Properties

Label 7623.2.a.bt.1.1
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $2$
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] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +1.56155 q^{10} -5.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} +4.12311 q^{17} -6.00000 q^{19} -0.438447 q^{20} +4.00000 q^{23} -4.00000 q^{25} +8.68466 q^{26} -0.438447 q^{28} +6.68466 q^{29} +8.24621 q^{31} +2.43845 q^{32} -6.43845 q^{34} +1.00000 q^{35} -2.68466 q^{37} +9.36932 q^{38} -2.43845 q^{40} -7.56155 q^{41} +5.68466 q^{43} -6.24621 q^{46} -3.43845 q^{47} +1.00000 q^{49} +6.24621 q^{50} -2.43845 q^{52} -7.80776 q^{53} -2.43845 q^{56} -10.4384 q^{58} +12.5616 q^{59} +13.3693 q^{61} -12.8769 q^{62} +5.56155 q^{64} +5.56155 q^{65} +8.80776 q^{67} +1.80776 q^{68} -1.56155 q^{70} -3.12311 q^{71} -7.12311 q^{73} +4.19224 q^{74} -2.63068 q^{76} -3.12311 q^{79} +4.68466 q^{80} +11.8078 q^{82} -8.80776 q^{83} -4.12311 q^{85} -8.87689 q^{86} +11.2462 q^{89} +5.56155 q^{91} +1.75379 q^{92} +5.36932 q^{94} +6.00000 q^{95} +1.31534 q^{97} -1.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8} - q^{10} - 7 q^{13} - q^{14} + 3 q^{16} - 12 q^{19} - 5 q^{20} + 8 q^{23} - 8 q^{25} + 5 q^{26} - 5 q^{28} + q^{29} + 9 q^{32} - 17 q^{34} + 2 q^{35} + 7 q^{37} - 6 q^{38} - 9 q^{40} - 11 q^{41} - q^{43} + 4 q^{46} - 11 q^{47} + 2 q^{49} - 4 q^{50} - 9 q^{52} + 5 q^{53} - 9 q^{56} - 25 q^{58} + 21 q^{59} + 2 q^{61} - 34 q^{62} + 7 q^{64} + 7 q^{65} - 3 q^{67} - 17 q^{68} + q^{70} + 2 q^{71} - 6 q^{73} + 29 q^{74} - 30 q^{76} + 2 q^{79} - 3 q^{80} + 3 q^{82} + 3 q^{83} - 26 q^{86} + 6 q^{89} + 7 q^{91} + 20 q^{92} - 14 q^{94} + 12 q^{95} + 15 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) 1.56155 0.493806
\(11\) 0 0
\(12\) 0 0
\(13\) −5.56155 −1.54250 −0.771249 0.636534i \(-0.780367\pi\)
−0.771249 + 0.636534i \(0.780367\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 4.12311 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −0.438447 −0.0980398
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 8.68466 1.70320
\(27\) 0 0
\(28\) −0.438447 −0.0828587
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) 8.24621 1.48106 0.740532 0.672022i \(-0.234574\pi\)
0.740532 + 0.672022i \(0.234574\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) −6.43845 −1.10418
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.68466 −0.441355 −0.220678 0.975347i \(-0.570827\pi\)
−0.220678 + 0.975347i \(0.570827\pi\)
\(38\) 9.36932 1.51990
\(39\) 0 0
\(40\) −2.43845 −0.385552
\(41\) −7.56155 −1.18092 −0.590458 0.807068i \(-0.701053\pi\)
−0.590458 + 0.807068i \(0.701053\pi\)
\(42\) 0 0
\(43\) 5.68466 0.866902 0.433451 0.901177i \(-0.357296\pi\)
0.433451 + 0.901177i \(0.357296\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.24621 −0.920954
\(47\) −3.43845 −0.501549 −0.250775 0.968046i \(-0.580685\pi\)
−0.250775 + 0.968046i \(0.580685\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.24621 0.883348
\(51\) 0 0
\(52\) −2.43845 −0.338152
\(53\) −7.80776 −1.07248 −0.536239 0.844066i \(-0.680156\pi\)
−0.536239 + 0.844066i \(0.680156\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.43845 −0.325851
\(57\) 0 0
\(58\) −10.4384 −1.37064
\(59\) 12.5616 1.63537 0.817687 0.575662i \(-0.195256\pi\)
0.817687 + 0.575662i \(0.195256\pi\)
\(60\) 0 0
\(61\) 13.3693 1.71177 0.855883 0.517170i \(-0.173014\pi\)
0.855883 + 0.517170i \(0.173014\pi\)
\(62\) −12.8769 −1.63537
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) 5.56155 0.689826
\(66\) 0 0
\(67\) 8.80776 1.07604 0.538020 0.842932i \(-0.319173\pi\)
0.538020 + 0.842932i \(0.319173\pi\)
\(68\) 1.80776 0.219224
\(69\) 0 0
\(70\) −1.56155 −0.186641
\(71\) −3.12311 −0.370644 −0.185322 0.982678i \(-0.559333\pi\)
−0.185322 + 0.982678i \(0.559333\pi\)
\(72\) 0 0
\(73\) −7.12311 −0.833696 −0.416848 0.908976i \(-0.636865\pi\)
−0.416848 + 0.908976i \(0.636865\pi\)
\(74\) 4.19224 0.487338
\(75\) 0 0
\(76\) −2.63068 −0.301760
\(77\) 0 0
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 4.68466 0.523761
\(81\) 0 0
\(82\) 11.8078 1.30395
\(83\) −8.80776 −0.966778 −0.483389 0.875406i \(-0.660594\pi\)
−0.483389 + 0.875406i \(0.660594\pi\)
\(84\) 0 0
