Properties

Label 9405.2.a.w.1.1
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56745\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56745 q^{2} +4.59179 q^{4} -1.00000 q^{5} +2.16848 q^{7} -6.65430 q^{8} +O(q^{10})\) \(q-2.56745 q^{2} +4.59179 q^{4} -1.00000 q^{5} +2.16848 q^{7} -6.65430 q^{8} +2.56745 q^{10} +1.00000 q^{11} -4.47137 q^{13} -5.56745 q^{14} +7.90099 q^{16} -2.71222 q^{17} -1.00000 q^{19} -4.59179 q^{20} -2.56745 q^{22} +2.85635 q^{23} +1.00000 q^{25} +11.4800 q^{26} +9.95719 q^{28} +2.17663 q^{29} -6.53387 q^{31} -6.97678 q^{32} +6.96349 q^{34} -2.16848 q^{35} +0.861702 q^{37} +2.56745 q^{38} +6.65430 q^{40} -10.7487 q^{41} +8.75038 q^{43} +4.59179 q^{44} -7.33354 q^{46} +0.665870 q^{47} -2.29772 q^{49} -2.56745 q^{50} -20.5316 q^{52} +13.9364 q^{53} -1.00000 q^{55} -14.4297 q^{56} -5.58839 q^{58} -8.66353 q^{59} +5.59008 q^{61} +16.7754 q^{62} +2.11055 q^{64} +4.47137 q^{65} +15.2043 q^{67} -12.4540 q^{68} +5.56745 q^{70} +5.99426 q^{71} -10.8568 q^{73} -2.21237 q^{74} -4.59179 q^{76} +2.16848 q^{77} +5.74408 q^{79} -7.90099 q^{80} +27.5967 q^{82} +12.3220 q^{83} +2.71222 q^{85} -22.4662 q^{86} -6.65430 q^{88} -4.83970 q^{89} -9.69605 q^{91} +13.1158 q^{92} -1.70959 q^{94} +1.00000 q^{95} +14.8355 q^{97} +5.89927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56745 −1.81546 −0.907730 0.419554i \(-0.862186\pi\)
−0.907730 + 0.419554i \(0.862186\pi\)
\(3\) 0 0
\(4\) 4.59179 2.29590
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.16848 0.819607 0.409803 0.912174i \(-0.365597\pi\)
0.409803 + 0.912174i \(0.365597\pi\)
\(8\) −6.65430 −2.35265
\(9\) 0 0
\(10\) 2.56745 0.811899
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.47137 −1.24013 −0.620067 0.784549i \(-0.712895\pi\)
−0.620067 + 0.784549i \(0.712895\pi\)
\(14\) −5.56745 −1.48796
\(15\) 0 0
\(16\) 7.90099 1.97525
\(17\) −2.71222 −0.657810 −0.328905 0.944363i \(-0.606680\pi\)
−0.328905 + 0.944363i \(0.606680\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −4.59179 −1.02676
\(21\) 0 0
\(22\) −2.56745 −0.547382
\(23\) 2.85635 0.595590 0.297795 0.954630i \(-0.403749\pi\)
0.297795 + 0.954630i \(0.403749\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 11.4800 2.25141
\(27\) 0 0
\(28\) 9.95719 1.88173
\(29\) 2.17663 0.404191 0.202095 0.979366i \(-0.435225\pi\)
0.202095 + 0.979366i \(0.435225\pi\)
\(30\) 0 0
\(31\) −6.53387 −1.17352 −0.586759 0.809762i \(-0.699596\pi\)
−0.586759 + 0.809762i \(0.699596\pi\)
\(32\) −6.97678 −1.23333
\(33\) 0 0
\(34\) 6.96349 1.19423
\(35\) −2.16848 −0.366539
\(36\) 0 0
\(37\) 0.861702 0.141663 0.0708314 0.997488i \(-0.477435\pi\)
0.0708314 + 0.997488i \(0.477435\pi\)
\(38\) 2.56745 0.416495
\(39\) 0 0
\(40\) 6.65430 1.05214
\(41\) −10.7487 −1.67866 −0.839332 0.543619i \(-0.817054\pi\)
−0.839332 + 0.543619i \(0.817054\pi\)
\(42\) 0 0
\(43\) 8.75038 1.33442 0.667210 0.744869i \(-0.267488\pi\)
0.667210 + 0.744869i \(0.267488\pi\)
\(44\) 4.59179 0.692239
\(45\) 0 0
\(46\) −7.33354 −1.08127
\(47\) 0.665870 0.0971272 0.0485636 0.998820i \(-0.484536\pi\)
0.0485636 + 0.998820i \(0.484536\pi\)
\(48\) 0 0
\(49\) −2.29772 −0.328245
\(50\) −2.56745 −0.363092
\(51\) 0 0
\(52\) −20.5316 −2.84722
\(53\) 13.9364 1.91431 0.957155 0.289576i \(-0.0935142\pi\)
0.957155 + 0.289576i \(0.0935142\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −14.4297 −1.92825
\(57\) 0 0
\(58\) −5.58839 −0.733792
\(59\) −8.66353 −1.12790 −0.563948 0.825810i \(-0.690718\pi\)
−0.563948 + 0.825810i \(0.690718\pi\)
\(60\) 0 0
\(61\) 5.59008 0.715736 0.357868 0.933772i \(-0.383504\pi\)
0.357868 + 0.933772i \(0.383504\pi\)
\(62\) 16.7754 2.13048
\(63\) 0 0
\(64\) 2.11055 0.263819
\(65\) 4.47137 0.554605
\(66\) 0 0
\(67\) 15.2043 1.85751 0.928753 0.370698i \(-0.120882\pi\)
0.928753 + 0.370698i \(0.120882\pi\)
\(68\) −12.4540 −1.51027
\(69\) 0 0
\(70\) 5.56745 0.665437
\(71\) 5.99426 0.711388 0.355694 0.934603i \(-0.384245\pi\)
0.355694 + 0.934603i \(0.384245\pi\)
\(72\) 0 0
\(73\) −10.8568 −1.27069 −0.635347 0.772226i \(-0.719143\pi\)
−0.635347 + 0.772226i \(0.719143\pi\)
\(74\) −2.21237 −0.257183
\(75\) 0 0
\(76\) −4.59179 −0.526715
\(77\) 2.16848 0.247121
\(78\) 0 0
\(79\) 5.74408 0.646260 0.323130 0.946355i \(-0.395265\pi\)
0.323130 + 0.946355i \(0.395265\pi\)
\(80\) −7.90099 −0.883357
\(81\) 0 0
\(82\) 27.5967 3.04755
\(83\) 12.3220 1.35252 0.676259 0.736664i \(-0.263600\pi\)
