Properties

Label 7623.2.a.ct.1.7
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.70716\) 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)
 
\(f(q)\) \(=\) \(q+1.70716 q^{2} +0.914391 q^{4} -4.06637 q^{5} +1.00000 q^{7} -1.85331 q^{8} +O(q^{10})\) \(q+1.70716 q^{2} +0.914391 q^{4} -4.06637 q^{5} +1.00000 q^{7} -1.85331 q^{8} -6.94194 q^{10} +3.27792 q^{13} +1.70716 q^{14} -4.99267 q^{16} +1.32088 q^{17} +2.16175 q^{19} -3.71825 q^{20} +1.86611 q^{23} +11.5354 q^{25} +5.59593 q^{26} +0.914391 q^{28} +0.244102 q^{29} -6.86988 q^{31} -4.81667 q^{32} +2.25495 q^{34} -4.06637 q^{35} +0.255619 q^{37} +3.69045 q^{38} +7.53623 q^{40} +5.73233 q^{41} -8.01781 q^{43} +3.18575 q^{46} +4.06768 q^{47} +1.00000 q^{49} +19.6927 q^{50} +2.99730 q^{52} +5.00585 q^{53} -1.85331 q^{56} +0.416720 q^{58} +0.983592 q^{59} +1.84671 q^{61} -11.7280 q^{62} +1.76252 q^{64} -13.3292 q^{65} -3.00700 q^{67} +1.20780 q^{68} -6.94194 q^{70} -6.47642 q^{71} -9.65164 q^{73} +0.436383 q^{74} +1.97668 q^{76} -5.53450 q^{79} +20.3020 q^{80} +9.78600 q^{82} -2.07023 q^{83} -5.37119 q^{85} -13.6877 q^{86} -16.6306 q^{89} +3.27792 q^{91} +1.70636 q^{92} +6.94417 q^{94} -8.79046 q^{95} -2.58559 q^{97} +1.70716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.70716 1.20714 0.603572 0.797309i \(-0.293744\pi\)
0.603572 + 0.797309i \(0.293744\pi\)
\(3\) 0 0
\(4\) 0.914391 0.457196
\(5\) −4.06637 −1.81854 −0.909268 0.416211i \(-0.863358\pi\)
−0.909268 + 0.416211i \(0.863358\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.85331 −0.655243
\(9\) 0 0
\(10\) −6.94194 −2.19523
\(11\) 0 0
\(12\) 0 0
\(13\) 3.27792 0.909132 0.454566 0.890713i \(-0.349794\pi\)
0.454566 + 0.890713i \(0.349794\pi\)
\(14\) 1.70716 0.456257
\(15\) 0 0
\(16\) −4.99267 −1.24817
\(17\) 1.32088 0.320361 0.160180 0.987088i \(-0.448792\pi\)
0.160180 + 0.987088i \(0.448792\pi\)
\(18\) 0 0
\(19\) 2.16175 0.495939 0.247969 0.968768i \(-0.420237\pi\)
0.247969 + 0.968768i \(0.420237\pi\)
\(20\) −3.71825 −0.831427
\(21\) 0 0
\(22\) 0 0
\(23\) 1.86611 0.389112 0.194556 0.980891i \(-0.437673\pi\)
0.194556 + 0.980891i \(0.437673\pi\)
\(24\) 0 0
\(25\) 11.5354 2.30707
\(26\) 5.59593 1.09745
\(27\) 0 0
\(28\) 0.914391 0.172804
\(29\) 0.244102 0.0453286 0.0226643 0.999743i \(-0.492785\pi\)
0.0226643 + 0.999743i \(0.492785\pi\)
\(30\) 0 0
\(31\) −6.86988 −1.23387 −0.616933 0.787015i \(-0.711625\pi\)
−0.616933 + 0.787015i \(0.711625\pi\)
\(32\) −4.81667 −0.851475
\(33\) 0 0
\(34\) 2.25495 0.386721
\(35\) −4.06637 −0.687342
\(36\) 0 0
\(37\) 0.255619 0.0420236 0.0210118 0.999779i \(-0.493311\pi\)
0.0210118 + 0.999779i \(0.493311\pi\)
\(38\) 3.69045 0.598669
\(39\) 0 0
\(40\) 7.53623 1.19158
\(41\) 5.73233 0.895239 0.447620 0.894224i \(-0.352272\pi\)
0.447620 + 0.894224i \(0.352272\pi\)
\(42\) 0 0
\(43\) −8.01781 −1.22270 −0.611352 0.791358i \(-0.709374\pi\)
−0.611352 + 0.791358i \(0.709374\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.18575 0.469714
\(47\) 4.06768 0.593332 0.296666 0.954981i \(-0.404125\pi\)
0.296666 + 0.954981i \(0.404125\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 19.6927 2.78497
\(51\) 0 0
\(52\) 2.99730 0.415651
\(53\) 5.00585 0.687607 0.343803 0.939042i \(-0.388285\pi\)
0.343803 + 0.939042i \(0.388285\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.85331 −0.247658
\(57\) 0 0
\(58\) 0.416720 0.0547181
\(59\) 0.983592 0.128053 0.0640264 0.997948i \(-0.479606\pi\)
0.0640264 + 0.997948i \(0.479606\pi\)
\(60\) 0 0
\(61\) 1.84671 0.236447 0.118223 0.992987i \(-0.462280\pi\)
0.118223 + 0.992987i \(0.462280\pi\)
\(62\) −11.7280 −1.48945
\(63\) 0 0
\(64\) 1.76252 0.220315
\(65\) −13.3292 −1.65329
\(66\) 0 0
\(67\) −3.00700 −0.367364 −0.183682 0.982986i \(-0.558802\pi\)
−0.183682 + 0.982986i \(0.558802\pi\)
\(68\) 1.20780 0.146468
\(69\) 0 0
\(70\) −6.94194 −0.829720
\(71\) −6.47642 −0.768609 −0.384305 0.923206i \(-0.625559\pi\)
−0.384305 + 0.923206i \(0.625559\pi\)
\(72\) 0 0
\(73\) −9.65164 −1.12964 −0.564820 0.825214i \(-0.691054\pi\)
−0.564820 + 0.825214i \(0.691054\pi\)
\(74\) 0.436383 0.0507285
\(75\) 0 0
\(76\) 1.97668 0.226741
\(77\) 0 0
\(78\) 0 0
\(79\) −5.53450 −0.622680 −0.311340 0.950299i \(-0.600778\pi\)
−0.311340 + 0.950299i \(0.600778\pi\)
\(80\) 20.3020 2.26984
\(81\) 0 0
\(82\) 9.78600 1.08068
\(83\) −2.07023 −0.227237 −0.113618 0.993524i \(-0.536244\pi\)
−0.113618 + 0.993524i \(0.536244\pi\)
\(84\) 0 0
\(85\) −5.37119 −0.582588
\(86\) −13.6877 −1.47598
\(87\) 0 0
\(88\) 0 0
\(89\) −16.6306 −1.76284 −0.881421 0.472331i \(-0.843413\pi\)