\(85\) −4.12311 −0.447214
\(86\) −8.87689 −0.957220
\(87\) 0 0
\(88\) 0 0
\(89\) 11.2462 1.19210 0.596048 0.802949i \(-0.296737\pi\)
0.596048 + 0.802949i \(0.296737\pi\)
\(90\) 0 0
\(91\) 5.56155 0.583009
\(92\) 1.75379 0.182845
\(93\) 0 0
\(94\) 5.36932 0.553803
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 1.31534 0.133553 0.0667764 0.997768i \(-0.478729\pi\)
0.0667764 + 0.997768i \(0.478729\pi\)
\(98\) −1.56155 −0.157741
\(99\) 0 0
\(100\) −1.75379 −0.175379
\(101\) 4.56155 0.453891 0.226946 0.973907i \(-0.427126\pi\)
0.226946 + 0.973907i \(0.427126\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −13.5616 −1.32982
\(105\) 0 0
\(106\) 12.1922 1.18421
\(107\) −3.12311 −0.301922 −0.150961 0.988540i \(-0.548237\pi\)
−0.150961 + 0.988540i \(0.548237\pi\)
\(108\) 0 0
\(109\) −3.87689 −0.371339 −0.185670 0.982612i \(-0.559445\pi\)
−0.185670 + 0.982612i \(0.559445\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.68466 0.442659
\(113\) 8.93087 0.840146 0.420073 0.907490i \(-0.362005\pi\)
0.420073 + 0.907490i \(0.362005\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 2.93087 0.272124
\(117\) 0 0
\(118\) −19.6155 −1.80576
\(119\) −4.12311 −0.377964
\(120\) 0 0
\(121\) 0 0
\(122\) −20.8769 −1.89011
\(123\) 0 0
\(124\) 3.61553 0.324684
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −21.9309 −1.94605 −0.973025 0.230700i \(-0.925899\pi\)
−0.973025 + 0.230700i \(0.925899\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) −8.68466 −0.761695
\(131\) −2.56155 −0.223804 −0.111902 0.993719i \(-0.535694\pi\)
−0.111902 + 0.993719i \(0.535694\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −13.7538 −1.18815
\(135\) 0 0
\(136\) 10.0540 0.862121
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) −6.24621 −0.529797 −0.264898 0.964276i \(-0.585338\pi\)
−0.264898 + 0.964276i \(0.585338\pi\)
\(140\) 0.438447 0.0370556
\(141\) 0 0
\(142\) 4.87689 0.409260
\(143\) 0 0
\(144\) 0 0
\(145\) −6.68466 −0.555131
\(146\) 11.1231 0.920555
\(147\) 0 0
\(148\) −1.17708 −0.0967555
\(149\) −5.31534 −0.435450 −0.217725 0.976010i \(-0.569864\pi\)
−0.217725 + 0.976010i \(0.569864\pi\)
\(150\) 0 0
\(151\) −11.4384 −0.930848 −0.465424 0.885088i \(-0.654098\pi\)
−0.465424 + 0.885088i \(0.654098\pi\)
\(152\) −14.6307 −1.18671
\(153\) 0 0
\(154\) 0 0
\(155\) −8.24621 −0.662352
\(156\) 0 0
\(157\) −5.36932 −0.428518 −0.214259 0.976777i \(-0.568734\pi\)
−0.214259 + 0.976777i \(0.568734\pi\)
\(158\) 4.87689 0.387985
\(159\) 0 0
\(160\) −2.43845 −0.192776
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 17.3693 1.36047 0.680235 0.732994i \(-0.261878\pi\)
0.680235 + 0.732994i \(0.261878\pi\)
\(164\) −3.31534 −0.258885
\(165\) 0 0
\(166\) 13.7538 1.06750
\(167\) −12.8078 −0.991095 −0.495547 0.868581i \(-0.665032\pi\)
−0.495547 + 0.868581i \(0.665032\pi\)
\(168\) 0 0
\(169\) 17.9309 1.37930
\(170\) 6.43845 0.493806
\(171\) 0 0
\(172\) 2.49242 0.190045
\(173\) 12.5616 0.955037 0.477519 0.878622i \(-0.341536\pi\)
0.477519 + 0.878622i \(0.341536\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −17.5616 −1.31629
\(179\) 4.87689 0.364516 0.182258 0.983251i \(-0.441659\pi\)
0.182258 + 0.983251i \(0.441659\pi\)
\(180\) 0 0
\(181\) −0.192236 −0.0142888 −0.00714439 0.999974i \(-0.502274\pi\)
−0.00714439 + 0.999974i \(0.502274\pi\)
\(182\) −8.68466 −0.643750
\(183\) 0 0
\(184\) 9.75379 0.719059
\(185\) 2.68466 0.197380
\(186\) 0 0
\(187\) 0 0
\(188\) −1.50758 −0.109951
\(189\) 0 0
\(190\) −9.36932 −0.679722
\(191\) 12.2462 0.886105 0.443052 0.896496i \(-0.353896\pi\)
0.443052 + 0.896496i \(0.353896\pi\)
\(192\) 0 0
\(193\) 15.2462 1.09745 0.548723 0.836004i \(-0.315114\pi\)
0.548723 + 0.836004i \(0.315114\pi\)
\(194\) −2.05398 −0.147467
\(195\) 0 0
\(196\) 0.438447 0.0313177
\(197\) 27.1771 1.93629 0.968143 0.250396i \(-0.0805609\pi\)
0.968143 + 0.250396i \(0.0805609\pi\)
\(198\) 0 0
\(199\) −7.12311 −0.504944 −0.252472 0.967604i \(-0.581243\pi\)
−0.252472 + 0.967604i \(0.581243\pi\)
\(200\) −9.75379 −0.689697
\(201\) 0 0
\(202\) −7.12311 −0.501180
\(203\) −6.68466 −0.469171
\(204\) 0 0
\(205\) 7.56155 0.528122
\(206\) 12.4924 0.870388
\(207\) 0 0
\(208\) 26.0540 1.80652
\(209\) 0 0
\(210\) 0 0