0.676259 + 0.736664i \(0.263600\pi\)
\(84\) 0 0
\(85\) 2.71222 0.294182
\(86\) −22.4662 −2.42259
\(87\) 0 0
\(88\) −6.65430 −0.709351
\(89\) −4.83970 −0.513007 −0.256503 0.966543i \(-0.582570\pi\)
−0.256503 + 0.966543i \(0.582570\pi\)
\(90\) 0 0
\(91\) −9.69605 −1.01642
\(92\) 13.1158 1.36741
\(93\) 0 0
\(94\) −1.70959 −0.176331
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 14.8355 1.50631 0.753157 0.657841i \(-0.228530\pi\)
0.753157 + 0.657841i \(0.228530\pi\)
\(98\) 5.89927 0.595916
\(99\) 0 0
\(100\) 4.59179 0.459179
\(101\) −9.35188 −0.930546 −0.465273 0.885167i \(-0.654044\pi\)
−0.465273 + 0.885167i \(0.654044\pi\)
\(102\) 0 0
\(103\) 8.08853 0.796987 0.398493 0.917171i \(-0.369533\pi\)
0.398493 + 0.917171i \(0.369533\pi\)
\(104\) 29.7538 2.91760
\(105\) 0 0
\(106\) −35.7810 −3.47535
\(107\) 2.71050 0.262034 0.131017 0.991380i \(-0.458176\pi\)
0.131017 + 0.991380i \(0.458176\pi\)
\(108\) 0 0
\(109\) 0.495056 0.0474178 0.0237089 0.999719i \(-0.492453\pi\)
0.0237089 + 0.999719i \(0.492453\pi\)
\(110\) 2.56745 0.244797
\(111\) 0 0
\(112\) 17.1331 1.61892
\(113\) −20.6374 −1.94140 −0.970701 0.240292i \(-0.922757\pi\)
−0.970701 + 0.240292i \(0.922757\pi\)
\(114\) 0 0
\(115\) −2.85635 −0.266356
\(116\) 9.99465 0.927980
\(117\) 0 0
\(118\) 22.2432 2.04765
\(119\) −5.88139 −0.539146
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.3522 −1.29939
\(123\) 0 0
\(124\) −30.0022 −2.69428
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.2674 −1.35477 −0.677383 0.735631i \(-0.736886\pi\)
−0.677383 + 0.735631i \(0.736886\pi\)
\(128\) 8.53482 0.754379
\(129\) 0 0
\(130\) −11.4800 −1.00686
\(131\) 14.5918 1.27489 0.637444 0.770497i \(-0.279992\pi\)
0.637444 + 0.770497i \(0.279992\pi\)
\(132\) 0 0
\(133\) −2.16848 −0.188031
\(134\) −39.0364 −3.37223
\(135\) 0 0
\(136\) 18.0479 1.54760
\(137\) 11.9896 1.02434 0.512169 0.858885i \(-0.328842\pi\)
0.512169 + 0.858885i \(0.328842\pi\)
\(138\) 0 0
\(139\) −7.16027 −0.607326 −0.303663 0.952779i \(-0.598210\pi\)
−0.303663 + 0.952779i \(0.598210\pi\)
\(140\) −9.95719 −0.841536
\(141\) 0 0
\(142\) −15.3900 −1.29150
\(143\) −4.47137 −0.373914
\(144\) 0 0
\(145\) −2.17663 −0.180760
\(146\) 27.8743 2.30690
\(147\) 0 0
\(148\) 3.95676 0.325243
\(149\) −3.73998 −0.306391 −0.153196 0.988196i \(-0.548956\pi\)
−0.153196 + 0.988196i \(0.548956\pi\)
\(150\) 0 0
\(151\) −7.70787 −0.627258 −0.313629 0.949546i \(-0.601545\pi\)
−0.313629 + 0.949546i \(0.601545\pi\)
\(152\) 6.65430 0.539735
\(153\) 0 0
\(154\) −5.56745 −0.448638
\(155\) 6.53387 0.524813
\(156\) 0 0
\(157\) −7.78415 −0.621242 −0.310621 0.950534i \(-0.600537\pi\)
−0.310621 + 0.950534i \(0.600537\pi\)
\(158\) −14.7476 −1.17326
\(159\) 0 0
\(160\) 6.97678 0.551563
\(161\) 6.19393 0.488150
\(162\) 0 0
\(163\) −3.60748 −0.282560 −0.141280 0.989970i \(-0.545122\pi\)
−0.141280 + 0.989970i \(0.545122\pi\)
\(164\) −49.3558 −3.85404
\(165\) 0 0
\(166\) −31.6362 −2.45544
\(167\) 0.588302 0.0455242 0.0227621 0.999741i \(-0.492754\pi\)
0.0227621 + 0.999741i \(0.492754\pi\)
\(168\) 0 0
\(169\) 6.99312 0.537932
\(170\) −6.96349 −0.534075
\(171\) 0 0
\(172\) 40.1800 3.06369
\(173\) −23.1505 −1.76010 −0.880050 0.474882i \(-0.842491\pi\)
−0.880050 + 0.474882i \(0.842491\pi\)
\(174\) 0 0
\(175\) 2.16848 0.163921
\(176\) 7.90099 0.595559
\(177\) 0 0
\(178\) 12.4257 0.931343
\(179\) −11.2340 −0.839666 −0.419833 0.907601i \(-0.637911\pi\)
−0.419833 + 0.907601i \(0.637911\pi\)
\(180\) 0 0
\(181\) −7.18465 −0.534031 −0.267015 0.963692i \(-0.586037\pi\)
−0.267015 + 0.963692i \(0.586037\pi\)
\(182\) 24.8941 1.84527
\(183\) 0 0
\(184\) −19.0070 −1.40122
\(185\) −0.861702 −0.0633536
\(186\) 0 0
\(187\) −2.71222 −0.198337
\(188\) 3.05754 0.222994
\(189\) 0 0
\(190\) −2.56745 −0.186262
\(191\) 13.9736 1.01109 0.505546 0.862800i \(-0.331291\pi\)
0.505546 + 0.862800i \(0.331291\pi\)
\(192\) 0 0
\(193\) −12.8048 −0.921710 −0.460855 0.887475i \(-0.652457\pi\)
−0.460855 + 0.887475i \(0.652457\pi\)
\(194\) −38.0893 −2.73465
\(195\) 0 0
\(196\) −10.5506 −0.753617
\(197\) −3.78816 −0.269895 −0.134948 0.990853i \(-0.543087\pi\)
−0.134948 + 0.990853i \(0.543087\pi\)
\(198\) 0 0
\(199\) 15.9061 1.12755 0.563777 0.825927i \(-0.309348\pi\)
0.563777 + 0.825927i \(0.309348\pi\)
\(200\) −6.65430 −0.470530
\(201\) 0 0
\(202\) 24.0105 1.68937
\(203\) 4.71997 0.331277
\(204\) 0 0
\(205\) 10.7487 0.750722
\(206\) −20.7669 −1.44690