−0.881421 + 0.472331i \(0.843413\pi\)
\(90\) 0 0
\(91\) 3.27792 0.343620
\(92\) 1.70636 0.177900
\(93\) 0 0
\(94\) 6.94417 0.716237
\(95\) −8.79046 −0.901883
\(96\) 0 0
\(97\) −2.58559 −0.262527 −0.131264 0.991348i \(-0.541903\pi\)
−0.131264 + 0.991348i \(0.541903\pi\)
\(98\) 1.70716 0.172449
\(99\) 0 0
\(100\) 10.5478 1.05478
\(101\) −10.5686 −1.05162 −0.525808 0.850604i \(-0.676237\pi\)
−0.525808 + 0.850604i \(0.676237\pi\)
\(102\) 0 0
\(103\) 13.8489 1.36457 0.682286 0.731086i \(-0.260986\pi\)
0.682286 + 0.731086i \(0.260986\pi\)
\(104\) −6.07499 −0.595702
\(105\) 0 0
\(106\) 8.54578 0.830040
\(107\) −12.4286 −1.20152 −0.600761 0.799428i \(-0.705136\pi\)
−0.600761 + 0.799428i \(0.705136\pi\)
\(108\) 0 0
\(109\) −7.36748 −0.705676 −0.352838 0.935684i \(-0.614783\pi\)
−0.352838 + 0.935684i \(0.614783\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.99267 −0.471763
\(113\) −11.4083 −1.07320 −0.536600 0.843837i \(-0.680292\pi\)
−0.536600 + 0.843837i \(0.680292\pi\)
\(114\) 0 0
\(115\) −7.58831 −0.707614
\(116\) 0.223205 0.0207240
\(117\) 0 0
\(118\) 1.67915 0.154578
\(119\) 1.32088 0.121085
\(120\) 0 0
\(121\) 0 0
\(122\) 3.15263 0.285425
\(123\) 0 0
\(124\) −6.28176 −0.564119
\(125\) −26.5752 −2.37696
\(126\) 0 0
\(127\) −8.74845 −0.776300 −0.388150 0.921596i \(-0.626886\pi\)
−0.388150 + 0.921596i \(0.626886\pi\)
\(128\) 12.6422 1.11743
\(129\) 0 0
\(130\) −22.7551 −1.99576
\(131\) 12.5516 1.09664 0.548319 0.836269i \(-0.315268\pi\)
0.548319 + 0.836269i \(0.315268\pi\)
\(132\) 0 0
\(133\) 2.16175 0.187447
\(134\) −5.13343 −0.443461
\(135\) 0 0
\(136\) −2.44800 −0.209914
\(137\) 6.32819 0.540654 0.270327 0.962769i \(-0.412868\pi\)
0.270327 + 0.962769i \(0.412868\pi\)
\(138\) 0 0
\(139\) 14.4429 1.22503 0.612517 0.790457i \(-0.290157\pi\)
0.612517 + 0.790457i \(0.290157\pi\)
\(140\) −3.71825 −0.314250
\(141\) 0 0
\(142\) −11.0563 −0.927822
\(143\) 0 0
\(144\) 0 0
\(145\) −0.992608 −0.0824316
\(146\) −16.4769 −1.36364
\(147\) 0 0
\(148\) 0.233736 0.0192130
\(149\) −24.0592 −1.97101 −0.985504 0.169653i \(-0.945735\pi\)
−0.985504 + 0.169653i \(0.945735\pi\)
\(150\) 0 0
\(151\) −1.32418 −0.107760 −0.0538802 0.998547i \(-0.517159\pi\)
−0.0538802 + 0.998547i \(0.517159\pi\)
\(152\) −4.00638 −0.324960
\(153\) 0 0
\(154\) 0 0
\(155\) 27.9355 2.24383
\(156\) 0 0
\(157\) 14.6141 1.16633 0.583165 0.812354i \(-0.301814\pi\)
0.583165 + 0.812354i \(0.301814\pi\)
\(158\) −9.44827 −0.751664
\(159\) 0 0
\(160\) 19.5864 1.54844
\(161\) 1.86611 0.147070
\(162\) 0 0
\(163\) −20.9215 −1.63870 −0.819349 0.573296i \(-0.805665\pi\)
−0.819349 + 0.573296i \(0.805665\pi\)
\(164\) 5.24159 0.409300
\(165\) 0 0
\(166\) −3.53421 −0.274308
\(167\) −4.91335 −0.380206 −0.190103 0.981764i \(-0.560882\pi\)
−0.190103 + 0.981764i \(0.560882\pi\)
\(168\) 0 0
\(169\) −2.25523 −0.173479
\(170\) −9.16948 −0.703267
\(171\) 0 0
\(172\) −7.33142 −0.559015
\(173\) −19.6578 −1.49456 −0.747279 0.664511i \(-0.768640\pi\)
−0.747279 + 0.664511i \(0.768640\pi\)
\(174\) 0 0
\(175\) 11.5354 0.871992
\(176\) 0 0
\(177\) 0 0
\(178\) −28.3911 −2.12800
\(179\) 25.3588 1.89541 0.947705 0.319149i \(-0.103397\pi\)
0.947705 + 0.319149i \(0.103397\pi\)
\(180\) 0 0
\(181\) −1.97569 −0.146852 −0.0734258 0.997301i \(-0.523393\pi\)
−0.0734258 + 0.997301i \(0.523393\pi\)
\(182\) 5.59593 0.414798
\(183\) 0 0
\(184\) −3.45848 −0.254963
\(185\) −1.03944 −0.0764214
\(186\) 0 0
\(187\) 0 0
\(188\) 3.71945 0.271269
\(189\) 0 0
\(190\) −15.0067 −1.08870
\(191\) −0.972995 −0.0704035 −0.0352017 0.999380i \(-0.511207\pi\)
−0.0352017 + 0.999380i \(0.511207\pi\)
\(192\) 0 0
\(193\) −23.1326 −1.66512 −0.832562 0.553932i \(-0.813127\pi\)
−0.832562 + 0.553932i \(0.813127\pi\)
\(194\) −4.41402 −0.316908
\(195\) 0 0
\(196\) 0.914391 0.0653137
\(197\) 9.91237 0.706227 0.353114 0.935580i \(-0.385123\pi\)
0.353114 + 0.935580i \(0.385123\pi\)
\(198\) 0 0
\(199\) 10.3847 0.736154 0.368077 0.929795i \(-0.380016\pi\)
0.368077 + 0.929795i \(0.380016\pi\)
\(200\) −21.3786 −1.51169
\(201\) 0 0
\(202\) −18.0423 −1.26945
\(203\) 0.244102 0.0171326
\(204\) 0 0
\(205\) −23.3098 −1.62803
\(206\) 23.6422 1.64723
\(207\) 0 0
\(208\) −16.3656 −1.13475
\(209\) 0 0
\(210\) 0 0
\(211\) −12.9916 −0.894378 −0.447189 0.894439i \(-0.647575\pi\)
−0.447189 + 0.894439i \(0.647575\pi\)
\(212\) 4.57731 0.314371
\(213\) 0 0
\(214\) −21.2177 −1.45041
\(215\) 32.6034 2.22353
\(216\) 0 0
\(217\) −6.86988 −0.466358
\(218\) −12.5775 −0.851853
\(219\) 0 0
\(220\) 0 0
\(221\) 4.32975 0.291250