\(211\) −27.0540 −1.86247 −0.931236 0.364416i \(-0.881269\pi\)
−0.931236 + 0.364416i \(0.881269\pi\)
\(212\) −3.42329 −0.235113
\(213\) 0 0
\(214\) 4.87689 0.333378
\(215\) −5.68466 −0.387690
\(216\) 0 0
\(217\) −8.24621 −0.559789
\(218\) 6.05398 0.410027
\(219\) 0 0
\(220\) 0 0
\(221\) −22.9309 −1.54250
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −2.43845 −0.162926
\(225\) 0 0
\(226\) −13.9460 −0.927676
\(227\) 5.68466 0.377304 0.188652 0.982044i \(-0.439588\pi\)
0.188652 + 0.982044i \(0.439588\pi\)
\(228\) 0 0
\(229\) −4.19224 −0.277031 −0.138515 0.990360i \(-0.544233\pi\)
−0.138515 + 0.990360i \(0.544233\pi\)
\(230\) 6.24621 0.411863
\(231\) 0 0
\(232\) 16.3002 1.07016
\(233\) −27.1771 −1.78043 −0.890215 0.455541i \(-0.849446\pi\)
−0.890215 + 0.455541i \(0.849446\pi\)
\(234\) 0 0
\(235\) 3.43845 0.224300
\(236\) 5.50758 0.358513
\(237\) 0 0
\(238\) 6.43845 0.417343
\(239\) −19.3693 −1.25290 −0.626448 0.779463i \(-0.715492\pi\)
−0.626448 + 0.779463i \(0.715492\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 5.86174 0.375259
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 33.3693 2.12324
\(248\) 20.1080 1.27686
\(249\) 0 0
\(250\) −14.0540 −0.888851
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 34.2462 2.14880
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −23.0000 −1.43470 −0.717350 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) 0 0
\(259\) 2.68466 0.166817
\(260\) 2.43845 0.151226
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 30.7386 1.89543 0.947713 0.319125i \(-0.103389\pi\)
0.947713 + 0.319125i \(0.103389\pi\)
\(264\) 0 0
\(265\) 7.80776 0.479627
\(266\) −9.36932 −0.574470
\(267\) 0 0
\(268\) 3.86174 0.235893
\(269\) 4.43845 0.270617 0.135308 0.990804i \(-0.456797\pi\)
0.135308 + 0.990804i \(0.456797\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −19.3153 −1.17116
\(273\) 0 0
\(274\) −21.8617 −1.32072
\(275\) 0 0
\(276\) 0 0
\(277\) 13.2462 0.795888 0.397944 0.917410i \(-0.369724\pi\)
0.397944 + 0.917410i \(0.369724\pi\)
\(278\) 9.75379 0.584993
\(279\) 0 0
\(280\) 2.43845 0.145725
\(281\) 20.8769 1.24541 0.622706 0.782456i \(-0.286033\pi\)
0.622706 + 0.782456i \(0.286033\pi\)
\(282\) 0 0
\(283\) −15.1231 −0.898975 −0.449488 0.893287i \(-0.648393\pi\)
−0.449488 + 0.893287i \(0.648393\pi\)
\(284\) −1.36932 −0.0812540
\(285\) 0 0
\(286\) 0 0
\(287\) 7.56155 0.446344
\(288\) 0 0
\(289\) 0 0
\(290\) 10.4384 0.612967
\(291\) 0 0
\(292\) −3.12311 −0.182766
\(293\) −22.3693 −1.30683 −0.653415 0.757000i \(-0.726664\pi\)
−0.653415 + 0.757000i \(0.726664\pi\)
\(294\) 0 0
\(295\) −12.5616 −0.731362
\(296\) −6.54640 −0.380502
\(297\) 0 0
\(298\) 8.30019 0.480817
\(299\) −22.2462 −1.28653
\(300\) 0 0
\(301\) −5.68466 −0.327658
\(302\) 17.8617 1.02783
\(303\) 0 0
\(304\) 28.1080 1.61210
\(305\) −13.3693 −0.765525
\(306\) 0 0
\(307\) −22.8769 −1.30565 −0.652827 0.757507i \(-0.726417\pi\)
−0.652827 + 0.757507i \(0.726417\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.8769 0.731358
\(311\) −4.56155 −0.258662 −0.129331 0.991601i \(-0.541283\pi\)
−0.129331 + 0.991601i \(0.541283\pi\)
\(312\) 0 0
\(313\) 2.43845 0.137829 0.0689146 0.997623i \(-0.478046\pi\)
0.0689146 + 0.997623i \(0.478046\pi\)
\(314\) 8.38447 0.473163
\(315\) 0 0
\(316\) −1.36932 −0.0770301
\(317\) 27.1231 1.52339 0.761693 0.647938i \(-0.224369\pi\)
0.761693 + 0.647938i \(0.224369\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −5.56155 −0.310900
\(321\) 0 0
\(322\) 6.24621 0.348088
\(323\) −24.7386 −1.37649
\(324\) 0 0
\(325\) 22.2462 1.23400
\(326\) −27.1231 −1.50221
\(327\) 0 0
\(328\) −18.4384 −1.01809
\(329\) 3.43845 0.189568
\(330\) 0 0
\(331\) −17.6847 −0.972037 −0.486018 0.873949i \(-0.661551\pi\)
−0.486018 + 0.873949i \(0.661551\pi\)
\(332\) −3.86174 −0.211940
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) −8.80776 −0.481220
\(336\) 0 0
\(337\) −27.1771 −1.48043 −0.740215 0.672370i \(-0.765276\pi\)
−0.740215 + 0.672370i \(0.765276\pi\)
\(338\) −28.0000 −1.52300
\(339\) 0 0
\(340\) −1.80776 −0.0980398
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 13.8617 0.747375
\(345\) 0 0
\(346\) −19.6155 −1.05454
\(347\) −14.0000 −0.751559 −0.375780 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) 0 0
