.271568 + 0.962419i \(0.412458\pi\)
\(450\) 0 0
\(451\) −12.4111 −0.584416
\(452\) 21.0036 0.987925
\(453\) 42.9200 2.01656
\(454\) 4.60303 0.216031
\(455\) 0 0
\(456\) 3.62721 0.169860
\(457\) −17.4600 −0.816743 −0.408372 0.912816i \(-0.633903\pi\)
−0.408372 + 0.912816i \(0.633903\pi\)
\(458\) 5.42166 0.253338
\(459\) 25.9547 1.21146
\(460\) 0 0
\(461\) 9.35363 0.435642 0.217821 0.975989i \(-0.430105\pi\)
0.217821 + 0.975989i \(0.430105\pi\)
\(462\) 2.81361 0.130901
\(463\) 16.3380 0.759293 0.379647 0.925132i \(-0.376046\pi\)
0.379647 + 0.925132i \(0.376046\pi\)
\(464\) −1.86751 −0.0866968
\(465\) 0 0
\(466\) −2.57834 −0.119439
\(467\) 3.55918 0.164699 0.0823496 0.996604i \(-0.473758\pi\)
0.0823496 + 0.996604i \(0.473758\pi\)
\(468\) 24.1708 1.11730
\(469\) −3.62721 −0.167489
\(470\) 0 0
\(471\) −23.9406 −1.10312
\(472\) −1.39194 −0.0640694
\(473\) 2.62721 0.120799
\(474\) −17.6952 −0.812770
\(475\) 0 0
\(476\) −4.81361 −0.220631
\(477\) 47.9305 2.19459
\(478\) 21.0630 0.963400
\(479\) −38.7144 −1.76891 −0.884453 0.466629i \(-0.845468\pi\)
−0.884453 + 0.466629i \(0.845468\pi\)
\(480\) 0 0
\(481\) −33.2333 −1.51531
\(482\) −21.0192 −0.957396
\(483\) 21.0872 0.959500
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −14.9894 −0.679935
\(487\) −14.8781 −0.674190 −0.337095 0.941471i \(-0.609444\pi\)
−0.337095 + 0.941471i \(0.609444\pi\)
\(488\) −6.57834 −0.297787
\(489\) 44.2439 2.00078
\(490\) 0 0
\(491\) 12.2927 0.554764 0.277382 0.960760i \(-0.410533\pi\)
0.277382 + 0.960760i \(0.410533\pi\)
\(492\) 34.9200 1.57431
\(493\) 8.98944 0.404864
\(494\) −6.33804 −0.285162
\(495\) 0 0
\(496\) 9.15165 0.410921
\(497\) −0.338044 −0.0151634
\(498\) −27.4600 −1.23051
\(499\) 20.0383 0.897038 0.448519 0.893773i \(-0.351952\pi\)
0.448519 + 0.893773i \(0.351952\pi\)
\(500\) 0 0
\(501\) −41.1255 −1.83735
\(502\) 27.4897 1.22693
\(503\) 37.6061 1.67677 0.838386 0.545077i \(-0.183499\pi\)
0.838386 + 0.545077i \(0.183499\pi\)
\(504\) −4.91638 −0.218993
\(505\) 0 0
\(506\) 7.49472 0.333181
\(507\) −31.4303 −1.39587
\(508\) −4.96526 −0.220298
\(509\) −6.04836 −0.268089 −0.134044 0.990975i \(-0.542796\pi\)
−0.134044 + 0.990975i \(0.542796\pi\)
\(510\) 0 0
\(511\) 13.6952 0.605842
\(512\) −1.00000 −0.0441942
\(513\) −6.95112 −0.306900
\(514\) −8.23025 −0.363021
\(515\) 0 0
\(516\) −7.39194 −0.325412
\(517\) −11.0192 −0.484622
\(518\) 6.75971 0.297004
\(519\) 8.67609 0.380838
\(520\) 0 0
\(521\) 7.90582 0.346360 0.173180 0.984890i \(-0.444596\pi\)
0.173180 + 0.984890i \(0.444596\pi\)
\(522\) 9.18137 0.401858
\(523\) −12.6413 −0.552767 −0.276384 0.961047i \(-0.589136\pi\)
−0.276384 + 0.961047i \(0.589136\pi\)
\(524\) 12.1325 0.530010
\(525\) 0 0
\(526\) 8.96526 0.390904
\(527\) −44.0524 −1.91895
