Properties

Label 2070.3.c.a.91.10
Level $2070$
Weight $3$
Character 2070.91
Analytic conductor $56.403$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,3,Mod(91,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2070.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(56.4034147226\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.10
Root \(0.0431371i\) of defining polynomial
Character \(\chi\) \(=\) 2070.91
Dual form 2070.3.c.a.91.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -8.24199i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -8.24199i q^{7} +2.82843 q^{8} -3.16228i q^{10} +15.8246i q^{11} -14.3219 q^{13} -11.6559i q^{14} +4.00000 q^{16} -10.1666i q^{17} +36.5359i q^{19} -4.47214i q^{20} +22.3793i q^{22} +(-22.2445 + 5.84663i) q^{23} -5.00000 q^{25} -20.2543 q^{26} -16.4840i q^{28} -6.46533 q^{29} -42.8526 q^{31} +5.65685 q^{32} -14.3777i q^{34} -18.4296 q^{35} +63.6379i q^{37} +51.6696i q^{38} -6.32456i q^{40} +37.0921 q^{41} -6.00126i q^{43} +31.6491i q^{44} +(-31.4584 + 8.26838i) q^{46} +32.4676 q^{47} -18.9303 q^{49} -7.07107 q^{50} -28.6438 q^{52} -36.6640i q^{53} +35.3848 q^{55} -23.3119i q^{56} -9.14336 q^{58} -6.65851 q^{59} +55.7093i q^{61} -60.6028 q^{62} +8.00000 q^{64} +32.0248i q^{65} -4.45984i q^{67} -20.3331i q^{68} -26.0634 q^{70} -118.412 q^{71} +82.2675 q^{73} +89.9976i q^{74} +73.0718i q^{76} +130.426 q^{77} +133.084i q^{79} -8.94427i q^{80} +52.4561 q^{82} -67.5614i q^{83} -22.7331 q^{85} -8.48707i q^{86} +44.7586i q^{88} +104.729i q^{89} +118.041i q^{91} +(-44.4890 + 11.6933i) q^{92} +45.9161 q^{94} +81.6968 q^{95} +98.6666i q^{97} -26.7715 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 8.24199i 1.17743i −0.808342 0.588713i \(-0.799635\pi\)
0.808342 0.588713i \(-0.200365\pi\)
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) 3.16228i 0.316228i
\(11\) 15.8246i 1.43860i 0.694702 + 0.719298i \(0.255536\pi\)
−0.694702 + 0.719298i \(0.744464\pi\)
\(12\) 0 0
\(13\) −14.3219 −1.10169 −0.550843 0.834609i \(-0.685694\pi\)
−0.550843 + 0.834609i \(0.685694\pi\)
\(14\) 11.6559i 0.832566i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 10.1666i 0.598034i −0.954248 0.299017i \(-0.903341\pi\)
0.954248 0.299017i \(-0.0966587\pi\)
\(18\) 0 0
\(19\) 36.5359i 1.92294i 0.274904 + 0.961472i \(0.411354\pi\)
−0.274904 + 0.961472i \(0.588646\pi\)
\(20\) 4.47214i 0.223607i
\(21\) 0 0
\(22\) 22.3793i 1.01724i
\(23\) −22.2445 + 5.84663i −0.967151 + 0.254201i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) −20.2543 −0.779010
\(27\) 0 0
\(28\) 16.4840i 0.588713i
\(29\) −6.46533 −0.222942 −0.111471 0.993768i \(-0.535556\pi\)
−0.111471 + 0.993768i \(0.535556\pi\)
\(30\) 0 0
\(31\) −42.8526 −1.38234 −0.691172 0.722691i \(-0.742905\pi\)
−0.691172 + 0.722691i \(0.742905\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) 14.3777i 0.422874i
\(35\) −18.4296 −0.526561
\(36\) 0 0
\(37\) 63.6379i 1.71994i 0.510341 + 0.859972i \(0.329519\pi\)
−0.510341 + 0.859972i \(0.670481\pi\)
\(38\) 51.6696i 1.35973i
\(39\) 0 0
\(40\) 6.32456i 0.158114i
\(41\) 37.0921 0.904685 0.452342 0.891844i \(-0.350588\pi\)
0.452342 + 0.891844i \(0.350588\pi\)
\(42\) 0 0
\(43\) 6.00126i 0.139564i −0.997562 0.0697821i \(-0.977770\pi\)
0.997562 0.0697821i \(-0.0222304\pi\)
\(44\) 31.6491i 0.719298i
\(45\) 0 0
\(46\) −31.4584 + 8.26838i −0.683879 + 0.179747i
\(47\) 32.4676 0.690800 0.345400 0.938455i \(-0.387743\pi\)
0.345400 + 0.938455i \(0.387743\pi\)
\(48\) 0 0
\(49\) −18.9303 −0.386333
\(50\) −7.07107 −0.141421
\(51\) 0 0
\(52\) −28.6438 −0.550843
\(53\) 36.6640i 0.691774i −0.938276 0.345887i \(-0.887578\pi\)
0.938276 0.345887i \(-0.112422\pi\)
\(54\) 0 0
\(55\) 35.3848 0.643360
\(56\) 23.3119i 0.416283i
\(57\) 0 0
\(58\) −9.14336 −0.157644
\(59\) −6.65851 −0.112856 −0.0564280 0.998407i \(-0.517971\pi\)
−0.0564280 + 0.998407i \(0.517971\pi\)
\(60\) 0 0
\(61\) 55.7093i 0.913268i 0.889655 + 0.456634i \(0.150945\pi\)
−0.889655 + 0.456634i \(0.849055\pi\)
\(62\) −60.6028 −0.977464
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 32.0248i 0.492689i
\(66\) 0 0
\(67\) 4.45984i 0.0665648i −0.999446 0.0332824i \(-0.989404\pi\)
0.999446 0.0332824i \(-0.0105961\pi\)
\(68\) 20.3331i 0.299017i
\(69\) 0 0
\(70\) −26.0634 −0.372335
\(71\) −118.412 −1.66777 −0.833886 0.551937i \(-0.813889\pi\)
−0.833886 + 0.551937i \(0.813889\pi\)
\(72\) 0 0
\(73\) 82.2675 1.12695 0.563476 0.826133i \(-0.309464\pi\)
0.563476 + 0.826133i \(0.309464\pi\)
\(74\) 89.9976i 1.21618i
\(75\) 0 0
\(76\) 73.0718i 0.961472i
\(77\) 130.426 1.69384
\(78\) 0 0
\(79\) 133.084i 1.68461i 0.538999 + 0.842307i \(0.318803\pi\)
−0.538999 + 0.842307i \(0.681197\pi\)