\(349\) 8.05398 0.431119 0.215560 0.976491i \(-0.430842\pi\)
0.215560 + 0.976491i \(0.430842\pi\)
\(350\) −6.24621 −0.333874
\(351\) 0 0
\(352\) 0 0
\(353\) −23.1771 −1.23359 −0.616796 0.787123i \(-0.711570\pi\)
−0.616796 + 0.787123i \(0.711570\pi\)
\(354\) 0 0
\(355\) 3.12311 0.165757
\(356\) 4.93087 0.261336
\(357\) 0 0
\(358\) −7.61553 −0.402493
\(359\) 28.0000 1.47778 0.738892 0.673824i \(-0.235349\pi\)
0.738892 + 0.673824i \(0.235349\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0.300187 0.0157775
\(363\) 0 0
\(364\) 2.43845 0.127809
\(365\) 7.12311 0.372840
\(366\) 0 0
\(367\) 14.4924 0.756498 0.378249 0.925704i \(-0.376526\pi\)
0.378249 + 0.925704i \(0.376526\pi\)
\(368\) −18.7386 −0.976819
\(369\) 0 0
\(370\) −4.19224 −0.217944
\(371\) 7.80776 0.405359
\(372\) 0 0
\(373\) −6.80776 −0.352493 −0.176246 0.984346i \(-0.556396\pi\)
−0.176246 + 0.984346i \(0.556396\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.38447 −0.432396
\(377\) −37.1771 −1.91472
\(378\) 0 0
\(379\) −14.5616 −0.747977 −0.373988 0.927433i \(-0.622010\pi\)
−0.373988 + 0.927433i \(0.622010\pi\)
\(380\) 2.63068 0.134951
\(381\) 0 0
\(382\) −19.1231 −0.978423
\(383\) −17.4384 −0.891063 −0.445532 0.895266i \(-0.646985\pi\)
−0.445532 + 0.895266i \(0.646985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −23.8078 −1.21178
\(387\) 0 0
\(388\) 0.576708 0.0292779
\(389\) −17.5616 −0.890406 −0.445203 0.895430i \(-0.646868\pi\)
−0.445203 + 0.895430i \(0.646868\pi\)
\(390\) 0 0
\(391\) 16.4924 0.834058
\(392\) 2.43845 0.123160
\(393\) 0 0
\(394\) −42.4384 −2.13802
\(395\) 3.12311 0.157140
\(396\) 0 0
\(397\) −13.3153 −0.668278 −0.334139 0.942524i \(-0.608445\pi\)
−0.334139 + 0.942524i \(0.608445\pi\)
\(398\) 11.1231 0.557551
\(399\) 0 0
\(400\) 18.7386 0.936932
\(401\) 15.3153 0.764812 0.382406 0.923994i \(-0.375096\pi\)
0.382406 + 0.923994i \(0.375096\pi\)
\(402\) 0 0
\(403\) −45.8617 −2.28454
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 10.4384 0.518051
\(407\) 0 0
\(408\) 0 0
\(409\) −34.3002 −1.69604 −0.848018 0.529968i \(-0.822204\pi\)
−0.848018 + 0.529968i \(0.822204\pi\)
\(410\) −11.8078 −0.583144
\(411\) 0 0
\(412\) −3.50758 −0.172806
\(413\) −12.5616 −0.618114
\(414\) 0 0
\(415\) 8.80776 0.432356
\(416\) −13.5616 −0.664910
\(417\) 0 0
\(418\) 0 0
\(419\) 21.3002 1.04058 0.520291 0.853989i \(-0.325823\pi\)
0.520291 + 0.853989i \(0.325823\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 42.2462 2.05651
\(423\) 0 0
\(424\) −19.0388 −0.924607
\(425\) −16.4924 −0.800000
\(426\) 0 0
\(427\) −13.3693 −0.646987
\(428\) −1.36932 −0.0661884
\(429\) 0 0
\(430\) 8.87689 0.428082
\(431\) 10.6307 0.512062 0.256031 0.966669i \(-0.417585\pi\)
0.256031 + 0.966669i \(0.417585\pi\)
\(432\) 0 0
\(433\) −12.9309 −0.621418 −0.310709 0.950505i \(-0.600566\pi\)
−0.310709 + 0.950505i \(0.600566\pi\)
\(434\) 12.8769 0.618111
\(435\) 0 0
\(436\) −1.69981 −0.0814063
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 1.12311 0.0536029 0.0268015 0.999641i \(-0.491468\pi\)
0.0268015 + 0.999641i \(0.491468\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 35.8078 1.70320
\(443\) 8.49242 0.403487 0.201744 0.979438i \(-0.435339\pi\)
0.201744 + 0.979438i \(0.435339\pi\)
\(444\) 0 0
\(445\) −11.2462 −0.533122
\(446\) −37.4773 −1.77460
\(447\) 0 0
\(448\) −5.56155 −0.262759
\(449\) −28.0540 −1.32395 −0.661974 0.749526i \(-0.730281\pi\)
−0.661974 + 0.749526i \(0.730281\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 3.91571 0.184180
\(453\) 0 0
\(454\) −8.87689 −0.416613
\(455\) −5.56155 −0.260730
\(456\) 0 0
\(457\) 10.6155 0.496573 0.248287 0.968687i \(-0.420132\pi\)
0.248287 + 0.968687i \(0.420132\pi\)
\(458\) 6.54640 0.305893
\(459\) 0 0
\(460\) −1.75379 −0.0817708
\(461\) −10.3693 −0.482947 −0.241474 0.970407i \(-0.577631\pi\)
−0.241474 + 0.970407i \(0.577631\pi\)
\(462\) 0 0
\(463\) 0.876894 0.0407527 0.0203764 0.999792i \(-0.493514\pi\)
0.0203764 + 0.999792i \(0.493514\pi\)
\(464\) −31.3153 −1.45378
\(465\) 0 0
\(466\) 42.4384 1.96592
\(467\) 21.7538 1.00665 0.503323 0.864099i \(-0.332111\pi\)
0.503323 + 0.864099i \(0.332111\pi\)
\(468\) 0 0
\(469\) −8.80776 −0.406705