\(528\) −2.81361 −0.122447
\(529\) 33.1708 1.44221
\(530\) 0 0
\(531\) 6.84333 0.296975
\(532\) 1.28917 0.0558925
\(533\) −61.0177 −2.64297
\(534\) −18.8816 −0.817088
\(535\) 0 0
\(536\) 3.62721 0.156672
\(537\) 47.9688 2.07001
\(538\) −7.56777 −0.326270
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −41.7980 −1.79704 −0.898519 0.438935i \(-0.855356\pi\)
−0.898519 + 0.438935i \(0.855356\pi\)
\(542\) −20.6167 −0.885561
\(543\) 31.1950 1.33871
\(544\) 4.81361 0.206382
\(545\) 0 0
\(546\) 13.8328 0.591988
\(547\) 3.63778 0.155540 0.0777700 0.996971i \(-0.475220\pi\)
0.0777700 + 0.996971i \(0.475220\pi\)
\(548\) 16.7491 0.715488
\(549\) 32.3416 1.38031
\(550\) 0 0
\(551\) −2.40753 −0.102564
\(552\) −21.0872 −0.897530
\(553\) −6.28917 −0.267443
\(554\) 22.3517 0.949631
\(555\) 0 0
\(556\) −22.5819 −0.957686
\(557\) −24.8852 −1.05442 −0.527210 0.849735i \(-0.676762\pi\)
−0.527210 + 0.849735i \(0.676762\pi\)
\(558\) −44.9930 −1.90471
\(559\) 12.9164 0.546305
\(560\) 0 0
\(561\) 13.5436 0.571811
\(562\) 6.20555 0.261765
\(563\) 5.10831 0.215290 0.107645 0.994189i \(-0.465669\pi\)
0.107645 + 0.994189i \(0.465669\pi\)
\(564\) 31.0036 1.30549
\(565\) 0 0
\(566\) −10.6761 −0.448749
\(567\) 0.421663 0.0177082
\(568\) 0.338044 0.0141840
\(569\) 21.4116 0.897622 0.448811 0.893627i \(-0.351848\pi\)
0.448811 + 0.893627i \(0.351848\pi\)
\(570\) 0 0
\(571\) −9.85746 −0.412522 −0.206261 0.978497i \(-0.566130\pi\)
−0.206261 + 0.978497i \(0.566130\pi\)
\(572\) 4.91638 0.205564
\(573\) 4.37279 0.182676
\(574\) 12.4111 0.518029
\(575\) 0 0
\(576\) 4.91638 0.204849
\(577\) −5.77332 −0.240347 −0.120173 0.992753i \(-0.538345\pi\)
−0.120173 + 0.992753i \(0.538345\pi\)
\(578\) −6.17081 −0.256672
\(579\) 36.5472 1.51885
\(580\) 0 0
\(581\) −9.75971 −0.404901
\(582\) 0.951124 0.0394254
\(583\) 9.74914 0.403768
\(584\) −13.6952 −0.566713
\(585\) 0 0
\(586\) −12.2736 −0.507017
\(587\) −28.1361 −1.16130 −0.580650 0.814153i \(-0.697201\pi\)
−0.580650 + 0.814153i \(0.697201\pi\)
\(588\) −2.81361 −0.116031
\(589\) 11.7980 0.486129
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −6.75971 −0.277822
\(593\) −20.8917 −0.857919 −0.428959 0.903324i \(-0.641120\pi\)
−0.428959 + 0.903324i \(0.641120\pi\)
\(594\) 5.39194 0.221234
\(595\) 0 0
\(596\) −13.4217 −0.549773
\(597\) −42.8222 −1.75260
\(598\) 36.8469 1.50678
\(599\) 38.7839 1.58467 0.792333 0.610088i \(-0.208866\pi\)
0.792333 + 0.610088i \(0.208866\pi\)
\(600\) 0 0
\(601\) 41.0857 1.67592 0.837961 0.545730i \(-0.183748\pi\)
0.837961 + 0.545730i \(0.183748\pi\)
\(602\) −2.62721 −0.107077
\(603\) −17.8328 −0.726206
\(604\) −15.2544 −0.620694
\(605\) 0 0
\(606\) 46.4635 1.88745
\(607\) 9.36222 0.380001 0.190000 0.981784i \(-0.439151\pi\)
0.190000 + 0.981784i \(0.439151\pi\)
