Properties

Label 1815.2.a.l.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $3$
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] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, 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("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.772866\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.772866 q^{2} +1.00000 q^{3} -1.40268 q^{4} +1.00000 q^{5} -0.772866 q^{6} +1.62981 q^{7} +2.62981 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.772866 q^{2} +1.00000 q^{3} -1.40268 q^{4} +1.00000 q^{5} -0.772866 q^{6} +1.62981 q^{7} +2.62981 q^{8} +1.00000 q^{9} -0.772866 q^{10} -1.40268 q^{12} -4.80536 q^{13} -1.25963 q^{14} +1.00000 q^{15} +0.772866 q^{16} -2.17554 q^{17} -0.772866 q^{18} +5.17554 q^{19} -1.40268 q^{20} +1.62981 q^{21} +4.62981 q^{23} +2.62981 q^{24} +1.00000 q^{25} +3.71390 q^{26} +1.00000 q^{27} -2.28610 q^{28} -2.45427 q^{29} -0.772866 q^{30} +5.80536 q^{31} -5.85695 q^{32} +1.68140 q^{34} +1.62981 q^{35} -1.40268 q^{36} +8.88944 q^{37} -4.00000 q^{38} -4.80536 q^{39} +2.62981 q^{40} +5.09146 q^{41} -1.25963 q^{42} -1.54573 q^{43} +1.00000 q^{45} -3.57822 q^{46} +9.72128 q^{47} +0.772866 q^{48} -4.34371 q^{49} -0.772866 q^{50} -2.17554 q^{51} +6.74037 q^{52} -5.26701 q^{53} -0.772866 q^{54} +4.28610 q^{56} +5.17554 q^{57} +1.89682 q^{58} -7.15645 q^{59} -1.40268 q^{60} -14.2596 q^{61} -4.48676 q^{62} +1.62981 q^{63} +2.98090 q^{64} -4.80536 q^{65} -4.72128 q^{67} +3.05159 q^{68} +4.62981 q^{69} -1.25963 q^{70} +13.4426 q^{71} +2.62981 q^{72} -1.17554 q^{73} -6.87034 q^{74} +1.00000 q^{75} -7.25963 q^{76} +3.71390 q^{78} +4.89682 q^{79} +0.772866 q^{80} +1.00000 q^{81} -3.93502 q^{82} -4.00000 q^{83} -2.28610 q^{84} -2.17554 q^{85} +1.19464 q^{86} -2.45427 q^{87} +11.5457 q^{89} -0.772866 q^{90} -7.83184 q^{91} -6.49414 q^{92} +5.80536 q^{93} -7.51324 q^{94} +5.17554 q^{95} -5.85695 q^{96} +4.72128 q^{97} +3.35710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} + 12 q^{14} + 3 q^{15} + q^{16} + 4 q^{17} - q^{18} + 5 q^{19} + 5 q^{20} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{27} - 20 q^{28} - 10 q^{29} - q^{30} - q^{31} - 11 q^{32} + 9 q^{34} - 3 q^{35} + 5 q^{36} + 3 q^{37} - 12 q^{38} + 4 q^{39} + 10 q^{41} + 12 q^{42} - 2 q^{43} + 3 q^{45} + 9 q^{46} + 16 q^{47} + q^{48} + 8 q^{49} - q^{50} + 4 q^{51} + 36 q^{52} - q^{54} + 26 q^{56} + 5 q^{57} - 18 q^{58} + 18 q^{59} + 5 q^{60} - 27 q^{61} + q^{62} - 3 q^{63} - 20 q^{64} + 4 q^{65} - q^{67} + 21 q^{68} + 6 q^{69} + 12 q^{70} + 14 q^{71} + 7 q^{73} + 32 q^{74} + 3 q^{75} - 6 q^{76} - 2 q^{78} - 9 q^{79} + q^{80} + 3 q^{81} - 46 q^{82} - 12 q^{83} - 20 q^{84} + 4 q^{85} + 22 q^{86} - 10 q^{87} + 32 q^{89} - q^{90} - 34 q^{91} - 5 q^{92} - q^{93} - 37 q^{94} + 5 q^{95} - 11 q^{96} + q^{97} - 57 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.772866 −0.546498 −0.273249 0.961943i \(-0.588098\pi\)
−0.273249 + 0.961943i \(0.588098\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.40268 −0.701339
\(5\) 1.00000 0.447214
\(6\) −0.772866 −0.315521
\(7\) 1.62981 0.616012 0.308006 0.951384i \(-0.400338\pi\)
0.308006 + 0.951384i \(0.400338\pi\)
\(8\) 2.62981 0.929779
\(9\) 1.00000 0.333333
\(10\) −0.772866 −0.244402
\(11\) 0 0
\(12\) −1.40268 −0.404918
\(13\) −4.80536 −1.33277 −0.666383 0.745609i \(-0.732158\pi\)
−0.666383 + 0.745609i \(0.732158\pi\)
\(14\) −1.25963 −0.336649
\(15\) 1.00000 0.258199
\(16\) 0.772866 0.193216
\(17\) −2.17554 −0.527647 −0.263824 0.964571i \(-0.584984\pi\)
−0.263824 + 0.964571i \(0.584984\pi\)
\(18\) −0.772866 −0.182166
\(19\) 5.17554 1.18735 0.593676 0.804704i \(-0.297676\pi\)
0.593676 + 0.804704i \(0.297676\pi\)
\(20\) −1.40268 −0.313649
\(21\) 1.62981 0.355654
\(22\) 0 0
\(23\) 4.62981 0.965383 0.482691 0.875791i \(-0.339659\pi\)
0.482691 + 0.875791i \(0.339659\pi\)
\(24\) 2.62981 0.536808
\(25\) 1.00000 0.200000
\(26\) 3.71390 0.728355
\(27\) 1.00000 0.192450
\(28\) −2.28610 −0.432033
\(29\) −2.45427 −0.455746 −0.227873 0.973691i \(-0.573177\pi\)
−0.227873 + 0.973691i \(0.573177\pi\)
\(30\) −0.772866 −0.141105
\(31\) 5.80536 1.04267 0.521337 0.853351i \(-0.325434\pi\)
0.521337 + 0.853351i \(0.325434\pi\)
\(32\) −5.85695 −1.03537
\(33\) 0 0
\(34\) 1.68140 0.288358
\(35\) 1.62981 0.275489
\(36\) −1.40268 −0.233780
\(37\) 8.88944 1.46141 0.730707 0.682691i \(-0.239190\pi\)
0.730707 + 0.682691i \(0.239190\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.80536 −0.769473
\(40\) 2.62981 0.415810
\(41\) 5.09146 0.795153 0.397576 0.917569i \(-0.369851\pi\)
0.397576 + 0.917569i \(0.369851\pi\)
\(42\) −1.25963 −0.194365
\(43\) −1.54573 −0.235722 −0.117861 0.993030i \(-0.537604\pi\)
−0.117861 + 0.993030i \(0.537604\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −3.57822 −0.527580
\(47\) 9.72128 1.41799 0.708997 0.705212i \(-0.249148\pi\)
0.708997 + 0.705212i \(0.249148\pi\)
\(48\) 0.772866 0.111554
\(49\) −4.34371 −0.620530
\(50\) −0.772866 −0.109300
\(51\) −2.17554 −0.304637
\(52\) 6.74037 0.934722