\(207\) 0 0
\(208\) −35.3282 −2.44957
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.20422 0.151745 0.0758723 0.997118i \(-0.475826\pi\)
0.0758723 + 0.997118i \(0.475826\pi\)
\(212\) 63.9930 4.39506
\(213\) 0 0
\(214\) −6.95908 −0.475713
\(215\) −8.75038 −0.596771
\(216\) 0 0
\(217\) −14.1685 −0.961823
\(218\) −1.27103 −0.0860851
\(219\) 0 0
\(220\) −4.59179 −0.309579
\(221\) 12.1273 0.815773
\(222\) 0 0
\(223\) −14.3670 −0.962086 −0.481043 0.876697i \(-0.659742\pi\)
−0.481043 + 0.876697i \(0.659742\pi\)
\(224\) −15.1290 −1.01085
\(225\) 0 0
\(226\) 52.9854 3.52454
\(227\) −21.6909 −1.43968 −0.719838 0.694142i \(-0.755784\pi\)
−0.719838 + 0.694142i \(0.755784\pi\)
\(228\) 0 0
\(229\) 9.39735 0.620995 0.310497 0.950574i \(-0.399504\pi\)
0.310497 + 0.950574i \(0.399504\pi\)
\(230\) 7.33354 0.483559
\(231\) 0 0
\(232\) −14.4840 −0.950919
\(233\) 22.6514 1.48394 0.741971 0.670432i \(-0.233891\pi\)
0.741971 + 0.670432i \(0.233891\pi\)
\(234\) 0 0
\(235\) −0.665870 −0.0434366
\(236\) −39.7812 −2.58953
\(237\) 0 0
\(238\) 15.1002 0.978798
\(239\) −12.3190 −0.796853 −0.398426 0.917200i \(-0.630444\pi\)
−0.398426 + 0.917200i \(0.630444\pi\)
\(240\) 0 0
\(241\) −27.5506 −1.77469 −0.887345 0.461106i \(-0.847453\pi\)
−0.887345 + 0.461106i \(0.847453\pi\)
\(242\) −2.56745 −0.165042
\(243\) 0 0
\(244\) 25.6685 1.64326
\(245\) 2.29772 0.146796
\(246\) 0 0
\(247\) 4.47137 0.284506
\(248\) 43.4783 2.76088
\(249\) 0 0
\(250\) 2.56745 0.162380
\(251\) −11.9051 −0.751443 −0.375722 0.926733i \(-0.622605\pi\)
−0.375722 + 0.926733i \(0.622605\pi\)
\(252\) 0 0
\(253\) 2.85635 0.179577
\(254\) 39.1983 2.45952
\(255\) 0 0
\(256\) −26.1338 −1.63336
\(257\) 8.19766 0.511356 0.255678 0.966762i \(-0.417701\pi\)
0.255678 + 0.966762i \(0.417701\pi\)
\(258\) 0 0
\(259\) 1.86858 0.116108
\(260\) 20.5316 1.27332
\(261\) 0 0
\(262\) −37.4636 −2.31451
\(263\) 13.1875 0.813175 0.406588 0.913612i \(-0.366719\pi\)
0.406588 + 0.913612i \(0.366719\pi\)
\(264\) 0 0
\(265\) −13.9364 −0.856106
\(266\) 5.56745 0.341362
\(267\) 0 0
\(268\) 69.8152 4.26464
\(269\) −19.3988 −1.18277 −0.591384 0.806390i \(-0.701418\pi\)
−0.591384 + 0.806390i \(0.701418\pi\)
\(270\) 0 0
\(271\) 3.57069 0.216904 0.108452 0.994102i \(-0.465411\pi\)
0.108452 + 0.994102i \(0.465411\pi\)
\(272\) −21.4292 −1.29934
\(273\) 0 0
\(274\) −30.7826 −1.85964
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −20.6671 −1.24176 −0.620882 0.783904i \(-0.713225\pi\)
−0.620882 + 0.783904i \(0.713225\pi\)
\(278\) 18.3836 1.10258
\(279\) 0 0
\(280\) 14.4297 0.862338
\(281\) 6.89894 0.411556 0.205778 0.978599i \(-0.434027\pi\)
0.205778 + 0.978599i \(0.434027\pi\)
\(282\) 0 0
\(283\) 23.5344 1.39898 0.699488 0.714644i \(-0.253411\pi\)
0.699488 + 0.714644i \(0.253411\pi\)
\(284\) 27.5244 1.63327
\(285\) 0 0
\(286\) 11.4800 0.678827
\(287\) −23.3083 −1.37584
\(288\) 0 0
\(289\) −9.64385 −0.567285
\(290\) 5.58839 0.328162
\(291\) 0 0
\(292\) −49.8523 −2.91738
\(293\) −8.70291 −0.508429 −0.254215 0.967148i \(-0.581817\pi\)
−0.254215 + 0.967148i \(0.581817\pi\)
\(294\) 0 0
\(295\) 8.66353 0.504410
\(296\) −5.73402 −0.333283
\(297\) 0 0
\(298\) 9.60221 0.556241
\(299\) −12.7718 −0.738612
\(300\) 0 0
\(301\) 18.9750 1.09370
\(302\) 19.7896 1.13876
\(303\) 0 0
\(304\) −7.90099 −0.453153
\(305\) −5.59008 −0.320087
\(306\) 0 0
\(307\) −0.0345724 −0.00197315 −0.000986576 1.00000i \(-0.500314\pi\)
−0.000986576 1.00000i \(0.500314\pi\)
\(308\) 9.95719 0.567364
\(309\) 0 0
\(310\) −16.7754 −0.952778
\(311\) 16.8869 0.957570 0.478785 0.877932i \(-0.341077\pi\)
0.478785 + 0.877932i \(0.341077\pi\)
\(312\) 0 0
\(313\) −1.47670 −0.0834682 −0.0417341 0.999129i \(-0.513288\pi\)
−0.0417341 + 0.999129i \(0.513288\pi\)
\(314\) 19.9854 1.12784
\(315\) 0 0
\(316\) 26.3756 1.48375
\(317\) −24.5279 −1.37763 −0.688813 0.724939i \(-0.741868\pi\)
−0.688813 + 0.724939i \(0.741868\pi\)
\(318\) 0 0
\(319\) 2.17663 0.121868
\(320\) −2.11055 −0.117983
\(321\) 0 0
\(322\) −15.9026 −0.886217
\(323\) 2.71222 0.150912
\(324\) 0 0
\(325\) −4.47137 −0.248027
\(326\) 9.26202 0.512976
\(327\) 0 0
\(328\) 71.5251 3.94931
\(329\) 1.44392 0.0796061
\(330\) 0 0
\(331\) 19.8183 1.08931 0.544657 0.838659i \(-0.316660\pi\)
0.544657 + 0.838659i \(0.316660\pi\)
\(332\) 56.5802 3.10524
\(333\) 0 0
\(334\) −1.51044 −0.0826474
\(335\) −15.2043 −0.830702
\(336\) 0 0