\(608\) −1.28917 −0.0522827
\(609\) 5.25443 0.212920
\(610\) 0 0
\(611\) −54.1744 −2.19166
\(612\) −23.6655 −0.956622
\(613\) 10.4705 0.422901 0.211451 0.977389i \(-0.432181\pi\)
0.211451 + 0.977389i \(0.432181\pi\)
\(614\) 19.2544 0.777045
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −15.9200 −0.640913 −0.320457 0.947263i \(-0.603836\pi\)
−0.320457 + 0.947263i \(0.603836\pi\)
\(618\) −16.4252 −0.660720
\(619\) 17.5280 0.704510 0.352255 0.935904i \(-0.385415\pi\)
0.352255 + 0.935904i \(0.385415\pi\)
\(620\) 0 0
\(621\) 40.4111 1.62164
\(622\) 1.36725 0.0548216
\(623\) −6.71083 −0.268864
\(624\) −13.8328 −0.553754
\(625\) 0 0
\(626\) −32.2822 −1.29026
\(627\) −3.62721 −0.144857
\(628\) 8.50885 0.339540
\(629\) 32.5386 1.29740
\(630\) 0 0
\(631\) 22.8122 0.908138 0.454069 0.890967i \(-0.349972\pi\)
0.454069 + 0.890967i \(0.349972\pi\)
\(632\) 6.28917 0.250170
\(633\) 33.8313 1.34467
\(634\) 11.3380 0.450291
\(635\) 0 0
\(636\) −27.4303 −1.08768
\(637\) 4.91638 0.194794
\(638\) 1.86751 0.0739353
\(639\) −1.66196 −0.0657459
\(640\) 0 0
\(641\) 10.7350 0.424007 0.212004 0.977269i \(-0.432001\pi\)
0.212004 + 0.977269i \(0.432001\pi\)
\(642\) 0.137518 0.00542739
\(643\) −14.2736 −0.562895 −0.281448 0.959577i \(-0.590815\pi\)
−0.281448 + 0.959577i \(0.590815\pi\)
\(644\) −7.49472 −0.295333
\(645\) 0 0
\(646\) 6.20555 0.244154
\(647\) −46.9597 −1.84618 −0.923089 0.384588i \(-0.874344\pi\)
−0.923089 + 0.384588i \(0.874344\pi\)
\(648\) −0.421663 −0.0165645
\(649\) 1.39194 0.0546386
\(650\) 0 0
\(651\) −25.7491 −1.00919
\(652\) −15.7250 −0.615837
\(653\) −21.9164 −0.857654 −0.428827 0.903387i \(-0.641073\pi\)
−0.428827 + 0.903387i \(0.641073\pi\)
\(654\) 0.676089 0.0264372
\(655\) 0 0
\(656\) −12.4111 −0.484572
\(657\) 67.3311 2.62683
\(658\) 11.0192 0.429571
\(659\) 12.5819 0.490122 0.245061 0.969508i \(-0.421192\pi\)
0.245061 + 0.969508i \(0.421192\pi\)
\(660\) 0 0
\(661\) 12.4705 0.485048 0.242524 0.970145i \(-0.422025\pi\)
0.242524 + 0.970145i \(0.422025\pi\)
\(662\) 32.7144 1.27148
\(663\) 66.5855 2.58597
\(664\) 9.75971 0.378750
\(665\) 0 0
\(666\) 33.2333 1.28776
\(667\) 13.9964 0.541944
\(668\) 14.6167 0.565535
\(669\) 16.9894 0.656850
\(670\) 0 0
\(671\) 6.57834 0.253954
\(672\) 2.81361 0.108537
\(673\) 28.4705 1.09746 0.548729 0.836000i \(-0.315112\pi\)
0.548729 + 0.836000i \(0.315112\pi\)
\(674\) 18.5472 0.714410
\(675\) 0 0
\(676\) 11.1708 0.429646
\(677\) 42.1411 1.61961 0.809807 0.586697i \(-0.199572\pi\)
0.809807 + 0.586697i \(0.199572\pi\)
\(678\) 59.0958 2.26956
\(679\) 0.338044 0.0129730
\(680\) 0 0
\(681\) 12.9511 0.496288
\(682\) −9.15165 −0.350435
\(683\) 1.21611 0.0465333 0.0232666 0.999729i \(-0.492593\pi\)
0.0232666 + 0.999729i \(0.492593\pi\)
\(684\) 6.33804 0.242341