\(222\) 0 0
\(223\) −14.3527 −0.961127 −0.480563 0.876960i \(-0.659568\pi\)
−0.480563 + 0.876960i \(0.659568\pi\)
\(224\) −4.81667 −0.321827
\(225\) 0 0
\(226\) −19.4757 −1.29551
\(227\) −6.37479 −0.423110 −0.211555 0.977366i \(-0.567853\pi\)
−0.211555 + 0.977366i \(0.567853\pi\)
\(228\) 0 0
\(229\) −8.00237 −0.528812 −0.264406 0.964412i \(-0.585176\pi\)
−0.264406 + 0.964412i \(0.585176\pi\)
\(230\) −12.9545 −0.854191
\(231\) 0 0
\(232\) −0.452395 −0.0297012
\(233\) 10.8022 0.707675 0.353837 0.935307i \(-0.384877\pi\)
0.353837 + 0.935307i \(0.384877\pi\)
\(234\) 0 0
\(235\) −16.5407 −1.07900
\(236\) 0.899388 0.0585452
\(237\) 0 0
\(238\) 2.25495 0.146167
\(239\) −25.1583 −1.62735 −0.813676 0.581318i \(-0.802537\pi\)
−0.813676 + 0.581318i \(0.802537\pi\)
\(240\) 0 0
\(241\) −18.2462 −1.17534 −0.587669 0.809101i \(-0.699954\pi\)
−0.587669 + 0.809101i \(0.699954\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.68862 0.108103
\(245\) −4.06637 −0.259791
\(246\) 0 0
\(247\) 7.08604 0.450874
\(248\) 12.7320 0.808482
\(249\) 0 0
\(250\) −45.3681 −2.86933
\(251\) −15.3284 −0.967520 −0.483760 0.875201i \(-0.660729\pi\)
−0.483760 + 0.875201i \(0.660729\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −14.9350 −0.937105
\(255\) 0 0
\(256\) 18.0573 1.12858
\(257\) −2.67980 −0.167161 −0.0835805 0.996501i \(-0.526636\pi\)
−0.0835805 + 0.996501i \(0.526636\pi\)
\(258\) 0 0
\(259\) 0.255619 0.0158834
\(260\) −12.1881 −0.755877
\(261\) 0 0
\(262\) 21.4276 1.32380
\(263\) 9.97733 0.615229 0.307614 0.951511i \(-0.400469\pi\)
0.307614 + 0.951511i \(0.400469\pi\)
\(264\) 0 0
\(265\) −20.3556 −1.25044
\(266\) 3.69045 0.226276
\(267\) 0 0
\(268\) −2.74958 −0.167957
\(269\) 21.5774 1.31560 0.657798 0.753194i \(-0.271488\pi\)
0.657798 + 0.753194i \(0.271488\pi\)
\(270\) 0 0
\(271\) −14.9862 −0.910349 −0.455175 0.890402i \(-0.650423\pi\)
−0.455175 + 0.890402i \(0.650423\pi\)
\(272\) −6.59473 −0.399864
\(273\) 0 0
\(274\) 10.8032 0.652647
\(275\) 0 0
\(276\) 0 0
\(277\) 5.20640 0.312822 0.156411 0.987692i \(-0.450008\pi\)
0.156411 + 0.987692i \(0.450008\pi\)
\(278\) 24.6564 1.47879
\(279\) 0 0
\(280\) 7.53623 0.450376
\(281\) −15.1601 −0.904374 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(282\) 0 0
\(283\) −5.66559 −0.336784 −0.168392 0.985720i \(-0.553857\pi\)
−0.168392 + 0.985720i \(0.553857\pi\)
\(284\) −5.92198 −0.351405
\(285\) 0 0
\(286\) 0 0
\(287\) 5.73233 0.338369
\(288\) 0 0
\(289\) −15.2553 −0.897369
\(290\) −1.69454 −0.0995068
\(291\) 0 0
\(292\) −8.82538 −0.516466
\(293\) 26.9403 1.57387 0.786935 0.617035i \(-0.211666\pi\)
0.786935 + 0.617035i \(0.211666\pi\)
\(294\) 0 0
\(295\) −3.99965 −0.232869
\(296\) −0.473741 −0.0275356
\(297\) 0 0
\(298\) −41.0729 −2.37929
\(299\) 6.11698 0.353754
\(300\) 0 0
\(301\) −8.01781 −0.462139
\(302\) −2.26059 −0.130082
\(303\) 0 0
\(304\) −10.7929 −0.619015
\(305\) −7.50941 −0.429987
\(306\) 0 0
\(307\) 24.6157 1.40489 0.702447 0.711736i \(-0.252091\pi\)
0.702447 + 0.711736i \(0.252091\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 47.6903 2.70863
\(311\) 10.2131 0.579132 0.289566 0.957158i \(-0.406489\pi\)
0.289566 + 0.957158i \(0.406489\pi\)
\(312\) 0 0
\(313\) 22.4003 1.26614 0.633069 0.774096i \(-0.281795\pi\)
0.633069 + 0.774096i \(0.281795\pi\)
\(314\) 24.9485 1.40793
\(315\) 0 0
\(316\) −5.06070 −0.284687
\(317\) −23.3595 −1.31200 −0.656000 0.754761i \(-0.727753\pi\)
−0.656000 + 0.754761i \(0.727753\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.16707 −0.400651
\(321\) 0 0
\(322\) 3.18575 0.177535
\(323\) 2.85541 0.158879
\(324\) 0 0
\(325\) 37.8120 2.09743
\(326\) −35.7163 −1.97814
\(327\) 0 0
\(328\) −10.6238 −0.586599
\(329\) 4.06768 0.224258
\(330\) 0 0
\(331\) 15.6444 0.859892 0.429946 0.902855i \(-0.358533\pi\)
0.429946 + 0.902855i \(0.358533\pi\)
\(332\) −1.89300 −0.103892
\(333\) 0 0
\(334\) −8.38787 −0.458964
\(335\) 12.2276 0.668064
\(336\) 0 0
\(337\) 32.5910 1.77534 0.887672 0.460477i \(-0.152322\pi\)
0.887672 + 0.460477i \(0.152322\pi\)
\(338\) −3.85004 −0.209414
\(339\) 0 0
\(340\) −4.91137 −0.266356
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 14.8595 0.801169
\(345\) 0 0
\(346\) −33.5590 −1.80415
\(347\) −5.75284 −0.308829 −0.154414 0.988006i \(-0.549349\pi\)
−0.154414 + 0.988006i \(0.549349\pi\)
\(348\) 0 0
\(349\) −0.183939 −0.00984606 −0.00492303 0.999988i \(-0.501567\pi\)
−0.00492303 + 0.999988i \(0.501567\pi\)
\(350\) 19.6927 1.05262
\(351\) 0 0
\(352\) 0 0
\(353\) −19.0211 −1.01239 −0.506195 0.862419i \(-0.668948\pi\)