\(53\) −5.26701 −0.723479 −0.361739 0.932279i \(-0.617817\pi\)
−0.361739 + 0.932279i \(0.617817\pi\)
\(54\) −0.772866 −0.105174
\(55\) 0 0
\(56\) 4.28610 0.572755
\(57\) 5.17554 0.685518
\(58\) 1.89682 0.249065
\(59\) −7.15645 −0.931690 −0.465845 0.884866i \(-0.654250\pi\)
−0.465845 + 0.884866i \(0.654250\pi\)
\(60\) −1.40268 −0.181085
\(61\) −14.2596 −1.82576 −0.912879 0.408230i \(-0.866146\pi\)
−0.912879 + 0.408230i \(0.866146\pi\)
\(62\) −4.48676 −0.569819
\(63\) 1.62981 0.205337
\(64\) 2.98090 0.372613
\(65\) −4.80536 −0.596031
\(66\) 0 0
\(67\) −4.72128 −0.576796 −0.288398 0.957511i \(-0.593123\pi\)
−0.288398 + 0.957511i \(0.593123\pi\)
\(68\) 3.05159 0.370060
\(69\) 4.62981 0.557364
\(70\) −1.25963 −0.150554
\(71\) 13.4426 1.59534 0.797669 0.603096i \(-0.206066\pi\)
0.797669 + 0.603096i \(0.206066\pi\)
\(72\) 2.62981 0.309926
\(73\) −1.17554 −0.137587 −0.0687935 0.997631i \(-0.521915\pi\)
−0.0687935 + 0.997631i \(0.521915\pi\)
\(74\) −6.87034 −0.798661
\(75\) 1.00000 0.115470
\(76\) −7.25963 −0.832736
\(77\) 0 0
\(78\) 3.71390 0.420516
\(79\) 4.89682 0.550935 0.275468 0.961310i \(-0.411167\pi\)
0.275468 + 0.961310i \(0.411167\pi\)
\(80\) 0.772866 0.0864090
\(81\) 1.00000 0.111111
\(82\) −3.93502 −0.434550
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.28610 −0.249434
\(85\) −2.17554 −0.235971
\(86\) 1.19464 0.128822
\(87\) −2.45427 −0.263125
\(88\) 0 0
\(89\) 11.5457 1.22385 0.611923 0.790918i \(-0.290396\pi\)
0.611923 + 0.790918i \(0.290396\pi\)
\(90\) −0.772866 −0.0814672
\(91\) −7.83184 −0.820999
\(92\) −6.49414 −0.677061
\(93\) 5.80536 0.601988
\(94\) −7.51324 −0.774931
\(95\) 5.17554 0.531000
\(96\) −5.85695 −0.597772
\(97\) 4.72128 0.479373 0.239686 0.970850i \(-0.422955\pi\)
0.239686 + 0.970850i \(0.422955\pi\)
\(98\) 3.35710 0.339119
\(99\) 0 0
\(100\) −1.40268 −0.140268
\(101\) 13.6107 1.35432 0.677158 0.735837i \(-0.263211\pi\)
0.677158 + 0.735837i \(0.263211\pi\)
\(102\) 1.68140 0.166484
\(103\) 14.6181 1.44036 0.720182 0.693785i \(-0.244058\pi\)
0.720182 + 0.693785i \(0.244058\pi\)
\(104\) −12.6372 −1.23918
\(105\) 1.62981 0.159053
\(106\) 4.07069 0.395380
\(107\) 19.3320 1.86889 0.934447 0.356102i \(-0.115894\pi\)
0.934447 + 0.356102i \(0.115894\pi\)
\(108\) −1.40268 −0.134973
\(109\) −8.78626 −0.841571 −0.420786 0.907160i \(-0.638246\pi\)
−0.420786 + 0.907160i \(0.638246\pi\)
\(110\) 0 0
\(111\) 8.88944 0.843748
\(112\) 1.25963 0.119024
\(113\) 12.5266 1.17841 0.589203 0.807985i \(-0.299442\pi\)
0.589203 + 0.807985i \(0.299442\pi\)
\(114\) −4.00000 −0.374634
\(115\) 4.62981 0.431732
\(116\) 3.44255 0.319633
\(117\) −4.80536 −0.444255
\(118\) 5.53097 0.509167
\(119\) −3.54573 −0.325037
\(120\) 2.62981 0.240068
\(121\) 0 0
\(122\) 11.0208 0.997774
\(123\) 5.09146 0.459082
\(124\) −8.14305 −0.731268
\(125\) 1.00000 0.0894427
\(126\) −1.25963 −0.112216
\(127\) 17.2405 1.52985 0.764925 0.644120i \(-0.222776\pi\)
0.764925 + 0.644120i \(0.222776\pi\)
\(128\) 9.41006 0.831740
\(129\) −1.54573 −0.136094
\(130\) 3.71390 0.325730
\(131\) 15.6107 1.36391 0.681957 0.731392i \(-0.261129\pi\)
0.681957 + 0.731392i \(0.261129\pi\)
\(132\) 0 0
\(133\) 8.43517 0.731422
\(134\) 3.64891 0.315218
\(135\) 1.00000 0.0860663
\(136\) −5.72128 −0.490595
\(137\) −10.6948 −0.913718 −0.456859 0.889539i \(-0.651026\pi\)
−0.456859 + 0.889539i \(0.651026\pi\)
\(138\) −3.57822 −0.304599
\(139\) 6.98090 0.592112 0.296056 0.955171i \(-0.404328\pi\)
0.296056 + 0.955171i \(0.404328\pi\)
\(140\) −2.28610 −0.193211
\(141\) 9.72128 0.818679
\(142\) −10.3893 −0.871849
\(143\) 0 0
\(144\) 0.772866 0.0644055
\(145\) −2.45427 −0.203816
\(146\) 0.908538 0.0751911
\(147\) −4.34371 −0.358263
\(148\) −12.4690 −1.02495
\(149\) 3.89682 0.319240 0.159620 0.987179i \(-0.448973\pi\)
0.159620 + 0.987179i \(0.448973\pi\)
\(150\) −0.772866 −0.0631042
\(151\) 5.72128 0.465591 0.232795 0.972526i \(-0.425213\pi\)
0.232795 + 0.972526i \(0.425213\pi\)
\(152\) 13.6107 1.10397
\(153\) −2.17554 −0.175882
\(154\) 0 0
\(155\) 5.80536 0.466298
\(156\) 6.74037 0.539662
\(157\) −15.2405 −1.21633 −0.608163 0.793812i \(-0.708093\pi\)
−0.608163 + 0.793812i \(0.708093\pi\)
\(158\) −3.78458 −0.301085
\(159\) −5.26701 −0.417701
\(160\) −5.85695 −0.463032
\(161\) 7.54573 0.594687
\(162\) −0.772866 −0.0607221
\(163\) −17.2405 −1.35038 −0.675191 0.737643i \(-0.735939\pi\)
−0.675191 + 0.737643i \(0.735939\pi\)
\(164\) −7.14169 −0.557672
\(165\) 0 0
\(166\) 3.09146 0.239944
\(167\) −5.53835 −0.428570 −0.214285 0.976771i \(-0.568742\pi\)
−0.214285 + 0.976771i \(0.568742\pi\)
\(168\) 4.28610 0.330680
\(169\) 10.0915 0.776266
\(170\) 1.68140 0.128958
\(171\) 5.17554 0.395784
\(172\) 2.16816 0.165321
\(173\) −18.7022 −1.42190 −0.710950 0.703242i \(-0.751735\pi\)
−0.710950 + 0.703242i \(0.751735\pi\)
\(174\) 1.89682 0.143798
\(175\) 1.62981 0.123202
\(176\) 0 0
\(177\) −7.15645 −0.537911
\(178\) −8.92330 −0.668829
\(179\) 1.19464 0.0892918 0.0446459 0.999003i \(-0.485784\pi\)
0.0446459 + 0.999003i \(0.485784\pi\)
\(180\) −1.40268 −0.104550
\(181\) 15.3437 1.14049 0.570244 0.821475i \(-0.306848\pi\)