\(80\) 8.94427i 0.111803i
\(81\) 0 0
\(82\) 52.4561 0.639709
\(83\) 67.5614i 0.813993i −0.913430 0.406996i \(-0.866576\pi\)
0.913430 0.406996i \(-0.133424\pi\)
\(84\) 0 0
\(85\) −22.7331 −0.267449
\(86\) 8.48707i 0.0986868i
\(87\) 0 0
\(88\) 44.7586i 0.508620i
\(89\) 104.729i 1.17673i 0.808595 + 0.588365i \(0.200228\pi\)
−0.808595 + 0.588365i \(0.799772\pi\)
\(90\) 0 0
\(91\) 118.041i 1.29715i
\(92\) −44.4890 + 11.6933i −0.483576 + 0.127101i
\(93\) 0 0
\(94\) 45.9161 0.488470
\(95\) 81.6968 0.859966
\(96\) 0 0
\(97\) 98.6666i 1.01718i 0.861008 + 0.508591i \(0.169833\pi\)
−0.861008 + 0.508591i \(0.830167\pi\)
\(98\) −26.7715 −0.273179
\(99\) 0 0
\(100\) −10.0000 −0.100000
\(101\) 75.6811 0.749318 0.374659 0.927163i \(-0.377760\pi\)
0.374659 + 0.927163i \(0.377760\pi\)
\(102\) 0 0
\(103\) 86.8499i 0.843203i −0.906781 0.421602i \(-0.861468\pi\)
0.906781 0.421602i \(-0.138532\pi\)
\(104\) −40.5085 −0.389505
\(105\) 0 0
\(106\) 51.8508i 0.489158i
\(107\) 5.55552i 0.0519208i −0.999663 0.0259604i \(-0.991736\pi\)
0.999663 0.0259604i \(-0.00826438\pi\)
\(108\) 0 0
\(109\) 71.0003i 0.651379i 0.945477 + 0.325689i \(0.105596\pi\)
−0.945477 + 0.325689i \(0.894404\pi\)
\(110\) 50.0416 0.454924
\(111\) 0 0
\(112\) 32.9679i 0.294357i
\(113\) 100.763i 0.891707i −0.895106 0.445854i \(-0.852900\pi\)
0.895106 0.445854i \(-0.147100\pi\)
\(114\) 0 0
\(115\) 13.0735 + 49.7402i 0.113682 + 0.432523i
\(116\) −12.9307 −0.111471
\(117\) 0 0
\(118\) −9.41655 −0.0798013
\(119\) −83.7927 −0.704141
\(120\) 0 0
\(121\) −129.417 −1.06956
\(122\) 78.7849i 0.645778i
\(123\) 0 0
\(124\) −85.7053 −0.691172
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −43.9602 −0.346143 −0.173072 0.984909i \(-0.555369\pi\)
−0.173072 + 0.984909i \(0.555369\pi\)
\(128\) 11.3137 0.0883883
\(129\) 0 0
\(130\) 45.2899i 0.348384i
\(131\) 44.4134 0.339033 0.169517 0.985527i \(-0.445779\pi\)
0.169517 + 0.985527i \(0.445779\pi\)
\(132\) 0 0
\(133\) 301.129 2.26412
\(134\) 6.30717i 0.0470684i
\(135\) 0 0
\(136\) 28.7554i 0.211437i
\(137\) 44.9295i 0.327953i 0.986464 + 0.163976i \(0.0524321\pi\)
−0.986464 + 0.163976i \(0.947568\pi\)
\(138\) 0 0
\(139\) −139.796 −1.00573 −0.502864 0.864365i \(-0.667720\pi\)
−0.502864 + 0.864365i \(0.667720\pi\)
\(140\) −36.8593 −0.263281
\(141\) 0 0
\(142\) −167.460 −1.17929
\(143\) 226.638i 1.58488i
\(144\) 0 0
\(145\) 14.4569i 0.0997029i
\(146\) 116.344 0.796875
\(147\) 0 0
\(148\) 127.276i 0.859972i
\(149\) 25.4227i 0.170622i −0.996354 0.0853111i \(-0.972812\pi\)
0.996354 0.0853111i \(-0.0271884\pi\)
\(150\) 0 0
\(151\) −52.5299 −0.347880 −0.173940 0.984756i \(-0.555650\pi\)
−0.173940 + 0.984756i \(0.555650\pi\)
\(152\) 103.339i 0.679863i
\(153\) 0 0
\(154\) 184.450 1.19773
\(155\) 95.8214i 0.618203i
\(156\) 0 0
\(157\) 74.5874i 0.475079i 0.971378 + 0.237539i \(0.0763409\pi\)
−0.971378 + 0.237539i \(0.923659\pi\)
\(158\) 188.210i 1.19120i
\(159\) 0 0
\(160\) 12.6491i 0.0790569i
\(161\) 48.1878 + 183.339i 0.299303 + 1.13875i
\(162\) 0 0
\(163\) −18.3238 −0.112416 −0.0562080 0.998419i \(-0.517901\pi\)
−0.0562080 + 0.998419i \(0.517901\pi\)
\(164\) 74.1842 0.452342
\(165\) 0 0
\(166\) 95.5462i 0.575580i
\(167\) −68.9768 −0.413035 −0.206517 0.978443i \(-0.566213\pi\)
−0.206517 + 0.978443i \(0.566213\pi\)
\(168\) 0 0
\(169\) 36.1174 0.213712
\(170\) −32.1495 −0.189115
\(171\) 0 0
\(172\) 12.0025i 0.0697821i
\(173\) −118.707 −0.686166 −0.343083 0.939305i \(-0.611471\pi\)
−0.343083 + 0.939305i \(0.611471\pi\)
\(174\) 0 0
\(175\) 41.2099i 0.235485i
\(176\) 63.2982i 0.359649i
\(177\) 0 0
\(178\) 148.109i 0.832074i
\(179\) 278.892 1.55806 0.779029 0.626988i \(-0.215712\pi\)
0.779029 + 0.626988i \(0.215712\pi\)
\(180\) 0 0
\(181\) 66.1123i 0.365261i −0.983182 0.182631i \(-0.941539\pi\)
0.983182 0.182631i \(-0.0584613\pi\)
\(182\) 166.935i 0.917227i
\(183\) 0 0
\(184\) −62.9169 + 16.5368i −0.341940 + 0.0898737i
\(185\) 142.299 0.769182
\(186\) 0 0
\(187\) 160.882 0.860329
\(188\) 64.9352 0.345400
\(189\) 0 0
\(190\) 115.537 0.608088
\(191\) 5.53406i 0.0289741i 0.999895 + 0.0144871i \(0.00461154\pi\)
−0.999895 + 0.0144871i \(0.995388\pi\)
\(192\) 0 0
\(193\) −72.0460 −0.373295 −0.186648 0.982427i \(-0.559762\pi\)
−0.186648 + 0.982427i \(0.559762\pi\)
\(194\) 139.536i 0.719256i
\(195\) 0 0
\(196\) −37.8606 −0.193167
\(197\) −191.143 −0.970271 −0.485136 0.874439i \(-0.661230\pi\)
−0.485136 + 0.874439i \(0.661230\pi\)
\(198\) 0 0
\(199\) 172.543i 0.867051i 0.901141 + 0.433525i \(0.142731\pi\)
−0.901141 + 0.433525i \(0.857269\pi\)
\(200\) −14.1421 −0.0707107
\(201\) 0 0
\(202\) 107.029 0.529848
\(203\) 53.2871i 0.262498i
\(204\) 0 0
\(205\) 82.9404i 0.404587i