\(470\) −5.36932 −0.247668
\(471\) 0 0
\(472\) 30.6307 1.40989
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) −1.80776 −0.0828587
\(477\) 0 0
\(478\) 30.2462 1.38343
\(479\) −29.6847 −1.35633 −0.678163 0.734911i \(-0.737224\pi\)
−0.678163 + 0.734911i \(0.737224\pi\)
\(480\) 0 0
\(481\) 14.9309 0.680789
\(482\) −19.1231 −0.871034
\(483\) 0 0
\(484\) 0 0
\(485\) −1.31534 −0.0597266
\(486\) 0 0
\(487\) 19.4384 0.880840 0.440420 0.897792i \(-0.354830\pi\)
0.440420 + 0.897792i \(0.354830\pi\)
\(488\) 32.6004 1.47575
\(489\) 0 0
\(490\) 1.56155 0.0705438
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 27.5616 1.24131
\(494\) −52.1080 −2.34445
\(495\) 0 0
\(496\) −38.6307 −1.73457
\(497\) 3.12311 0.140090
\(498\) 0 0
\(499\) −39.5464 −1.77034 −0.885170 0.465268i \(-0.845958\pi\)
−0.885170 + 0.465268i \(0.845958\pi\)
\(500\) 3.94602 0.176472
\(501\) 0 0
\(502\) 6.24621 0.278782
\(503\) −19.6847 −0.877696 −0.438848 0.898561i \(-0.644613\pi\)
−0.438848 + 0.898561i \(0.644613\pi\)
\(504\) 0 0
\(505\) −4.56155 −0.202986
\(506\) 0 0
\(507\) 0 0
\(508\) −9.61553 −0.426620
\(509\) −25.6847 −1.13845 −0.569226 0.822181i \(-0.692757\pi\)
−0.569226 + 0.822181i \(0.692757\pi\)
\(510\) 0 0
\(511\) 7.12311 0.315108
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) 35.9157 1.58417
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) −4.19224 −0.184196
\(519\) 0 0
\(520\) 13.5616 0.594713
\(521\) −43.9309 −1.92465 −0.962323 0.271908i \(-0.912345\pi\)
−0.962323 + 0.271908i \(0.912345\pi\)
\(522\) 0 0
\(523\) 25.1231 1.09856 0.549278 0.835639i \(-0.314903\pi\)
0.549278 + 0.835639i \(0.314903\pi\)
\(524\) −1.12311 −0.0490631
\(525\) 0 0
\(526\) −48.0000 −2.09290
\(527\) 34.0000 1.48106
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −12.1922 −0.529597
\(531\) 0 0
\(532\) 2.63068 0.114055
\(533\) 42.0540 1.82156
\(534\) 0 0
\(535\) 3.12311 0.135024
\(536\) 21.4773 0.927677
\(537\) 0 0
\(538\) −6.93087 −0.298811
\(539\) 0 0
\(540\) 0 0
\(541\) −11.9309 −0.512948 −0.256474 0.966551i \(-0.582561\pi\)
−0.256474 + 0.966551i \(0.582561\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 10.0540 0.431061
\(545\) 3.87689 0.166068
\(546\) 0 0
\(547\) 21.3693 0.913686 0.456843 0.889547i \(-0.348980\pi\)
0.456843 + 0.889547i \(0.348980\pi\)
\(548\) 6.13826 0.262213
\(549\) 0 0
\(550\) 0 0
\(551\) −40.1080 −1.70866
\(552\) 0 0
\(553\) 3.12311 0.132808
\(554\) −20.6847 −0.878807
\(555\) 0 0
\(556\) −2.73863 −0.116144
\(557\) −42.4924 −1.80046 −0.900231 0.435413i \(-0.856602\pi\)
−0.900231 + 0.435413i \(0.856602\pi\)
\(558\) 0 0
\(559\) −31.6155 −1.33719
\(560\) −4.68466 −0.197963
\(561\) 0 0
\(562\) −32.6004 −1.37516
\(563\) 6.80776 0.286913 0.143457 0.989657i \(-0.454178\pi\)
0.143457 + 0.989657i \(0.454178\pi\)
\(564\) 0 0
\(565\) −8.93087 −0.375725
\(566\) 23.6155 0.992635
\(567\) 0 0
\(568\) −7.61553 −0.319540
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 10.2462 0.428791 0.214395 0.976747i \(-0.431222\pi\)
0.214395 + 0.976747i \(0.431222\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −11.8078 −0.492847
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −27.8078 −1.15765 −0.578826 0.815451i \(-0.696489\pi\)
−0.578826 + 0.815451i \(0.696489\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −2.93087 −0.121698
\(581\) 8.80776 0.365408
\(582\) 0 0
\(583\) 0 0
\(584\) −17.3693 −0.718747
\(585\) 0 0
\(586\) 34.9309 1.44298
\(587\) 3.43845 0.141920 0.0709600 0.997479i \(-0.477394\pi\)
0.0709600 + 0.997479i \(0.477394\pi\)
\(588\) 0 0
\(589\) −49.4773 −2.03868
\(590\) 19.6155 0.807559
\(591\) 0 0
\(592\) 12.5767 0.516900
\(593\) −44.3693 −1.82203 −0.911015 0.412374i \(-0.864700\pi\)
−0.911015 + 0.412374i \(0.864700\pi\)
\(594\) 0 0
\(595\) 4.12311 0.169031
\(596\) −2.33050 −0.0954609
\(597\) 0 0
\(598\) 34.7386 1.42057
\(599\) 41.2311 1.68466 0.842328 0.538966i \(-0.181185\pi\)
0.842328 + 0.538966i \(0.181185\pi\)
\(600\) 0 0
\(601\) −4.68466 −0.191091 −0.0955456 0.995425i \(-0.530460\pi\)
−0.0955456 + 0.995425i \(0.530460\pi\)
\(602\) 8.87689 0.361795
\(603\) 0 0
\(604\) −5.01515 −0.204064
\(605\) 0 0
\(606\) 0 0
\(607\) 14.2462 0.578236 0.289118 0.957293i \(-0.406638\pi\)
