Properties

Label 1071.2.a.m.1.3
Level $1071$
Weight $2$
Character 1071.1
Self dual yes
Analytic conductor $8.552$
Analytic rank $0$
Dimension $5$
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] = [1071,2,Mod(1,1071)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1071, 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("1071.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1071.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: \(8.55197805648\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.453749.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.32183\) of defining polynomial
Character \(\chi\) \(=\) 1071.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +O(q^{10})\) \(q-0.877834 q^{2} -1.22941 q^{4} -3.03818 q^{5} -1.00000 q^{7} +2.83488 q^{8} +2.66702 q^{10} -4.78178 q^{11} -4.39933 q^{13} +0.877834 q^{14} -0.0297440 q^{16} -1.00000 q^{17} -2.64366 q^{19} +3.73516 q^{20} +4.19761 q^{22} +8.45881 q^{23} +4.23054 q^{25} +3.86188 q^{26} +1.22941 q^{28} -7.04298 q^{29} -3.42063 q^{31} -5.64366 q^{32} +0.877834 q^{34} +3.03818 q^{35} +9.66977 q^{37} +2.32069 q^{38} -8.61289 q^{40} -1.33675 q^{41} +2.52513 q^{43} +5.87875 q^{44} -7.42544 q^{46} +5.57082 q^{47} +1.00000 q^{49} -3.71372 q^{50} +5.40856 q^{52} -5.17630 q^{53} +14.5279 q^{55} -2.83488 q^{56} +6.18257 q^{58} -9.28732 q^{59} +6.66325 q^{61} +3.00275 q^{62} +5.01368 q^{64} +13.3659 q^{65} +5.28251 q^{67} +1.22941 q^{68} -2.66702 q^{70} +11.9310 q^{71} +12.7155 q^{73} -8.48845 q^{74} +3.25013 q^{76} +4.78178 q^{77} +1.94051 q^{79} +0.0903677 q^{80} +1.17345 q^{82} +4.58417 q^{83} +3.03818 q^{85} -2.21664 q^{86} -13.5558 q^{88} +13.3956 q^{89} +4.39933 q^{91} -10.3993 q^{92} -4.89026 q^{94} +8.03191 q^{95} -15.1668 q^{97} -0.877834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 10 q^{4} - 5 q^{7} - 6 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{14} + 4 q^{16} - 5 q^{17} + 6 q^{19} + 19 q^{20} + 6 q^{22} + 10 q^{23} + 21 q^{25} + 26 q^{26} - 10 q^{28} + 8 q^{29} - 9 q^{32} + 2 q^{34} + 8 q^{37} - 14 q^{38} + 5 q^{40} - 18 q^{41} + 8 q^{43} + 14 q^{44} + 8 q^{46} + 10 q^{47} + 5 q^{49} - 27 q^{50} + 4 q^{52} - 4 q^{53} - 24 q^{55} + 6 q^{56} + 12 q^{58} - 8 q^{59} + 22 q^{61} - 16 q^{62} - 16 q^{64} + 30 q^{65} + 16 q^{67} - 10 q^{68} - 4 q^{70} + 2 q^{71} + 10 q^{73} + 40 q^{74} + 24 q^{76} - 2 q^{77} + 18 q^{79} + 4 q^{80} - 31 q^{82} + 12 q^{83} - 23 q^{86} - 46 q^{88} - 20 q^{89} - 2 q^{91} - 28 q^{92} - 42 q^{94} + 22 q^{95} + 12 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.877834 −0.620723 −0.310361 0.950619i \(-0.600450\pi\)
−0.310361 + 0.950619i \(0.600450\pi\)
\(3\) 0 0
\(4\) −1.22941 −0.614704
\(5\) −3.03818 −1.35872 −0.679358 0.733807i \(-0.737742\pi\)
−0.679358 + 0.733807i \(0.737742\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.83488 1.00228
\(9\) 0 0
\(10\) 2.66702 0.843385
\(11\) −4.78178 −1.44176 −0.720880 0.693060i \(-0.756262\pi\)
−0.720880 + 0.693060i \(0.756262\pi\)
\(12\) 0 0
\(13\) −4.39933 −1.22015 −0.610077 0.792342i \(-0.708861\pi\)
−0.610077 + 0.792342i \(0.708861\pi\)
\(14\) 0.877834 0.234611
\(15\) 0 0
\(16\) −0.0297440 −0.00743600
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −2.64366 −0.606497 −0.303248 0.952912i \(-0.598071\pi\)
−0.303248 + 0.952912i \(0.598071\pi\)
\(20\) 3.73516 0.835207
\(21\) 0 0
\(22\) 4.19761 0.894933
\(23\) 8.45881 1.76378 0.881892 0.471451i \(-0.156270\pi\)
0.881892 + 0.471451i \(0.156270\pi\)
\(24\) 0 0
\(25\) 4.23054 0.846109
\(26\) 3.86188 0.757377
\(27\) 0 0
\(28\) 1.22941 0.232336
\(29\) −7.04298 −1.30785 −0.653925 0.756560i \(-0.726879\pi\)
−0.653925 + 0.756560i \(0.726879\pi\)
\(30\) 0 0
\(31\) −3.42063 −0.614364 −0.307182 0.951651i \(-0.599386\pi\)
−0.307182 + 0.951651i \(0.599386\pi\)
\(32\) −5.64366 −0.997667
\(33\) 0 0
\(34\) 0.877834 0.150547
\(35\) 3.03818 0.513546
\(36\) 0 0
\(37\) 9.66977 1.58970 0.794850 0.606806i \(-0.207549\pi\)
0.794850 + 0.606806i \(0.207549\pi\)
\(38\) 2.32069 0.376466
\(39\) 0 0
\(40\) −8.61289 −1.36182
\(41\) −1.33675 −0.208766 −0.104383 0.994537i \(-0.533287\pi\)
−0.104383 + 0.994537i \(0.533287\pi\)
\(42\) 0 0
\(43\) 2.52513 0.385078 0.192539 0.981289i \(-0.438328\pi\)
0.192539 + 0.981289i \(0.438328\pi\)
\(44\) 5.87875 0.886255
\(45\) 0 0
\(46\) −7.42544 −1.09482
\(47\) 5.57082 0.812588 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.71372 −0.525199
\(51\) 0 0
\(52\) 5.40856 0.750033
\(53\) −5.17630 −0.711020 −0.355510 0.934673i \(-0.615693\pi\)
−0.355510 + 0.934673i \(0.615693\pi\)
\(54\) 0 0
\(55\) 14.5279 1.95894
\(56\) −2.83488 −0.378827
\(57\) 0 0
\(58\) 6.18257 0.811812