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 15.2544 0.581993
\(688\) 2.62721 0.100162
\(689\) 47.9305 1.82601
\(690\) 0 0
\(691\) −21.4514 −0.816049 −0.408025 0.912971i \(-0.633782\pi\)
−0.408025 + 0.912971i \(0.633782\pi\)
\(692\) −3.08362 −0.117222
\(693\) 4.91638 0.186758
\(694\) 6.28917 0.238734
\(695\) 0 0
\(696\) −5.25443 −0.199169
\(697\) 59.7422 2.26290
\(698\) 23.2283 0.879203
\(699\) −7.25443 −0.274388
\(700\) 0 0
\(701\) −37.3275 −1.40984 −0.704920 0.709287i \(-0.749017\pi\)
−0.704920 + 0.709287i \(0.749017\pi\)
\(702\) 26.5089 1.00051
\(703\) −8.71440 −0.328670
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −9.21968 −0.346987
\(707\) 16.5139 0.621068
\(708\) −3.91638 −0.147187
\(709\) 24.1602 0.907357 0.453679 0.891165i \(-0.350111\pi\)
0.453679 + 0.891165i \(0.350111\pi\)
\(710\) 0 0
\(711\) −30.9200 −1.15959
\(712\) 6.71083 0.251499
\(713\) −68.5890 −2.56868
\(714\) −13.5436 −0.506856
\(715\) 0 0
\(716\) −17.0489 −0.637146
\(717\) 59.2630 2.21322
\(718\) −28.0766 −1.04781
\(719\) 20.8816 0.778754 0.389377 0.921079i \(-0.372690\pi\)
0.389377 + 0.921079i \(0.372690\pi\)
\(720\) 0 0
\(721\) −5.83779 −0.217411
\(722\) 17.3380 0.645255
\(723\) −59.1396 −2.19943
\(724\) −11.0872 −0.412052
\(725\) 0 0
\(726\) 2.81361 0.104423
\(727\) −48.8172 −1.81053 −0.905264 0.424849i \(-0.860327\pi\)
−0.905264 + 0.424849i \(0.860327\pi\)
\(728\) −4.91638 −0.182213
\(729\) −43.4394 −1.60887
\(730\) 0 0
\(731\) −12.6464 −0.467743
\(732\) −18.5089 −0.684107
\(733\) −20.6600 −0.763094 −0.381547 0.924349i \(-0.624609\pi\)
−0.381547 + 0.924349i \(0.624609\pi\)
\(734\) 6.87807 0.253874
\(735\) 0 0
\(736\) 7.49472 0.276259
\(737\) −3.62721 −0.133610
\(738\) 61.0177 2.24609
\(739\) −8.46355 −0.311337 −0.155668 0.987809i \(-0.549753\pi\)
−0.155668 + 0.987809i \(0.549753\pi\)
\(740\) 0 0
\(741\) −17.8328 −0.655103
\(742\) −9.74914 −0.357902
\(743\) 9.27861 0.340399 0.170200 0.985410i \(-0.445559\pi\)
0.170200 + 0.985410i \(0.445559\pi\)
\(744\) 25.7491 0.944009
\(745\) 0 0
\(746\) 10.3627 0.379407
\(747\) −47.9824 −1.75559
\(748\) −4.81361 −0.176003
\(749\) 0.0488759 0.00178589
\(750\) 0 0
\(751\) 14.9164 0.544306 0.272153 0.962254i \(-0.412264\pi\)
0.272153 + 0.962254i \(0.412264\pi\)
\(752\) −11.0192 −0.401827
\(753\) 77.3452 2.81862
\(754\) 9.18137 0.334366
\(755\) 0 0
\(756\) −5.39194 −0.196103
\(757\) 22.4736 0.816817 0.408408 0.912799i \(-0.366084\pi\)
0.408408 + 0.912799i \(0.366084\pi\)
\(758\) 32.4494 1.17862
\(759\) 21.0872 0.765416
\(760\) 0 0
\(761\) 16.2836 0.590281 0.295141 0.955454i \(-0.404634\pi\)
0.295141 + 0.955454i \(0.404634\pi\)
\(762\) −13.9703 −0.506090
\(763\) 0.240293 0.00869918
\(764\) −1.55416 −0.0562274
\(765\) 0 0
\(766\) 30.7194 1.10994