0.289118 + 0.957293i \(0.406638\pi\)
\(608\) −14.6307 −0.593353
\(609\) 0 0
\(610\) 20.8769 0.845281
\(611\) 19.1231 0.773638
\(612\) 0 0
\(613\) 27.4924 1.11041 0.555204 0.831714i \(-0.312640\pi\)
0.555204 + 0.831714i \(0.312640\pi\)
\(614\) 35.7235 1.44168
\(615\) 0 0
\(616\) 0 0
\(617\) 47.1771 1.89928 0.949639 0.313346i \(-0.101450\pi\)
0.949639 + 0.313346i \(0.101450\pi\)
\(618\) 0 0
\(619\) −46.4924 −1.86869 −0.934344 0.356372i \(-0.884014\pi\)
−0.934344 + 0.356372i \(0.884014\pi\)
\(620\) −3.61553 −0.145203
\(621\) 0 0
\(622\) 7.12311 0.285611
\(623\) −11.2462 −0.450570
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −3.80776 −0.152189
\(627\) 0 0
\(628\) −2.35416 −0.0939413
\(629\) −11.0691 −0.441355
\(630\) 0 0
\(631\) −47.9309 −1.90810 −0.954049 0.299651i \(-0.903130\pi\)
−0.954049 + 0.299651i \(0.903130\pi\)
\(632\) −7.61553 −0.302929
\(633\) 0 0
\(634\) −42.3542 −1.68210
\(635\) 21.9309 0.870300
\(636\) 0 0
\(637\) −5.56155 −0.220357
\(638\) 0 0
\(639\) 0 0
\(640\) 13.5616 0.536067
\(641\) 14.6847 0.580009 0.290005 0.957025i \(-0.406343\pi\)
0.290005 + 0.957025i \(0.406343\pi\)
\(642\) 0 0
\(643\) −10.7386 −0.423490 −0.211745 0.977325i \(-0.567915\pi\)
−0.211745 + 0.977325i \(0.567915\pi\)
\(644\) −1.75379 −0.0691090
\(645\) 0 0
\(646\) 38.6307 1.51990
\(647\) −13.9309 −0.547679 −0.273840 0.961775i \(-0.588294\pi\)
−0.273840 + 0.961775i \(0.588294\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −34.7386 −1.36256
\(651\) 0 0
\(652\) 7.61553 0.298247
\(653\) −8.24621 −0.322699 −0.161350 0.986897i \(-0.551585\pi\)
−0.161350 + 0.986897i \(0.551585\pi\)
\(654\) 0 0
\(655\) 2.56155 0.100088
\(656\) 35.4233 1.38305
\(657\) 0 0
\(658\) −5.36932 −0.209318
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) −31.4233 −1.22222 −0.611112 0.791544i \(-0.709278\pi\)
−0.611112 + 0.791544i \(0.709278\pi\)
\(662\) 27.6155 1.07331
\(663\) 0 0
\(664\) −21.4773 −0.833480
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 26.7386 1.03532
\(668\) −5.61553 −0.217271
\(669\) 0 0
\(670\) 13.7538 0.531355
\(671\) 0 0
\(672\) 0 0
\(673\) −27.9309 −1.07666 −0.538328 0.842735i \(-0.680944\pi\)
−0.538328 + 0.842735i \(0.680944\pi\)
\(674\) 42.4384 1.63467
\(675\) 0 0
\(676\) 7.86174 0.302375
\(677\) −22.8617 −0.878648 −0.439324 0.898329i \(-0.644782\pi\)
−0.439324 + 0.898329i \(0.644782\pi\)
\(678\) 0 0
\(679\) −1.31534 −0.0504782
\(680\) −10.0540 −0.385552
\(681\) 0 0
\(682\) 0 0
\(683\) −13.6155 −0.520984 −0.260492 0.965476i \(-0.583885\pi\)
−0.260492 + 0.965476i \(0.583885\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 1.56155 0.0596204
\(687\) 0 0
\(688\) −26.6307 −1.01529
\(689\) 43.4233 1.65430
\(690\) 0 0
\(691\) 36.1080 1.37361 0.686806 0.726841i \(-0.259012\pi\)
0.686806 + 0.726841i \(0.259012\pi\)
\(692\) 5.50758 0.209367
\(693\) 0 0
\(694\) 21.8617 0.829860
\(695\) 6.24621 0.236932
\(696\) 0 0
\(697\) −31.1771 −1.18092
\(698\) −12.5767 −0.476035
\(699\) 0 0
\(700\) 1.75379 0.0662870
\(701\) 7.17708 0.271075 0.135537 0.990772i \(-0.456724\pi\)
0.135537 + 0.990772i \(0.456724\pi\)
\(702\) 0 0
\(703\) 16.1080 0.607523
\(704\) 0 0
\(705\) 0 0
\(706\) 36.1922 1.36211
\(707\) −4.56155 −0.171555
\(708\) 0 0
\(709\) −35.9309 −1.34941 −0.674706 0.738087i \(-0.735730\pi\)
−0.674706 + 0.738087i \(0.735730\pi\)
\(710\) −4.87689 −0.183027
\(711\) 0 0
\(712\) 27.4233 1.02773
\(713\) 32.9848 1.23529
\(714\) 0 0
\(715\) 0 0
\(716\) 2.13826 0.0799106
\(717\) 0 0
\(718\) −43.7235 −1.63175
\(719\) 21.8617 0.815305 0.407653 0.913137i \(-0.366347\pi\)
0.407653 + 0.913137i \(0.366347\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −26.5464 −0.987955
\(723\) 0 0
\(724\) −0.0842853 −0.00313244
\(725\) −26.7386 −0.993048
\(726\) 0 0
\(727\) −49.2311 −1.82588 −0.912939 0.408095i \(-0.866193\pi\)
−0.912939 + 0.408095i \(0.866193\pi\)
\(728\) 13.5616 0.502625
\(729\) 0 0
\(730\) −11.1231 −0.411685
\(731\) 23.4384 0.866902
\(732\) 0 0
\(733\) 34.4384 1.27201 0.636007 0.771684i \(-0.280585\pi\)
0.636007 + 0.771684i \(0.280585\pi\)
\(734\) −22.6307 −0.835314
\(735\) 0 0
\(736\) 9.75379 0.359529
\(737\) 0 0
\(738\) 0 0
\(739\) −8.38447 −0.308428 −0.154214 0.988037i \(-0.549284\pi\)
−0.154214 + 0.988037i \(0.549284\pi\)