0.570244 + 0.821475i \(0.306848\pi\)
\(182\) 6.05296 0.448675
\(183\) −14.2596 −1.05410
\(184\) 12.1755 0.897593
\(185\) 8.88944 0.653565
\(186\) −4.48676 −0.328985
\(187\) 0 0
\(188\) −13.6358 −0.994495
\(189\) 1.62981 0.118551
\(190\) −4.00000 −0.290191
\(191\) 15.7139 1.13702 0.568509 0.822677i \(-0.307521\pi\)
0.568509 + 0.822677i \(0.307521\pi\)
\(192\) 2.98090 0.215128
\(193\) −11.2405 −0.809111 −0.404555 0.914513i \(-0.632574\pi\)
−0.404555 + 0.914513i \(0.632574\pi\)
\(194\) −3.64891 −0.261977
\(195\) −4.80536 −0.344119
\(196\) 6.09283 0.435202
\(197\) 2.74037 0.195244 0.0976218 0.995224i \(-0.468876\pi\)
0.0976218 + 0.995224i \(0.468876\pi\)
\(198\) 0 0
\(199\) −25.0650 −1.77681 −0.888405 0.459061i \(-0.848186\pi\)
−0.888405 + 0.459061i \(0.848186\pi\)
\(200\) 2.62981 0.185956
\(201\) −4.72128 −0.333013
\(202\) −10.5193 −0.740132
\(203\) −4.00000 −0.280745
\(204\) 3.05159 0.213654
\(205\) 5.09146 0.355603
\(206\) −11.2978 −0.787157
\(207\) 4.62981 0.321794
\(208\) −3.71390 −0.257512
\(209\) 0 0
\(210\) −1.25963 −0.0869225
\(211\) −17.9735 −1.23735 −0.618674 0.785648i \(-0.712330\pi\)
−0.618674 + 0.785648i \(0.712330\pi\)
\(212\) 7.38792 0.507404
\(213\) 13.4426 0.921068
\(214\) −14.9410 −1.02135
\(215\) −1.54573 −0.105418
\(216\) 2.62981 0.178936
\(217\) 9.46165 0.642299
\(218\) 6.79060 0.459917
\(219\) −1.17554 −0.0794359
\(220\) 0 0
\(221\) 10.4543 0.703230
\(222\) −6.87034 −0.461107
\(223\) −12.7863 −0.856231 −0.428116 0.903724i \(-0.640822\pi\)
−0.428116 + 0.903724i \(0.640822\pi\)
\(224\) −9.54573 −0.637801
\(225\) 1.00000 0.0666667
\(226\) −9.68140 −0.643997
\(227\) −14.2787 −0.947712 −0.473856 0.880602i \(-0.657138\pi\)
−0.473856 + 0.880602i \(0.657138\pi\)
\(228\) −7.25963 −0.480781
\(229\) −25.9692 −1.71609 −0.858046 0.513573i \(-0.828322\pi\)
−0.858046 + 0.513573i \(0.828322\pi\)
\(230\) −3.57822 −0.235941
\(231\) 0 0
\(232\) −6.45427 −0.423744
\(233\) 10.5648 0.692125 0.346062 0.938211i \(-0.387519\pi\)
0.346062 + 0.938211i \(0.387519\pi\)
\(234\) 3.71390 0.242785
\(235\) 9.72128 0.634146
\(236\) 10.0382 0.653431
\(237\) 4.89682 0.318083
\(238\) 2.74037 0.177632
\(239\) 6.45427 0.417492 0.208746 0.977970i \(-0.433062\pi\)
0.208746 + 0.977970i \(0.433062\pi\)
\(240\) 0.772866 0.0498883
\(241\) 25.6181 1.65021 0.825103 0.564982i \(-0.191117\pi\)
0.825103 + 0.564982i \(0.191117\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 20.0017 1.28048
\(245\) −4.34371 −0.277509
\(246\) −3.93502 −0.250887
\(247\) −24.8703 −1.58246
\(248\) 15.2670 0.969456
\(249\) −4.00000 −0.253490
\(250\) −0.772866 −0.0488803
\(251\) 4.80536 0.303311 0.151656 0.988433i \(-0.451539\pi\)
0.151656 + 0.988433i \(0.451539\pi\)
\(252\) −2.28610 −0.144011
\(253\) 0 0
\(254\) −13.3246 −0.836060
\(255\) −2.17554 −0.136238
\(256\) −13.2345 −0.827157
\(257\) 11.7863 0.735207 0.367603 0.929983i \(-0.380178\pi\)
0.367603 + 0.929983i \(0.380178\pi\)
\(258\) 1.19464 0.0743752
\(259\) 14.4881 0.900248
\(260\) 6.74037 0.418020
\(261\) −2.45427 −0.151915
\(262\) −12.0650 −0.745377
\(263\) −12.4235 −0.766063 −0.383031 0.923735i \(-0.625120\pi\)
−0.383031 + 0.923735i \(0.625120\pi\)
\(264\) 0 0
\(265\) −5.26701 −0.323550
\(266\) −6.51925 −0.399721
\(267\) 11.5457 0.706587
\(268\) 6.62243 0.404529
\(269\) 31.2214 1.90360 0.951802 0.306713i \(-0.0992291\pi\)
0.951802 + 0.306713i \(0.0992291\pi\)
\(270\) −0.772866 −0.0470351
\(271\) 25.5002 1.54902 0.774512 0.632559i \(-0.217995\pi\)
0.774512 + 0.632559i \(0.217995\pi\)
\(272\) −1.68140 −0.101950
\(273\) −7.83184 −0.474004
\(274\) 8.26564 0.499346
\(275\) 0 0
\(276\) −6.49414 −0.390901
\(277\) −24.6033 −1.47827 −0.739136 0.673557i \(-0.764766\pi\)
−0.739136 + 0.673557i \(0.764766\pi\)
\(278\) −5.39530 −0.323589
\(279\) 5.80536 0.347558
\(280\) 4.28610 0.256144
\(281\) −11.5457 −0.688761 −0.344380 0.938830i \(-0.611911\pi\)
−0.344380 + 0.938830i \(0.611911\pi\)
\(282\) −7.51324 −0.447407
\(283\) −6.09884 −0.362539 −0.181269 0.983434i \(-0.558021\pi\)
−0.181269 + 0.983434i \(0.558021\pi\)
\(284\) −18.8556 −1.11887
\(285\) 5.17554 0.306573
\(286\) 0 0
\(287\) 8.29813 0.489823
\(288\) −5.85695 −0.345124
\(289\) −12.2670 −0.721589
\(290\) 1.89682 0.111385
\(291\) 4.72128 0.276766
\(292\) 1.64891 0.0964952
\(293\) −5.82446 −0.340268 −0.170134 0.985421i \(-0.554420\pi\)
−0.170134 + 0.985421i \(0.554420\pi\)
\(294\) 3.35710 0.195790
\(295\) −7.15645 −0.416664
\(296\) 23.3776 1.35879
\(297\) 0 0
\(298\) −3.01172 −0.174464
\(299\) −22.2479 −1.28663
\(300\) −1.40268 −0.0809837
\(301\) −2.51925 −0.145207
\(302\) −4.42178 −0.254445
\(303\) 13.6107 0.781915
\(304\) 4.00000 0.229416
\(305\) −14.2596 −0.816504
\(306\) 1.68140 0.0961194
\(307\) 10.2670 0.585969 0.292985 0.956117i \(-0.405352\pi\)
0.292985 + 0.956117i \(0.405352\pi\)
\(308\) 0 0
\(309\) 14.6181 0.831594
\(310\) −4.48676 −0.254831
\(311\) −0.454269 −0.0257592 −0.0128796 0.999917i \(-0.504100\pi\)
−0.0128796 + 0.999917i \(0.504100\pi\)
\(312\) −12.6372 −0.715440
\(313\) 0.103180 0.00583208 0.00291604 0.999996i \(-0.499072\pi\)