\(59\) −9.28732 −1.20911 −0.604553 0.796565i \(-0.706648\pi\)
−0.604553 + 0.796565i \(0.706648\pi\)
\(60\) 0 0
\(61\) 6.66325 0.853141 0.426571 0.904454i \(-0.359722\pi\)
0.426571 + 0.904454i \(0.359722\pi\)
\(62\) 3.00275 0.381350
\(63\) 0 0
\(64\) 5.01368 0.626710
\(65\) 13.3659 1.65784
\(66\) 0 0
\(67\) 5.28251 0.645362 0.322681 0.946508i \(-0.395416\pi\)
0.322681 + 0.946508i \(0.395416\pi\)
\(68\) 1.22941 0.149088
\(69\) 0 0
\(70\) −2.66702 −0.318770
\(71\) 11.9310 1.41595 0.707973 0.706239i \(-0.249610\pi\)
0.707973 + 0.706239i \(0.249610\pi\)
\(72\) 0 0
\(73\) 12.7155 1.48823 0.744116 0.668050i \(-0.232871\pi\)
0.744116 + 0.668050i \(0.232871\pi\)
\(74\) −8.48845 −0.986763
\(75\) 0 0
\(76\) 3.25013 0.372816
\(77\) 4.78178 0.544934
\(78\) 0 0
\(79\) 1.94051 0.218325 0.109162 0.994024i \(-0.465183\pi\)
0.109162 + 0.994024i \(0.465183\pi\)
\(80\) 0.0903677 0.0101034
\(81\) 0 0
\(82\) 1.17345 0.129586
\(83\) 4.58417 0.503178 0.251589 0.967834i \(-0.419047\pi\)
0.251589 + 0.967834i \(0.419047\pi\)
\(84\) 0 0
\(85\) 3.03818 0.329537
\(86\) −2.21664 −0.239027
\(87\) 0 0
\(88\) −13.5558 −1.44505
\(89\) 13.3956 1.41993 0.709965 0.704237i \(-0.248711\pi\)
0.709965 + 0.704237i \(0.248711\pi\)
\(90\) 0 0
\(91\) 4.39933 0.461175
\(92\) −10.3993 −1.08420
\(93\) 0 0
\(94\) −4.89026 −0.504392
\(95\) 8.03191 0.824057
\(96\) 0 0
\(97\) −15.1668 −1.53995 −0.769976 0.638073i \(-0.779732\pi\)
−0.769976 + 0.638073i \(0.779732\pi\)
\(98\) −0.877834 −0.0886746
\(99\) 0 0
\(100\) −5.20106 −0.520106
\(101\) −3.24786 −0.323174 −0.161587 0.986858i \(-0.551661\pi\)
−0.161587 + 0.986858i \(0.551661\pi\)
\(102\) 0 0
\(103\) −12.7031 −1.25168 −0.625839 0.779952i \(-0.715243\pi\)
−0.625839 + 0.779952i \(0.715243\pi\)
\(104\) −12.4716 −1.22294
\(105\) 0 0
\(106\) 4.54393 0.441346
\(107\) −3.91410 −0.378390 −0.189195 0.981940i \(-0.560588\pi\)
−0.189195 + 0.981940i \(0.560588\pi\)
\(108\) 0 0
\(109\) −3.56356 −0.341327 −0.170663 0.985329i \(-0.554591\pi\)
−0.170663 + 0.985329i \(0.554591\pi\)
\(110\) −12.7531 −1.21596
\(111\) 0 0
\(112\) 0.0297440 0.00281054
\(113\) −0.966622 −0.0909322 −0.0454661 0.998966i \(-0.514477\pi\)
−0.0454661 + 0.998966i \(0.514477\pi\)
\(114\) 0 0
\(115\) −25.6994 −2.39648
\(116\) 8.65869 0.803940
\(117\) 0 0
\(118\) 8.15272 0.750519
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 11.8654 1.07867
\(122\) −5.84923 −0.529564
\(123\) 0 0
\(124\) 4.20535 0.377652
\(125\) 2.33775 0.209095
\(126\) 0 0
\(127\) −5.55875 −0.493260 −0.246630 0.969110i \(-0.579323\pi\)
−0.246630 + 0.969110i \(0.579323\pi\)
\(128\) 6.88613 0.608654
\(129\) 0 0
\(130\) −11.7331 −1.02906
\(131\) 3.51134 0.306787 0.153393 0.988165i \(-0.450980\pi\)
0.153393 + 0.988165i \(0.450980\pi\)
\(132\) 0 0
\(133\) 2.64366 0.229234
\(134\) −4.63717 −0.400590
\(135\) 0 0
\(136\) −2.83488 −0.243089
\(137\) 1.93780 0.165557 0.0827786 0.996568i \(-0.473621\pi\)
0.0827786 + 0.996568i \(0.473621\pi\)
\(138\) 0 0
\(139\) 4.69834 0.398508 0.199254 0.979948i \(-0.436148\pi\)
0.199254 + 0.979948i \(0.436148\pi\)
\(140\) −3.73516 −0.315679
\(141\) 0 0
\(142\) −10.4734 −0.878910
\(143\) 21.0366 1.75917
\(144\) 0 0
\(145\) 21.3979 1.77700
\(146\) −11.1621 −0.923780
\(147\) 0 0
\(148\) −11.8881 −0.977194
\(149\) −4.24162 −0.347487 −0.173743 0.984791i \(-0.555586\pi\)
−0.173743 + 0.984791i \(0.555586\pi\)
\(150\) 0 0
\(151\) 12.4584 1.01385 0.506924 0.861991i \(-0.330783\pi\)
0.506924 + 0.861991i \(0.330783\pi\)
\(152\) −7.49446 −0.607881
\(153\) 0 0
\(154\) −4.19761 −0.338253
\(155\) 10.3925 0.834746
\(156\) 0 0
\(157\) −17.6472 −1.40840 −0.704199 0.710002i \(-0.748694\pi\)
−0.704199 + 0.710002i \(0.748694\pi\)
\(158\) −1.70345 −0.135519
\(159\) 0 0
\(160\) 17.1465 1.35555
\(161\) −8.45881 −0.666648
\(162\) 0 0
\(163\) −2.67557 −0.209567 −0.104783 0.994495i \(-0.533415\pi\)
−0.104783 + 0.994495i \(0.533415\pi\)
\(164\) 1.64341 0.128329
\(165\) 0 0
\(166\) −4.02414 −0.312334
\(167\) 1.95783 0.151501 0.0757507 0.997127i \(-0.475865\pi\)
0.0757507 + 0.997127i \(0.475865\pi\)
\(168\) 0 0
\(169\) 6.35407 0.488775
\(170\) −2.66702 −0.204551
\(171\) 0 0
\(172\) −3.10441 −0.236709
\(173\) −6.01732 −0.457488 −0.228744 0.973487i \(-0.573462\pi\)
−0.228744 + 0.973487i \(0.573462\pi\)
\(174\) 0 0
\(175\) −4.23054 −0.319799
\(176\) 0.142229 0.0107209
\(177\) 0 0
\(178\) −11.7591 −0.881382
\(179\) −1.75112 −0.130885 −0.0654423 0.997856i \(-0.520846\pi\)
−0.0654423 + 0.997856i \(0.520846\pi\)
\(180\) 0 0
\(181\) 2.76491 0.205514 0.102757 0.994707i \(-0.467234\pi\)
0.102757 + 0.994707i \(0.467234\pi\)
\(182\) −3.86188 −0.286262
\(183\) 0 0
\(184\) 23.9798 1.76781