−0.506195 + 0.862419i \(0.668948\pi\)
\(354\) 0 0
\(355\) 26.3355 1.39774
\(356\) −15.2069 −0.805964
\(357\) 0 0
\(358\) 43.2916 2.28803
\(359\) 0.718928 0.0379435 0.0189718 0.999820i \(-0.493961\pi\)
0.0189718 + 0.999820i \(0.493961\pi\)
\(360\) 0 0
\(361\) −14.3268 −0.754045
\(362\) −3.37281 −0.177271
\(363\) 0 0
\(364\) 2.99730 0.157101
\(365\) 39.2471 2.05429
\(366\) 0 0
\(367\) −7.89295 −0.412009 −0.206004 0.978551i \(-0.566046\pi\)
−0.206004 + 0.978551i \(0.566046\pi\)
\(368\) −9.31689 −0.485677
\(369\) 0 0
\(370\) −1.77449 −0.0922516
\(371\) 5.00585 0.259891
\(372\) 0 0
\(373\) 15.6686 0.811292 0.405646 0.914030i \(-0.367047\pi\)
0.405646 + 0.914030i \(0.367047\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −7.53865 −0.388776
\(377\) 0.800146 0.0412096
\(378\) 0 0
\(379\) −13.3144 −0.683914 −0.341957 0.939716i \(-0.611090\pi\)
−0.341957 + 0.939716i \(0.611090\pi\)
\(380\) −8.03792 −0.412337
\(381\) 0 0
\(382\) −1.66106 −0.0849871
\(383\) −11.8484 −0.605426 −0.302713 0.953082i \(-0.597892\pi\)
−0.302713 + 0.953082i \(0.597892\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −39.4911 −2.01004
\(387\) 0 0
\(388\) −2.36424 −0.120026
\(389\) 6.04972 0.306733 0.153366 0.988169i \(-0.450989\pi\)
0.153366 + 0.988169i \(0.450989\pi\)
\(390\) 0 0
\(391\) 2.46491 0.124656
\(392\) −1.85331 −0.0936061
\(393\) 0 0
\(394\) 16.9220 0.852518
\(395\) 22.5053 1.13237
\(396\) 0 0
\(397\) −11.5763 −0.580996 −0.290498 0.956876i \(-0.593821\pi\)
−0.290498 + 0.956876i \(0.593821\pi\)
\(398\) 17.7284 0.888643
\(399\) 0 0
\(400\) −57.5923 −2.87961
\(401\) −33.7924 −1.68751 −0.843755 0.536728i \(-0.819660\pi\)
−0.843755 + 0.536728i \(0.819660\pi\)
\(402\) 0 0
\(403\) −22.5189 −1.12175
\(404\) −9.66384 −0.480794
\(405\) 0 0
\(406\) 0.416720 0.0206815
\(407\) 0 0
\(408\) 0 0
\(409\) 18.0520 0.892613 0.446306 0.894880i \(-0.352739\pi\)
0.446306 + 0.894880i \(0.352739\pi\)
\(410\) −39.7935 −1.96526
\(411\) 0 0
\(412\) 12.6633 0.623876
\(413\) 0.983592 0.0483994
\(414\) 0 0
\(415\) 8.41831 0.413239
\(416\) −15.7887 −0.774103
\(417\) 0 0
\(418\) 0 0
\(419\) −2.58559 −0.126314 −0.0631571 0.998004i \(-0.520117\pi\)
−0.0631571 + 0.998004i \(0.520117\pi\)
\(420\) 0 0
\(421\) 13.9948 0.682064 0.341032 0.940052i \(-0.389224\pi\)
0.341032 + 0.940052i \(0.389224\pi\)
\(422\) −22.1787 −1.07964
\(423\) 0 0
\(424\) −9.27738 −0.450549
\(425\) 15.2368 0.739096
\(426\) 0 0
\(427\) 1.84671 0.0893686
\(428\) −11.3646 −0.549331
\(429\) 0 0
\(430\) 55.6592 2.68412
\(431\) 33.6682 1.62174 0.810870 0.585227i \(-0.198995\pi\)
0.810870 + 0.585227i \(0.198995\pi\)
\(432\) 0 0
\(433\) 7.52292 0.361529 0.180764 0.983526i \(-0.442143\pi\)
0.180764 + 0.983526i \(0.442143\pi\)
\(434\) −11.7280 −0.562961
\(435\) 0 0
\(436\) −6.73676 −0.322632
\(437\) 4.03407 0.192976
\(438\) 0 0
\(439\) −22.0123 −1.05059 −0.525294 0.850921i \(-0.676045\pi\)
−0.525294 + 0.850921i \(0.676045\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.39156 0.351581
\(443\) −23.6893 −1.12551 −0.562756 0.826623i \(-0.690259\pi\)
−0.562756 + 0.826623i \(0.690259\pi\)
\(444\) 0 0
\(445\) 67.6263 3.20579
\(446\) −24.5023 −1.16022
\(447\) 0 0
\(448\) 1.76252 0.0832713
\(449\) −2.27513 −0.107370 −0.0536851 0.998558i \(-0.517097\pi\)
−0.0536851 + 0.998558i \(0.517097\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −10.4316 −0.490663
\(453\) 0 0
\(454\) −10.8828 −0.510754
\(455\) −13.3292 −0.624885
\(456\) 0 0
\(457\) −27.3134 −1.27767 −0.638833 0.769345i \(-0.720583\pi\)
−0.638833 + 0.769345i \(0.720583\pi\)
\(458\) −13.6613 −0.638351
\(459\) 0 0
\(460\) −6.93868 −0.323518
\(461\) 8.21908 0.382801 0.191400 0.981512i \(-0.438697\pi\)
0.191400 + 0.981512i \(0.438697\pi\)
\(462\) 0 0
\(463\) −27.9839 −1.30052 −0.650261 0.759711i \(-0.725340\pi\)
−0.650261 + 0.759711i \(0.725340\pi\)
\(464\) −1.21872 −0.0565776
\(465\) 0 0
\(466\) 18.4411 0.854265
\(467\) −26.6745 −1.23435 −0.617175 0.786826i \(-0.711723\pi\)
−0.617175 + 0.786826i \(0.711723\pi\)
\(468\) 0 0
\(469\) −3.00700 −0.138850
\(470\) −28.2376 −1.30250
\(471\) 0 0
\(472\) −1.82290 −0.0839057
\(473\) 0 0
\(474\) 0 0
\(475\) 24.9365 1.14417
\(476\) 1.20780 0.0553595
\(477\) 0 0
\(478\) −42.9491 −1.96445
\(479\) 23.2240 1.06113 0.530566 0.847644i \(-0.321980\pi\)
0.530566 + 0.847644i \(0.321980\pi\)
\(480\) 0 0
\(481\) 0.837900 0.0382050
\(482\) −31.1491 −1.41880
\(483\) 0 0
\(484\) 0 0
\(485\) 10.5140 0.477415
\(486\) 0 0
\(487\) −18.7920 −0.851548 −0.425774 0.904830i \(-0.639998\pi\)