\(206\) 122.824i 0.596235i
\(207\) 0 0
\(208\) −57.2877 −0.275422
\(209\) −578.165 −2.76634
\(210\) 0 0
\(211\) −334.039 −1.58312 −0.791561 0.611090i \(-0.790731\pi\)
−0.791561 + 0.611090i \(0.790731\pi\)
\(212\) 73.3281i 0.345887i
\(213\) 0 0
\(214\) 7.85670i 0.0367135i
\(215\) −13.4192 −0.0624150
\(216\) 0 0
\(217\) 353.191i 1.62761i
\(218\) 100.410i 0.460594i
\(219\) 0 0
\(220\) 70.7696 0.321680
\(221\) 145.605i 0.658845i
\(222\) 0 0
\(223\) −441.580 −1.98018 −0.990089 0.140440i \(-0.955148\pi\)
−0.990089 + 0.140440i \(0.955148\pi\)
\(224\) 46.6237i 0.208142i
\(225\) 0 0
\(226\) 142.500i 0.630532i
\(227\) 54.9222i 0.241948i −0.992656 0.120974i \(-0.961398\pi\)
0.992656 0.120974i \(-0.0386018\pi\)
\(228\) 0 0
\(229\) 322.300i 1.40743i −0.710485 0.703713i \(-0.751524\pi\)
0.710485 0.703713i \(-0.248476\pi\)
\(230\) 18.4887 + 70.3432i 0.0803855 + 0.305840i
\(231\) 0 0
\(232\) −18.2867 −0.0788220
\(233\) −159.208 −0.683296 −0.341648 0.939828i \(-0.610985\pi\)
−0.341648 + 0.939828i \(0.610985\pi\)
\(234\) 0 0
\(235\) 72.5998i 0.308935i
\(236\) −13.3170 −0.0564280
\(237\) 0 0
\(238\) −118.501 −0.497903
\(239\) −32.1990 −0.134724 −0.0673620 0.997729i \(-0.521458\pi\)
−0.0673620 + 0.997729i \(0.521458\pi\)
\(240\) 0 0
\(241\) 39.8259i 0.165253i −0.996581 0.0826263i \(-0.973669\pi\)
0.996581 0.0826263i \(-0.0263308\pi\)
\(242\) −183.023 −0.756292
\(243\) 0 0
\(244\) 111.419i 0.456634i
\(245\) 42.3295i 0.172773i
\(246\) 0 0
\(247\) 523.265i 2.11848i
\(248\) −121.206 −0.488732
\(249\) 0 0
\(250\) 15.8114i 0.0632456i
\(251\) 232.529i 0.926410i 0.886251 + 0.463205i \(0.153301\pi\)
−0.886251 + 0.463205i \(0.846699\pi\)
\(252\) 0 0
\(253\) −92.5203 352.009i −0.365693 1.39134i
\(254\) −62.1691 −0.244760
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −325.160 −1.26521 −0.632606 0.774473i \(-0.718015\pi\)
−0.632606 + 0.774473i \(0.718015\pi\)
\(258\) 0 0
\(259\) 524.503 2.02511
\(260\) 64.0496i 0.246344i
\(261\) 0 0
\(262\) 62.8100 0.239733
\(263\) 4.94890i 0.0188171i 0.999956 + 0.00940855i \(0.00299488\pi\)
−0.999956 + 0.00940855i \(0.997005\pi\)
\(264\) 0 0
\(265\) −81.9833 −0.309371
\(266\) 425.860 1.60098
\(267\) 0 0
\(268\) 8.91969i 0.0332824i
\(269\) −235.047 −0.873781 −0.436891 0.899515i \(-0.643920\pi\)
−0.436891 + 0.899515i \(0.643920\pi\)
\(270\) 0 0
\(271\) −53.5224 −0.197500 −0.0987498 0.995112i \(-0.531484\pi\)
−0.0987498 + 0.995112i \(0.531484\pi\)
\(272\) 40.6663i 0.149508i
\(273\) 0 0
\(274\) 63.5399i 0.231898i
\(275\) 79.1228i 0.287719i
\(276\) 0 0
\(277\) −143.576 −0.518326 −0.259163 0.965834i \(-0.583447\pi\)
−0.259163 + 0.965834i \(0.583447\pi\)
\(278\) −197.702 −0.711157
\(279\) 0 0
\(280\) −52.1269 −0.186167
\(281\) 187.330i 0.666656i −0.942811 0.333328i \(-0.891828\pi\)
0.942811 0.333328i \(-0.108172\pi\)
\(282\) 0 0
\(283\) 486.156i 1.71787i 0.512087 + 0.858934i \(0.328873\pi\)
−0.512087 + 0.858934i \(0.671127\pi\)
\(284\) −236.824 −0.833886
\(285\) 0 0
\(286\) 320.515i 1.12068i
\(287\) 305.712i 1.06520i
\(288\) 0 0
\(289\) 185.641 0.642356
\(290\) 20.4452i 0.0705006i
\(291\) 0 0
\(292\) 164.535 0.563476
\(293\) 240.733i 0.821616i 0.911722 + 0.410808i \(0.134753\pi\)
−0.911722 + 0.410808i \(0.865247\pi\)
\(294\) 0 0
\(295\) 14.8889i 0.0504708i
\(296\) 179.995i 0.608092i
\(297\) 0 0
\(298\) 35.9531i 0.120648i
\(299\) 318.584 83.7350i 1.06550 0.280050i
\(300\) 0 0
\(301\) −49.4623 −0.164327
\(302\) −74.2885 −0.245988
\(303\) 0 0
\(304\) 146.144i 0.480736i
\(305\) 124.570 0.408426
\(306\) 0 0
\(307\) 338.091 1.10127 0.550636 0.834745i \(-0.314385\pi\)
0.550636 + 0.834745i \(0.314385\pi\)
\(308\) 260.852 0.846920
\(309\) 0 0
\(310\) 135.512i 0.437135i
\(311\) 131.884 0.424066 0.212033 0.977263i \(-0.431992\pi\)
0.212033 + 0.977263i \(0.431992\pi\)
\(312\) 0 0
\(313\) 160.254i 0.511994i 0.966678 + 0.255997i \(0.0824038\pi\)
−0.966678 + 0.255997i \(0.917596\pi\)
\(314\) 105.482i 0.335931i
\(315\) 0 0
\(316\) 266.169i 0.842307i
\(317\) 525.687 1.65832 0.829160 0.559012i \(-0.188819\pi\)
0.829160 + 0.559012i \(0.188819\pi\)
\(318\) 0 0
\(319\) 102.311i 0.320724i
\(320\) 17.8885i 0.0559017i
\(321\) 0 0
\(322\) 68.1479 + 259.280i 0.211639 + 0.805218i
\(323\) 371.445 1.14998
\(324\) 0 0
\(325\) 71.6096 0.220337
\(326\) −25.9138 −0.0794901
\(327\) 0 0
\(328\) 104.912 0.319854
\(329\) 267.598i 0.813367i
\(330\) 0 0
\(331\) 0.120137 0.000362953 0.000181477 1.00000i \(-0.499942\pi\)
0.000181477 1.00000i \(0.499942\pi\)
\(332\) 135.123i 0.406996i
\(333\) 0 0
\(334\) −97.5480 −0.292060
\(335\) −9.97251 −0.0297687
\(336\) 0 0
\(337\) 652.946i 1.93752i −0.247992 0.968762i \(-0.579771\pi\)
0.247992 0.968762i \(-0.420229\pi\)