\(337\) 12.6702 0.690188 0.345094 0.938568i \(-0.387847\pi\)
0.345094 + 0.938568i \(0.387847\pi\)
\(338\) −17.9545 −0.976595
\(339\) 0 0
\(340\) 12.4540 0.675411
\(341\) −6.53387 −0.353829
\(342\) 0 0
\(343\) −20.1619 −1.08864
\(344\) −58.2277 −3.13943
\(345\) 0 0
\(346\) 59.4377 3.19539
\(347\) −6.29698 −0.338040 −0.169020 0.985613i \(-0.554060\pi\)
−0.169020 + 0.985613i \(0.554060\pi\)
\(348\) 0 0
\(349\) −4.90512 −0.262565 −0.131283 0.991345i \(-0.541910\pi\)
−0.131283 + 0.991345i \(0.541910\pi\)
\(350\) −5.56745 −0.297593
\(351\) 0 0
\(352\) −6.97678 −0.371864
\(353\) −22.7868 −1.21282 −0.606409 0.795153i \(-0.707391\pi\)
−0.606409 + 0.795153i \(0.707391\pi\)
\(354\) 0 0
\(355\) −5.99426 −0.318142
\(356\) −22.2229 −1.17781
\(357\) 0 0
\(358\) 28.8426 1.52438
\(359\) −26.4835 −1.39774 −0.698872 0.715246i \(-0.746314\pi\)
−0.698872 + 0.715246i \(0.746314\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 18.4462 0.969512
\(363\) 0 0
\(364\) −44.5223 −2.33360
\(365\) 10.8568 0.568272
\(366\) 0 0
\(367\) 14.4411 0.753818 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(368\) 22.5680 1.17644
\(369\) 0 0
\(370\) 2.21237 0.115016
\(371\) 30.2207 1.56898
\(372\) 0 0
\(373\) −3.81445 −0.197505 −0.0987525 0.995112i \(-0.531485\pi\)
−0.0987525 + 0.995112i \(0.531485\pi\)
\(374\) 6.96349 0.360074
\(375\) 0 0
\(376\) −4.43090 −0.228506
\(377\) −9.73252 −0.501250
\(378\) 0 0
\(379\) −8.00109 −0.410988 −0.205494 0.978658i \(-0.565880\pi\)
−0.205494 + 0.978658i \(0.565880\pi\)
\(380\) 4.59179 0.235554
\(381\) 0 0
\(382\) −35.8764 −1.83560
\(383\) −13.5609 −0.692932 −0.346466 0.938063i \(-0.612618\pi\)
−0.346466 + 0.938063i \(0.612618\pi\)
\(384\) 0 0
\(385\) −2.16848 −0.110516
\(386\) 32.8757 1.67333
\(387\) 0 0
\(388\) 68.1214 3.45834
\(389\) 1.63547 0.0829216 0.0414608 0.999140i \(-0.486799\pi\)
0.0414608 + 0.999140i \(0.486799\pi\)
\(390\) 0 0
\(391\) −7.74706 −0.391786
\(392\) 15.2897 0.772246
\(393\) 0 0
\(394\) 9.72590 0.489984
\(395\) −5.74408 −0.289016
\(396\) 0 0
\(397\) 9.24683 0.464085 0.232043 0.972706i \(-0.425459\pi\)
0.232043 + 0.972706i \(0.425459\pi\)
\(398\) −40.8381 −2.04703
\(399\) 0 0
\(400\) 7.90099 0.395049
\(401\) 0.299914 0.0149770 0.00748851 0.999972i \(-0.497616\pi\)
0.00748851 + 0.999972i \(0.497616\pi\)
\(402\) 0 0
\(403\) 29.2153 1.45532
\(404\) −42.9419 −2.13644
\(405\) 0 0
\(406\) −12.1183 −0.601421
\(407\) 0.861702 0.0427130
\(408\) 0 0
\(409\) 9.28447 0.459088 0.229544 0.973298i \(-0.426277\pi\)
0.229544 + 0.973298i \(0.426277\pi\)
\(410\) −27.5967 −1.36291
\(411\) 0 0
\(412\) 37.1409 1.82980
\(413\) −18.7867 −0.924431
\(414\) 0 0
\(415\) −12.3220 −0.604865
\(416\) 31.1957 1.52950
\(417\) 0 0
\(418\) 2.56745 0.125578
\(419\) 9.33196 0.455896 0.227948 0.973673i \(-0.426798\pi\)
0.227948 + 0.973673i \(0.426798\pi\)
\(420\) 0 0
\(421\) −25.1456 −1.22552 −0.612760 0.790269i \(-0.709941\pi\)
−0.612760 + 0.790269i \(0.709941\pi\)
\(422\) −5.65921 −0.275486
\(423\) 0 0
\(424\) −92.7369 −4.50370
\(425\) −2.71222 −0.131562
\(426\) 0 0
\(427\) 12.1219 0.586622
\(428\) 12.4461 0.601604
\(429\) 0 0
\(430\) 22.4662 1.08341
\(431\) −37.5660 −1.80949 −0.904744 0.425955i \(-0.859938\pi\)
−0.904744 + 0.425955i \(0.859938\pi\)
\(432\) 0 0
\(433\) 9.24282 0.444181 0.222091 0.975026i \(-0.428712\pi\)
0.222091 + 0.975026i \(0.428712\pi\)
\(434\) 36.3770 1.74615
\(435\) 0 0
\(436\) 2.27320 0.108866
\(437\) −2.85635 −0.136638
\(438\) 0 0
\(439\) 0.941826 0.0449509 0.0224754 0.999747i \(-0.492845\pi\)
0.0224754 + 0.999747i \(0.492845\pi\)
\(440\) 6.65430 0.317231
\(441\) 0 0
\(442\) −31.1363 −1.48100
\(443\) −17.2727 −0.820649 −0.410324 0.911940i \(-0.634585\pi\)
−0.410324 + 0.911940i \(0.634585\pi\)
\(444\) 0 0
\(445\) 4.83970 0.229424
\(446\) 36.8866 1.74663
\(447\) 0 0
\(448\) 4.57668 0.216228
\(449\) 20.9917 0.990661 0.495330 0.868705i \(-0.335047\pi\)
0.495330 + 0.868705i \(0.335047\pi\)
\(450\) 0 0
\(451\) −10.7487 −0.506136
\(452\) −94.7626 −4.45726
\(453\) 0 0
\(454\) 55.6903 2.61368
\(455\) 9.69605 0.454558
\(456\) 0 0
\(457\) −28.9893 −1.35606 −0.678032 0.735032i \(-0.737167\pi\)
−0.678032 + 0.735032i \(0.737167\pi\)
\(458\) −24.1272 −1.12739
\(459\) 0 0
\(460\) −13.1158 −0.611526
\(461\) 24.7915 1.15465 0.577327 0.816513i \(-0.304096\pi\)
0.577327 + 0.816513i \(0.304096\pi\)
\(462\) 0 0
\(463\) −0.757122 −0.0351864 −0.0175932 0.999845i \(-0.505600\pi\)