−0.425774 + 0.904830i \(0.639998\pi\)
\(488\) −3.42252 −0.154930
\(489\) 0 0
\(490\) −6.94194 −0.313605
\(491\) −7.40573 −0.334216 −0.167108 0.985939i \(-0.553443\pi\)
−0.167108 + 0.985939i \(0.553443\pi\)
\(492\) 0 0
\(493\) 0.322429 0.0145215
\(494\) 12.0970 0.544269
\(495\) 0 0
\(496\) 34.2991 1.54007
\(497\) −6.47642 −0.290507
\(498\) 0 0
\(499\) 23.6703 1.05963 0.529815 0.848113i \(-0.322261\pi\)
0.529815 + 0.848113i \(0.322261\pi\)
\(500\) −24.3001 −1.08674
\(501\) 0 0
\(502\) −26.1680 −1.16794
\(503\) 8.39731 0.374418 0.187209 0.982320i \(-0.440056\pi\)
0.187209 + 0.982320i \(0.440056\pi\)
\(504\) 0 0
\(505\) 42.9758 1.91240
\(506\) 0 0
\(507\) 0 0
\(508\) −7.99951 −0.354921
\(509\) −14.3065 −0.634123 −0.317061 0.948405i \(-0.602696\pi\)
−0.317061 + 0.948405i \(0.602696\pi\)
\(510\) 0 0
\(511\) −9.65164 −0.426963
\(512\) 5.54215 0.244931
\(513\) 0 0
\(514\) −4.57484 −0.201787
\(515\) −56.3147 −2.48152
\(516\) 0 0
\(517\) 0 0
\(518\) 0.436383 0.0191736
\(519\) 0 0
\(520\) 24.7032 1.08331
\(521\) 33.3684 1.46189 0.730947 0.682434i \(-0.239079\pi\)
0.730947 + 0.682434i \(0.239079\pi\)
\(522\) 0 0
\(523\) 13.3817 0.585143 0.292571 0.956244i \(-0.405489\pi\)
0.292571 + 0.956244i \(0.405489\pi\)
\(524\) 11.4771 0.501378
\(525\) 0 0
\(526\) 17.0329 0.742669
\(527\) −9.07430 −0.395283
\(528\) 0 0
\(529\) −19.5176 −0.848592
\(530\) −34.7503 −1.50946
\(531\) 0 0
\(532\) 1.97668 0.0857001
\(533\) 18.7901 0.813891
\(534\) 0 0
\(535\) 50.5395 2.18501
\(536\) 5.57289 0.240712
\(537\) 0 0
\(538\) 36.8360 1.58811
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0639 −1.46452 −0.732262 0.681023i \(-0.761535\pi\)
−0.732262 + 0.681023i \(0.761535\pi\)
\(542\) −25.5839 −1.09892
\(543\) 0 0
\(544\) −6.36225 −0.272779
\(545\) 29.9589 1.28330
\(546\) 0 0
\(547\) −15.7614 −0.673909 −0.336954 0.941521i \(-0.609397\pi\)
−0.336954 + 0.941521i \(0.609397\pi\)
\(548\) 5.78644 0.247185
\(549\) 0 0
\(550\) 0 0
\(551\) 0.527686 0.0224802
\(552\) 0 0
\(553\) −5.53450 −0.235351
\(554\) 8.88815 0.377621
\(555\) 0 0
\(556\) 13.2065 0.560080
\(557\) −16.1024 −0.682282 −0.341141 0.940012i \(-0.610813\pi\)
−0.341141 + 0.940012i \(0.610813\pi\)
\(558\) 0 0
\(559\) −26.2818 −1.11160
\(560\) 20.3020 0.857918
\(561\) 0 0
\(562\) −25.8806 −1.09171
\(563\) 17.2305 0.726177 0.363089 0.931755i \(-0.381722\pi\)
0.363089 + 0.931755i \(0.381722\pi\)
\(564\) 0 0
\(565\) 46.3903 1.95165
\(566\) −9.67206 −0.406547
\(567\) 0 0
\(568\) 12.0028 0.503626
\(569\) 35.6143 1.49303 0.746514 0.665370i \(-0.231726\pi\)
0.746514 + 0.665370i \(0.231726\pi\)
\(570\) 0 0
\(571\) 26.6026 1.11329 0.556643 0.830752i \(-0.312089\pi\)
0.556643 + 0.830752i \(0.312089\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.78600 0.408460
\(575\) 21.5263 0.897709
\(576\) 0 0
\(577\) 28.8725 1.20198 0.600989 0.799257i \(-0.294773\pi\)
0.600989 + 0.799257i \(0.294773\pi\)
\(578\) −26.0432 −1.08325
\(579\) 0 0
\(580\) −0.907632 −0.0376874
\(581\) −2.07023 −0.0858875
\(582\) 0 0
\(583\) 0 0
\(584\) 17.8874 0.740188
\(585\) 0 0
\(586\) 45.9914 1.89989
\(587\) −2.43414 −0.100468 −0.0502339 0.998737i \(-0.515997\pi\)
−0.0502339 + 0.998737i \(0.515997\pi\)
\(588\) 0 0
\(589\) −14.8509 −0.611922
\(590\) −6.82804 −0.281106
\(591\) 0 0
\(592\) −1.27622 −0.0524525
\(593\) −15.6870 −0.644187 −0.322093 0.946708i \(-0.604387\pi\)
−0.322093 + 0.946708i \(0.604387\pi\)
\(594\) 0 0
\(595\) −5.37119 −0.220197
\(596\) −21.9995 −0.901136
\(597\) 0 0
\(598\) 10.4426 0.427032
\(599\) 13.2218 0.540229 0.270115 0.962828i \(-0.412938\pi\)
0.270115 + 0.962828i \(0.412938\pi\)
\(600\) 0 0
\(601\) −11.2303 −0.458093 −0.229047 0.973415i \(-0.573561\pi\)
−0.229047 + 0.973415i \(0.573561\pi\)
\(602\) −13.6877 −0.557868
\(603\) 0 0
\(604\) −1.21082 −0.0492676
\(605\) 0 0
\(606\) 0 0
\(607\) 19.3477 0.785298 0.392649 0.919688i \(-0.371559\pi\)
0.392649 + 0.919688i \(0.371559\pi\)
\(608\) −10.4124 −0.422279
\(609\) 0 0
\(610\) −12.8197 −0.519056
\(611\) 13.3335 0.539417
\(612\) 0 0
\(613\) 21.0549 0.850398 0.425199 0.905100i \(-0.360204\pi\)
0.425199 + 0.905100i \(0.360204\pi\)
\(614\) 42.0230 1.69591
\(615\) 0 0
\(616\) 0 0
\(617\) 24.1496 0.972228 0.486114 0.873895i \(-0.338414\pi\)
0.486114 + 0.873895i \(0.338414\pi\)
\(618\) 0 0
\(619\) 2.62475 0.105497 0.0527487 0.998608i \(-0.483202\pi\)
0.0527487 + 0.998608i \(0.483202\pi\)
\(620\) 25.5440 1.02587
\(621\) 0 0
\(622\) 17.4354 0.699095
\(623\) −16.6306 −0.666292
\(624\) 0 0
\(625\) 50.3878 2.01551
\(626\) 38.2408 1.52841
\(627\) 0 0