\(338\) 51.0777 0.151117
\(339\) 0 0
\(340\) −45.4663 −0.133724
\(341\) 678.124i 1.98863i
\(342\) 0 0
\(343\) 247.834i 0.722548i
\(344\) 16.9741i 0.0493434i
\(345\) 0 0
\(346\) −167.877 −0.485193
\(347\) 468.304 1.34958 0.674789 0.738010i \(-0.264234\pi\)
0.674789 + 0.738010i \(0.264234\pi\)
\(348\) 0 0
\(349\) 182.288 0.522315 0.261157 0.965296i \(-0.415896\pi\)
0.261157 + 0.965296i \(0.415896\pi\)
\(350\) 58.2796i 0.166513i
\(351\) 0 0
\(352\) 89.5172i 0.254310i
\(353\) 301.039 0.852802 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(354\) 0 0
\(355\) 264.777i 0.745850i
\(356\) 209.458i 0.588365i
\(357\) 0 0
\(358\) 394.413 1.10171
\(359\) 24.6772i 0.0687388i 0.999409 + 0.0343694i \(0.0109423\pi\)
−0.999409 + 0.0343694i \(0.989058\pi\)
\(360\) 0 0
\(361\) −973.874 −2.69771
\(362\) 93.4969i 0.258279i
\(363\) 0 0
\(364\) 236.082i 0.648577i
\(365\) 183.956i 0.503988i
\(366\) 0 0
\(367\) 427.508i 1.16487i 0.812877 + 0.582435i \(0.197900\pi\)
−0.812877 + 0.582435i \(0.802100\pi\)
\(368\) −88.9779 + 23.3865i −0.241788 + 0.0635503i
\(369\) 0 0
\(370\) 201.241 0.543894
\(371\) −302.184 −0.814513
\(372\) 0 0
\(373\) 338.652i 0.907913i 0.891024 + 0.453957i \(0.149988\pi\)
−0.891024 + 0.453957i \(0.850012\pi\)
\(374\) 227.521 0.608344
\(375\) 0 0
\(376\) 91.8323 0.244235
\(377\) 92.5959 0.245612
\(378\) 0 0
\(379\) 287.157i 0.757671i 0.925464 + 0.378835i \(0.123675\pi\)
−0.925464 + 0.378835i \(0.876325\pi\)
\(380\) 163.394 0.429983
\(381\) 0 0
\(382\) 7.82634i 0.0204878i
\(383\) 682.661i 1.78240i 0.453606 + 0.891202i \(0.350137\pi\)
−0.453606 + 0.891202i \(0.649863\pi\)
\(384\) 0 0
\(385\) 291.641i 0.757509i
\(386\) −101.888 −0.263960
\(387\) 0 0
\(388\) 197.333i 0.508591i
\(389\) 543.886i 1.39816i −0.715041 0.699082i \(-0.753592\pi\)
0.715041 0.699082i \(-0.246408\pi\)
\(390\) 0 0
\(391\) 59.4402 + 226.150i 0.152021 + 0.578389i
\(392\) −53.5430 −0.136589
\(393\) 0 0
\(394\) −270.318 −0.686085
\(395\) 297.586 0.753382
\(396\) 0 0
\(397\) 99.6609 0.251035 0.125517 0.992091i \(-0.459941\pi\)
0.125517 + 0.992091i \(0.459941\pi\)
\(398\) 244.013i 0.613097i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 404.668i 1.00915i −0.863369 0.504573i \(-0.831650\pi\)
0.863369 0.504573i \(-0.168350\pi\)
\(402\) 0 0
\(403\) 613.732 1.52291
\(404\) 151.362 0.374659
\(405\) 0 0
\(406\) 75.3594i 0.185614i
\(407\) −1007.04 −2.47430
\(408\) 0 0
\(409\) 701.478 1.71510 0.857552 0.514397i \(-0.171984\pi\)
0.857552 + 0.514397i \(0.171984\pi\)
\(410\) 117.295i 0.286087i
\(411\) 0 0
\(412\) 173.700i 0.421602i
\(413\) 54.8793i 0.132880i
\(414\) 0 0
\(415\) −151.072 −0.364029
\(416\) −81.0170 −0.194752
\(417\) 0 0
\(418\) −817.648 −1.95610
\(419\) 685.089i 1.63506i 0.575888 + 0.817529i \(0.304657\pi\)
−0.575888 + 0.817529i \(0.695343\pi\)
\(420\) 0 0
\(421\) 205.326i 0.487711i −0.969812 0.243856i \(-0.921588\pi\)
0.969812 0.243856i \(-0.0784123\pi\)
\(422\) −472.402 −1.11944
\(423\) 0 0
\(424\) 103.702i 0.244579i
\(425\) 50.8329i 0.119607i
\(426\) 0 0
\(427\) 459.156 1.07531
\(428\) 11.1110i 0.0259604i
\(429\) 0 0
\(430\) −18.9777 −0.0441341
\(431\) 355.926i 0.825814i −0.910773 0.412907i \(-0.864513\pi\)
0.910773 0.412907i \(-0.135487\pi\)
\(432\) 0 0
\(433\) 17.2870i 0.0399238i −0.999801 0.0199619i \(-0.993646\pi\)
0.999801 0.0199619i \(-0.00635449\pi\)
\(434\) 499.487i 1.15089i
\(435\) 0 0
\(436\) 142.001i 0.325689i
\(437\) −213.612 812.723i −0.488815 1.85978i
\(438\) 0 0
\(439\) −458.708 −1.04489 −0.522447 0.852672i \(-0.674981\pi\)
−0.522447 + 0.852672i \(0.674981\pi\)
\(440\) 100.083 0.227462
\(441\) 0 0
\(442\) 205.916i 0.465874i
\(443\) 196.292 0.443096 0.221548 0.975149i \(-0.428889\pi\)
0.221548 + 0.975149i \(0.428889\pi\)
\(444\) 0 0
\(445\) 234.181 0.526250
\(446\) −624.488 −1.40020
\(447\) 0 0
\(448\) 65.9359i 0.147178i
\(449\) 344.536 0.767341 0.383671 0.923470i \(-0.374660\pi\)
0.383671 + 0.923470i \(0.374660\pi\)
\(450\) 0 0
\(451\) 586.966i 1.30148i
\(452\) 201.526i 0.445854i
\(453\) 0 0
\(454\) 77.6717i 0.171083i
\(455\) 263.948 0.580105
\(456\) 0 0
\(457\) 122.214i 0.267426i −0.991020 0.133713i \(-0.957310\pi\)
0.991020 0.133713i \(-0.0426900\pi\)
\(458\) 455.802i 0.995200i
\(459\) 0 0
\(460\) 26.1469 + 99.4803i 0.0568411 + 0.216262i
\(461\) −218.326 −0.473593 −0.236796 0.971559i \(-0.576097\pi\)
−0.236796 + 0.971559i \(0.576097\pi\)
\(462\) 0 0
\(463\) −455.380 −0.983542 −0.491771 0.870725i \(-0.663650\pi\)
−0.491771 + 0.870725i \(0.663650\pi\)
\(464\) −25.8613 −0.0557356
\(465\) 0 0
\(466\) −225.154 −0.483163
\(467\) 350.332i 0.750176i −0.926989 0.375088i \(-0.877612\pi\)
0.926989 0.375088i \(-0.122388\pi\)
\(468\) 0 0