−0.0175932 + 0.999845i \(0.505600\pi\)
\(464\) 17.1975 0.798376
\(465\) 0 0
\(466\) −58.1563 −2.69404
\(467\) 5.54461 0.256574 0.128287 0.991737i \(-0.459052\pi\)
0.128287 + 0.991737i \(0.459052\pi\)
\(468\) 0 0
\(469\) 32.9702 1.52242
\(470\) 1.70959 0.0788574
\(471\) 0 0
\(472\) 57.6497 2.65354
\(473\) 8.75038 0.402343
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −27.0061 −1.23782
\(477\) 0 0
\(478\) 31.6285 1.44665
\(479\) 8.78759 0.401515 0.200758 0.979641i \(-0.435660\pi\)
0.200758 + 0.979641i \(0.435660\pi\)
\(480\) 0 0
\(481\) −3.85298 −0.175681
\(482\) 70.7348 3.22188
\(483\) 0 0
\(484\) 4.59179 0.208718
\(485\) −14.8355 −0.673644
\(486\) 0 0
\(487\) 36.8929 1.67178 0.835889 0.548898i \(-0.184953\pi\)
0.835889 + 0.548898i \(0.184953\pi\)
\(488\) −37.1980 −1.68388
\(489\) 0 0
\(490\) −5.89927 −0.266502
\(491\) 6.12358 0.276353 0.138177 0.990408i \(-0.455876\pi\)
0.138177 + 0.990408i \(0.455876\pi\)
\(492\) 0 0
\(493\) −5.90351 −0.265881
\(494\) −11.4800 −0.516510
\(495\) 0 0
\(496\) −51.6240 −2.31799
\(497\) 12.9984 0.583058
\(498\) 0 0
\(499\) −16.9154 −0.757239 −0.378619 0.925552i \(-0.623601\pi\)
−0.378619 + 0.925552i \(0.623601\pi\)
\(500\) −4.59179 −0.205351
\(501\) 0 0
\(502\) 30.5657 1.36422
\(503\) −4.84680 −0.216108 −0.108054 0.994145i \(-0.534462\pi\)
−0.108054 + 0.994145i \(0.534462\pi\)
\(504\) 0 0
\(505\) 9.35188 0.416153
\(506\) −7.33354 −0.326015
\(507\) 0 0
\(508\) −70.1049 −3.11040
\(509\) 18.1418 0.804122 0.402061 0.915613i \(-0.368294\pi\)
0.402061 + 0.915613i \(0.368294\pi\)
\(510\) 0 0
\(511\) −23.5427 −1.04147
\(512\) 50.0276 2.21093
\(513\) 0 0
\(514\) −21.0471 −0.928346
\(515\) −8.08853 −0.356423
\(516\) 0 0
\(517\) 0.665870 0.0292849
\(518\) −4.79748 −0.210789
\(519\) 0 0
\(520\) −29.7538 −1.30479
\(521\) −36.0419 −1.57902 −0.789511 0.613736i \(-0.789666\pi\)
−0.789511 + 0.613736i \(0.789666\pi\)
\(522\) 0 0
\(523\) 38.6617 1.69056 0.845279 0.534325i \(-0.179434\pi\)
0.845279 + 0.534325i \(0.179434\pi\)
\(524\) 67.0024 2.92701
\(525\) 0 0
\(526\) −33.8582 −1.47629
\(527\) 17.7213 0.771952
\(528\) 0 0
\(529\) −14.8413 −0.645272
\(530\) 35.7810 1.55423
\(531\) 0 0
\(532\) −9.95719 −0.431699
\(533\) 48.0614 2.08177
\(534\) 0 0
\(535\) −2.71050 −0.117185
\(536\) −101.174 −4.37006
\(537\) 0 0
\(538\) 49.8056 2.14727
\(539\) −2.29772 −0.0989696
\(540\) 0 0
\(541\) −39.8208 −1.71203 −0.856015 0.516951i \(-0.827067\pi\)
−0.856015 + 0.516951i \(0.827067\pi\)
\(542\) −9.16756 −0.393780
\(543\) 0 0
\(544\) 18.9226 0.811299
\(545\) −0.495056 −0.0212059
\(546\) 0 0
\(547\) 34.9397 1.49392 0.746958 0.664871i \(-0.231514\pi\)
0.746958 + 0.664871i \(0.231514\pi\)
\(548\) 55.0536 2.35177
\(549\) 0 0
\(550\) −2.56745 −0.109476
\(551\) −2.17663 −0.0927277
\(552\) 0 0
\(553\) 12.4559 0.529679
\(554\) 53.0616 2.25437
\(555\) 0 0
\(556\) −32.8785 −1.39436
\(557\) 1.09011 0.0461896 0.0230948 0.999733i \(-0.492648\pi\)
0.0230948 + 0.999733i \(0.492648\pi\)
\(558\) 0 0
\(559\) −39.1262 −1.65486
\(560\) −17.1331 −0.724005
\(561\) 0 0
\(562\) −17.7127 −0.747164
\(563\) 2.39110 0.100773 0.0503864 0.998730i \(-0.483955\pi\)
0.0503864 + 0.998730i \(0.483955\pi\)
\(564\) 0 0
\(565\) 20.6374 0.868221
\(566\) −60.4234 −2.53979
\(567\) 0 0
\(568\) −39.8876 −1.67365
\(569\) 27.9054 1.16985 0.584927 0.811086i \(-0.301123\pi\)
0.584927 + 0.811086i \(0.301123\pi\)
\(570\) 0 0
\(571\) 9.33966 0.390853 0.195426 0.980718i \(-0.437391\pi\)
0.195426 + 0.980718i \(0.437391\pi\)
\(572\) −20.5316 −0.858469
\(573\) 0 0
\(574\) 59.8428 2.49779
\(575\) 2.85635 0.119118
\(576\) 0 0
\(577\) 24.2224 1.00839 0.504195 0.863590i \(-0.331789\pi\)
0.504195 + 0.863590i \(0.331789\pi\)
\(578\) 24.7601 1.02988
\(579\) 0 0
\(580\) −9.99465 −0.415005
\(581\) 26.7200 1.10853
\(582\) 0 0
\(583\) 13.9364 0.577186
\(584\) 72.2445 2.98950
\(585\) 0 0
\(586\) 22.3443 0.923033
\(587\) −3.72615 −0.153795 −0.0768973 0.997039i \(-0.524501\pi\)
−0.0768973 + 0.997039i \(0.524501\pi\)
\(588\) 0 0
\(589\) 6.53387 0.269224
\(590\) −22.2432 −0.915737
\(591\) 0 0
\(592\) 6.80829 0.279819
\(593\) −14.1469 −0.580944 −0.290472 0.956883i \(-0.593812\pi\)
−0.290472 + 0.956883i \(0.593812\pi\)
\(594\) 0 0
\(595\) 5.88139 0.241113
\(596\) −17.1732 −0.703442
\(597\) 0 0
\(598\) 32.7909 1.34092
\(599\) 37.7856 1.54388 0.771940 0.635696i \(-0.219287\pi\)
0.771940 + 0.635696i \(0.219287\pi\)
\(600\) 0 0