\(185\) −29.3785 −2.15995
\(186\) 0 0
\(187\) 4.78178 0.349678
\(188\) −6.84881 −0.499501
\(189\) 0 0
\(190\) −7.05069 −0.511511
\(191\) −18.8461 −1.36365 −0.681827 0.731514i \(-0.738814\pi\)
−0.681827 + 0.731514i \(0.738814\pi\)
\(192\) 0 0
\(193\) 11.9107 0.857348 0.428674 0.903459i \(-0.358981\pi\)
0.428674 + 0.903459i \(0.358981\pi\)
\(194\) 13.3139 0.955883
\(195\) 0 0
\(196\) −1.22941 −0.0878148
\(197\) 17.2682 1.23031 0.615153 0.788408i \(-0.289094\pi\)
0.615153 + 0.788408i \(0.289094\pi\)
\(198\) 0 0
\(199\) −15.2451 −1.08070 −0.540350 0.841440i \(-0.681708\pi\)
−0.540350 + 0.841440i \(0.681708\pi\)
\(200\) 11.9931 0.848040
\(201\) 0 0
\(202\) 2.85108 0.200601
\(203\) 7.04298 0.494321
\(204\) 0 0
\(205\) 4.06130 0.283653
\(206\) 11.1513 0.776945
\(207\) 0 0
\(208\) 0.130854 0.00907306
\(209\) 12.6414 0.874423
\(210\) 0 0
\(211\) 8.51331 0.586080 0.293040 0.956100i \(-0.405333\pi\)
0.293040 + 0.956100i \(0.405333\pi\)
\(212\) 6.36378 0.437066
\(213\) 0 0
\(214\) 3.43593 0.234875
\(215\) −7.67179 −0.523212
\(216\) 0 0
\(217\) 3.42063 0.232208
\(218\) 3.12821 0.211869
\(219\) 0 0
\(220\) −17.8607 −1.20417
\(221\) 4.39933 0.295931
\(222\) 0 0
\(223\) −2.49093 −0.166805 −0.0834027 0.996516i \(-0.526579\pi\)
−0.0834027 + 0.996516i \(0.526579\pi\)
\(224\) 5.64366 0.377083
\(225\) 0 0
\(226\) 0.848534 0.0564437
\(227\) 16.9842 1.12728 0.563640 0.826020i \(-0.309400\pi\)
0.563640 + 0.826020i \(0.309400\pi\)
\(228\) 0 0
\(229\) −16.3303 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(230\) 22.5598 1.48755
\(231\) 0 0
\(232\) −19.9660 −1.31083
\(233\) 5.77825 0.378546 0.189273 0.981925i \(-0.439387\pi\)
0.189273 + 0.981925i \(0.439387\pi\)
\(234\) 0 0
\(235\) −16.9252 −1.10408
\(236\) 11.4179 0.743241
\(237\) 0 0
\(238\) −0.877834 −0.0569015
\(239\) −7.44275 −0.481432 −0.240716 0.970596i \(-0.577382\pi\)
−0.240716 + 0.970596i \(0.577382\pi\)
\(240\) 0 0
\(241\) 21.9036 1.41093 0.705467 0.708743i \(-0.250737\pi\)
0.705467 + 0.708743i \(0.250737\pi\)
\(242\) −10.4159 −0.669557
\(243\) 0 0
\(244\) −8.19184 −0.524429
\(245\) −3.03818 −0.194102
\(246\) 0 0
\(247\) 11.6303 0.740019
\(248\) −9.69710 −0.615766
\(249\) 0 0
\(250\) −2.05216 −0.129790
\(251\) −16.5769 −1.04632 −0.523162 0.852233i \(-0.675248\pi\)
−0.523162 + 0.852233i \(0.675248\pi\)
\(252\) 0 0
\(253\) −40.4482 −2.54296
\(254\) 4.87966 0.306177
\(255\) 0 0
\(256\) −16.0723 −1.00452
\(257\) −12.1158 −0.755764 −0.377882 0.925854i \(-0.623347\pi\)
−0.377882 + 0.925854i \(0.623347\pi\)
\(258\) 0 0
\(259\) −9.66977 −0.600850
\(260\) −16.4322 −1.01908
\(261\) 0 0
\(262\) −3.08237 −0.190430
\(263\) −3.20135 −0.197404 −0.0987018 0.995117i \(-0.531469\pi\)
−0.0987018 + 0.995117i \(0.531469\pi\)
\(264\) 0 0
\(265\) 15.7265 0.966074
\(266\) −2.32069 −0.142291
\(267\) 0 0
\(268\) −6.49436 −0.396706
\(269\) −15.4400 −0.941396 −0.470698 0.882294i \(-0.655998\pi\)
−0.470698 + 0.882294i \(0.655998\pi\)
\(270\) 0 0
\(271\) −24.8878 −1.51182 −0.755912 0.654673i \(-0.772806\pi\)
−0.755912 + 0.654673i \(0.772806\pi\)
\(272\) 0.0297440 0.00180349
\(273\) 0 0
\(274\) −1.70106 −0.102765
\(275\) −20.2295 −1.21989
\(276\) 0 0
\(277\) −11.2870 −0.678172 −0.339086 0.940755i \(-0.610118\pi\)
−0.339086 + 0.940755i \(0.610118\pi\)
\(278\) −4.12437 −0.247363
\(279\) 0 0
\(280\) 8.61289 0.514719
\(281\) −13.0332 −0.777495 −0.388747 0.921344i \(-0.627092\pi\)
−0.388747 + 0.921344i \(0.627092\pi\)
\(282\) 0 0
\(283\) 21.7190 1.29106 0.645531 0.763734i \(-0.276636\pi\)
0.645531 + 0.763734i \(0.276636\pi\)
\(284\) −14.6680 −0.870387
\(285\) 0 0
\(286\) −18.4667 −1.09196
\(287\) 1.33675 0.0789061
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) −18.7838 −1.10302
\(291\) 0 0
\(292\) −15.6325 −0.914822
\(293\) 28.9400 1.69069 0.845346 0.534219i \(-0.179394\pi\)
0.845346 + 0.534219i \(0.179394\pi\)
\(294\) 0 0
\(295\) 28.2165 1.64283
\(296\) 27.4127 1.59333
\(297\) 0 0
\(298\) 3.72344 0.215693
\(299\) −37.2131 −2.15209
\(300\) 0 0
\(301\) −2.52513 −0.145546
\(302\) −10.9364 −0.629318
\(303\) 0 0
\(304\) 0.0786330 0.00450991
\(305\) −20.2441 −1.15918
\(306\) 0 0
\(307\) −11.5836 −0.661110 −0.330555 0.943787i \(-0.607236\pi\)
−0.330555 + 0.943787i \(0.607236\pi\)
\(308\) −5.87875 −0.334973
\(309\) 0 0
\(310\) −9.12289 −0.518146
\(311\) 15.7999 0.895932 0.447966 0.894051i \(-0.352149\pi\)
0.447966 + 0.894051i \(0.352149\pi\)
\(312\) 0 0
\(313\) 20.2658 1.14549 0.572745 0.819733i \(-0.305878\pi\)
0.572745 + 0.819733i \(0.305878\pi\)
\(314\) 15.4913 0.874225
\(315\) 0 0
\(316\) −2.38568 −0.134205
\(317\) −6.36537 −0.357515 −0.178757 0.983893i \(-0.557208\pi\)
−0.178757 + 0.983893i \(0.557208\pi\)