\(469\) −36.7580 −0.0783752
\(470\) 102.672i 0.218450i
\(471\) 0 0
\(472\) −18.8331 −0.0399006
\(473\) 94.9673 0.200777
\(474\) 0 0
\(475\) 182.680i 0.384589i
\(476\) −167.585 −0.352070
\(477\) 0 0
\(478\) −45.5363 −0.0952643
\(479\) 743.178i 1.55152i −0.631028 0.775760i \(-0.717367\pi\)
0.631028 0.775760i \(-0.282633\pi\)
\(480\) 0 0
\(481\) 911.417i 1.89484i
\(482\) 56.3223i 0.116851i
\(483\) 0 0
\(484\) −258.833 −0.534779
\(485\) 220.625 0.454897
\(486\) 0 0
\(487\) −366.770 −0.753122 −0.376561 0.926392i \(-0.622893\pi\)
−0.376561 + 0.926392i \(0.622893\pi\)
\(488\) 157.570i 0.322889i
\(489\) 0 0
\(490\) 59.8629i 0.122169i
\(491\) −127.024 −0.258704 −0.129352 0.991599i \(-0.541290\pi\)
−0.129352 + 0.991599i \(0.541290\pi\)
\(492\) 0 0
\(493\) 65.7302i 0.133327i
\(494\) 740.008i 1.49799i
\(495\) 0 0
\(496\) −171.411 −0.345586
\(497\) 975.948i 1.96368i
\(498\) 0 0
\(499\) 523.599 1.04930 0.524648 0.851319i \(-0.324197\pi\)
0.524648 + 0.851319i \(0.324197\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 0 0
\(502\) 328.845i 0.655071i
\(503\) 646.170i 1.28463i −0.766440 0.642316i \(-0.777974\pi\)
0.766440 0.642316i \(-0.222026\pi\)
\(504\) 0 0
\(505\) 169.228i 0.335105i
\(506\) −130.844 497.816i −0.258584 0.983826i
\(507\) 0 0
\(508\) −87.9203 −0.173072
\(509\) 605.436 1.18946 0.594731 0.803925i \(-0.297259\pi\)
0.594731 + 0.803925i \(0.297259\pi\)
\(510\) 0 0
\(511\) 678.047i 1.32690i
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) −459.845 −0.894641
\(515\) −194.202 −0.377092
\(516\) 0 0
\(517\) 513.786i 0.993783i
\(518\) 741.759 1.43197
\(519\) 0 0
\(520\) 90.5798i 0.174192i
\(521\) 655.952i 1.25902i 0.776990 + 0.629512i \(0.216745\pi\)
−0.776990 + 0.629512i \(0.783255\pi\)
\(522\) 0 0
\(523\) 317.109i 0.606328i −0.952938 0.303164i \(-0.901957\pi\)
0.952938 0.303164i \(-0.0980430\pi\)
\(524\) 88.8267 0.169517
\(525\) 0 0
\(526\) 6.99880i 0.0133057i
\(527\) 435.665i 0.826688i
\(528\) 0 0
\(529\) 460.634 260.111i 0.870763 0.491702i
\(530\) −115.942 −0.218758
\(531\) 0 0
\(532\) 602.257 1.13206
\(533\) −531.230 −0.996679
\(534\) 0 0
\(535\) −12.4225 −0.0232197
\(536\) 12.6143i 0.0235342i
\(537\) 0 0
\(538\) −332.407 −0.617856
\(539\) 299.564i 0.555777i
\(540\) 0 0
\(541\) −222.888 −0.411992 −0.205996 0.978553i \(-0.566043\pi\)
−0.205996 + 0.978553i \(0.566043\pi\)
\(542\) −75.6921 −0.139653
\(543\) 0 0
\(544\) 57.5108i 0.105718i
\(545\) 158.762 0.291306
\(546\) 0 0
\(547\) −552.627 −1.01029 −0.505144 0.863035i \(-0.668561\pi\)
−0.505144 + 0.863035i \(0.668561\pi\)
\(548\) 89.8590i 0.163976i
\(549\) 0 0
\(550\) 111.897i 0.203448i
\(551\) 236.217i 0.428706i
\(552\) 0 0
\(553\) 1096.88 1.98351
\(554\) −203.048 −0.366512
\(555\) 0 0
\(556\) −279.592 −0.502864
\(557\) 110.000i 0.197486i −0.995113 0.0987430i \(-0.968518\pi\)
0.995113 0.0987430i \(-0.0314822\pi\)
\(558\) 0 0
\(559\) 85.9496i 0.153756i
\(560\) −73.7186 −0.131640
\(561\) 0 0
\(562\) 264.925i 0.471397i
\(563\) 360.367i 0.640084i −0.947403 0.320042i \(-0.896303\pi\)
0.947403 0.320042i \(-0.103697\pi\)
\(564\) 0 0
\(565\) −225.313 −0.398784
\(566\) 687.529i 1.21472i
\(567\) 0 0
\(568\) −334.919 −0.589646
\(569\) 532.981i 0.936698i −0.883544 0.468349i \(-0.844849\pi\)
0.883544 0.468349i \(-0.155151\pi\)
\(570\) 0 0
\(571\) 582.697i 1.02048i −0.860031 0.510242i \(-0.829556\pi\)
0.860031 0.510242i \(-0.170444\pi\)
\(572\) 453.276i 0.792441i
\(573\) 0 0
\(574\) 432.343i 0.753210i
\(575\) 111.222 29.2332i 0.193430 0.0508403i
\(576\) 0 0
\(577\) −869.422 −1.50680 −0.753399 0.657564i \(-0.771587\pi\)
−0.753399 + 0.657564i \(0.771587\pi\)
\(578\) 262.536 0.454214
\(579\) 0 0
\(580\) 28.9138i 0.0498514i
\(581\) −556.840 −0.958416
\(582\) 0 0
\(583\) 580.192 0.995183
\(584\) 232.688 0.398438
\(585\) 0 0
\(586\) 340.449i 0.580970i
\(587\) 403.118 0.686742 0.343371 0.939200i \(-0.388431\pi\)
0.343371 + 0.939200i \(0.388431\pi\)
\(588\) 0 0
\(589\) 1565.66i 2.65817i
\(590\) 21.0560i 0.0356882i
\(591\) 0 0
\(592\) 254.552i 0.429986i
\(593\) 668.332 1.12703 0.563517 0.826104i \(-0.309448\pi\)
0.563517 + 0.826104i \(0.309448\pi\)
\(594\) 0 0
\(595\) 187.366i 0.314901i
\(596\) 50.8454i 0.0853111i
\(597\) 0 0
\(598\) 450.545 118.419i 0.753420 0.198025i
\(599\) −866.946 −1.44732 −0.723661 0.690155i \(-0.757542\pi\)
−0.723661 + 0.690155i \(0.757542\pi\)
\(600\) 0 0
\(601\) 544.425 0.905866 0.452933 0.891545i \(-0.350378\pi\)
0.452933 + 0.891545i \(0.350378\pi\)
\(602\) −69.9503 −0.116196
\(603\) 0 0
\(604\) −105.060 −0.173940
\(605\) 289.384i 0.478321i
\(606\) 0 0
\(607\) 23.2820 0.0383559 0.0191779 0.999816i \(-0.493895\pi\)
0.0191779 + 0.999816i \(0.493895\pi\)
\(608\) 206.678i 0.339932i