0.00291604 + 0.999996i \(0.499072\pi\)
\(314\) 11.7789 0.664721
\(315\) 1.62981 0.0918296
\(316\) −6.86867 −0.386393
\(317\) −19.0841 −1.07187 −0.535934 0.844260i \(-0.680041\pi\)
−0.535934 + 0.844260i \(0.680041\pi\)
\(318\) 4.07069 0.228273
\(319\) 0 0
\(320\) 2.98090 0.166637
\(321\) 19.3320 1.07901
\(322\) −5.83184 −0.324995
\(323\) −11.2596 −0.626502
\(324\) −1.40268 −0.0779266
\(325\) −4.80536 −0.266553
\(326\) 13.3246 0.737982
\(327\) −8.78626 −0.485881
\(328\) 13.3896 0.739317
\(329\) 15.8439 0.873500
\(330\) 0 0
\(331\) −9.45427 −0.519654 −0.259827 0.965655i \(-0.583665\pi\)
−0.259827 + 0.965655i \(0.583665\pi\)
\(332\) 5.61072 0.307928
\(333\) 8.88944 0.487138
\(334\) 4.28040 0.234213
\(335\) −4.72128 −0.257951
\(336\) 1.25963 0.0687183
\(337\) −16.9544 −0.923566 −0.461783 0.886993i \(-0.652790\pi\)
−0.461783 + 0.886993i \(0.652790\pi\)
\(338\) −7.79934 −0.424228
\(339\) 12.5266 0.680353
\(340\) 3.05159 0.165496
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −18.4881 −0.998265
\(344\) −4.06498 −0.219169
\(345\) 4.62981 0.249261
\(346\) 14.4543 0.777066
\(347\) 28.0576 1.50621 0.753106 0.657900i \(-0.228555\pi\)
0.753106 + 0.657900i \(0.228555\pi\)
\(348\) 3.44255 0.184540
\(349\) 17.5340 0.938574 0.469287 0.883046i \(-0.344511\pi\)
0.469287 + 0.883046i \(0.344511\pi\)
\(350\) −1.25963 −0.0673299
\(351\) −4.80536 −0.256491
\(352\) 0 0
\(353\) −20.4884 −1.09049 −0.545245 0.838277i \(-0.683563\pi\)
−0.545245 + 0.838277i \(0.683563\pi\)
\(354\) 5.53097 0.293968
\(355\) 13.4426 0.713457
\(356\) −16.1950 −0.858331
\(357\) −3.54573 −0.187660
\(358\) −0.923298 −0.0487978
\(359\) −28.6640 −1.51283 −0.756414 0.654094i \(-0.773050\pi\)
−0.756414 + 0.654094i \(0.773050\pi\)
\(360\) 2.62981 0.138603
\(361\) 7.78626 0.409803
\(362\) −11.8586 −0.623275
\(363\) 0 0
\(364\) 10.9856 0.575799
\(365\) −1.17554 −0.0615308
\(366\) 11.0208 0.576065
\(367\) −22.5990 −1.17966 −0.589829 0.807528i \(-0.700805\pi\)
−0.589829 + 0.807528i \(0.700805\pi\)
\(368\) 3.57822 0.186528
\(369\) 5.09146 0.265051
\(370\) −6.87034 −0.357172
\(371\) −8.58424 −0.445671
\(372\) −8.14305 −0.422198
\(373\) 26.0841 1.35058 0.675291 0.737551i \(-0.264018\pi\)
0.675291 + 0.737551i \(0.264018\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 25.5651 1.31842
\(377\) 11.7936 0.607403
\(378\) −1.25963 −0.0647882
\(379\) 9.16383 0.470714 0.235357 0.971909i \(-0.424374\pi\)
0.235357 + 0.971909i \(0.424374\pi\)
\(380\) −7.25963 −0.372411
\(381\) 17.2405 0.883259
\(382\) −12.1447 −0.621378
\(383\) −18.3511 −0.937697 −0.468848 0.883279i \(-0.655331\pi\)
−0.468848 + 0.883279i \(0.655331\pi\)
\(384\) 9.41006 0.480205
\(385\) 0 0
\(386\) 8.68742 0.442178
\(387\) −1.54573 −0.0785739
\(388\) −6.62243 −0.336203
\(389\) −12.5990 −0.638795 −0.319397 0.947621i \(-0.603480\pi\)
−0.319397 + 0.947621i \(0.603480\pi\)
\(390\) 3.71390 0.188060
\(391\) −10.0724 −0.509381
\(392\) −11.4231 −0.576956
\(393\) 15.6107 0.787456
\(394\) −2.11794 −0.106700
\(395\) 4.89682 0.246386
\(396\) 0 0
\(397\) −29.6948 −1.49034 −0.745170 0.666875i \(-0.767631\pi\)
−0.745170 + 0.666875i \(0.767631\pi\)
\(398\) 19.3719 0.971024
\(399\) 8.43517 0.422287
\(400\) 0.772866 0.0386433
\(401\) 39.0533 1.95023 0.975114 0.221706i \(-0.0711626\pi\)
0.975114 + 0.221706i \(0.0711626\pi\)
\(402\) 3.64891 0.181991
\(403\) −27.8968 −1.38964
\(404\) −19.0915 −0.949836
\(405\) 1.00000 0.0496904
\(406\) 3.09146 0.153427
\(407\) 0 0
\(408\) −5.72128 −0.283245
\(409\) −29.5193 −1.45963 −0.729817 0.683643i \(-0.760395\pi\)
−0.729817 + 0.683643i \(0.760395\pi\)
\(410\) −3.93502 −0.194337
\(411\) −10.6948 −0.527535
\(412\) −20.5045 −1.01018
\(413\) −11.6637 −0.573932
\(414\) −3.57822 −0.175860
\(415\) −4.00000 −0.196352
\(416\) 28.1447 1.37991
\(417\) 6.98090 0.341856
\(418\) 0 0
\(419\) −22.8703 −1.11729 −0.558645 0.829407i \(-0.688678\pi\)
−0.558645 + 0.829407i \(0.688678\pi\)
\(420\) −2.28610 −0.111550
\(421\) −21.6033 −1.05288 −0.526441 0.850212i \(-0.676474\pi\)
−0.526441 + 0.850212i \(0.676474\pi\)
\(422\) 13.8911 0.676209
\(423\) 9.72128 0.472665
\(424\) −13.8512 −0.672676
\(425\) −2.17554 −0.105529
\(426\) −10.3893 −0.503362
\(427\) −23.2405 −1.12469
\(428\) −27.1166 −1.31073
\(429\) 0 0
\(430\) 1.19464 0.0576108
\(431\) −34.4161 −1.65776 −0.828882 0.559424i \(-0.811023\pi\)
−0.828882 + 0.559424i \(0.811023\pi\)
\(432\) 0.772866 0.0371845
\(433\) −20.2938 −0.975258 −0.487629 0.873051i \(-0.662138\pi\)
−0.487629 + 0.873051i \(0.662138\pi\)
\(434\) −7.31258 −0.351015
\(435\) −2.45427 −0.117673
\(436\) 12.3243 0.590227
\(437\) 23.9618 1.14625
\(438\) 0.908538 0.0434116
\(439\) −7.28610 −0.347747 −0.173873 0.984768i \(-0.555628\pi\)
−0.173873 + 0.984768i \(0.555628\pi\)
\(440\) 0 0
\(441\) −4.34371 −0.206843
\(442\) −8.07974 −0.384314
\(443\) 7.25963 0.344915 0.172458 0.985017i \(-0.444829\pi\)
0.172458 + 0.985017i \(0.444829\pi\)
\(444\) −12.4690 −0.591754
\(445\) 11.5457 0.547320
\(446\) 9.88206 0.467929
\(447\) 3.89682 0.184313
\(448\) 4.85831 0.229534
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) −0.772866 −0.0364332