\(318\) 0 0
\(319\) 33.6780 1.88561
\(320\) −15.2325 −0.851521
\(321\) 0 0
\(322\) 7.42544 0.413803
\(323\) 2.64366 0.147097
\(324\) 0 0
\(325\) −18.6115 −1.03238
\(326\) 2.34871 0.130083
\(327\) 0 0
\(328\) −3.78954 −0.209242
\(329\) −5.57082 −0.307130
\(330\) 0 0
\(331\) 24.5910 1.35165 0.675823 0.737064i \(-0.263788\pi\)
0.675823 + 0.737064i \(0.263788\pi\)
\(332\) −5.63581 −0.309305
\(333\) 0 0
\(334\) −1.71865 −0.0940403
\(335\) −16.0492 −0.876863
\(336\) 0 0
\(337\) 7.11956 0.387827 0.193913 0.981019i \(-0.437882\pi\)
0.193913 + 0.981019i \(0.437882\pi\)
\(338\) −5.57782 −0.303393
\(339\) 0 0
\(340\) −3.73516 −0.202568
\(341\) 16.3567 0.885766
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 7.15844 0.385957
\(345\) 0 0
\(346\) 5.28221 0.283973
\(347\) −1.51477 −0.0813173 −0.0406586 0.999173i \(-0.512946\pi\)
−0.0406586 + 0.999173i \(0.512946\pi\)
\(348\) 0 0
\(349\) 24.7937 1.32717 0.663587 0.748099i \(-0.269033\pi\)
0.663587 + 0.748099i \(0.269033\pi\)
\(350\) 3.71372 0.198506
\(351\) 0 0
\(352\) 26.9867 1.43840
\(353\) 11.8416 0.630267 0.315133 0.949047i \(-0.397951\pi\)
0.315133 + 0.949047i \(0.397951\pi\)
\(354\) 0 0
\(355\) −36.2485 −1.92387
\(356\) −16.4686 −0.872836
\(357\) 0 0
\(358\) 1.53719 0.0812430
\(359\) 26.6743 1.40781 0.703907 0.710292i \(-0.251437\pi\)
0.703907 + 0.710292i \(0.251437\pi\)
\(360\) 0 0
\(361\) −12.0111 −0.632162
\(362\) −2.42713 −0.127567
\(363\) 0 0
\(364\) −5.40856 −0.283486
\(365\) −38.6319 −2.02209
\(366\) 0 0
\(367\) −28.6897 −1.49759 −0.748796 0.662801i \(-0.769368\pi\)
−0.748796 + 0.662801i \(0.769368\pi\)
\(368\) −0.251599 −0.0131155
\(369\) 0 0
\(370\) 25.7895 1.34073
\(371\) 5.17630 0.268740
\(372\) 0 0
\(373\) −26.8946 −1.39255 −0.696275 0.717775i \(-0.745161\pi\)
−0.696275 + 0.717775i \(0.745161\pi\)
\(374\) −4.19761 −0.217053
\(375\) 0 0
\(376\) 15.7926 0.814443
\(377\) 30.9844 1.59578
\(378\) 0 0
\(379\) 0.960453 0.0493351 0.0246676 0.999696i \(-0.492147\pi\)
0.0246676 + 0.999696i \(0.492147\pi\)
\(380\) −9.87449 −0.506551
\(381\) 0 0
\(382\) 16.5437 0.846451
\(383\) 10.9913 0.561627 0.280814 0.959762i \(-0.409396\pi\)
0.280814 + 0.959762i \(0.409396\pi\)
\(384\) 0 0
\(385\) −14.5279 −0.740411
\(386\) −10.4556 −0.532175
\(387\) 0 0
\(388\) 18.6461 0.946614
\(389\) 25.4239 1.28904 0.644520 0.764587i \(-0.277057\pi\)
0.644520 + 0.764587i \(0.277057\pi\)
\(390\) 0 0
\(391\) −8.45881 −0.427781
\(392\) 2.83488 0.143183
\(393\) 0 0
\(394\) −15.1586 −0.763679
\(395\) −5.89563 −0.296641
\(396\) 0 0
\(397\) −15.0140 −0.753533 −0.376767 0.926308i \(-0.622964\pi\)
−0.376767 + 0.926308i \(0.622964\pi\)
\(398\) 13.3827 0.670815
\(399\) 0 0
\(400\) −0.125833 −0.00629166
\(401\) 11.0656 0.552588 0.276294 0.961073i \(-0.410894\pi\)
0.276294 + 0.961073i \(0.410894\pi\)
\(402\) 0 0
\(403\) 15.0485 0.749618
\(404\) 3.99294 0.198656
\(405\) 0 0
\(406\) −6.18257 −0.306836
\(407\) −46.2387 −2.29197
\(408\) 0 0
\(409\) 28.5650 1.41245 0.706225 0.707988i \(-0.250397\pi\)
0.706225 + 0.707988i \(0.250397\pi\)
\(410\) −3.56515 −0.176070
\(411\) 0 0
\(412\) 15.6173 0.769411
\(413\) 9.28732 0.456999
\(414\) 0 0
\(415\) −13.9275 −0.683676
\(416\) 24.8283 1.21731
\(417\) 0 0
\(418\) −11.0970 −0.542774
\(419\) 1.98566 0.0970057 0.0485029 0.998823i \(-0.484555\pi\)
0.0485029 + 0.998823i \(0.484555\pi\)
\(420\) 0 0
\(421\) 25.1984 1.22810 0.614048 0.789269i \(-0.289540\pi\)
0.614048 + 0.789269i \(0.289540\pi\)
\(422\) −7.47327 −0.363793
\(423\) 0 0
\(424\) −14.6742 −0.712643
\(425\) −4.23054 −0.205211
\(426\) 0 0
\(427\) −6.66325 −0.322457
\(428\) 4.81202 0.232598
\(429\) 0 0
\(430\) 6.73456 0.324769
\(431\) 27.4984 1.32455 0.662275 0.749261i \(-0.269591\pi\)
0.662275 + 0.749261i \(0.269591\pi\)
\(432\) 0 0
\(433\) 28.1977 1.35509 0.677547 0.735480i \(-0.263043\pi\)
0.677547 + 0.735480i \(0.263043\pi\)
\(434\) −3.00275 −0.144137
\(435\) 0 0
\(436\) 4.38106 0.209815
\(437\) −22.3622 −1.06973
\(438\) 0 0
\(439\) 6.31298 0.301302 0.150651 0.988587i \(-0.451863\pi\)
0.150651 + 0.988587i \(0.451863\pi\)
\(440\) 41.1849 1.96341
\(441\) 0 0
\(442\) −3.86188 −0.183691
\(443\) −26.4557 −1.25695 −0.628473 0.777831i \(-0.716320\pi\)
−0.628473 + 0.777831i \(0.716320\pi\)
\(444\) 0 0
\(445\) −40.6982 −1.92928
\(446\) 2.18663 0.103540
\(447\) 0 0
\(448\) −5.01368 −0.236874
\(449\) 34.5555 1.63077 0.815387 0.578916i \(-0.196524\pi\)
0.815387 + 0.578916i \(0.196524\pi\)
\(450\) 0 0
\(451\) 6.39206 0.300990
\(452\) 1.18837 0.0558963
\(453\) 0 0
\(454\) −14.9093 −0.699728
\(455\) −13.3659 −0.626605
\(456\) 0 0
\(457\) 26.5922 1.24393 0.621965 0.783045i \(-0.286334\pi\)
0.621965 + 0.783045i \(0.286334\pi\)