\(609\) 0 0
\(610\) 176.168 0.288801
\(611\) −464.999 −0.761045
\(612\) 0 0
\(613\) 559.419i 0.912592i −0.889828 0.456296i \(-0.849176\pi\)
0.889828 0.456296i \(-0.150824\pi\)
\(614\) 478.133 0.778718
\(615\) 0 0
\(616\) 368.900 0.598863
\(617\) 1080.07i 1.75052i 0.483649 + 0.875262i \(0.339311\pi\)
−0.483649 + 0.875262i \(0.660689\pi\)
\(618\) 0 0
\(619\) 729.225i 1.17807i 0.808107 + 0.589035i \(0.200492\pi\)
−0.808107 + 0.589035i \(0.799508\pi\)
\(620\) 191.643i 0.309101i
\(621\) 0 0
\(622\) 186.513 0.299860
\(623\) 863.175 1.38551
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 226.634i 0.362035i
\(627\) 0 0
\(628\) 149.175i 0.237539i
\(629\) 646.980 1.02858
\(630\) 0 0
\(631\) 386.633i 0.612731i 0.951914 + 0.306365i \(0.0991129\pi\)
−0.951914 + 0.306365i \(0.900887\pi\)
\(632\) 376.420i 0.595601i
\(633\) 0 0
\(634\) 743.434 1.17261
\(635\) 98.2979i 0.154800i
\(636\) 0 0
\(637\) 271.118 0.425618
\(638\) 144.690i 0.226786i
\(639\) 0 0
\(640\) 25.2982i 0.0395285i
\(641\) 676.123i 1.05479i 0.849619 + 0.527397i \(0.176832\pi\)
−0.849619 + 0.527397i \(0.823168\pi\)
\(642\) 0 0
\(643\) 1063.72i 1.65430i −0.561980 0.827151i \(-0.689960\pi\)
0.561980 0.827151i \(-0.310040\pi\)
\(644\) 96.3757 + 366.677i 0.149652 + 0.569375i
\(645\) 0 0
\(646\) 525.303 0.813162
\(647\) 157.776 0.243857 0.121929 0.992539i \(-0.461092\pi\)
0.121929 + 0.992539i \(0.461092\pi\)
\(648\) 0 0
\(649\) 105.368i 0.162354i
\(650\) 101.271 0.155802
\(651\) 0 0
\(652\) −36.6476 −0.0562080
\(653\) −41.1895 −0.0630773 −0.0315386 0.999503i \(-0.510041\pi\)
−0.0315386 + 0.999503i \(0.510041\pi\)
\(654\) 0 0
\(655\) 99.3113i 0.151620i
\(656\) 148.368 0.226171
\(657\) 0 0
\(658\) 378.440i 0.575137i
\(659\) 19.2526i 0.0292149i −0.999893 0.0146075i \(-0.995350\pi\)
0.999893 0.0146075i \(-0.00464986\pi\)
\(660\) 0 0
\(661\) 75.8322i 0.114724i −0.998353 0.0573618i \(-0.981731\pi\)
0.998353 0.0573618i \(-0.0182688\pi\)
\(662\) 0.169900 0.000256647
\(663\) 0 0
\(664\) 191.092i 0.287790i
\(665\) 673.344i 1.01255i
\(666\) 0 0
\(667\) 143.818 37.8004i 0.215619 0.0566722i
\(668\) −137.954 −0.206517
\(669\) 0 0
\(670\) −14.1033 −0.0210497
\(671\) −881.576 −1.31382
\(672\) 0 0
\(673\) 12.6902 0.0188561 0.00942807 0.999956i \(-0.496999\pi\)
0.00942807 + 0.999956i \(0.496999\pi\)
\(674\) 923.405i 1.37004i
\(675\) 0 0
\(676\) 72.2347 0.106856
\(677\) 233.185i 0.344439i 0.985059 + 0.172220i \(0.0550939\pi\)
−0.985059 + 0.172220i \(0.944906\pi\)
\(678\) 0 0
\(679\) 813.209 1.19766
\(680\) −64.2991 −0.0945574
\(681\) 0 0
\(682\) 959.012i 1.40618i
\(683\) −367.020 −0.537364 −0.268682 0.963229i \(-0.586588\pi\)
−0.268682 + 0.963229i \(0.586588\pi\)
\(684\) 0 0
\(685\) 100.465 0.146665
\(686\) 350.490i 0.510918i
\(687\) 0 0
\(688\) 24.0051i 0.0348911i
\(689\) 525.099i 0.762118i
\(690\) 0 0
\(691\) −1186.21 −1.71666 −0.858332 0.513095i \(-0.828499\pi\)
−0.858332 + 0.513095i \(0.828499\pi\)
\(692\) −237.413 −0.343083
\(693\) 0 0
\(694\) 662.282 0.954296
\(695\) 312.594i 0.449775i
\(696\) 0 0
\(697\) 377.099i 0.541032i
\(698\) 257.794 0.369332
\(699\) 0 0
\(700\) 82.4199i 0.117743i
\(701\) 848.508i 1.21043i −0.796064 0.605213i \(-0.793088\pi\)
0.796064 0.605213i \(-0.206912\pi\)
\(702\) 0 0
\(703\) −2325.07 −3.30736
\(704\) 126.596i 0.179825i
\(705\) 0 0
\(706\) 425.733 0.603022
\(707\) 623.763i 0.882267i
\(708\) 0 0
\(709\) 533.757i 0.752831i 0.926451 + 0.376416i \(0.122844\pi\)
−0.926451 + 0.376416i \(0.877156\pi\)
\(710\) 374.451i 0.527396i
\(711\) 0 0
\(712\) 296.218i 0.416037i
\(713\) 953.235 250.544i 1.33694 0.351394i
\(714\) 0 0
\(715\) −506.778 −0.708780
\(716\) 557.785 0.779029
\(717\) 0 0
\(718\) 34.8989i 0.0486057i
\(719\) −1210.11 −1.68304 −0.841520 0.540225i \(-0.818339\pi\)
−0.841520 + 0.540225i \(0.818339\pi\)
\(720\) 0 0
\(721\) −715.816 −0.992810
\(722\) −1377.27 −1.90757
\(723\) 0 0
\(724\) 132.225i 0.182631i
\(725\) 32.3266 0.0445885
\(726\) 0 0
\(727\) 2.43604i 0.00335081i −0.999999 0.00167540i \(-0.999467\pi\)
0.999999 0.00167540i \(-0.000533298\pi\)
\(728\) 333.870i 0.458613i
\(729\) 0 0
\(730\) 260.153i 0.356373i
\(731\) −61.0123 −0.0834641
\(732\) 0 0
\(733\) 495.146i 0.675506i 0.941235 + 0.337753i \(0.109667\pi\)
−0.941235 + 0.337753i \(0.890333\pi\)
\(734\) 604.587i 0.823688i
\(735\) 0 0
\(736\) −125.834 + 33.0735i −0.170970 + 0.0449369i
\(737\) 70.5751 0.0957599
\(738\) 0 0
\(739\) 1235.77 1.67222 0.836112 0.548559i \(-0.184823\pi\)
0.836112 + 0.548559i \(0.184823\pi\)
\(740\) 284.597 0.384591
\(741\) 0 0
\(742\) −427.353 −0.575948
\(743\) 1030.67i 1.38717i −0.720373 0.693587i \(-0.756029\pi\)
0.720373 0.693587i \(-0.243971\pi\)
\(744\) 0 0