\(451\) 0 0
\(452\) −17.5708 −0.826463
\(453\) 5.72128 0.268809
\(454\) 11.0355 0.517923
\(455\) −7.83184 −0.367162
\(456\) 13.6107 0.637380
\(457\) 3.61072 0.168902 0.0844511 0.996428i \(-0.473086\pi\)
0.0844511 + 0.996428i \(0.473086\pi\)
\(458\) 20.0707 0.937842
\(459\) −2.17554 −0.101546
\(460\) −6.49414 −0.302791
\(461\) 20.1032 0.936298 0.468149 0.883649i \(-0.344921\pi\)
0.468149 + 0.883649i \(0.344921\pi\)
\(462\) 0 0
\(463\) −20.6874 −0.961426 −0.480713 0.876878i \(-0.659622\pi\)
−0.480713 + 0.876878i \(0.659622\pi\)
\(464\) −1.89682 −0.0880577
\(465\) 5.80536 0.269217
\(466\) −8.16519 −0.378245
\(467\) −0.850934 −0.0393765 −0.0196883 0.999806i \(-0.506267\pi\)
−0.0196883 + 0.999806i \(0.506267\pi\)
\(468\) 6.74037 0.311574
\(469\) −7.69480 −0.355313
\(470\) −7.51324 −0.346560
\(471\) −15.2405 −0.702246
\(472\) −18.8201 −0.866266
\(473\) 0 0
\(474\) −3.78458 −0.173832
\(475\) 5.17554 0.237470
\(476\) 4.97352 0.227961
\(477\) −5.26701 −0.241160
\(478\) −4.98828 −0.228159
\(479\) 28.4013 1.29769 0.648845 0.760921i \(-0.275253\pi\)
0.648845 + 0.760921i \(0.275253\pi\)
\(480\) −5.85695 −0.267332
\(481\) −42.7169 −1.94772
\(482\) −19.7993 −0.901835
\(483\) 7.54573 0.343343
\(484\) 0 0
\(485\) 4.72128 0.214382
\(486\) −0.772866 −0.0350579
\(487\) 4.06498 0.184202 0.0921010 0.995750i \(-0.470642\pi\)
0.0921010 + 0.995750i \(0.470642\pi\)
\(488\) −37.5002 −1.69755
\(489\) −17.2405 −0.779644
\(490\) 3.35710 0.151658
\(491\) −27.9618 −1.26190 −0.630949 0.775824i \(-0.717334\pi\)
−0.630949 + 0.775824i \(0.717334\pi\)
\(492\) −7.14169 −0.321972
\(493\) 5.33937 0.240473
\(494\) 19.2214 0.864813
\(495\) 0 0
\(496\) 4.48676 0.201462
\(497\) 21.9088 0.982746
\(498\) 3.09146 0.138532
\(499\) 1.13735 0.0509147 0.0254574 0.999676i \(-0.491896\pi\)
0.0254574 + 0.999676i \(0.491896\pi\)
\(500\) −1.40268 −0.0627297
\(501\) −5.53835 −0.247435
\(502\) −3.71390 −0.165759
\(503\) −21.8512 −0.974299 −0.487149 0.873319i \(-0.661963\pi\)
−0.487149 + 0.873319i \(0.661963\pi\)
\(504\) 4.28610 0.190918
\(505\) 13.6107 0.605669
\(506\) 0 0
\(507\) 10.0915 0.448178
\(508\) −24.1829 −1.07294
\(509\) 18.2331 0.808170 0.404085 0.914721i \(-0.367590\pi\)
0.404085 + 0.914721i \(0.367590\pi\)
\(510\) 1.68140 0.0744538
\(511\) −1.91592 −0.0847552
\(512\) −8.59162 −0.379699
\(513\) 5.17554 0.228506
\(514\) −9.10919 −0.401789
\(515\) 14.6181 0.644150
\(516\) 2.16816 0.0954481
\(517\) 0 0
\(518\) −11.1974 −0.491984
\(519\) −18.7022 −0.820934
\(520\) −12.6372 −0.554178
\(521\) 37.1035 1.62553 0.812767 0.582589i \(-0.197960\pi\)
0.812767 + 0.582589i \(0.197960\pi\)
\(522\) 1.89682 0.0830216
\(523\) 15.1106 0.660739 0.330369 0.943852i \(-0.392827\pi\)
0.330369 + 0.943852i \(0.392827\pi\)
\(524\) −21.8968 −0.956567
\(525\) 1.62981 0.0711309
\(526\) 9.60166 0.418652
\(527\) −12.6298 −0.550163
\(528\) 0 0
\(529\) −1.56483 −0.0680360
\(530\) 4.07069 0.176819
\(531\) −7.15645 −0.310563
\(532\) −11.8318 −0.512975
\(533\) −24.4663 −1.05975
\(534\) −8.92330 −0.386149
\(535\) 19.3320 0.835795
\(536\) −12.4161 −0.536293
\(537\) 1.19464 0.0515526
\(538\) −24.1300 −1.04032
\(539\) 0 0
\(540\) −1.40268 −0.0603617
\(541\) −3.27439 −0.140777 −0.0703884 0.997520i \(-0.522424\pi\)
−0.0703884 + 0.997520i \(0.522424\pi\)
\(542\) −19.7082 −0.846539
\(543\) 15.3437 0.658462
\(544\) 12.7420 0.546311
\(545\) −8.78626 −0.376362
\(546\) 6.05296 0.259043
\(547\) 18.2479 0.780224 0.390112 0.920767i \(-0.372436\pi\)
0.390112 + 0.920767i \(0.372436\pi\)
\(548\) 15.0014 0.640827
\(549\) −14.2596 −0.608586
\(550\) 0 0
\(551\) −12.7022 −0.541131
\(552\) 12.1755 0.518226
\(553\) 7.98090 0.339382
\(554\) 19.0151 0.807873
\(555\) 8.88944 0.377336
\(556\) −9.79196 −0.415272
\(557\) −16.3585 −0.693131 −0.346565 0.938026i \(-0.612652\pi\)
−0.346565 + 0.938026i \(0.612652\pi\)
\(558\) −4.48676 −0.189940
\(559\) 7.42779 0.314162
\(560\) 1.25963 0.0532289
\(561\) 0 0
\(562\) 8.92330 0.376407
\(563\) 31.2214 1.31583 0.657913 0.753094i \(-0.271439\pi\)
0.657913 + 0.753094i \(0.271439\pi\)
\(564\) −13.6358 −0.574172
\(565\) 12.5266 0.526999
\(566\) 4.71359 0.198127
\(567\) 1.62981 0.0684457
\(568\) 35.3514 1.48331
\(569\) 8.97352 0.376190 0.188095 0.982151i \(-0.439769\pi\)
0.188095 + 0.982151i \(0.439769\pi\)
\(570\) −4.00000 −0.167542
\(571\) −19.5990 −0.820193 −0.410096 0.912042i \(-0.634505\pi\)
−0.410096 + 0.912042i \(0.634505\pi\)
\(572\) 0 0
\(573\) 15.7139 0.656457
\(574\) −6.41334 −0.267688
\(575\) 4.62981 0.193077
\(576\) 2.98090 0.124204
\(577\) 10.4204 0.433807 0.216904 0.976193i \(-0.430404\pi\)
0.216904 + 0.976193i \(0.430404\pi\)
\(578\) 9.48075 0.394347
\(579\) −11.2405 −0.467140
\(580\) 3.44255 0.142944
\(581\) −6.51925 −0.270464
\(582\) −3.64891 −0.151252
\(583\) 0 0
\(584\) −3.09146 −0.127926
\(585\) −4.80536 −0.198677
\(586\) 4.50152 0.185956
\(587\) −4.59162 −0.189516 −0.0947582 0.995500i \(-0.530208\pi\)
−0.0947582 + 0.995500i \(0.530208\pi\)
\(588\) 6.09283 0.251264
\(589\) 30.0459 1.23802
\(590\) 5.53097 0.227706
\(591\) 2.74037 0.112724