\(601\) −29.5251 −1.20436 −0.602178 0.798362i \(-0.705700\pi\)
−0.602178 + 0.798362i \(0.705700\pi\)
\(602\) −48.7173 −1.98557
\(603\) 0 0
\(604\) −35.3930 −1.44012
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −5.19284 −0.210771 −0.105385 0.994431i \(-0.533608\pi\)
−0.105385 + 0.994431i \(0.533608\pi\)
\(608\) 6.97678 0.282946
\(609\) 0 0
\(610\) 14.3522 0.581105
\(611\) −2.97735 −0.120451
\(612\) 0 0
\(613\) −44.9254 −1.81452 −0.907260 0.420569i \(-0.861830\pi\)
−0.907260 + 0.420569i \(0.861830\pi\)
\(614\) 0.0887629 0.00358218
\(615\) 0 0
\(616\) −14.4297 −0.581388
\(617\) 8.39687 0.338045 0.169023 0.985612i \(-0.445939\pi\)
0.169023 + 0.985612i \(0.445939\pi\)
\(618\) 0 0
\(619\) 4.27492 0.171823 0.0859117 0.996303i \(-0.472620\pi\)
0.0859117 + 0.996303i \(0.472620\pi\)
\(620\) 30.0022 1.20492
\(621\) 0 0
\(622\) −43.3563 −1.73843
\(623\) −10.4948 −0.420464
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.79136 0.151533
\(627\) 0 0
\(628\) −35.7432 −1.42631
\(629\) −2.33713 −0.0931873
\(630\) 0 0
\(631\) 7.33731 0.292094 0.146047 0.989278i \(-0.453345\pi\)
0.146047 + 0.989278i \(0.453345\pi\)
\(632\) −38.2228 −1.52042
\(633\) 0 0
\(634\) 62.9742 2.50103
\(635\) 15.2674 0.605869
\(636\) 0 0
\(637\) 10.2739 0.407068
\(638\) −5.58839 −0.221247
\(639\) 0 0
\(640\) −8.53482 −0.337368
\(641\) −13.0203 −0.514272 −0.257136 0.966375i \(-0.582779\pi\)
−0.257136 + 0.966375i \(0.582779\pi\)
\(642\) 0 0
\(643\) 33.7766 1.33202 0.666010 0.745943i \(-0.268001\pi\)
0.666010 + 0.745943i \(0.268001\pi\)
\(644\) 28.4412 1.12074
\(645\) 0 0
\(646\) −6.96349 −0.273975
\(647\) −36.6351 −1.44028 −0.720138 0.693831i \(-0.755921\pi\)
−0.720138 + 0.693831i \(0.755921\pi\)
\(648\) 0 0
\(649\) −8.66353 −0.340073
\(650\) 11.4800 0.450283
\(651\) 0 0
\(652\) −16.5648 −0.648728
\(653\) −0.637706 −0.0249553 −0.0124777 0.999922i \(-0.503972\pi\)
−0.0124777 + 0.999922i \(0.503972\pi\)
\(654\) 0 0
\(655\) −14.5918 −0.570147
\(656\) −84.9253 −3.31578
\(657\) 0 0
\(658\) −3.70720 −0.144522
\(659\) 5.35534 0.208614 0.104307 0.994545i \(-0.466737\pi\)
0.104307 + 0.994545i \(0.466737\pi\)
\(660\) 0 0
\(661\) −38.4599 −1.49592 −0.747958 0.663746i \(-0.768966\pi\)
−0.747958 + 0.663746i \(0.768966\pi\)
\(662\) −50.8825 −1.97761
\(663\) 0 0
\(664\) −81.9945 −3.18200
\(665\) 2.16848 0.0840899
\(666\) 0 0
\(667\) 6.21723 0.240732
\(668\) 2.70136 0.104519
\(669\) 0 0
\(670\) 39.0364 1.50811
\(671\) 5.59008 0.215803
\(672\) 0 0
\(673\) −34.8302 −1.34261 −0.671303 0.741183i \(-0.734265\pi\)
−0.671303 + 0.741183i \(0.734265\pi\)
\(674\) −32.5300 −1.25301
\(675\) 0 0
\(676\) 32.1110 1.23504
\(677\) −24.7488 −0.951174 −0.475587 0.879669i \(-0.657764\pi\)
−0.475587 + 0.879669i \(0.657764\pi\)
\(678\) 0 0
\(679\) 32.1703 1.23458
\(680\) −18.0479 −0.692107
\(681\) 0 0
\(682\) 16.7754 0.642363
\(683\) −7.74417 −0.296322 −0.148161 0.988963i \(-0.547335\pi\)
−0.148161 + 0.988963i \(0.547335\pi\)
\(684\) 0 0
\(685\) −11.9896 −0.458098
\(686\) 51.7646 1.97638
\(687\) 0 0
\(688\) 69.1366 2.63581
\(689\) −62.3147 −2.37400
\(690\) 0 0
\(691\) −29.3966 −1.11830 −0.559150 0.829066i \(-0.688873\pi\)
−0.559150 + 0.829066i \(0.688873\pi\)
\(692\) −106.302 −4.04101
\(693\) 0 0
\(694\) 16.1672 0.613698
\(695\) 7.16027 0.271605
\(696\) 0 0
\(697\) 29.1529 1.10424
\(698\) 12.5936 0.476677
\(699\) 0 0
\(700\) 9.95719 0.376346
\(701\) 16.3181 0.616324 0.308162 0.951334i \(-0.400286\pi\)
0.308162 + 0.951334i \(0.400286\pi\)
\(702\) 0 0
\(703\) −0.861702 −0.0324997
\(704\) 2.11055 0.0795444
\(705\) 0 0
\(706\) 58.5039 2.20182
\(707\) −20.2793 −0.762682
\(708\) 0 0
\(709\) −17.0214 −0.639251 −0.319625 0.947544i \(-0.603557\pi\)
−0.319625 + 0.947544i \(0.603557\pi\)
\(710\) 15.3900 0.577575
\(711\) 0 0
\(712\) 32.2048 1.20693
\(713\) −18.6630 −0.698936
\(714\) 0 0
\(715\) 4.47137 0.167220
\(716\) −51.5840 −1.92779
\(717\) 0 0
\(718\) 67.9950 2.53755
\(719\) 13.3760 0.498842 0.249421 0.968395i \(-0.419760\pi\)
0.249421 + 0.968395i \(0.419760\pi\)
\(720\) 0 0
\(721\) 17.5398 0.653216
\(722\) −2.56745 −0.0955506
\(723\) 0 0
\(724\) −32.9904 −1.22608
\(725\) 2.17663 0.0808381
\(726\) 0 0
\(727\) 34.6984 1.28689 0.643446 0.765492i \(-0.277504\pi\)
0.643446 + 0.765492i \(0.277504\pi\)
\(728\) 64.5204 2.39129
\(729\) 0 0
\(730\) −27.8743 −1.03168
\(731\) −23.7330 −0.877796
\(732\) 0 0
\(733\) −8.67688 −0.320488 −0.160244 0.987077i \(-0.551228\pi\)