\(745\) −56.8469 −0.0763045
\(746\) 478.926i 0.641992i
\(747\) 0 0
\(748\) 321.763 0.430164
\(749\) −45.7885 −0.0611329
\(750\) 0 0
\(751\) 1112.44i 1.48128i 0.671903 + 0.740639i \(0.265477\pi\)
−0.671903 + 0.740639i \(0.734523\pi\)
\(752\) 129.870 0.172700
\(753\) 0 0
\(754\) 130.950 0.173674
\(755\) 117.460i 0.155577i
\(756\) 0 0
\(757\) 483.738i 0.639019i −0.947583 0.319510i \(-0.896482\pi\)
0.947583 0.319510i \(-0.103518\pi\)
\(758\) 406.102i 0.535754i
\(759\) 0 0
\(760\) 231.073 0.304044
\(761\) −833.341 −1.09506 −0.547530 0.836786i \(-0.684432\pi\)
−0.547530 + 0.836786i \(0.684432\pi\)
\(762\) 0 0
\(763\) 585.183 0.766951
\(764\) 11.0681i 0.0144871i
\(765\) 0 0
\(766\) 965.428i 1.26035i
\(767\) 95.3626 0.124332
\(768\) 0 0
\(769\) 356.518i 0.463613i 0.972762 + 0.231806i \(0.0744636\pi\)
−0.972762 + 0.231806i \(0.925536\pi\)
\(770\) 412.442i 0.535640i
\(771\) 0 0
\(772\) −144.092 −0.186648
\(773\) 286.633i 0.370805i 0.982663 + 0.185403i \(0.0593589\pi\)
−0.982663 + 0.185403i \(0.940641\pi\)
\(774\) 0 0
\(775\) 214.263 0.276469
\(776\) 279.071i 0.359628i
\(777\) 0 0
\(778\) 769.171i 0.988651i
\(779\) 1355.19i 1.73966i
\(780\) 0 0
\(781\) 1873.81i 2.39925i
\(782\) 84.0611 + 319.825i 0.107495 + 0.408983i
\(783\) 0 0
\(784\) −75.7213 −0.0965833
\(785\) 166.782 0.212462
\(786\) 0 0
\(787\) 930.798i 1.18272i −0.806409 0.591359i \(-0.798592\pi\)
0.806409 0.591359i \(-0.201408\pi\)
\(788\) −382.287 −0.485136
\(789\) 0 0
\(790\) 420.850 0.532721
\(791\) −830.487 −1.04992
\(792\) 0 0
\(793\) 797.865i 1.00613i
\(794\) 140.942 0.177509
\(795\) 0 0
\(796\) 345.086i 0.433525i
\(797\) 1493.90i 1.87441i −0.348785 0.937203i \(-0.613406\pi\)
0.348785 0.937203i \(-0.386594\pi\)
\(798\) 0 0
\(799\) 330.084i 0.413122i
\(800\) −28.2843 −0.0353553
\(801\) 0 0
\(802\) 572.286i 0.713574i
\(803\) 1301.85i 1.62123i
\(804\) 0 0
\(805\) 409.958 107.751i 0.509264 0.133853i
\(806\) 867.948 1.07686
\(807\) 0 0
\(808\) 214.059 0.264924
\(809\) 850.318 1.05107 0.525537 0.850771i \(-0.323865\pi\)
0.525537 + 0.850771i \(0.323865\pi\)
\(810\) 0 0
\(811\) 1050.33 1.29510 0.647552 0.762022i \(-0.275793\pi\)
0.647552 + 0.762022i \(0.275793\pi\)
\(812\) 106.574i 0.131249i
\(813\) 0 0
\(814\) −1424.17 −1.74960
\(815\) 40.9733i 0.0502739i
\(816\) 0 0
\(817\) 219.262 0.268374
\(818\) 992.039 1.21276
\(819\) 0 0
\(820\) 165.881i 0.202294i
\(821\) 461.343 0.561929 0.280964 0.959718i \(-0.409346\pi\)
0.280964 + 0.959718i \(0.409346\pi\)
\(822\) 0 0
\(823\) 101.176 0.122936 0.0614680 0.998109i \(-0.480422\pi\)
0.0614680 + 0.998109i \(0.480422\pi\)
\(824\) 245.649i 0.298117i
\(825\) 0 0
\(826\) 77.6111i 0.0939601i
\(827\) 718.169i 0.868403i 0.900816 + 0.434202i \(0.142969\pi\)
−0.900816 + 0.434202i \(0.857031\pi\)
\(828\) 0 0
\(829\) 634.298 0.765137 0.382568 0.923927i \(-0.375040\pi\)
0.382568 + 0.923927i \(0.375040\pi\)
\(830\) −213.648 −0.257407
\(831\) 0 0
\(832\) −114.575 −0.137711
\(833\) 192.456i 0.231040i
\(834\) 0 0
\(835\) 154.237i 0.184715i
\(836\) −1156.33 −1.38317
\(837\) 0 0
\(838\) 968.862i 1.15616i
\(839\) 133.138i 0.158687i −0.996847 0.0793434i \(-0.974718\pi\)
0.996847 0.0793434i \(-0.0252824\pi\)
\(840\) 0 0
\(841\) −799.200 −0.950297
\(842\) 290.375i 0.344864i
\(843\) 0 0
\(844\) −668.078 −0.791561
\(845\) 80.7609i 0.0955750i
\(846\) 0 0
\(847\) 1066.65i 1.25933i
\(848\) 146.656i 0.172944i
\(849\) 0 0
\(850\) 71.8885i 0.0845747i
\(851\) −372.068 1415.59i −0.437212 1.66345i
\(852\) 0 0
\(853\) 728.808 0.854406 0.427203 0.904156i \(-0.359499\pi\)
0.427203 + 0.904156i \(0.359499\pi\)
\(854\) 649.344 0.760356
\(855\) 0 0
\(856\) 15.7134i 0.0183568i
\(857\) −1137.95 −1.32783 −0.663914 0.747809i \(-0.731106\pi\)
−0.663914 + 0.747809i \(0.731106\pi\)
\(858\) 0 0
\(859\) −782.016 −0.910379 −0.455190 0.890395i \(-0.650429\pi\)
−0.455190 + 0.890395i \(0.650429\pi\)
\(860\) −26.8385 −0.0312075
\(861\) 0 0
\(862\) 503.355i 0.583939i
\(863\) 324.868 0.376440 0.188220 0.982127i \(-0.439728\pi\)
0.188220 + 0.982127i \(0.439728\pi\)
\(864\) 0 0
\(865\) 265.436i 0.306863i
\(866\) 24.4475i 0.0282304i
\(867\) 0 0
\(868\) 706.382i 0.813804i
\(869\) −2106.00 −2.42348
\(870\) 0 0
\(871\) 63.8735i 0.0733336i
\(872\) 200.819i 0.230297i
\(873\) 0 0
\(874\) −302.093 1149.36i −0.345644 1.31506i
\(875\) 92.1482 0.105312
\(876\) 0 0
\(877\) −411.966 −0.469745 −0.234872 0.972026i \(-0.575467\pi\)
−0.234872 + 0.972026i \(0.575467\pi\)
\(878\) −648.712 −0.738852
\(879\) 0 0
\(880\) 141.539 0.160840
\(881\) 1082.39i 1.22859i −0.789076 0.614296i \(-0.789440\pi\)
0.789076 0.614296i \(-0.210560\pi\)
\(882\) 0 0
\(883\) 1343.48 1.52149 0.760746 0.649049i \(-0.224833\pi\)