\(767\) 6.84333 0.247098
\(768\) −2.81361 −0.101527
\(769\) 10.5783 0.381465 0.190732 0.981642i \(-0.438914\pi\)
0.190732 + 0.981642i \(0.438914\pi\)
\(770\) 0 0
\(771\) −23.1567 −0.833967
\(772\) −12.9894 −0.467500
\(773\) −6.90276 −0.248275 −0.124138 0.992265i \(-0.539616\pi\)
−0.124138 + 0.992265i \(0.539616\pi\)
\(774\) −12.9164 −0.464270
\(775\) 0 0
\(776\) −0.338044 −0.0121351
\(777\) 19.0192 0.682308
\(778\) 29.3169 1.05106
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −0.338044 −0.0120962
\(782\) −36.0766 −1.29010
\(783\) 10.0695 0.359854
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 34.1361 1.21759
\(787\) −7.25443 −0.258592 −0.129296 0.991606i \(-0.541272\pi\)
−0.129296 + 0.991606i \(0.541272\pi\)
\(788\) 0.710831 0.0253223
\(789\) 25.2247 0.898023
\(790\) 0 0
\(791\) 21.0036 0.746801
\(792\) −4.91638 −0.174696
\(793\) 32.3416 1.14848
\(794\) 22.1955 0.787689
\(795\) 0 0
\(796\) 15.2197 0.539447
\(797\) 35.7733 1.26716 0.633578 0.773679i \(-0.281586\pi\)
0.633578 + 0.773679i \(0.281586\pi\)
\(798\) 3.62721 0.128402
\(799\) 53.0419 1.87649
\(800\) 0 0
\(801\) −32.9930 −1.16575
\(802\) 20.2544 0.715209
\(803\) 13.6952 0.483295
\(804\) 10.2056 0.359922
\(805\) 0 0
\(806\) −44.9930 −1.58481
\(807\) −21.2927 −0.749540
\(808\) −16.5139 −0.580956
\(809\) −32.8122 −1.15361 −0.576807 0.816880i \(-0.695702\pi\)
−0.576807 + 0.816880i \(0.695702\pi\)
\(810\) 0 0
\(811\) 6.31335 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(812\) −1.86751 −0.0655366
\(813\) −58.0071 −2.03440
\(814\) 6.75971 0.236928
\(815\) 0 0
\(816\) 13.5436 0.474121
\(817\) 3.38692 0.118493
\(818\) 22.1658 0.775008
\(819\) 24.1708 0.844596
\(820\) 0 0
\(821\) −30.5225 −1.06524 −0.532621 0.846354i \(-0.678793\pi\)
−0.532621 + 0.846354i \(0.678793\pi\)
\(822\) 47.1255 1.64369
\(823\) 9.57885 0.333898 0.166949 0.985966i \(-0.446609\pi\)
0.166949 + 0.985966i \(0.446609\pi\)
\(824\) 5.83779 0.203369
\(825\) 0 0
\(826\) −1.39194 −0.0484319
\(827\) 44.7250 1.55524 0.777620 0.628735i \(-0.216427\pi\)
0.777620 + 0.628735i \(0.216427\pi\)
\(828\) −36.8469 −1.28052
\(829\) −38.7910 −1.34727 −0.673634 0.739065i \(-0.735268\pi\)
−0.673634 + 0.739065i \(0.735268\pi\)
\(830\) 0 0
\(831\) 62.8888 2.18159
\(832\) 4.91638 0.170445
\(833\) −4.81361 −0.166782
\(834\) −63.5366 −2.20009
\(835\) 0 0
\(836\) 1.28917 0.0445868
\(837\) −49.3452 −1.70562
\(838\) −36.5769 −1.26353
\(839\) 24.6861 0.852260 0.426130 0.904662i \(-0.359877\pi\)
0.426130 + 0.904662i \(0.359877\pi\)
\(840\) 0 0
\(841\) −25.5124 −0.879739
\(842\) 2.94108 0.101356
\(843\) 17.4600 0.601354
\(844\) −12.0242 −0.413889
\(845\) 0 0
\(846\) 54.1744 1.86255
\(847\) 1.00000 0.0343604
\(848\) 9.74914 0.334787
\(849\) −30.0383 −1.03091
\(850\) 0 0
\(851\) 50.6621 1.73667