\(740\) 1.17708 0.0432704
\(741\) 0 0
\(742\) −12.1922 −0.447591
\(743\) −32.8769 −1.20614 −0.603068 0.797690i \(-0.706055\pi\)
−0.603068 + 0.797690i \(0.706055\pi\)
\(744\) 0 0
\(745\) 5.31534 0.194739
\(746\) 10.6307 0.389217
\(747\) 0 0
\(748\) 0 0
\(749\) 3.12311 0.114116
\(750\) 0 0
\(751\) 12.5616 0.458378 0.229189 0.973382i \(-0.426393\pi\)
0.229189 + 0.973382i \(0.426393\pi\)
\(752\) 16.1080 0.587397
\(753\) 0 0
\(754\) 58.0540 2.11420
\(755\) 11.4384 0.416288
\(756\) 0 0
\(757\) −0.615528 −0.0223718 −0.0111859 0.999937i \(-0.503561\pi\)
−0.0111859 + 0.999937i \(0.503561\pi\)
\(758\) 22.7386 0.825904
\(759\) 0 0
\(760\) 14.6307 0.530711
\(761\) −13.9848 −0.506950 −0.253475 0.967342i \(-0.581574\pi\)
−0.253475 + 0.967342i \(0.581574\pi\)
\(762\) 0 0
\(763\) 3.87689 0.140353
\(764\) 5.36932 0.194255
\(765\) 0 0
\(766\) 27.2311 0.983898
\(767\) −69.8617 −2.52256
\(768\) 0 0
\(769\) 20.3002 0.732043 0.366022 0.930606i \(-0.380720\pi\)
0.366022 + 0.930606i \(0.380720\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.68466 0.240586
\(773\) −29.5464 −1.06271 −0.531355 0.847149i \(-0.678317\pi\)
−0.531355 + 0.847149i \(0.678317\pi\)
\(774\) 0 0
\(775\) −32.9848 −1.18485
\(776\) 3.20739 0.115139
\(777\) 0 0
\(778\) 27.4233 0.983173
\(779\) 45.3693 1.62552
\(780\) 0 0
\(781\) 0 0
\(782\) −25.7538 −0.920954
\(783\) 0 0
\(784\) −4.68466 −0.167309
\(785\) 5.36932 0.191639
\(786\) 0 0
\(787\) −37.1231 −1.32330 −0.661648 0.749815i \(-0.730143\pi\)
−0.661648 + 0.749815i \(0.730143\pi\)
\(788\) 11.9157 0.424480
\(789\) 0 0
\(790\) −4.87689 −0.173512
\(791\) −8.93087 −0.317545
\(792\) 0 0
\(793\) −74.3542 −2.64039
\(794\) 20.7926 0.737902
\(795\) 0 0
\(796\) −3.12311 −0.110696
\(797\) −39.3002 −1.39208 −0.696042 0.718001i \(-0.745057\pi\)
−0.696042 + 0.718001i \(0.745057\pi\)
\(798\) 0 0
\(799\) −14.1771 −0.501549
\(800\) −9.75379 −0.344849
\(801\) 0 0
\(802\) −23.9157 −0.844493
\(803\) 0 0
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 71.6155 2.52255
\(807\) 0 0
\(808\) 11.1231 0.391309
\(809\) 45.8617 1.61241 0.806206 0.591634i \(-0.201517\pi\)
0.806206 + 0.591634i \(0.201517\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) −2.93087 −0.102853
\(813\) 0 0
\(814\) 0 0
\(815\) −17.3693 −0.608421
\(816\) 0 0
\(817\) −34.1080 −1.19329
\(818\) 53.5616 1.87274
\(819\) 0 0
\(820\) 3.31534 0.115777
\(821\) −1.50758 −0.0526148 −0.0263074 0.999654i \(-0.508375\pi\)
−0.0263074 + 0.999654i \(0.508375\pi\)
\(822\) 0 0
\(823\) −38.2462 −1.33318 −0.666590 0.745425i \(-0.732247\pi\)
−0.666590 + 0.745425i \(0.732247\pi\)
\(824\) −19.5076 −0.679579
\(825\) 0 0
\(826\) 19.6155 0.682512
\(827\) −39.1231 −1.36044 −0.680222 0.733006i \(-0.738117\pi\)
−0.680222 + 0.733006i \(0.738117\pi\)
\(828\) 0 0
\(829\) 47.8078 1.66043 0.830216 0.557442i \(-0.188217\pi\)
0.830216 + 0.557442i \(0.188217\pi\)
\(830\) −13.7538 −0.477401
\(831\) 0 0
\(832\) −30.9309 −1.07233
\(833\) 4.12311 0.142857
\(834\) 0 0
\(835\) 12.8078 0.443231
\(836\) 0 0
\(837\) 0 0
\(838\) −33.2614 −1.14899
\(839\) −57.0540 −1.96972 −0.984861 0.173346i \(-0.944542\pi\)
−0.984861 + 0.173346i \(0.944542\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 10.9309 0.376703
\(843\) 0 0
\(844\) −11.8617 −0.408298
\(845\) −17.9309 −0.616841
\(846\) 0 0
\(847\) 0 0
\(848\) 36.5767 1.25605
\(849\) 0 0
\(850\) 25.7538 0.883348
\(851\) −10.7386 −0.368116
\(852\) 0 0
\(853\) −52.7926 −1.80758 −0.903792 0.427971i \(-0.859228\pi\)
−0.903792 + 0.427971i \(0.859228\pi\)
\(854\) 20.8769 0.714393
\(855\) 0 0
\(856\) −7.61553 −0.260293
\(857\) −9.19224 −0.314001 −0.157000 0.987599i \(-0.550182\pi\)
−0.157000 + 0.987599i \(0.550182\pi\)
\(858\) 0 0
\(859\) 16.6307 0.567432 0.283716 0.958908i \(-0.408433\pi\)
0.283716 + 0.958908i \(0.408433\pi\)
\(860\) −2.49242 −0.0849909
\(861\) 0 0
\(862\) −16.6004 −0.565411
\(863\) −36.2462 −1.23384 −0.616918 0.787028i \(-0.711619\pi\)
−0.616918 + 0.787028i \(0.711619\pi\)
\(864\) 0 0
\(865\) −12.5616 −0.427106
\(866\) 20.1922 0.686160
\(867\) 0 0
\(868\) −3.61553 −0.122719
\(869\) 0 0
\(870\) 0 0
\(871\) −48.9848 −1.65979
\(872\) −9.45360 −0.320139
\(873\) 0 0
\(874\) 37.4773 1.26769