\(628\) 13.3630 0.533241
\(629\) 0.337643 0.0134627
\(630\) 0 0
\(631\) −25.1176 −0.999914 −0.499957 0.866050i \(-0.666651\pi\)
−0.499957 + 0.866050i \(0.666651\pi\)
\(632\) 10.2571 0.408006
\(633\) 0 0
\(634\) −39.8784 −1.58377
\(635\) 35.5744 1.41173
\(636\) 0 0
\(637\) 3.27792 0.129876
\(638\) 0 0
\(639\) 0 0
\(640\) −51.4080 −2.03208
\(641\) −20.6584 −0.815958 −0.407979 0.912991i \(-0.633766\pi\)
−0.407979 + 0.912991i \(0.633766\pi\)
\(642\) 0 0
\(643\) 16.4632 0.649246 0.324623 0.945844i \(-0.394763\pi\)
0.324623 + 0.945844i \(0.394763\pi\)
\(644\) 1.70636 0.0672399
\(645\) 0 0
\(646\) 4.87464 0.191790
\(647\) 34.9584 1.37435 0.687177 0.726490i \(-0.258850\pi\)
0.687177 + 0.726490i \(0.258850\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 64.5511 2.53190
\(651\) 0 0
\(652\) −19.1304 −0.749205
\(653\) −2.41649 −0.0945645 −0.0472823 0.998882i \(-0.515056\pi\)
−0.0472823 + 0.998882i \(0.515056\pi\)
\(654\) 0 0
\(655\) −51.0395 −1.99428
\(656\) −28.6196 −1.11741
\(657\) 0 0
\(658\) 6.94417 0.270712
\(659\) −16.9733 −0.661186 −0.330593 0.943773i \(-0.607249\pi\)
−0.330593 + 0.943773i \(0.607249\pi\)
\(660\) 0 0
\(661\) 7.96946 0.309976 0.154988 0.987916i \(-0.450466\pi\)
0.154988 + 0.987916i \(0.450466\pi\)
\(662\) 26.7074 1.03801
\(663\) 0 0
\(664\) 3.83677 0.148895
\(665\) −8.79046 −0.340880
\(666\) 0 0
\(667\) 0.455522 0.0176379
\(668\) −4.49272 −0.173829
\(669\) 0 0
\(670\) 20.8744 0.806449
\(671\) 0 0
\(672\) 0 0
\(673\) 16.0250 0.617717 0.308859 0.951108i \(-0.400053\pi\)
0.308859 + 0.951108i \(0.400053\pi\)
\(674\) 55.6380 2.14309
\(675\) 0 0
\(676\) −2.06216 −0.0793140
\(677\) −15.2366 −0.585589 −0.292794 0.956175i \(-0.594585\pi\)
−0.292794 + 0.956175i \(0.594585\pi\)
\(678\) 0 0
\(679\) −2.58559 −0.0992259
\(680\) 9.95446 0.381736
\(681\) 0 0
\(682\) 0 0
\(683\) −14.3742 −0.550013 −0.275007 0.961442i \(-0.588680\pi\)
−0.275007 + 0.961442i \(0.588680\pi\)
\(684\) 0 0
\(685\) −25.7328 −0.983198
\(686\) 1.70716 0.0651796
\(687\) 0 0
\(688\) 40.0303 1.52614
\(689\) 16.4088 0.625125
\(690\) 0 0
\(691\) −44.7978 −1.70419 −0.852095 0.523387i \(-0.824668\pi\)
−0.852095 + 0.523387i \(0.824668\pi\)
\(692\) −17.9750 −0.683305
\(693\) 0 0
\(694\) −9.82101 −0.372800
\(695\) −58.7303 −2.22777
\(696\) 0 0
\(697\) 7.57173 0.286800
\(698\) −0.314014 −0.0118856
\(699\) 0 0
\(700\) 10.5478 0.398671
\(701\) 0.240330 0.00907713 0.00453857 0.999990i \(-0.498555\pi\)
0.00453857 + 0.999990i \(0.498555\pi\)
\(702\) 0 0
\(703\) 0.552584 0.0208411
\(704\) 0 0
\(705\) 0 0
\(706\) −32.4720 −1.22210
\(707\) −10.5686 −0.397473
\(708\) 0 0
\(709\) 10.7534 0.403852 0.201926 0.979401i \(-0.435280\pi\)
0.201926 + 0.979401i \(0.435280\pi\)
\(710\) 44.9589 1.68728
\(711\) 0 0
\(712\) 30.8216 1.15509
\(713\) −12.8200 −0.480112
\(714\) 0 0
\(715\) 0 0
\(716\) 23.1879 0.866573
\(717\) 0 0
\(718\) 1.22732 0.0458033
\(719\) 25.7748 0.961239 0.480620 0.876929i \(-0.340412\pi\)
0.480620 + 0.876929i \(0.340412\pi\)
\(720\) 0 0
\(721\) 13.8489 0.515759
\(722\) −24.4582 −0.910240
\(723\) 0 0
\(724\) −1.80655 −0.0671399
\(725\) 2.81580 0.104576
\(726\) 0 0
\(727\) −5.47160 −0.202930 −0.101465 0.994839i \(-0.532353\pi\)
−0.101465 + 0.994839i \(0.532353\pi\)
\(728\) −6.07499 −0.225154
\(729\) 0 0
\(730\) 67.0011 2.47982
\(731\) −10.5906 −0.391707
\(732\) 0 0
\(733\) −19.8563 −0.733411 −0.366705 0.930337i \(-0.619514\pi\)
−0.366705 + 0.930337i \(0.619514\pi\)
\(734\) −13.4745 −0.497354
\(735\) 0 0
\(736\) −8.98845 −0.331319
\(737\) 0 0
\(738\) 0 0
\(739\) 1.80077 0.0662423 0.0331212 0.999451i \(-0.489455\pi\)
0.0331212 + 0.999451i \(0.489455\pi\)
\(740\) −0.950458 −0.0349395
\(741\) 0 0
\(742\) 8.54578 0.313726
\(743\) −48.0579 −1.76307 −0.881537 0.472115i \(-0.843491\pi\)
−0.881537 + 0.472115i \(0.843491\pi\)
\(744\) 0 0
\(745\) 97.8337 3.58435
\(746\) 26.7489 0.979346
\(747\) 0 0
\(748\) 0 0
\(749\) −12.4286 −0.454133
\(750\) 0 0
\(751\) −25.0465 −0.913961 −0.456980 0.889477i \(-0.651069\pi\)
−0.456980 + 0.889477i \(0.651069\pi\)
\(752\) −20.3086 −0.740578
\(753\) 0 0
\(754\) 1.36598 0.0497459
\(755\) 5.38461 0.195966
\(756\) 0 0
\(757\) 10.6487 0.387032 0.193516 0.981097i \(-0.438011\pi\)
0.193516 + 0.981097i \(0.438011\pi\)
\(758\) −22.7298 −0.825583
\(759\) 0 0
\(760\) 16.2914 0.590952
\(761\) 29.8160 1.08083 0.540414 0.841399i \(-0.318268\pi\)
0.540414 + 0.841399i \(0.318268\pi\)
\(762\) 0 0
\(763\) −7.36748 −0.266721
\(764\) −0.889698 −0.0321882
\(765\) 0 0
\(766\) −20.2271 −0.730836
\(767\) 3.22414 0.116417
\(768\) 0 0