\(852\) 0.951124 0.0325850
\(853\) −4.68468 −0.160400 −0.0802002 0.996779i \(-0.525556\pi\)
−0.0802002 + 0.996779i \(0.525556\pi\)
\(854\) −6.57834 −0.225106
\(855\) 0 0
\(856\) −0.0488759 −0.00167054
\(857\) −13.9022 −0.474892 −0.237446 0.971401i \(-0.576310\pi\)
−0.237446 + 0.971401i \(0.576310\pi\)
\(858\) 13.8328 0.472243
\(859\) 25.4585 0.868634 0.434317 0.900760i \(-0.356990\pi\)
0.434317 + 0.900760i \(0.356990\pi\)
\(860\) 0 0
\(861\) 34.9200 1.19007
\(862\) −6.53303 −0.222516
\(863\) −11.1814 −0.380618 −0.190309 0.981724i \(-0.560949\pi\)
−0.190309 + 0.981724i \(0.560949\pi\)
\(864\) 5.39194 0.183438
\(865\) 0 0
\(866\) −16.0347 −0.544883
\(867\) −17.3622 −0.589652
\(868\) 9.15165 0.310627
\(869\) −6.28917 −0.213345
\(870\) 0 0
\(871\) −17.8328 −0.604240
\(872\) −0.240293 −0.00813734
\(873\) 1.66196 0.0562487
\(874\) 9.66196 0.326821
\(875\) 0 0
\(876\) −38.5330 −1.30191
\(877\) −9.11189 −0.307687 −0.153843 0.988095i \(-0.549165\pi\)
−0.153843 + 0.988095i \(0.549165\pi\)
\(878\) 14.2650 0.481420
\(879\) −34.5330 −1.16477
\(880\) 0 0
\(881\) 28.2786 0.952730 0.476365 0.879248i \(-0.341954\pi\)
0.476365 + 0.879248i \(0.341954\pi\)
\(882\) −4.91638 −0.165543
\(883\) 43.6061 1.46746 0.733731 0.679440i \(-0.237777\pi\)
0.733731 + 0.679440i \(0.237777\pi\)
\(884\) −23.6655 −0.795958
\(885\) 0 0
\(886\) 13.9022 0.467055
\(887\) −22.8505 −0.767244 −0.383622 0.923490i \(-0.625323\pi\)
−0.383622 + 0.923490i \(0.625323\pi\)
\(888\) −19.0192 −0.638241
\(889\) −4.96526 −0.166529
\(890\) 0 0
\(891\) 0.421663 0.0141262
\(892\) −6.03831 −0.202178
\(893\) −14.2056 −0.475371
\(894\) −37.7633 −1.26299
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 103.673 3.46153
\(898\) −11.5089 −0.384055
\(899\) −17.0908 −0.570009
\(900\) 0 0
\(901\) −46.9285 −1.56342
\(902\) 12.4111 0.413244
\(903\) −7.39194 −0.245988
\(904\) −21.0036 −0.698568
\(905\) 0 0
\(906\) −42.9200 −1.42592
\(907\) −1.78440 −0.0592501 −0.0296251 0.999561i \(-0.509431\pi\)
−0.0296251 + 0.999561i \(0.509431\pi\)
\(908\) −4.60303 −0.152757
\(909\) 81.1885 2.69285
\(910\) 0 0
\(911\) −13.8539 −0.459000 −0.229500 0.973309i \(-0.573709\pi\)
−0.229500 + 0.973309i \(0.573709\pi\)
\(912\) −3.62721 −0.120109
\(913\) −9.75971 −0.322999
\(914\) 17.4600 0.577525
\(915\) 0 0
\(916\) −5.42166 −0.179137
\(917\) 12.1325 0.400650
\(918\) −25.9547 −0.856633
\(919\) −31.4741 −1.03824 −0.519118 0.854703i \(-0.673739\pi\)
−0.519118 + 0.854703i \(0.673739\pi\)
\(920\) 0 0
\(921\) 54.1744 1.78511
\(922\) −9.35363 −0.308045
\(923\) −1.66196 −0.0547039
\(924\) −2.81361 −0.0925609
\(925\) 0 0
\(926\) −16.3380 −0.536901
\(927\) −28.7008 −0.942657
\(928\) 1.86751 0.0613039
\(929\) −45.9341 −1.50705 −0.753524 0.657420i \(-0.771648\pi\)