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 53.4233 1.80398 0.901988 0.431761i \(-0.142107\pi\)
0.901988 + 0.431761i \(0.142107\pi\)
\(878\) −1.75379 −0.0591875
\(879\) 0 0
\(880\) 0 0
\(881\) 19.0691 0.642455 0.321228 0.947002i \(-0.395905\pi\)
0.321228 + 0.947002i \(0.395905\pi\)
\(882\) 0 0
\(883\) −2.17708 −0.0732646 −0.0366323 0.999329i \(-0.511663\pi\)
−0.0366323 + 0.999329i \(0.511663\pi\)
\(884\) −10.0540 −0.338152
\(885\) 0 0
\(886\) −13.2614 −0.445524
\(887\) 7.19224 0.241492 0.120746 0.992683i \(-0.461471\pi\)
0.120746 + 0.992683i \(0.461471\pi\)
\(888\) 0 0
\(889\) 21.9309 0.735538
\(890\) 17.5616 0.588665
\(891\) 0 0
\(892\) 10.5227 0.352327
\(893\) 20.6307 0.690379
\(894\) 0 0
\(895\) −4.87689 −0.163017
\(896\) 13.5616 0.453060
\(897\) 0 0
\(898\) 43.8078 1.46188
\(899\) 55.1231 1.83846
\(900\) 0 0
\(901\) −32.1922 −1.07248
\(902\) 0 0
\(903\) 0 0
\(904\) 21.7775 0.724307
\(905\) 0.192236 0.00639014
\(906\) 0 0
\(907\) 18.6307 0.618622 0.309311 0.950961i \(-0.399902\pi\)
0.309311 + 0.950961i \(0.399902\pi\)
\(908\) 2.49242 0.0827139
\(909\) 0 0
\(910\) 8.68466 0.287894
\(911\) 19.3693 0.641734 0.320867 0.947124i \(-0.396026\pi\)
0.320867 + 0.947124i \(0.396026\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −16.5767 −0.548309
\(915\) 0 0
\(916\) −1.83807 −0.0607317
\(917\) 2.56155 0.0845899
\(918\) 0 0
\(919\) −11.6155 −0.383161 −0.191580 0.981477i \(-0.561361\pi\)
−0.191580 + 0.981477i \(0.561361\pi\)
\(920\) −9.75379 −0.321573
\(921\) 0 0
\(922\) 16.1922 0.533263
\(923\) 17.3693 0.571718
\(924\) 0 0
\(925\) 10.7386 0.353084
\(926\) −1.36932 −0.0449985
\(927\) 0 0
\(928\) 16.3002 0.535080
\(929\) −2.50758 −0.0822709 −0.0411355 0.999154i \(-0.513098\pi\)
−0.0411355 + 0.999154i \(0.513098\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −11.9157 −0.390312
\(933\) 0 0
\(934\) −33.9697 −1.11152
\(935\) 0 0
\(936\) 0 0
\(937\) 4.93087 0.161084 0.0805422 0.996751i \(-0.474335\pi\)
0.0805422 + 0.996751i \(0.474335\pi\)
\(938\) 13.7538 0.449077
\(939\) 0 0
\(940\) 1.50758 0.0491718
\(941\) −28.4384 −0.927067 −0.463533 0.886079i \(-0.653419\pi\)
−0.463533 + 0.886079i \(0.653419\pi\)
\(942\) 0 0
\(943\) −30.2462 −0.984952
\(944\) −58.8466 −1.91529
\(945\) 0 0
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) 39.6155 1.28597
\(950\) −37.4773 −1.21592
\(951\) 0 0
\(952\) −10.0540 −0.325851
\(953\) −43.6695 −1.41459 −0.707297 0.706917i \(-0.750086\pi\)
−0.707297 + 0.706917i \(0.750086\pi\)
\(954\) 0 0
\(955\) −12.2462 −0.396278
\(956\) −8.49242 −0.274665
\(957\) 0 0
\(958\) 46.3542 1.49763
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 37.0000 1.19355
\(962\) −23.3153 −0.751717
\(963\) 0 0
\(964\) 5.36932 0.172934
\(965\) −15.2462 −0.490793
\(966\) 0 0
\(967\) −19.3002 −0.620652 −0.310326 0.950630i \(-0.600438\pi\)
−0.310326 + 0.950630i \(0.600438\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 2.05398 0.0659492
\(971\) −16.3153 −0.523584 −0.261792 0.965124i \(-0.584313\pi\)
−0.261792 + 0.965124i \(0.584313\pi\)
\(972\) 0 0
\(973\) 6.24621 0.200244
\(974\) −30.3542 −0.972610
\(975\) 0 0
\(976\) −62.6307 −2.00476
\(977\) −57.6695 −1.84501 −0.922505 0.385984i \(-0.873862\pi\)
−0.922505 + 0.385984i \(0.873862\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.438447 −0.0140057
\(981\) 0 0
\(982\) −31.2311 −0.996623
\(983\) 6.63068 0.211486 0.105743 0.994393i \(-0.466278\pi\)
0.105743 + 0.994393i \(0.466278\pi\)
\(984\) 0 0
\(985\) −27.1771 −0.865934
\(986\) −43.0388 −1.37064
\(987\) 0 0
\(988\) 14.6307 0.465464
\(989\) 22.7386 0.723046
\(990\) 0 0
\(991\) 61.8617 1.96510 0.982551 0.185991i \(-0.0595495\pi\)
0.982551 + 0.185991i \(0.0595495\pi\)
\(992\) 20.1080 0.638428
\(993\) 0 0
\(994\) −4.87689 −0.154686
\(995\) 7.12311 0.225818
\(996\) 0 0
\(997\) 41.0388 1.29971 0.649856 0.760057i \(-0.274829\pi\)
0.649856 + 0.760057i \(0.274829\pi\)
\(998\) 61.7538 1.95478
\(999\) 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 7623.2.a.bt.1.1 2
3.2 odd 2 2541.2.a.u.1.2 2
11.10 odd 2 7623.2.a.be.1.2 2
33.32 even 2 2541.2.a.bc.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.u.1.2 2 3.2 odd 2
2541.2.a.bc.1.1 yes 2 33.32 even 2
7623.2.a.be.1.2 2 11.10 odd 2
7623.2.a.bt.1.1 2 1.1 even 1 trivial