−0.160244 + 0.987077i \(0.551228\pi\)
\(734\) −37.0767 −1.36853
\(735\) 0 0
\(736\) −19.9281 −0.734561
\(737\) 15.2043 0.560059
\(738\) 0 0
\(739\) 50.7472 1.86677 0.933383 0.358882i \(-0.116842\pi\)
0.933383 + 0.358882i \(0.116842\pi\)
\(740\) −3.95676 −0.145453
\(741\) 0 0
\(742\) −77.5901 −2.84842
\(743\) −24.0074 −0.880745 −0.440372 0.897815i \(-0.645154\pi\)
−0.440372 + 0.897815i \(0.645154\pi\)
\(744\) 0 0
\(745\) 3.73998 0.137022
\(746\) 9.79342 0.358563
\(747\) 0 0
\(748\) −12.4540 −0.455362
\(749\) 5.87766 0.214765
\(750\) 0 0
\(751\) −6.31971 −0.230610 −0.115305 0.993330i \(-0.536784\pi\)
−0.115305 + 0.993330i \(0.536784\pi\)
\(752\) 5.26103 0.191850
\(753\) 0 0
\(754\) 24.9878 0.910000
\(755\) 7.70787 0.280518
\(756\) 0 0
\(757\) 1.33647 0.0485748 0.0242874 0.999705i \(-0.492268\pi\)
0.0242874 + 0.999705i \(0.492268\pi\)
\(758\) 20.5424 0.746133
\(759\) 0 0
\(760\) −6.65430 −0.241377
\(761\) −26.0134 −0.942986 −0.471493 0.881870i \(-0.656285\pi\)
−0.471493 + 0.881870i \(0.656285\pi\)
\(762\) 0 0
\(763\) 1.07352 0.0388639
\(764\) 64.1637 2.32136
\(765\) 0 0
\(766\) 34.8170 1.25799
\(767\) 38.7378 1.39874
\(768\) 0 0
\(769\) 14.6805 0.529394 0.264697 0.964332i \(-0.414728\pi\)
0.264697 + 0.964332i \(0.414728\pi\)
\(770\) 5.56745 0.200637
\(771\) 0 0
\(772\) −58.7970 −2.11615
\(773\) 47.7853 1.71872 0.859358 0.511374i \(-0.170863\pi\)
0.859358 + 0.511374i \(0.170863\pi\)
\(774\) 0 0
\(775\) −6.53387 −0.234704
\(776\) −98.7197 −3.54383
\(777\) 0 0
\(778\) −4.19898 −0.150541
\(779\) 10.7487 0.385112
\(780\) 0 0
\(781\) 5.99426 0.214491
\(782\) 19.8902 0.711271
\(783\) 0 0
\(784\) −18.1542 −0.648365
\(785\) 7.78415 0.277828
\(786\) 0 0
\(787\) −11.9838 −0.427177 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(788\) −17.3944 −0.619651
\(789\) 0 0
\(790\) 14.7476 0.524697
\(791\) −44.7516 −1.59119
\(792\) 0 0
\(793\) −24.9953 −0.887609
\(794\) −23.7408 −0.842528
\(795\) 0 0
\(796\) 73.0375 2.58875
\(797\) −33.0078 −1.16920 −0.584598 0.811323i \(-0.698748\pi\)
−0.584598 + 0.811323i \(0.698748\pi\)
\(798\) 0 0
\(799\) −1.80599 −0.0638913
\(800\) −6.97678 −0.246666
\(801\) 0 0
\(802\) −0.770015 −0.0271902
\(803\) −10.8568 −0.383129
\(804\) 0 0
\(805\) −6.19393 −0.218307
\(806\) −75.0089 −2.64208
\(807\) 0 0
\(808\) 62.2302 2.18925
\(809\) 29.9288 1.05224 0.526120 0.850411i \(-0.323646\pi\)
0.526120 + 0.850411i \(0.323646\pi\)
\(810\) 0 0
\(811\) −27.3858 −0.961647 −0.480824 0.876817i \(-0.659662\pi\)
−0.480824 + 0.876817i \(0.659662\pi\)
\(812\) 21.6731 0.760578
\(813\) 0 0
\(814\) −2.21237 −0.0775437
\(815\) 3.60748 0.126364
\(816\) 0 0
\(817\) −8.75038 −0.306137
\(818\) −23.8374 −0.833456
\(819\) 0 0
\(820\) 49.3558 1.72358
\(821\) −17.1910 −0.599971 −0.299986 0.953944i \(-0.596982\pi\)
−0.299986 + 0.953944i \(0.596982\pi\)
\(822\) 0 0
\(823\) 5.25893 0.183315 0.0916575 0.995791i \(-0.470784\pi\)
0.0916575 + 0.995791i \(0.470784\pi\)
\(824\) −53.8235 −1.87503
\(825\) 0 0
\(826\) 48.2338 1.67827
\(827\) 18.3379 0.637672 0.318836 0.947810i \(-0.396708\pi\)
0.318836 + 0.947810i \(0.396708\pi\)
\(828\) 0 0
\(829\) 37.5846 1.30537 0.652684 0.757630i \(-0.273643\pi\)
0.652684 + 0.757630i \(0.273643\pi\)
\(830\) 31.6362 1.09811
\(831\) 0 0
\(832\) −9.43705 −0.327171
\(833\) 6.23192 0.215923
\(834\) 0 0
\(835\) −0.588302 −0.0203590
\(836\) −4.59179 −0.158811
\(837\) 0 0
\(838\) −23.9593 −0.827662
\(839\) 52.4722 1.81154 0.905770 0.423769i \(-0.139293\pi\)
0.905770 + 0.423769i \(0.139293\pi\)
\(840\) 0 0
\(841\) −24.2623 −0.836630
\(842\) 64.5599 2.22488
\(843\) 0 0
\(844\) 10.1213 0.348390
\(845\) −6.99312 −0.240571
\(846\) 0 0
\(847\) 2.16848 0.0745097
\(848\) 110.111 3.78123
\(849\) 0 0
\(850\) 6.96349 0.238846
\(851\) 2.46132 0.0843730
\(852\) 0 0
\(853\) −29.5852 −1.01298 −0.506489 0.862246i \(-0.669057\pi\)
−0.506489 + 0.862246i \(0.669057\pi\)
\(854\) −31.1225 −1.06499
\(855\) 0 0
\(856\) −18.0365 −0.616475
\(857\) 23.6715 0.808601 0.404301 0.914626i \(-0.367515\pi\)
0.404301 + 0.914626i \(0.367515\pi\)
\(858\) 0 0
\(859\) −29.0072 −0.989714 −0.494857 0.868974i \(-0.664780\pi\)
−0.494857 + 0.868974i \(0.664780\pi\)
\(860\) −40.1800 −1.37013
\(861\) 0 0
\(862\) 96.4487 3.28505
\(863\) 36.3796 1.23837 0.619187 0.785243i \(-0.287462\pi\)
0.619187 + 0.785243i \(0.287462\pi\)
\(864\) 0 0
\(865\) 23.1505 0.787140
\(866\) −23.7305 −0.806394
\(867\) 0 0
\(868\) −65.0590 −2.20825