−0.753524 + 0.657420i \(0.771648\pi\)
\(930\) 0 0
\(931\) 1.28917 0.0422508
\(932\) 2.57834 0.0844562
\(933\) 3.84690 0.125942
\(934\) −3.55918 −0.116460
\(935\) 0 0
\(936\) −24.1708 −0.790048
\(937\) −20.2141 −0.660367 −0.330184 0.943917i \(-0.607111\pi\)
−0.330184 + 0.943917i \(0.607111\pi\)
\(938\) 3.62721 0.118433
\(939\) −90.8293 −2.96410
\(940\) 0 0
\(941\) 26.3713 0.859681 0.429840 0.902905i \(-0.358570\pi\)
0.429840 + 0.902905i \(0.358570\pi\)
\(942\) 23.9406 0.780026
\(943\) 93.0177 3.02907
\(944\) 1.39194 0.0453039
\(945\) 0 0
\(946\) −2.62721 −0.0854181
\(947\) 24.1149 0.783630 0.391815 0.920044i \(-0.371847\pi\)
0.391815 + 0.920044i \(0.371847\pi\)
\(948\) 17.6952 0.574715
\(949\) 67.3311 2.18566
\(950\) 0 0
\(951\) 31.9008 1.03445
\(952\) 4.81361 0.156010
\(953\) −50.1361 −1.62407 −0.812033 0.583611i \(-0.801639\pi\)
−0.812033 + 0.583611i \(0.801639\pi\)
\(954\) −47.9305 −1.55181
\(955\) 0 0
\(956\) −21.0630 −0.681226
\(957\) 5.25443 0.169851
\(958\) 38.7144 1.25081
\(959\) 16.7491 0.540858
\(960\) 0 0
\(961\) 52.7527 1.70170
\(962\) 33.2333 1.07148
\(963\) 0.240293 0.00774332
\(964\) 21.0192 0.676981
\(965\) 0 0
\(966\) −21.0872 −0.678469
\(967\) 39.9305 1.28408 0.642039 0.766672i \(-0.278089\pi\)
0.642039 + 0.766672i \(0.278089\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 17.4600 0.560895
\(970\) 0 0
\(971\) 7.38335 0.236943 0.118471 0.992957i \(-0.462201\pi\)
0.118471 + 0.992957i \(0.462201\pi\)
\(972\) 14.9894 0.480786
\(973\) −22.5819 −0.723943
\(974\) 14.8781 0.476724
\(975\) 0 0
\(976\) 6.57834 0.210567
\(977\) −10.4111 −0.333081 −0.166540 0.986035i \(-0.553260\pi\)
−0.166540 + 0.986035i \(0.553260\pi\)
\(978\) −44.2439 −1.41476
\(979\) −6.71083 −0.214479
\(980\) 0 0
\(981\) 1.18137 0.0377183
\(982\) −12.2927 −0.392277
\(983\) −6.10278 −0.194648 −0.0973241 0.995253i \(-0.531028\pi\)
−0.0973241 + 0.995253i \(0.531028\pi\)
\(984\) −34.9200 −1.11321
\(985\) 0 0
\(986\) −8.98944 −0.286282
\(987\) 31.0036 0.986855
\(988\) 6.33804 0.201640
\(989\) −19.6902 −0.626113
\(990\) 0 0
\(991\) −40.6550 −1.29145 −0.645724 0.763571i \(-0.723444\pi\)
−0.645724 + 0.763571i \(0.723444\pi\)
\(992\) −9.15165 −0.290565
\(993\) 92.0455 2.92097
\(994\) 0.338044 0.0107221
\(995\) 0 0
\(996\) 27.4600 0.870103
\(997\) 51.0575 1.61701 0.808503 0.588492i \(-0.200278\pi\)
0.808503 + 0.588492i \(0.200278\pi\)
\(998\) −20.0383 −0.634302
\(999\) 36.4480 1.15316
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 3850.2.a.bs.1.1 3
5.2 odd 4 3850.2.c.bb.1849.3 6
5.3 odd 4 3850.2.c.bb.1849.4 6
5.4 even 2 3850.2.a.bv.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3850.2.a.bs.1.1 3 1.1 even 1 trivial
3850.2.a.bv.1.3 yes 3 5.4 even 2
3850.2.c.bb.1849.3 6 5.2 odd 4
3850.2.c.bb.1849.4 6 5.3 odd 4