\(458\) 14.3353 0.669844
\(459\) 0 0
\(460\) 31.5950 1.47313
\(461\) 0.310199 0.0144474 0.00722371 0.999974i \(-0.497701\pi\)
0.00722371 + 0.999974i \(0.497701\pi\)
\(462\) 0 0
\(463\) 22.4654 1.04406 0.522028 0.852928i \(-0.325175\pi\)
0.522028 + 0.852928i \(0.325175\pi\)
\(464\) 0.209487 0.00972517
\(465\) 0 0
\(466\) −5.07235 −0.234972
\(467\) 11.6126 0.537365 0.268682 0.963229i \(-0.413412\pi\)
0.268682 + 0.963229i \(0.413412\pi\)
\(468\) 0 0
\(469\) −5.28251 −0.243924
\(470\) 14.8575 0.685325
\(471\) 0 0
\(472\) −26.3285 −1.21187
\(473\) −12.0746 −0.555190
\(474\) 0 0
\(475\) −11.1841 −0.513162
\(476\) −1.22941 −0.0563498
\(477\) 0 0
\(478\) 6.53350 0.298836
\(479\) −31.7976 −1.45287 −0.726435 0.687235i \(-0.758824\pi\)
−0.726435 + 0.687235i \(0.758824\pi\)
\(480\) 0 0
\(481\) −42.5405 −1.93968
\(482\) −19.2277 −0.875799
\(483\) 0 0
\(484\) −14.5874 −0.663064
\(485\) 46.0794 2.09236
\(486\) 0 0
\(487\) −23.0166 −1.04298 −0.521490 0.853257i \(-0.674624\pi\)
−0.521490 + 0.853257i \(0.674624\pi\)
\(488\) 18.8895 0.855089
\(489\) 0 0
\(490\) 2.66702 0.120484
\(491\) 33.3176 1.50360 0.751802 0.659389i \(-0.229185\pi\)
0.751802 + 0.659389i \(0.229185\pi\)
\(492\) 0 0
\(493\) 7.04298 0.317200
\(494\) −10.2095 −0.459347
\(495\) 0 0
\(496\) 0.101743 0.00456841
\(497\) −11.9310 −0.535177
\(498\) 0 0
\(499\) 37.0236 1.65741 0.828703 0.559689i \(-0.189079\pi\)
0.828703 + 0.559689i \(0.189079\pi\)
\(500\) −2.87405 −0.128531
\(501\) 0 0
\(502\) 14.5518 0.649477
\(503\) 12.4172 0.553654 0.276827 0.960920i \(-0.410717\pi\)
0.276827 + 0.960920i \(0.410717\pi\)
\(504\) 0 0
\(505\) 9.86759 0.439102
\(506\) 35.5068 1.57847
\(507\) 0 0
\(508\) 6.83397 0.303208
\(509\) −11.9257 −0.528597 −0.264299 0.964441i \(-0.585140\pi\)
−0.264299 + 0.964441i \(0.585140\pi\)
\(510\) 0 0
\(511\) −12.7155 −0.562499
\(512\) 0.336507 0.0148716
\(513\) 0 0
\(514\) 10.6357 0.469120
\(515\) 38.5945 1.70067
\(516\) 0 0
\(517\) −26.6385 −1.17156
\(518\) 8.48845 0.372961
\(519\) 0 0
\(520\) 37.8909 1.66163
\(521\) −26.3980 −1.15652 −0.578259 0.815853i \(-0.696268\pi\)
−0.578259 + 0.815853i \(0.696268\pi\)
\(522\) 0 0
\(523\) 13.6272 0.595874 0.297937 0.954586i \(-0.403701\pi\)
0.297937 + 0.954586i \(0.403701\pi\)
\(524\) −4.31686 −0.188583
\(525\) 0 0
\(526\) 2.81025 0.122533
\(527\) 3.42063 0.149005
\(528\) 0 0
\(529\) 48.5515 2.11094
\(530\) −13.8053 −0.599664
\(531\) 0 0
\(532\) −3.25013 −0.140911
\(533\) 5.88081 0.254726
\(534\) 0 0
\(535\) 11.8917 0.514125
\(536\) 14.9753 0.646835
\(537\) 0 0
\(538\) 13.5538 0.584346
\(539\) −4.78178 −0.205966
\(540\) 0 0
\(541\) 26.3358 1.13226 0.566132 0.824314i \(-0.308439\pi\)
0.566132 + 0.824314i \(0.308439\pi\)
\(542\) 21.8473 0.938424
\(543\) 0 0
\(544\) 5.64366 0.241970
\(545\) 10.8267 0.463766
\(546\) 0 0
\(547\) −2.25139 −0.0962624 −0.0481312 0.998841i \(-0.515327\pi\)
−0.0481312 + 0.998841i \(0.515327\pi\)
\(548\) −2.38234 −0.101769
\(549\) 0 0
\(550\) 17.7582 0.757211
\(551\) 18.6192 0.793206
\(552\) 0 0
\(553\) −1.94051 −0.0825190
\(554\) 9.90815 0.420957
\(555\) 0 0
\(556\) −5.77618 −0.244964
\(557\) −30.3355 −1.28536 −0.642679 0.766136i \(-0.722177\pi\)
−0.642679 + 0.766136i \(0.722177\pi\)
\(558\) 0 0
\(559\) −11.1089 −0.469854
\(560\) −0.0903677 −0.00381873
\(561\) 0 0
\(562\) 11.4410 0.482608
\(563\) −10.9867 −0.463035 −0.231518 0.972831i \(-0.574369\pi\)
−0.231518 + 0.972831i \(0.574369\pi\)
\(564\) 0 0
\(565\) 2.93677 0.123551
\(566\) −19.0657 −0.801391
\(567\) 0 0
\(568\) 33.8229 1.41918
\(569\) 28.8479 1.20937 0.604684 0.796465i \(-0.293299\pi\)
0.604684 + 0.796465i \(0.293299\pi\)
\(570\) 0 0
\(571\) −10.5165 −0.440100 −0.220050 0.975489i \(-0.570622\pi\)
−0.220050 + 0.975489i \(0.570622\pi\)
\(572\) −25.8626 −1.08137
\(573\) 0 0
\(574\) −1.17345 −0.0489788
\(575\) 35.7854 1.49235
\(576\) 0 0
\(577\) −36.8288 −1.53320 −0.766601 0.642123i \(-0.778054\pi\)
−0.766601 + 0.642123i \(0.778054\pi\)
\(578\) −0.877834 −0.0365131
\(579\) 0 0
\(580\) −26.3067 −1.09233
\(581\) −4.58417 −0.190183
\(582\) 0 0
\(583\) 24.7519 1.02512
\(584\) 36.0469 1.49163
\(585\) 0 0
\(586\) −25.4045 −1.04945
\(587\) −30.1420 −1.24409 −0.622047 0.782980i \(-0.713699\pi\)
−0.622047 + 0.782980i \(0.713699\pi\)
\(588\) 0 0
\(589\) 9.04298 0.372610
\(590\) −24.7694 −1.01974
\(591\) 0 0
\(592\) −0.287618 −0.0118210
\(593\) 7.88925 0.323973 0.161986 0.986793i \(-0.448210\pi\)
0.161986 + 0.986793i \(0.448210\pi\)
\(594\) 0 0
\(595\) −3.03818 −0.124553
\(596\) 5.21467 0.213601
\(597\) 0 0
\(598\) 32.6669 1.33585
\(599\) −29.8059 −1.21783 −0.608917 0.793234i \(-0.708396\pi\)
−0.608917 + 0.793234i \(0.708396\pi\)