\(869\) 5.74408 0.194855
\(870\) 0 0
\(871\) −67.9842 −2.30356
\(872\) −3.29425 −0.111557
\(873\) 0 0
\(874\) 7.33354 0.248061
\(875\) −2.16848 −0.0733078
\(876\) 0 0
\(877\) −56.9075 −1.92163 −0.960815 0.277192i \(-0.910596\pi\)
−0.960815 + 0.277192i \(0.910596\pi\)
\(878\) −2.41809 −0.0816066
\(879\) 0 0
\(880\) −7.90099 −0.266342
\(881\) −53.1897 −1.79201 −0.896004 0.444047i \(-0.853542\pi\)
−0.896004 + 0.444047i \(0.853542\pi\)
\(882\) 0 0
\(883\) 22.9447 0.772151 0.386075 0.922467i \(-0.373831\pi\)
0.386075 + 0.922467i \(0.373831\pi\)
\(884\) 55.6862 1.87293
\(885\) 0 0
\(886\) 44.3467 1.48986
\(887\) −35.0993 −1.17852 −0.589260 0.807943i \(-0.700581\pi\)
−0.589260 + 0.807943i \(0.700581\pi\)
\(888\) 0 0
\(889\) −33.1070 −1.11037
\(890\) −12.4257 −0.416509
\(891\) 0 0
\(892\) −65.9704 −2.20885
\(893\) −0.665870 −0.0222825
\(894\) 0 0
\(895\) 11.2340 0.375510
\(896\) 18.5075 0.618294
\(897\) 0 0
\(898\) −53.8952 −1.79851
\(899\) −14.2218 −0.474325
\(900\) 0 0
\(901\) −37.7986 −1.25925
\(902\) 27.5967 0.918871
\(903\) 0 0
\(904\) 137.327 4.56744
\(905\) 7.18465 0.238826
\(906\) 0 0
\(907\) 49.6640 1.64906 0.824532 0.565815i \(-0.191438\pi\)
0.824532 + 0.565815i \(0.191438\pi\)
\(908\) −99.6002 −3.30535
\(909\) 0 0
\(910\) −24.8941 −0.825232
\(911\) −44.2360 −1.46560 −0.732802 0.680442i \(-0.761788\pi\)
−0.732802 + 0.680442i \(0.761788\pi\)
\(912\) 0 0
\(913\) 12.3220 0.407800
\(914\) 74.4286 2.46188
\(915\) 0 0
\(916\) 43.1507 1.42574
\(917\) 31.6419 1.04491
\(918\) 0 0
\(919\) −36.2925 −1.19718 −0.598590 0.801055i \(-0.704272\pi\)
−0.598590 + 0.801055i \(0.704272\pi\)
\(920\) 19.0070 0.626643
\(921\) 0 0
\(922\) −63.6508 −2.09623
\(923\) −26.8025 −0.882216
\(924\) 0 0
\(925\) 0.861702 0.0283326
\(926\) 1.94387 0.0638796
\(927\) 0 0
\(928\) −15.1859 −0.498501
\(929\) −20.6900 −0.678818 −0.339409 0.940639i \(-0.610227\pi\)
−0.339409 + 0.940639i \(0.610227\pi\)
\(930\) 0 0
\(931\) 2.29772 0.0753046
\(932\) 104.010 3.40698
\(933\) 0 0
\(934\) −14.2355 −0.465800
\(935\) 2.71222 0.0886991
\(936\) 0 0
\(937\) −58.8084 −1.92119 −0.960594 0.277957i \(-0.910343\pi\)
−0.960594 + 0.277957i \(0.910343\pi\)
\(938\) −84.6494 −2.76390
\(939\) 0 0
\(940\) −3.05754 −0.0997260
\(941\) 18.1745 0.592471 0.296235 0.955115i \(-0.404269\pi\)
0.296235 + 0.955115i \(0.404269\pi\)
\(942\) 0 0
\(943\) −30.7021 −0.999797
\(944\) −68.4504 −2.22787
\(945\) 0 0
\(946\) −22.4662 −0.730438
\(947\) −9.22596 −0.299803 −0.149902 0.988701i \(-0.547896\pi\)
−0.149902 + 0.988701i \(0.547896\pi\)
\(948\) 0 0
\(949\) 48.5448 1.57583
\(950\) 2.56745 0.0832990
\(951\) 0 0
\(952\) 39.1365 1.26842
\(953\) −34.8382 −1.12852 −0.564260 0.825597i \(-0.690838\pi\)
−0.564260 + 0.825597i \(0.690838\pi\)
\(954\) 0 0
\(955\) −13.9736 −0.452174
\(956\) −56.5665 −1.82949
\(957\) 0 0
\(958\) −22.5617 −0.728935
\(959\) 25.9991 0.839554
\(960\) 0 0
\(961\) 11.6915 0.377145
\(962\) 9.89234 0.318942
\(963\) 0 0
\(964\) −126.507 −4.07451
\(965\) 12.8048 0.412201
\(966\) 0 0
\(967\) −21.5696 −0.693631 −0.346816 0.937933i \(-0.612737\pi\)
−0.346816 + 0.937933i \(0.612737\pi\)
\(968\) −6.65430 −0.213877
\(969\) 0 0
\(970\) 38.0893 1.22297
\(971\) −52.0792 −1.67130 −0.835650 0.549262i \(-0.814909\pi\)
−0.835650 + 0.549262i \(0.814909\pi\)
\(972\) 0 0
\(973\) −15.5269 −0.497769
\(974\) −94.7207 −3.03505
\(975\) 0 0
\(976\) 44.1671 1.41376
\(977\) 14.7908 0.473200 0.236600 0.971607i \(-0.423967\pi\)
0.236600 + 0.971607i \(0.423967\pi\)
\(978\) 0 0
\(979\) −4.83970 −0.154677
\(980\) 10.5506 0.337028
\(981\) 0 0
\(982\) −15.7220 −0.501708
\(983\) 35.1347 1.12062 0.560311 0.828282i \(-0.310682\pi\)
0.560311 + 0.828282i \(0.310682\pi\)
\(984\) 0 0
\(985\) 3.78816 0.120701
\(986\) 15.1570 0.482696
\(987\) 0 0
\(988\) 20.5316 0.653197
\(989\) 24.9942 0.794768
\(990\) 0 0
\(991\) −50.5723 −1.60648 −0.803241 0.595654i \(-0.796893\pi\)
−0.803241 + 0.595654i \(0.796893\pi\)
\(992\) 45.5854 1.44734
\(993\) 0 0
\(994\) −33.3727 −1.05852
\(995\) −15.9061 −0.504257
\(996\) 0 0
\(997\) −16.9360 −0.536370 −0.268185 0.963367i \(-0.586424\pi\)
−0.268185 + 0.963367i \(0.586424\pi\)
\(998\) 43.4295 1.37474
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9405.2.a.w.1.1 6
3.2 odd 2 1045.2.a.g.1.6 6
15.14 odd 2 5225.2.a.k.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.6 6 3.2 odd 2
5225.2.a.k.1.1 6 15.14 odd 2
9405.2.a.w.1.1 6 1.1 even 1 trivial