\(592\) 6.87034 0.282369
\(593\) −16.1829 −0.664553 −0.332277 0.943182i \(-0.607817\pi\)
−0.332277 + 0.943182i \(0.607817\pi\)
\(594\) 0 0
\(595\) −3.54573 −0.145361
\(596\) −5.46599 −0.223896
\(597\) −25.0650 −1.02584
\(598\) 17.1946 0.703141
\(599\) −40.0120 −1.63485 −0.817424 0.576037i \(-0.804598\pi\)
−0.817424 + 0.576037i \(0.804598\pi\)
\(600\) 2.62981 0.107362
\(601\) 36.9162 1.50584 0.752922 0.658110i \(-0.228644\pi\)
0.752922 + 0.658110i \(0.228644\pi\)
\(602\) 1.94704 0.0793556
\(603\) −4.72128 −0.192265
\(604\) −8.02511 −0.326537
\(605\) 0 0
\(606\) −10.5193 −0.427315
\(607\) 19.7139 0.800162 0.400081 0.916480i \(-0.368982\pi\)
0.400081 + 0.916480i \(0.368982\pi\)
\(608\) −30.3129 −1.22935
\(609\) −4.00000 −0.162088
\(610\) 11.0208 0.446218
\(611\) −46.7142 −1.88985
\(612\) 3.05159 0.123353
\(613\) −3.73299 −0.150774 −0.0753871 0.997154i \(-0.524019\pi\)
−0.0753871 + 0.997154i \(0.524019\pi\)
\(614\) −7.93502 −0.320231
\(615\) 5.09146 0.205308
\(616\) 0 0
\(617\) −24.8851 −1.00184 −0.500918 0.865495i \(-0.667004\pi\)
−0.500918 + 0.865495i \(0.667004\pi\)
\(618\) −11.2978 −0.454465
\(619\) 29.9470 1.20367 0.601837 0.798619i \(-0.294436\pi\)
0.601837 + 0.798619i \(0.294436\pi\)
\(620\) −8.14305 −0.327033
\(621\) 4.62981 0.185788
\(622\) 0.351089 0.0140774
\(623\) 18.8174 0.753903
\(624\) −3.71390 −0.148675
\(625\) 1.00000 0.0400000
\(626\) −0.0797444 −0.00318723
\(627\) 0 0
\(628\) 21.3776 0.853058
\(629\) −19.3394 −0.771111
\(630\) −1.25963 −0.0501847
\(631\) −10.2787 −0.409190 −0.204595 0.978847i \(-0.565588\pi\)
−0.204595 + 0.978847i \(0.565588\pi\)
\(632\) 12.8777 0.512248
\(633\) −17.9735 −0.714383
\(634\) 14.7494 0.585775
\(635\) 17.2405 0.684170
\(636\) 7.38792 0.292950
\(637\) 20.8731 0.827021
\(638\) 0 0
\(639\) 13.4426 0.531779
\(640\) 9.41006 0.371965
\(641\) −13.8586 −0.547383 −0.273691 0.961818i \(-0.588245\pi\)
−0.273691 + 0.961818i \(0.588245\pi\)
\(642\) −14.9410 −0.589675
\(643\) −16.4499 −0.648722 −0.324361 0.945933i \(-0.605149\pi\)
−0.324361 + 0.945933i \(0.605149\pi\)
\(644\) −10.5842 −0.417077
\(645\) −1.54573 −0.0608631
\(646\) 8.70218 0.342383
\(647\) −15.3172 −0.602182 −0.301091 0.953595i \(-0.597351\pi\)
−0.301091 + 0.953595i \(0.597351\pi\)
\(648\) 2.62981 0.103309
\(649\) 0 0
\(650\) 3.71390 0.145671
\(651\) 9.46165 0.370831
\(652\) 24.1829 0.947076
\(653\) −37.2214 −1.45659 −0.728294 0.685265i \(-0.759686\pi\)
−0.728294 + 0.685265i \(0.759686\pi\)
\(654\) 6.79060 0.265533
\(655\) 15.6107 0.609961
\(656\) 3.93502 0.153637
\(657\) −1.17554 −0.0458624
\(658\) −12.2452 −0.477367
\(659\) −33.1564 −1.29159 −0.645796 0.763510i \(-0.723474\pi\)
−0.645796 + 0.763510i \(0.723474\pi\)
\(660\) 0 0
\(661\) 14.2670 0.554922 0.277461 0.960737i \(-0.410507\pi\)
0.277461 + 0.960737i \(0.410507\pi\)
\(662\) 7.30688 0.283990
\(663\) 10.4543 0.406010
\(664\) −10.5193 −0.408226
\(665\) 8.43517 0.327102
\(666\) −6.87034 −0.266220
\(667\) −11.3628 −0.439970
\(668\) 7.76853 0.300573
\(669\) −12.7863 −0.494345
\(670\) 3.64891 0.140970
\(671\) 0 0
\(672\) −9.54573 −0.368235
\(673\) −19.9427 −0.768735 −0.384367 0.923180i \(-0.625580\pi\)
−0.384367 + 0.923180i \(0.625580\pi\)
\(674\) 13.1035 0.504728
\(675\) 1.00000 0.0384900
\(676\) −14.1551 −0.544426
\(677\) −25.6255 −0.984867 −0.492434 0.870350i \(-0.663893\pi\)
−0.492434 + 0.870350i \(0.663893\pi\)
\(678\) −9.68140 −0.371812
\(679\) 7.69480 0.295299
\(680\) −5.72128 −0.219401
\(681\) −14.2787 −0.547162
\(682\) 0 0
\(683\) −0.870342 −0.0333027 −0.0166514 0.999861i \(-0.505301\pi\)
−0.0166514 + 0.999861i \(0.505301\pi\)
\(684\) −7.25963 −0.277579
\(685\) −10.6948 −0.408627
\(686\) 14.2888 0.545550
\(687\) −25.9692 −0.990786
\(688\) −1.19464 −0.0455453
\(689\) 25.3099 0.964229
\(690\) −3.57822 −0.136221
\(691\) 4.32461 0.164516 0.0822580 0.996611i \(-0.473787\pi\)
0.0822580 + 0.996611i \(0.473787\pi\)
\(692\) 26.2331 0.997235
\(693\) 0 0
\(694\) −21.6848 −0.823142
\(695\) 6.98090 0.264801
\(696\) −6.45427 −0.244648
\(697\) −11.0767 −0.419560
\(698\) −13.5514 −0.512929
\(699\) 10.5648 0.399599
\(700\) −2.28610 −0.0864066
\(701\) −5.02648 −0.189847 −0.0949237 0.995485i \(-0.530261\pi\)
−0.0949237 + 0.995485i \(0.530261\pi\)
\(702\) 3.71390 0.140172
\(703\) 46.0077 1.73521
\(704\) 0 0
\(705\) 9.72128 0.366124
\(706\) 15.8348 0.595951
\(707\) 22.1829 0.834275
\(708\) 10.0382 0.377259
\(709\) 32.1373 1.20694 0.603472 0.797384i \(-0.293784\pi\)
0.603472 + 0.797384i \(0.293784\pi\)
\(710\) −10.3893 −0.389903
\(711\) 4.89682 0.183645
\(712\) 30.3631 1.13791
\(713\) 26.8777 1.00658
\(714\) 2.74037 0.102556
\(715\) 0 0
\(716\) −1.67570 −0.0626238
\(717\) 6.45427 0.241039
\(718\) 22.1534 0.826758
\(719\) 42.6640 1.59110 0.795549 0.605889i \(-0.207183\pi\)
0.795549 + 0.605889i \(0.207183\pi\)
\(720\) 0.772866 0.0288030
\(721\) 23.8248 0.887281
\(722\) −6.01773 −0.223957
\(723\) 25.6181 0.952747
\(724\) −21.5223 −0.799870
\(725\) −2.45427 −0.0911493
\(726\) 0 0
\(727\) −0.351089 −0.0130212 −0.00651058 0.999979i \(-0.502072\pi\)
−0.00651058 + 0.999979i \(0.502072\pi\)