\(600\) 0 0
\(601\) 11.9432 0.487175 0.243587 0.969879i \(-0.421676\pi\)
0.243587 + 0.969879i \(0.421676\pi\)
\(602\) 2.21664 0.0903436
\(603\) 0 0
\(604\) −15.3164 −0.623216
\(605\) −36.0493 −1.46561
\(606\) 0 0
\(607\) −4.67281 −0.189664 −0.0948318 0.995493i \(-0.530231\pi\)
−0.0948318 + 0.995493i \(0.530231\pi\)
\(608\) 14.9199 0.605082
\(609\) 0 0
\(610\) 17.7710 0.719527
\(611\) −24.5079 −0.991483
\(612\) 0 0
\(613\) 42.3894 1.71209 0.856046 0.516900i \(-0.172914\pi\)
0.856046 + 0.516900i \(0.172914\pi\)
\(614\) 10.1685 0.410366
\(615\) 0 0
\(616\) 13.5558 0.546178
\(617\) 23.2809 0.937255 0.468628 0.883396i \(-0.344749\pi\)
0.468628 + 0.883396i \(0.344749\pi\)
\(618\) 0 0
\(619\) −43.2201 −1.73716 −0.868582 0.495545i \(-0.834968\pi\)
−0.868582 + 0.495545i \(0.834968\pi\)
\(620\) −12.7766 −0.513121
\(621\) 0 0
\(622\) −13.8697 −0.556125
\(623\) −13.3956 −0.536683
\(624\) 0 0
\(625\) −28.2552 −1.13021
\(626\) −17.7900 −0.711032
\(627\) 0 0
\(628\) 21.6956 0.865748
\(629\) −9.66977 −0.385559
\(630\) 0 0
\(631\) 30.6825 1.22145 0.610725 0.791843i \(-0.290878\pi\)
0.610725 + 0.791843i \(0.290878\pi\)
\(632\) 5.50113 0.218823
\(633\) 0 0
\(634\) 5.58774 0.221917
\(635\) 16.8885 0.670200
\(636\) 0 0
\(637\) −4.39933 −0.174308
\(638\) −29.5637 −1.17044
\(639\) 0 0
\(640\) −20.9213 −0.826988
\(641\) 21.3561 0.843517 0.421758 0.906708i \(-0.361413\pi\)
0.421758 + 0.906708i \(0.361413\pi\)
\(642\) 0 0
\(643\) 22.7013 0.895253 0.447626 0.894221i \(-0.352269\pi\)
0.447626 + 0.894221i \(0.352269\pi\)
\(644\) 10.3993 0.409791
\(645\) 0 0
\(646\) −2.32069 −0.0913065
\(647\) −35.2630 −1.38633 −0.693165 0.720779i \(-0.743784\pi\)
−0.693165 + 0.720779i \(0.743784\pi\)
\(648\) 0 0
\(649\) 44.4099 1.74324
\(650\) 16.3378 0.640823
\(651\) 0 0
\(652\) 3.28936 0.128821
\(653\) 30.0593 1.17631 0.588155 0.808748i \(-0.299854\pi\)
0.588155 + 0.808748i \(0.299854\pi\)
\(654\) 0 0
\(655\) −10.6681 −0.416836
\(656\) 0.0397604 0.00155238
\(657\) 0 0
\(658\) 4.89026 0.190642
\(659\) 18.9747 0.739148 0.369574 0.929201i \(-0.379504\pi\)
0.369574 + 0.929201i \(0.379504\pi\)
\(660\) 0 0
\(661\) 25.8038 1.00365 0.501825 0.864969i \(-0.332662\pi\)
0.501825 + 0.864969i \(0.332662\pi\)
\(662\) −21.5869 −0.838997
\(663\) 0 0
\(664\) 12.9956 0.504327
\(665\) −8.03191 −0.311464
\(666\) 0 0
\(667\) −59.5753 −2.30676
\(668\) −2.40697 −0.0931284
\(669\) 0 0
\(670\) 14.0886 0.544289
\(671\) −31.8622 −1.23003
\(672\) 0 0
\(673\) 4.38401 0.168991 0.0844956 0.996424i \(-0.473072\pi\)
0.0844956 + 0.996424i \(0.473072\pi\)
\(674\) −6.24979 −0.240733
\(675\) 0 0
\(676\) −7.81174 −0.300452
\(677\) 32.1115 1.23415 0.617073 0.786906i \(-0.288318\pi\)
0.617073 + 0.786906i \(0.288318\pi\)
\(678\) 0 0
\(679\) 15.1668 0.582047
\(680\) 8.61289 0.330289
\(681\) 0 0
\(682\) −14.3585 −0.549815
\(683\) 18.7093 0.715892 0.357946 0.933742i \(-0.383477\pi\)
0.357946 + 0.933742i \(0.383477\pi\)
\(684\) 0 0
\(685\) −5.88738 −0.224945
\(686\) 0.877834 0.0335159
\(687\) 0 0
\(688\) −0.0751073 −0.00286344
\(689\) 22.7722 0.867553
\(690\) 0 0
\(691\) −40.9131 −1.55641 −0.778205 0.628011i \(-0.783869\pi\)
−0.778205 + 0.628011i \(0.783869\pi\)
\(692\) 7.39773 0.281220
\(693\) 0 0
\(694\) 1.32972 0.0504755
\(695\) −14.2744 −0.541459
\(696\) 0 0
\(697\) 1.33675 0.0506331
\(698\) −21.7647 −0.823807
\(699\) 0 0
\(700\) 5.20106 0.196582
\(701\) 17.6240 0.665649 0.332825 0.942989i \(-0.391998\pi\)
0.332825 + 0.942989i \(0.391998\pi\)
\(702\) 0 0
\(703\) −25.5636 −0.964148
\(704\) −23.9743 −0.903566
\(705\) 0 0
\(706\) −10.3950 −0.391221
\(707\) 3.24786 0.122148
\(708\) 0 0
\(709\) −8.18690 −0.307466 −0.153733 0.988112i \(-0.549130\pi\)
−0.153733 + 0.988112i \(0.549130\pi\)
\(710\) 31.8201 1.19419
\(711\) 0 0
\(712\) 37.9749 1.42317
\(713\) −28.9345 −1.08361
\(714\) 0 0
\(715\) −63.9130 −2.39021
\(716\) 2.15283 0.0804552
\(717\) 0 0
\(718\) −23.4156 −0.873862
\(719\) −42.6093 −1.58906 −0.794530 0.607225i \(-0.792283\pi\)
−0.794530 + 0.607225i \(0.792283\pi\)
\(720\) 0 0
\(721\) 12.7031 0.473090
\(722\) 10.5437 0.392397
\(723\) 0 0
\(724\) −3.39919 −0.126330
\(725\) −29.7956 −1.10658
\(726\) 0 0
\(727\) −10.5470 −0.391166 −0.195583 0.980687i \(-0.562660\pi\)
−0.195583 + 0.980687i \(0.562660\pi\)
\(728\) 12.4716 0.462227
\(729\) 0 0
\(730\) 33.9124 1.25515
\(731\) −2.52513 −0.0933951
\(732\) 0 0
\(733\) 6.61491 0.244327 0.122164 0.992510i \(-0.461017\pi\)
0.122164 + 0.992510i \(0.461017\pi\)
\(734\) 25.1848 0.929589
\(735\) 0 0
\(736\) −47.7387 −1.75967
\(737\) −25.2598 −0.930457
\(738\) 0 0
\(739\) −0.834376 −0.0306930 −0.0153465 0.999882i \(-0.504885\pi\)