0.760746 + 0.649049i \(0.224833\pi\)
\(884\) 291.210i 0.329423i
\(885\) 0 0
\(886\) 277.598 0.313316
\(887\) −377.176 −0.425227 −0.212613 0.977136i \(-0.568197\pi\)
−0.212613 + 0.977136i \(0.568197\pi\)
\(888\) 0 0
\(889\) 362.319i 0.407558i
\(890\) 331.182 0.372115
\(891\) 0 0
\(892\) −883.160 −0.990089
\(893\) 1186.23i 1.32837i
\(894\) 0 0
\(895\) 623.622i 0.696785i
\(896\) 93.2474i 0.104071i
\(897\) 0 0
\(898\) 487.248 0.542592
\(899\) 277.056 0.308183
\(900\) 0 0
\(901\) −372.748 −0.413704
\(902\) 830.095i 0.920283i
\(903\) 0 0
\(904\) 285.001i 0.315266i
\(905\) −147.832 −0.163350
\(906\) 0 0
\(907\) 295.704i 0.326024i 0.986624 + 0.163012i \(0.0521209\pi\)
−0.986624 + 0.163012i \(0.947879\pi\)
\(908\) 109.844i 0.120974i
\(909\) 0 0
\(910\) 373.279 0.410196
\(911\) 419.625i 0.460620i 0.973117 + 0.230310i \(0.0739740\pi\)
−0.973117 + 0.230310i \(0.926026\pi\)
\(912\) 0 0
\(913\) 1069.13 1.17101
\(914\) 172.836i 0.189099i
\(915\) 0 0
\(916\) 644.601i 0.703713i
\(917\) 366.054i 0.399187i
\(918\) 0 0
\(919\) 1654.94i 1.80080i 0.435062 + 0.900401i \(0.356726\pi\)
−0.435062 + 0.900401i \(0.643274\pi\)
\(920\) 36.9773 + 140.686i 0.0401928 + 0.152920i
\(921\) 0 0
\(922\) −308.760 −0.334881
\(923\) 1695.88 1.83736
\(924\) 0 0
\(925\) 318.190i 0.343989i
\(926\) −644.004 −0.695469
\(927\) 0 0
\(928\) −36.5734 −0.0394110
\(929\) 1291.79 1.39052 0.695260 0.718758i \(-0.255289\pi\)
0.695260 + 0.718758i \(0.255289\pi\)
\(930\) 0 0
\(931\) 691.637i 0.742897i
\(932\) −318.416 −0.341648
\(933\) 0 0
\(934\) 495.445i 0.530455i
\(935\) 359.742i 0.384751i
\(936\) 0 0
\(937\) 212.428i 0.226711i 0.993554 + 0.113355i \(0.0361599\pi\)
−0.993554 + 0.113355i \(0.963840\pi\)
\(938\) −51.9836 −0.0554196
\(939\) 0 0
\(940\) 145.200i 0.154468i
\(941\) 980.930i 1.04243i 0.853424 + 0.521217i \(0.174522\pi\)
−0.853424 + 0.521217i \(0.825478\pi\)
\(942\) 0 0
\(943\) −825.094 + 216.864i −0.874967 + 0.229972i
\(944\) −26.6340 −0.0282140
\(945\) 0 0
\(946\) 134.304 0.141970
\(947\) 953.755 1.00713 0.503567 0.863956i \(-0.332021\pi\)
0.503567 + 0.863956i \(0.332021\pi\)
\(948\) 0 0
\(949\) −1178.23 −1.24155
\(950\) 258.348i 0.271945i
\(951\) 0 0
\(952\) −237.002 −0.248951
\(953\) 248.522i 0.260779i −0.991463 0.130389i \(-0.958377\pi\)
0.991463 0.130389i \(-0.0416227\pi\)
\(954\) 0 0
\(955\) 12.3745 0.0129576
\(956\) −64.3981 −0.0673620
\(957\) 0 0
\(958\) 1051.01i 1.09709i
\(959\) 370.308 0.386140
\(960\) 0 0
\(961\) 875.349 0.910873
\(962\) 1288.94i 1.33985i
\(963\) 0 0
\(964\) 79.6517i 0.0826263i
\(965\) 161.100i 0.166943i
\(966\) 0 0
\(967\) −1112.21 −1.15016 −0.575081 0.818096i \(-0.695030\pi\)
−0.575081 + 0.818096i \(0.695030\pi\)
\(968\) −366.045 −0.378146
\(969\) 0 0
\(970\) 312.011 0.321661
\(971\) 548.535i 0.564918i −0.959279 0.282459i \(-0.908850\pi\)
0.959279 0.282459i \(-0.0911500\pi\)
\(972\) 0 0
\(973\) 1152.20i 1.18417i
\(974\) −518.692 −0.532538
\(975\) 0 0
\(976\) 222.837i 0.228317i
\(977\) 262.933i 0.269122i −0.990905 0.134561i \(-0.957038\pi\)
0.990905 0.134561i \(-0.0429625\pi\)
\(978\) 0 0
\(979\) −1657.29 −1.69284
\(980\) 84.6590i 0.0863867i
\(981\) 0 0
\(982\) −179.639 −0.182932
\(983\) 1516.23i 1.54245i 0.636562 + 0.771226i \(0.280356\pi\)
−0.636562 + 0.771226i \(0.719644\pi\)
\(984\) 0 0
\(985\) 427.410i 0.433918i
\(986\) 92.9566i 0.0942765i
\(987\) 0 0
\(988\) 1046.53i 1.05924i
\(989\) 35.0872 + 133.495i 0.0354774 + 0.134980i
\(990\) 0 0
\(991\) 705.633 0.712042 0.356021 0.934478i \(-0.384133\pi\)
0.356021 + 0.934478i \(0.384133\pi\)
\(992\) −242.411 −0.244366
\(993\) 0 0
\(994\) 1380.20i 1.38853i
\(995\) 385.818 0.387757
\(996\) 0 0
\(997\) 480.892 0.482339 0.241169 0.970483i \(-0.422469\pi\)
0.241169 + 0.970483i \(0.422469\pi\)
\(998\) 740.481 0.741965
\(999\) 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 2070.3.c.a.91.10 16
3.2 odd 2 230.3.d.a.91.2 yes 16
12.11 even 2 1840.3.k.d.321.16 16
15.2 even 4 1150.3.c.c.1149.3 32
15.8 even 4 1150.3.c.c.1149.30 32
15.14 odd 2 1150.3.d.b.551.16 16
23.22 odd 2 inner 2070.3.c.a.91.15 16
69.68 even 2 230.3.d.a.91.1 16
276.275 odd 2 1840.3.k.d.321.15 16
345.68 odd 4 1150.3.c.c.1149.4 32
345.137 odd 4 1150.3.c.c.1149.29 32
345.344 even 2 1150.3.d.b.551.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.d.a.91.1 16 69.68 even 2
230.3.d.a.91.2 yes 16 3.2 odd 2
1150.3.c.c.1149.3 32 15.2 even 4
1150.3.c.c.1149.4 32 345.68 odd 4
1150.3.c.c.1149.29 32 345.137 odd 4
1150.3.c.c.1149.30 32 15.8 even 4
1150.3.d.b.551.15 16 345.344 even 2
1150.3.d.b.551.16 16 15.14 odd 2
1840.3.k.d.321.15 16 276.275 odd 2
1840.3.k.d.321.16 16 12.11 even 2
2070.3.c.a.91.10 16 1.1 even 1 trivial
2070.3.c.a.91.15 16 23.22 odd 2 inner