\(728\) −20.5963 −0.763348
\(729\) 1.00000 0.0370370
\(730\) 0.908538 0.0336265
\(731\) 3.36281 0.124378
\(732\) 20.0017 0.739283
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 17.4660 0.644681
\(735\) −4.34371 −0.160220
\(736\) −27.1166 −0.999530
\(737\) 0 0
\(738\) −3.93502 −0.144850
\(739\) 15.2479 0.560903 0.280452 0.959868i \(-0.409516\pi\)
0.280452 + 0.959868i \(0.409516\pi\)
\(740\) −12.4690 −0.458371
\(741\) −24.8703 −0.913635
\(742\) 6.63446 0.243559
\(743\) 4.11056 0.150802 0.0754009 0.997153i \(-0.475976\pi\)
0.0754009 + 0.997153i \(0.475976\pi\)
\(744\) 15.2670 0.559716
\(745\) 3.89682 0.142768
\(746\) −20.1595 −0.738091
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 31.5075 1.15126
\(750\) −0.772866 −0.0282211
\(751\) −24.4928 −0.893754 −0.446877 0.894595i \(-0.647464\pi\)
−0.446877 + 0.894595i \(0.647464\pi\)
\(752\) 7.51324 0.273980
\(753\) 4.80536 0.175117
\(754\) −9.11490 −0.331945
\(755\) 5.72128 0.208219
\(756\) −2.28610 −0.0831448
\(757\) −1.03082 −0.0374656 −0.0187328 0.999825i \(-0.505963\pi\)
−0.0187328 + 0.999825i \(0.505963\pi\)
\(758\) −7.08241 −0.257245
\(759\) 0 0
\(760\) 13.6107 0.493713
\(761\) −14.4161 −0.522582 −0.261291 0.965260i \(-0.584148\pi\)
−0.261291 + 0.965260i \(0.584148\pi\)
\(762\) −13.3246 −0.482700
\(763\) −14.3200 −0.518417
\(764\) −22.0415 −0.797435
\(765\) −2.17554 −0.0786570
\(766\) 14.1829 0.512450
\(767\) 34.3893 1.24173
\(768\) −13.2345 −0.477559
\(769\) −29.1447 −1.05099 −0.525493 0.850798i \(-0.676119\pi\)
−0.525493 + 0.850798i \(0.676119\pi\)
\(770\) 0 0
\(771\) 11.7863 0.424472
\(772\) 15.7669 0.567461
\(773\) −44.3203 −1.59409 −0.797045 0.603920i \(-0.793605\pi\)
−0.797045 + 0.603920i \(0.793605\pi\)
\(774\) 1.19464 0.0429405
\(775\) 5.80536 0.208535
\(776\) 12.4161 0.445711
\(777\) 14.4881 0.519759
\(778\) 9.73733 0.349100
\(779\) 26.3511 0.944126
\(780\) 6.74037 0.241344
\(781\) 0 0
\(782\) 7.78458 0.278376
\(783\) −2.45427 −0.0877084
\(784\) −3.35710 −0.119897
\(785\) −15.2405 −0.543958
\(786\) −12.0650 −0.430344
\(787\) −9.03851 −0.322188 −0.161094 0.986939i \(-0.551502\pi\)
−0.161094 + 0.986939i \(0.551502\pi\)
\(788\) −3.84386 −0.136932
\(789\) −12.4235 −0.442287
\(790\) −3.78458 −0.134649
\(791\) 20.4161 0.725912
\(792\) 0 0
\(793\) 68.5226 2.43331
\(794\) 22.9501 0.814468
\(795\) −5.26701 −0.186801
\(796\) 35.1581 1.24615
\(797\) −53.5873 −1.89816 −0.949079 0.315037i \(-0.897983\pi\)
−0.949079 + 0.315037i \(0.897983\pi\)
\(798\) −6.51925 −0.230779
\(799\) −21.1491 −0.748200
\(800\) −5.85695 −0.207074
\(801\) 11.5457 0.407948
\(802\) −30.1829 −1.06580
\(803\) 0 0
\(804\) 6.62243 0.233555
\(805\) 7.54573 0.265952
\(806\) 21.5605 0.759436
\(807\) 31.2214 1.09905
\(808\) 35.7936 1.25922
\(809\) −24.8703 −0.874395 −0.437197 0.899366i \(-0.644029\pi\)
−0.437197 + 0.899366i \(0.644029\pi\)
\(810\) −0.772866 −0.0271557
\(811\) −35.0415 −1.23048 −0.615238 0.788342i \(-0.710940\pi\)
−0.615238 + 0.788342i \(0.710940\pi\)
\(812\) 5.61072 0.196898
\(813\) 25.5002 0.894329
\(814\) 0 0
\(815\) −17.2405 −0.603909
\(816\) −1.68140 −0.0588609
\(817\) −8.00000 −0.279885
\(818\) 22.8144 0.797687
\(819\) −7.83184 −0.273666
\(820\) −7.14169 −0.249399
\(821\) −35.4044 −1.23562 −0.617810 0.786327i \(-0.711980\pi\)
−0.617810 + 0.786327i \(0.711980\pi\)
\(822\) 8.26564 0.288297
\(823\) 27.4502 0.956855 0.478428 0.878127i \(-0.341207\pi\)
0.478428 + 0.878127i \(0.341207\pi\)
\(824\) 38.4429 1.33922
\(825\) 0 0
\(826\) 9.01445 0.313653
\(827\) −21.7407 −0.755998 −0.377999 0.925806i \(-0.623388\pi\)
−0.377999 + 0.925806i \(0.623388\pi\)
\(828\) −6.49414 −0.225687
\(829\) 35.5193 1.23363 0.616817 0.787106i \(-0.288422\pi\)
0.616817 + 0.787106i \(0.288422\pi\)
\(830\) 3.09146 0.107306
\(831\) −24.6033 −0.853480
\(832\) −14.3243 −0.496606
\(833\) 9.44993 0.327421
\(834\) −5.39530 −0.186824
\(835\) −5.53835 −0.191663
\(836\) 0 0
\(837\) 5.80536 0.200663
\(838\) 17.6757 0.610597
\(839\) −18.6492 −0.643843 −0.321921 0.946766i \(-0.604329\pi\)
−0.321921 + 0.946766i \(0.604329\pi\)
\(840\) 4.28610 0.147885
\(841\) −22.9766 −0.792295
\(842\) 16.6965 0.575398
\(843\) −11.5457 −0.397656
\(844\) 25.2111 0.867801
\(845\) 10.0915 0.347157
\(846\) −7.51324 −0.258310
\(847\) 0 0
\(848\) −4.07069 −0.139788
\(849\) −6.09884 −0.209312
\(850\) 1.68140 0.0576717
\(851\) 41.1564 1.41082
\(852\) −18.8556 −0.645982
\(853\) 5.32027 0.182163 0.0910813 0.995843i \(-0.470968\pi\)
0.0910813 + 0.995843i \(0.470968\pi\)
\(854\) 17.9618 0.614640
\(855\) 5.17554 0.177000
\(856\) 50.8395 1.73766
\(857\) 17.6181 0.601823 0.300911 0.953652i \(-0.402709\pi\)
0.300911 + 0.953652i \(0.402709\pi\)
\(858\) 0 0
\(859\) −11.9012 −0.406062 −0.203031 0.979172i \(-0.565079\pi\)
−0.203031 + 0.979172i \(0.565079\pi\)
\(860\) 2.16816 0.0739338
\(861\) 8.29813 0.282800
\(862\) 26.5990 0.905965
\(863\) 21.7407 0.740061 0.370031 0.929020i \(-0.379347\pi\)
0.370031 + 0.929020i \(0.379347\pi\)
\(864\) −5.85695 −0.199257
\(865\) −18.7022 −0.635893
\(866\) 15.6844 0.532977
\(867\) −12.2670 −0.416609