−0.0153465 + 0.999882i \(0.504885\pi\)
\(740\) 36.1181 1.32773
\(741\) 0 0
\(742\) −4.54393 −0.166813
\(743\) −52.9082 −1.94101 −0.970506 0.241077i \(-0.922499\pi\)
−0.970506 + 0.241077i \(0.922499\pi\)
\(744\) 0 0
\(745\) 12.8868 0.472136
\(746\) 23.6090 0.864387
\(747\) 0 0
\(748\) −5.87875 −0.214948
\(749\) 3.91410 0.143018
\(750\) 0 0
\(751\) 35.3681 1.29060 0.645300 0.763929i \(-0.276732\pi\)
0.645300 + 0.763929i \(0.276732\pi\)
\(752\) −0.165699 −0.00604241
\(753\) 0 0
\(754\) −27.1992 −0.990535
\(755\) −37.8508 −1.37753
\(756\) 0 0
\(757\) −19.6550 −0.714372 −0.357186 0.934033i \(-0.616264\pi\)
−0.357186 + 0.934033i \(0.616264\pi\)
\(758\) −0.843118 −0.0306234
\(759\) 0 0
\(760\) 22.7695 0.825938
\(761\) 6.80776 0.246781 0.123390 0.992358i \(-0.460623\pi\)
0.123390 + 0.992358i \(0.460623\pi\)
\(762\) 0 0
\(763\) 3.56356 0.129009
\(764\) 23.1695 0.838243
\(765\) 0 0
\(766\) −9.64851 −0.348615
\(767\) 40.8579 1.47529
\(768\) 0 0
\(769\) 44.3645 1.59983 0.799913 0.600116i \(-0.204879\pi\)
0.799913 + 0.600116i \(0.204879\pi\)
\(770\) 12.7531 0.459590
\(771\) 0 0
\(772\) −14.6431 −0.527015
\(773\) 25.6974 0.924270 0.462135 0.886810i \(-0.347084\pi\)
0.462135 + 0.886810i \(0.347084\pi\)
\(774\) 0 0
\(775\) −14.4711 −0.519819
\(776\) −42.9960 −1.54347
\(777\) 0 0
\(778\) −22.3179 −0.800137
\(779\) 3.53392 0.126616
\(780\) 0 0
\(781\) −57.0513 −2.04146
\(782\) 7.42544 0.265533
\(783\) 0 0
\(784\) −0.0297440 −0.00106229
\(785\) 53.6153 1.91361
\(786\) 0 0
\(787\) 25.6566 0.914558 0.457279 0.889323i \(-0.348824\pi\)
0.457279 + 0.889323i \(0.348824\pi\)
\(788\) −21.2296 −0.756274
\(789\) 0 0
\(790\) 5.17538 0.184132
\(791\) 0.966622 0.0343691
\(792\) 0 0
\(793\) −29.3138 −1.04096
\(794\) 13.1798 0.467735
\(795\) 0 0
\(796\) 18.7425 0.664310
\(797\) −20.3683 −0.721483 −0.360741 0.932666i \(-0.617476\pi\)
−0.360741 + 0.932666i \(0.617476\pi\)
\(798\) 0 0
\(799\) −5.57082 −0.197082
\(800\) −23.8757 −0.844135
\(801\) 0 0
\(802\) −9.71373 −0.343004
\(803\) −60.8026 −2.14568
\(804\) 0 0
\(805\) 25.6994 0.905785
\(806\) −13.2101 −0.465305
\(807\) 0 0
\(808\) −9.20731 −0.323912
\(809\) 36.5480 1.28496 0.642480 0.766303i \(-0.277906\pi\)
0.642480 + 0.766303i \(0.277906\pi\)
\(810\) 0 0
\(811\) 7.33037 0.257404 0.128702 0.991683i \(-0.458919\pi\)
0.128702 + 0.991683i \(0.458919\pi\)
\(812\) −8.65869 −0.303861
\(813\) 0 0
\(814\) 40.5899 1.42268
\(815\) 8.12886 0.284742
\(816\) 0 0
\(817\) −6.67557 −0.233549
\(818\) −25.0754 −0.876739
\(819\) 0 0
\(820\) −4.99299 −0.174363
\(821\) −1.76321 −0.0615366 −0.0307683 0.999527i \(-0.509795\pi\)
−0.0307683 + 0.999527i \(0.509795\pi\)
\(822\) 0 0
\(823\) 0.636323 0.0221808 0.0110904 0.999938i \(-0.496470\pi\)
0.0110904 + 0.999938i \(0.496470\pi\)
\(824\) −36.0119 −1.25454
\(825\) 0 0
\(826\) −8.15272 −0.283670
\(827\) 6.81869 0.237109 0.118554 0.992948i \(-0.462174\pi\)
0.118554 + 0.992948i \(0.462174\pi\)
\(828\) 0 0
\(829\) 50.9480 1.76950 0.884749 0.466068i \(-0.154330\pi\)
0.884749 + 0.466068i \(0.154330\pi\)
\(830\) 12.2261 0.424373
\(831\) 0 0
\(832\) −22.0568 −0.764683
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −5.94824 −0.205847
\(836\) −15.5414 −0.537511
\(837\) 0 0
\(838\) −1.74308 −0.0602137
\(839\) −2.95064 −0.101867 −0.0509336 0.998702i \(-0.516220\pi\)
−0.0509336 + 0.998702i \(0.516220\pi\)
\(840\) 0 0
\(841\) 20.6036 0.710470
\(842\) −22.1200 −0.762307
\(843\) 0 0
\(844\) −10.4663 −0.360266
\(845\) −19.3048 −0.664106
\(846\) 0 0
\(847\) −11.8654 −0.407700
\(848\) 0.153964 0.00528714
\(849\) 0 0
\(850\) 3.71372 0.127379
\(851\) 81.7948 2.80389
\(852\) 0 0
\(853\) −28.8764 −0.988709 −0.494355 0.869260i \(-0.664596\pi\)
−0.494355 + 0.869260i \(0.664596\pi\)
\(854\) 5.84923 0.200156
\(855\) 0 0
\(856\) −11.0960 −0.379254
\(857\) 9.83977 0.336120 0.168060 0.985777i \(-0.446250\pi\)
0.168060 + 0.985777i \(0.446250\pi\)
\(858\) 0 0
\(859\) −11.1344 −0.379900 −0.189950 0.981794i \(-0.560833\pi\)
−0.189950 + 0.981794i \(0.560833\pi\)
\(860\) 9.43175 0.321620
\(861\) 0 0
\(862\) −24.1390 −0.822178
\(863\) −22.7030 −0.772818 −0.386409 0.922328i \(-0.626285\pi\)
−0.386409 + 0.922328i \(0.626285\pi\)
\(864\) 0 0
\(865\) 18.2817 0.621596
\(866\) −24.7529 −0.841137
\(867\) 0 0
\(868\) −4.20535 −0.142739
\(869\) −9.27910 −0.314772
\(870\) 0 0
\(871\) −23.2395 −0.787440
\(872\) −10.1023 −0.342106
\(873\) 0 0
\(874\) 19.6303 0.664005
\(875\) −2.33775 −0.0790304
\(876\) 0 0
\(877\) 1.55717 0.0525819 0.0262910 0.999654i \(-0.491630\pi\)
0.0262910 + 0.999654i \(0.491630\pi\)
\(878\) −5.54175 −0.187025
\(879\) 0 0
\(880\) −0.432118 −0.0145667
\(881\) 10.3326 0.348113 0.174056 0.984736i \(-0.444313\pi\)