\(769\) −28.3115 −1.02094 −0.510470 0.859896i \(-0.670529\pi\)
−0.510470 + 0.859896i \(0.670529\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −21.1523 −0.761287
\(773\) −13.2564 −0.476800 −0.238400 0.971167i \(-0.576623\pi\)
−0.238400 + 0.971167i \(0.576623\pi\)
\(774\) 0 0
\(775\) −79.2466 −2.84662
\(776\) 4.79189 0.172019
\(777\) 0 0
\(778\) 10.3278 0.370271
\(779\) 12.3918 0.443984
\(780\) 0 0
\(781\) 0 0
\(782\) 4.20800 0.150478
\(783\) 0 0
\(784\) −4.99267 −0.178310
\(785\) −59.4262 −2.12101
\(786\) 0 0
\(787\) 52.6771 1.87774 0.938868 0.344279i \(-0.111876\pi\)
0.938868 + 0.344279i \(0.111876\pi\)
\(788\) 9.06379 0.322884
\(789\) 0 0
\(790\) 38.4202 1.36693
\(791\) −11.4083 −0.405632
\(792\) 0 0
\(793\) 6.05337 0.214961
\(794\) −19.7625 −0.701346
\(795\) 0 0
\(796\) 9.49570 0.336566
\(797\) 7.33480 0.259812 0.129906 0.991526i \(-0.458532\pi\)
0.129906 + 0.991526i \(0.458532\pi\)
\(798\) 0 0
\(799\) 5.37292 0.190080
\(800\) −55.5620 −1.96442
\(801\) 0 0
\(802\) −57.6890 −2.03707
\(803\) 0 0
\(804\) 0 0
\(805\) −7.58831 −0.267453
\(806\) −38.4434 −1.35411
\(807\) 0 0
\(808\) 19.5869 0.689063
\(809\) −8.66809 −0.304754 −0.152377 0.988322i \(-0.548693\pi\)
−0.152377 + 0.988322i \(0.548693\pi\)
\(810\) 0 0
\(811\) 19.9487 0.700494 0.350247 0.936657i \(-0.386098\pi\)
0.350247 + 0.936657i \(0.386098\pi\)
\(812\) 0.223205 0.00783294
\(813\) 0 0
\(814\) 0 0
\(815\) 85.0745 2.98003
\(816\) 0 0
\(817\) −17.3325 −0.606387
\(818\) 30.8176 1.07751
\(819\) 0 0
\(820\) −21.3143 −0.744326
\(821\) −53.5648 −1.86943 −0.934713 0.355404i \(-0.884343\pi\)
−0.934713 + 0.355404i \(0.884343\pi\)
\(822\) 0 0
\(823\) −8.44842 −0.294493 −0.147247 0.989100i \(-0.547041\pi\)
−0.147247 + 0.989100i \(0.547041\pi\)
\(824\) −25.6662 −0.894125
\(825\) 0 0
\(826\) 1.67915 0.0584250
\(827\) 3.18260 0.110670 0.0553349 0.998468i \(-0.482377\pi\)
0.0553349 + 0.998468i \(0.482377\pi\)
\(828\) 0 0
\(829\) 57.5089 1.99737 0.998684 0.0512831i \(-0.0163311\pi\)
0.998684 + 0.0512831i \(0.0163311\pi\)
\(830\) 14.3714 0.498838
\(831\) 0 0
\(832\) 5.77741 0.200296
\(833\) 1.32088 0.0457658
\(834\) 0 0
\(835\) 19.9795 0.691419
\(836\) 0 0
\(837\) 0 0
\(838\) −4.41401 −0.152479
\(839\) −12.5594 −0.433600 −0.216800 0.976216i \(-0.569562\pi\)
−0.216800 + 0.976216i \(0.569562\pi\)
\(840\) 0 0
\(841\) −28.9404 −0.997945
\(842\) 23.8913 0.823349
\(843\) 0 0
\(844\) −11.8794 −0.408906
\(845\) 9.17060 0.315478
\(846\) 0 0
\(847\) 0 0
\(848\) −24.9926 −0.858248
\(849\) 0 0
\(850\) 26.0117 0.892195
\(851\) 0.477015 0.0163519
\(852\) 0 0
\(853\) 33.3315 1.14125 0.570625 0.821211i \(-0.306701\pi\)
0.570625 + 0.821211i \(0.306701\pi\)
\(854\) 3.15263 0.107881
\(855\) 0 0
\(856\) 23.0341 0.787289
\(857\) −54.0291 −1.84560 −0.922800 0.385279i \(-0.874105\pi\)
−0.922800 + 0.385279i \(0.874105\pi\)
\(858\) 0 0
\(859\) −12.8624 −0.438860 −0.219430 0.975628i \(-0.570420\pi\)
−0.219430 + 0.975628i \(0.570420\pi\)
\(860\) 29.8123 1.01659
\(861\) 0 0
\(862\) 57.4769 1.95767
\(863\) 33.2271 1.13106 0.565531 0.824727i \(-0.308671\pi\)
0.565531 + 0.824727i \(0.308671\pi\)
\(864\) 0 0
\(865\) 79.9360 2.71791
\(866\) 12.8428 0.436417
\(867\) 0 0
\(868\) −6.28176 −0.213217
\(869\) 0 0
\(870\) 0 0
\(871\) −9.85671 −0.333982
\(872\) 13.6542 0.462389
\(873\) 0 0
\(874\) 6.88679 0.232949
\(875\) −26.5752 −0.898406
\(876\) 0 0
\(877\) 27.6709 0.934379 0.467189 0.884157i \(-0.345267\pi\)
0.467189 + 0.884157i \(0.345267\pi\)
\(878\) −37.5784 −1.26821
\(879\) 0 0
\(880\) 0 0
\(881\) −48.9636 −1.64963 −0.824813 0.565406i \(-0.808720\pi\)
−0.824813 + 0.565406i \(0.808720\pi\)
\(882\) 0 0
\(883\) 21.3223 0.717554 0.358777 0.933423i \(-0.383194\pi\)
0.358777 + 0.933423i \(0.383194\pi\)
\(884\) 3.95908 0.133158
\(885\) 0 0
\(886\) −40.4414 −1.35865
\(887\) 45.2483 1.51929 0.759645 0.650338i \(-0.225373\pi\)
0.759645 + 0.650338i \(0.225373\pi\)
\(888\) 0 0
\(889\) −8.74845 −0.293414
\(890\) 115.449 3.86985
\(891\) 0 0
\(892\) −13.1240 −0.439423
\(893\) 8.79329 0.294256
\(894\) 0 0
\(895\) −103.118 −3.44687
\(896\) 12.6422 0.422348
\(897\) 0 0
\(898\) −3.88401 −0.129611
\(899\) −1.67695 −0.0559294
\(900\) 0 0
\(901\) 6.61213 0.220282
\(902\) 0 0
\(903\) 0 0
\(904\) 21.1430 0.703207
\(905\) 8.03388 0.267055
\(906\) 0 0
\(907\) −45.0665 −1.49641 −0.748205 0.663468i \(-0.769084\pi\)
−0.748205 + 0.663468i \(0.769084\pi\)
\(908\) −5.82906 −0.193444
\(909\) 0 0
\(910\) −22.7551 −0.754325
\(911\) 9.21469 0.305296 0.152648 0.988281i \(-0.451220\pi\)