\(868\) −13.2717 −0.450469
\(869\) 0 0
\(870\) 1.89682 0.0643082
\(871\) 22.6874 0.768734
\(872\) −23.1062 −0.782475
\(873\) 4.72128 0.159791
\(874\) −18.5193 −0.626423
\(875\) 1.62981 0.0550977
\(876\) 1.64891 0.0557115
\(877\) −39.8777 −1.34657 −0.673287 0.739381i \(-0.735118\pi\)
−0.673287 + 0.739381i \(0.735118\pi\)
\(878\) 5.63118 0.190043
\(879\) −5.82446 −0.196454
\(880\) 0 0
\(881\) 51.5873 1.73802 0.869010 0.494795i \(-0.164757\pi\)
0.869010 + 0.494795i \(0.164757\pi\)
\(882\) 3.35710 0.113040
\(883\) 24.1343 0.812184 0.406092 0.913832i \(-0.366891\pi\)
0.406092 + 0.913832i \(0.366891\pi\)
\(884\) −14.6640 −0.493203
\(885\) −7.15645 −0.240561
\(886\) −5.61072 −0.188496
\(887\) −43.3896 −1.45688 −0.728440 0.685110i \(-0.759754\pi\)
−0.728440 + 0.685110i \(0.759754\pi\)
\(888\) 23.3776 0.784500
\(889\) 28.0988 0.942405
\(890\) −8.92330 −0.299110
\(891\) 0 0
\(892\) 17.9350 0.600509
\(893\) 50.3129 1.68366
\(894\) −3.01172 −0.100727
\(895\) 1.19464 0.0399325
\(896\) 15.3366 0.512361
\(897\) −22.2479 −0.742836
\(898\) −17.0030 −0.567399
\(899\) −14.2479 −0.475194
\(900\) −1.40268 −0.0467560
\(901\) 11.4586 0.381742
\(902\) 0 0
\(903\) −2.51925 −0.0838355
\(904\) 32.9427 1.09566
\(905\) 15.3437 0.510042
\(906\) −4.42178 −0.146904
\(907\) −28.6181 −0.950248 −0.475124 0.879919i \(-0.657597\pi\)
−0.475124 + 0.879919i \(0.657597\pi\)
\(908\) 20.0285 0.664668
\(909\) 13.6107 0.451439
\(910\) 6.05296 0.200654
\(911\) −19.3776 −0.642007 −0.321004 0.947078i \(-0.604020\pi\)
−0.321004 + 0.947078i \(0.604020\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −2.79060 −0.0923048
\(915\) −14.2596 −0.471409
\(916\) 36.4264 1.20356
\(917\) 25.4426 0.840187
\(918\) 1.68140 0.0554946
\(919\) 29.7096 0.980028 0.490014 0.871715i \(-0.336992\pi\)
0.490014 + 0.871715i \(0.336992\pi\)
\(920\) 12.1755 0.401416
\(921\) 10.2670 0.338309
\(922\) −15.5371 −0.511686
\(923\) −64.5963 −2.12621
\(924\) 0 0
\(925\) 8.88944 0.292283
\(926\) 15.9886 0.525418
\(927\) 14.6181 0.480121
\(928\) 14.3745 0.471867
\(929\) 7.11490 0.233432 0.116716 0.993165i \(-0.462763\pi\)
0.116716 + 0.993165i \(0.462763\pi\)
\(930\) −4.48676 −0.147127
\(931\) −22.4811 −0.736787
\(932\) −14.8191 −0.485415
\(933\) −0.454269 −0.0148721
\(934\) 0.657657 0.0215192
\(935\) 0 0
\(936\) −12.6372 −0.413060
\(937\) 8.68308 0.283664 0.141832 0.989891i \(-0.454701\pi\)
0.141832 + 0.989891i \(0.454701\pi\)
\(938\) 5.94704 0.194178
\(939\) 0.103180 0.00336716
\(940\) −13.6358 −0.444752
\(941\) −14.2861 −0.465714 −0.232857 0.972511i \(-0.574807\pi\)
−0.232857 + 0.972511i \(0.574807\pi\)
\(942\) 11.7789 0.383777
\(943\) 23.5725 0.767627
\(944\) −5.53097 −0.180018
\(945\) 1.62981 0.0530178
\(946\) 0 0
\(947\) 48.5916 1.57902 0.789508 0.613741i \(-0.210336\pi\)
0.789508 + 0.613741i \(0.210336\pi\)
\(948\) −6.86867 −0.223084
\(949\) 5.64891 0.183371
\(950\) −4.00000 −0.129777
\(951\) −19.0841 −0.618844
\(952\) −9.32461 −0.302212
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 4.07069 0.131793
\(955\) 15.7139 0.508490
\(956\) −9.05327 −0.292804
\(957\) 0 0
\(958\) −21.9504 −0.709185
\(959\) −17.4305 −0.562861
\(960\) 2.98090 0.0962082
\(961\) 2.70218 0.0871670
\(962\) 33.0144 1.06443
\(963\) 19.3320 0.622965
\(964\) −35.9340 −1.15735
\(965\) −11.2405 −0.361845
\(966\) −5.83184 −0.187636
\(967\) −17.3819 −0.558964 −0.279482 0.960151i \(-0.590163\pi\)
−0.279482 + 0.960151i \(0.590163\pi\)
\(968\) 0 0
\(969\) −11.2596 −0.361711
\(970\) −3.64891 −0.117159
\(971\) 7.69046 0.246799 0.123399 0.992357i \(-0.460620\pi\)
0.123399 + 0.992357i \(0.460620\pi\)
\(972\) −1.40268 −0.0449909
\(973\) 11.3776 0.364748
\(974\) −3.14169 −0.100666
\(975\) −4.80536 −0.153895
\(976\) −11.0208 −0.352766
\(977\) −3.78626 −0.121133 −0.0605666 0.998164i \(-0.519291\pi\)
−0.0605666 + 0.998164i \(0.519291\pi\)
\(978\) 13.3246 0.426074
\(979\) 0 0
\(980\) 6.09283 0.194628
\(981\) −8.78626 −0.280524
\(982\) 21.6107 0.689626
\(983\) −20.6446 −0.658460 −0.329230 0.944250i \(-0.606789\pi\)
−0.329230 + 0.944250i \(0.606789\pi\)
\(984\) 13.3896 0.426845
\(985\) 2.74037 0.0873156
\(986\) −4.12662 −0.131418
\(987\) 15.8439 0.504316
\(988\) 34.8851 1.10984
\(989\) −7.15645 −0.227562
\(990\) 0 0
\(991\) 27.8130 0.883511 0.441755 0.897136i \(-0.354356\pi\)
0.441755 + 0.897136i \(0.354356\pi\)
\(992\) −34.0017 −1.07955
\(993\) −9.45427 −0.300022
\(994\) −16.9326 −0.537069
\(995\) −25.0650 −0.794613
\(996\) 5.61072 0.177782
\(997\) 30.7213 0.972953 0.486476 0.873694i \(-0.338282\pi\)
0.486476 + 0.873694i \(0.338282\pi\)
\(998\) −0.879017 −0.0278248
\(999\) 8.88944 0.281249
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 1815.2.a.l.1.2 3
3.2 odd 2 5445.2.a.bc.1.2 3
5.4 even 2 9075.2.a.ci.1.2 3
11.10 odd 2 1815.2.a.n.1.2 yes 3
33.32 even 2 5445.2.a.ba.1.2 3
55.54 odd 2 9075.2.a.ce.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.l.1.2 3 1.1 even 1 trivial
1815.2.a.n.1.2 yes 3 11.10 odd 2
5445.2.a.ba.1.2 3 33.32 even 2
5445.2.a.bc.1.2 3 3.2 odd 2
9075.2.a.ce.1.2 3 55.54 odd 2
9075.2.a.ci.1.2 3 5.4 even 2