0.174056 + 0.984736i \(0.444313\pi\)
\(882\) 0 0
\(883\) 23.6160 0.794743 0.397371 0.917658i \(-0.369922\pi\)
0.397371 + 0.917658i \(0.369922\pi\)
\(884\) −5.40856 −0.181910
\(885\) 0 0
\(886\) 23.2237 0.780215
\(887\) −13.9260 −0.467588 −0.233794 0.972286i \(-0.575114\pi\)
−0.233794 + 0.972286i \(0.575114\pi\)
\(888\) 0 0
\(889\) 5.55875 0.186435
\(890\) 35.7263 1.19755
\(891\) 0 0
\(892\) 3.06237 0.102536
\(893\) −14.7274 −0.492832
\(894\) 0 0
\(895\) 5.32021 0.177835
\(896\) −6.88613 −0.230050
\(897\) 0 0
\(898\) −30.3340 −1.01226
\(899\) 24.0915 0.803495
\(900\) 0 0
\(901\) 5.17630 0.172448
\(902\) −5.61117 −0.186831
\(903\) 0 0
\(904\) −2.74026 −0.0911398
\(905\) −8.40028 −0.279235
\(906\) 0 0
\(907\) 11.3628 0.377295 0.188648 0.982045i \(-0.439590\pi\)
0.188648 + 0.982045i \(0.439590\pi\)
\(908\) −20.8805 −0.692943
\(909\) 0 0
\(910\) 11.7331 0.388948
\(911\) −4.75507 −0.157543 −0.0787713 0.996893i \(-0.525100\pi\)
−0.0787713 + 0.996893i \(0.525100\pi\)
\(912\) 0 0
\(913\) −21.9205 −0.725462
\(914\) −23.3435 −0.772136
\(915\) 0 0
\(916\) 20.0766 0.663349
\(917\) −3.51134 −0.115955
\(918\) 0 0
\(919\) −32.6303 −1.07637 −0.538187 0.842825i \(-0.680891\pi\)
−0.538187 + 0.842825i \(0.680891\pi\)
\(920\) −72.8548 −2.40195
\(921\) 0 0
\(922\) −0.272303 −0.00896784
\(923\) −52.4882 −1.72767
\(924\) 0 0
\(925\) 40.9084 1.34506
\(926\) −19.7209 −0.648070
\(927\) 0 0
\(928\) 39.7482 1.30480
\(929\) 47.3546 1.55365 0.776827 0.629714i \(-0.216828\pi\)
0.776827 + 0.629714i \(0.216828\pi\)
\(930\) 0 0
\(931\) −2.64366 −0.0866424
\(932\) −7.10382 −0.232693
\(933\) 0 0
\(934\) −10.1939 −0.333554
\(935\) −14.5279 −0.475113
\(936\) 0 0
\(937\) −51.6525 −1.68741 −0.843707 0.536803i \(-0.819632\pi\)
−0.843707 + 0.536803i \(0.819632\pi\)
\(938\) 4.63717 0.151409
\(939\) 0 0
\(940\) 20.8079 0.678680
\(941\) 54.7240 1.78395 0.891975 0.452084i \(-0.149319\pi\)
0.891975 + 0.452084i \(0.149319\pi\)
\(942\) 0 0
\(943\) −11.3073 −0.368218
\(944\) 0.276242 0.00899091
\(945\) 0 0
\(946\) 10.5995 0.344619
\(947\) 36.9728 1.20145 0.600727 0.799454i \(-0.294878\pi\)
0.600727 + 0.799454i \(0.294878\pi\)
\(948\) 0 0
\(949\) −55.9395 −1.81587
\(950\) 9.81779 0.318531
\(951\) 0 0
\(952\) 2.83488 0.0918791
\(953\) 7.91313 0.256331 0.128166 0.991753i \(-0.459091\pi\)
0.128166 + 0.991753i \(0.459091\pi\)
\(954\) 0 0
\(955\) 57.2578 1.85282
\(956\) 9.15017 0.295938
\(957\) 0 0
\(958\) 27.9130 0.901830
\(959\) −1.93780 −0.0625747
\(960\) 0 0
\(961\) −19.2993 −0.622557
\(962\) 37.3435 1.20400
\(963\) 0 0
\(964\) −26.9284 −0.867306
\(965\) −36.1867 −1.16489
\(966\) 0 0
\(967\) 8.05906 0.259162 0.129581 0.991569i \(-0.458637\pi\)
0.129581 + 0.991569i \(0.458637\pi\)
\(968\) 33.6371 1.08114
\(969\) 0 0
\(970\) −40.4500 −1.29877
\(971\) 1.81193 0.0581476 0.0290738 0.999577i \(-0.490744\pi\)
0.0290738 + 0.999577i \(0.490744\pi\)
\(972\) 0 0
\(973\) −4.69834 −0.150622
\(974\) 20.2047 0.647401
\(975\) 0 0
\(976\) −0.198192 −0.00634396
\(977\) −15.1934 −0.486079 −0.243040 0.970016i \(-0.578145\pi\)
−0.243040 + 0.970016i \(0.578145\pi\)
\(978\) 0 0
\(979\) −64.0547 −2.04720
\(980\) 3.73516 0.119315
\(981\) 0 0
\(982\) −29.2474 −0.933321
\(983\) −11.6472 −0.371488 −0.185744 0.982598i \(-0.559470\pi\)
−0.185744 + 0.982598i \(0.559470\pi\)
\(984\) 0 0
\(985\) −52.4638 −1.67164
\(986\) −6.18257 −0.196893
\(987\) 0 0
\(988\) −14.2984 −0.454892
\(989\) 21.3596 0.679195
\(990\) 0 0
\(991\) −47.1759 −1.49859 −0.749296 0.662235i \(-0.769608\pi\)
−0.749296 + 0.662235i \(0.769608\pi\)
\(992\) 19.3049 0.612931
\(993\) 0 0
\(994\) 10.4734 0.332197
\(995\) 46.3175 1.46836
\(996\) 0 0
\(997\) 39.7574 1.25913 0.629564 0.776949i \(-0.283234\pi\)
0.629564 + 0.776949i \(0.283234\pi\)
\(998\) −32.5006 −1.02879
\(999\) 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 1071.2.a.m.1.3 5
3.2 odd 2 119.2.a.b.1.3 5
7.6 odd 2 7497.2.a.br.1.3 5
12.11 even 2 1904.2.a.t.1.2 5
15.14 odd 2 2975.2.a.m.1.3 5
21.2 odd 6 833.2.e.i.18.3 10
21.5 even 6 833.2.e.h.18.3 10
21.11 odd 6 833.2.e.i.324.3 10
21.17 even 6 833.2.e.h.324.3 10
21.20 even 2 833.2.a.g.1.3 5
24.5 odd 2 7616.2.a.bt.1.2 5
24.11 even 2 7616.2.a.bq.1.4 5
51.50 odd 2 2023.2.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
119.2.a.b.1.3 5 3.2 odd 2
833.2.a.g.1.3 5 21.20 even 2
833.2.e.h.18.3 10 21.5 even 6
833.2.e.h.324.3 10 21.17 even 6
833.2.e.i.18.3 10 21.2 odd 6
833.2.e.i.324.3 10 21.11 odd 6
1071.2.a.m.1.3 5 1.1 even 1 trivial
1904.2.a.t.1.2 5 12.11 even 2
2023.2.a.j.1.3 5 51.50 odd 2
2975.2.a.m.1.3 5 15.14 odd 2
7497.2.a.br.1.3 5 7.6 odd 2
7616.2.a.bq.1.4 5 24.11 even 2
7616.2.a.bt.1.2 5 24.5 odd 2