0.152648 + 0.988281i \(0.451220\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −46.6283 −1.54233
\(915\) 0 0
\(916\) −7.31730 −0.241770
\(917\) 12.5516 0.414491
\(918\) 0 0
\(919\) 49.7144 1.63993 0.819963 0.572417i \(-0.193994\pi\)
0.819963 + 0.572417i \(0.193994\pi\)
\(920\) 14.0635 0.463659
\(921\) 0 0
\(922\) 14.0313 0.462095
\(923\) −21.2292 −0.698767
\(924\) 0 0
\(925\) 2.94866 0.0969514
\(926\) −47.7729 −1.56992
\(927\) 0 0
\(928\) −1.17576 −0.0385961
\(929\) −49.5105 −1.62439 −0.812193 0.583389i \(-0.801726\pi\)
−0.812193 + 0.583389i \(0.801726\pi\)
\(930\) 0 0
\(931\) 2.16175 0.0708484
\(932\) 9.87743 0.323546
\(933\) 0 0
\(934\) −45.5377 −1.49004
\(935\) 0 0
\(936\) 0 0
\(937\) 6.98790 0.228285 0.114142 0.993464i \(-0.463588\pi\)
0.114142 + 0.993464i \(0.463588\pi\)
\(938\) −5.13343 −0.167612
\(939\) 0 0
\(940\) −15.1247 −0.493312
\(941\) −46.8241 −1.52642 −0.763211 0.646150i \(-0.776378\pi\)
−0.763211 + 0.646150i \(0.776378\pi\)
\(942\) 0 0
\(943\) 10.6972 0.348348
\(944\) −4.91075 −0.159831
\(945\) 0 0
\(946\) 0 0
\(947\) −15.1286 −0.491614 −0.245807 0.969319i \(-0.579053\pi\)
−0.245807 + 0.969319i \(0.579053\pi\)
\(948\) 0 0
\(949\) −31.6373 −1.02699
\(950\) 42.5706 1.38117
\(951\) 0 0
\(952\) −2.44800 −0.0793401
\(953\) 56.2273 1.82138 0.910690 0.413090i \(-0.135550\pi\)
0.910690 + 0.413090i \(0.135550\pi\)
\(954\) 0 0
\(955\) 3.95656 0.128031
\(956\) −23.0045 −0.744019
\(957\) 0 0
\(958\) 39.6470 1.28094
\(959\) 6.32819 0.204348
\(960\) 0 0
\(961\) 16.1952 0.522427
\(962\) 1.43043 0.0461189
\(963\) 0 0
\(964\) −16.6841 −0.537359
\(965\) 94.0658 3.02809
\(966\) 0 0
\(967\) 46.3761 1.49136 0.745678 0.666307i \(-0.232126\pi\)
0.745678 + 0.666307i \(0.232126\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 17.9490 0.576308
\(971\) −15.5400 −0.498701 −0.249351 0.968413i \(-0.580217\pi\)
−0.249351 + 0.968413i \(0.580217\pi\)
\(972\) 0 0
\(973\) 14.4429 0.463019
\(974\) −32.0810 −1.02794
\(975\) 0 0
\(976\) −9.22002 −0.295125
\(977\) −32.8287 −1.05028 −0.525141 0.851015i \(-0.675988\pi\)
−0.525141 + 0.851015i \(0.675988\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.71825 −0.118775
\(981\) 0 0
\(982\) −12.6428 −0.403447
\(983\) −24.3184 −0.775635 −0.387817 0.921736i \(-0.626771\pi\)
−0.387817 + 0.921736i \(0.626771\pi\)
\(984\) 0 0
\(985\) −40.3074 −1.28430
\(986\) 0.550438 0.0175295
\(987\) 0 0
\(988\) 6.47941 0.206138
\(989\) −14.9622 −0.475769
\(990\) 0 0
\(991\) −47.7104 −1.51557 −0.757786 0.652503i \(-0.773719\pi\)
−0.757786 + 0.652503i \(0.773719\pi\)
\(992\) 33.0899 1.05061
\(993\) 0 0
\(994\) −11.0563 −0.350684
\(995\) −42.2281 −1.33872
\(996\) 0 0
\(997\) 19.8449 0.628494 0.314247 0.949341i \(-0.398248\pi\)
0.314247 + 0.949341i \(0.398248\pi\)
\(998\) 40.4090 1.27913
\(999\) 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.ct.1.7 8
3.2 odd 2 847.2.a.p.1.2 8
11.5 even 5 693.2.m.i.190.1 16
11.9 even 5 693.2.m.i.631.1 16
11.10 odd 2 7623.2.a.cw.1.2 8
21.20 even 2 5929.2.a.bt.1.2 8
33.2 even 10 847.2.f.x.323.1 16
33.5 odd 10 77.2.f.b.36.4 yes 16
33.8 even 10 847.2.f.v.372.4 16
33.14 odd 10 847.2.f.w.372.1 16
33.17 even 10 847.2.f.x.729.1 16
33.20 odd 10 77.2.f.b.15.4 16
33.26 odd 10 847.2.f.w.148.1 16
33.29 even 10 847.2.f.v.148.4 16
33.32 even 2 847.2.a.o.1.7 8
231.5 even 30 539.2.q.f.410.1 32
231.20 even 10 539.2.f.e.246.4 16
231.38 even 30 539.2.q.f.520.4 32
231.53 odd 30 539.2.q.g.422.1 32
231.86 odd 30 539.2.q.g.312.4 32
231.104 even 10 539.2.f.e.344.4 16
231.137 odd 30 539.2.q.g.520.4 32
231.152 even 30 539.2.q.f.312.4 32
231.170 odd 30 539.2.q.g.410.1 32
231.185 even 30 539.2.q.f.422.1 32
231.230 odd 2 5929.2.a.bs.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.4 16 33.20 odd 10
77.2.f.b.36.4 yes 16 33.5 odd 10
539.2.f.e.246.4 16 231.20 even 10
539.2.f.e.344.4 16 231.104 even 10
539.2.q.f.312.4 32 231.152 even 30
539.2.q.f.410.1 32 231.5 even 30
539.2.q.f.422.1 32 231.185 even 30
539.2.q.f.520.4 32 231.38 even 30
539.2.q.g.312.4 32 231.86 odd 30
539.2.q.g.410.1 32 231.170 odd 30
539.2.q.g.422.1 32 231.53 odd 30
539.2.q.g.520.4 32 231.137 odd 30
693.2.m.i.190.1 16 11.5 even 5
693.2.m.i.631.1 16 11.9 even 5
847.2.a.o.1.7 8 33.32 even 2
847.2.a.p.1.2 8 3.2 odd 2
847.2.f.v.148.4 16 33.29 even 10
847.2.f.v.372.4 16 33.8 even 10
847.2.f.w.148.1 16 33.26 odd 10
847.2.f.w.372.1 16 33.14 odd 10
847.2.f.x.323.1 16 33.2 even 10
847.2.f.x.729.1 16 33.17 even 10
5929.2.a.bs.1.7 8 231.230 odd 2
5929.2.a.bt.1.2 8 21.20 even 2
7623.2.a.ct.1.7 8 1.1 even 1 trivial
7623.2.a.cw.1.2 8 11.10 odd 2