Properties

Label 171.3.c.f.37.1
Level $171$
Weight $3$
Character 171.37
Analytic conductor $4.659$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,3,Mod(37,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("171.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.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: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.1
Root \(-1.63746 - 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 171.37
Dual form 171.3.c.f.37.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.04547i q^{2} -5.27492 q^{4} +6.27492 q^{5} +12.2749 q^{7} +3.88273i q^{8} +O(q^{10})\) \(q-3.04547i q^{2} -5.27492 q^{4} +6.27492 q^{5} +12.2749 q^{7} +3.88273i q^{8} -19.1101i q^{10} -0.274917 q^{11} +13.0192i q^{13} -37.3830i q^{14} -9.27492 q^{16} -17.3746 q^{17} +(7.54983 - 17.4356i) q^{19} -33.0997 q^{20} +0.837253i q^{22} -20.5498 q^{23} +14.3746 q^{25} +39.6495 q^{26} -64.7492 q^{28} +26.0383i q^{29} +23.5265i q^{31} +43.7774i q^{32} +52.9139i q^{34} +77.0241 q^{35} -66.7703i q^{37} +(-53.0997 - 22.9928i) q^{38} +24.3638i q^{40} +3.57919i q^{41} -48.1238 q^{43} +1.45017 q^{44} +62.5840i q^{46} +12.4743 q^{47} +101.674 q^{49} -43.7774i q^{50} -68.6750i q^{52} -25.8081i q^{53} -1.72508 q^{55} +47.6602i q^{56} +79.2990 q^{58} +0.230175i q^{59} -28.8248 q^{61} +71.6495 q^{62} +96.2234 q^{64} +81.6941i q^{65} +102.939i q^{67} +91.6495 q^{68} -234.575i q^{70} +107.732i q^{71} +11.1752 q^{73} -203.347 q^{74} +(-39.8248 + 91.9713i) q^{76} -3.37459 q^{77} +26.6454i q^{79} -58.1993 q^{80} +10.9003 q^{82} +89.6977 q^{83} -109.024 q^{85} +146.560i q^{86} -1.06743i q^{88} -139.255i q^{89} +159.809i q^{91} +108.399 q^{92} -37.9900i q^{94} +(47.3746 - 109.407i) q^{95} +41.3390i q^{97} -309.644i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} + 10 q^{5} + 34 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} + 10 q^{5} + 34 q^{7} + 14 q^{11} - 22 q^{16} + 6 q^{17} - 72 q^{20} - 52 q^{23} - 18 q^{25} + 68 q^{26} - 108 q^{28} + 142 q^{35} - 152 q^{38} + 34 q^{43} + 36 q^{44} - 86 q^{47} + 150 q^{49} - 22 q^{55} + 136 q^{58} - 70 q^{61} + 196 q^{62} + 98 q^{64} + 276 q^{68} + 90 q^{73} - 300 q^{74} - 114 q^{76} + 62 q^{77} - 112 q^{80} + 104 q^{82} - 64 q^{83} - 270 q^{85} + 192 q^{92} + 114 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.04547i 1.52274i −0.648319 0.761369i \(-0.724528\pi\)
0.648319 0.761369i \(-0.275472\pi\)
\(3\) 0 0
\(4\) −5.27492 −1.31873
\(5\) 6.27492 1.25498 0.627492 0.778623i \(-0.284082\pi\)
0.627492 + 0.778623i \(0.284082\pi\)
\(6\) 0 0
\(7\) 12.2749 1.75356 0.876780 0.480892i \(-0.159687\pi\)
0.876780 + 0.480892i \(0.159687\pi\)
\(8\) 3.88273i 0.485341i
\(9\) 0 0
\(10\) 19.1101i 1.91101i
\(11\) −0.274917 −0.0249925 −0.0124962 0.999922i \(-0.503978\pi\)
−0.0124962 + 0.999922i \(0.503978\pi\)
\(12\) 0 0
\(13\) 13.0192i 1.00147i 0.865600 + 0.500737i \(0.166937\pi\)
−0.865600 + 0.500737i \(0.833063\pi\)
\(14\) 37.3830i 2.67021i
\(15\) 0 0
\(16\) −9.27492 −0.579682
\(17\) −17.3746 −1.02203 −0.511017 0.859570i \(-0.670731\pi\)
−0.511017 + 0.859570i \(0.670731\pi\)
\(18\) 0 0
\(19\) 7.54983 17.4356i 0.397360 0.917663i
\(20\) −33.0997 −1.65498
\(21\) 0 0
\(22\) 0.837253i 0.0380570i
\(23\) −20.5498 −0.893471 −0.446736 0.894666i \(-0.647414\pi\)
−0.446736 + 0.894666i \(0.647414\pi\)
\(24\) 0 0
\(25\) 14.3746 0.574983
\(26\) 39.6495 1.52498
\(27\) 0 0
\(28\) −64.7492 −2.31247
\(29\) 26.0383i 0.897873i 0.893564 + 0.448936i \(0.148197\pi\)
−0.893564 + 0.448936i \(0.851803\pi\)
\(30\) 0 0
\(31\) 23.5265i 0.758921i 0.925208 + 0.379460i \(0.123890\pi\)
−0.925208 + 0.379460i \(0.876110\pi\)
\(32\) 43.7774i 1.36805i
\(33\) 0 0
\(34\) 52.9139i 1.55629i
\(35\) 77.0241 2.20069
\(36\) 0 0
\(37\) 66.7703i 1.80460i −0.431107 0.902301i \(-0.641877\pi\)
0.431107 0.902301i \(-0.358123\pi\)
\(38\) −53.0997 22.9928i −1.39736 0.605075i
\(39\) 0 0
\(40\) 24.3638i 0.609095i
\(41\) 3.57919i 0.0872973i 0.999047 + 0.0436487i \(0.0138982\pi\)
−0.999047 + 0.0436487i \(0.986102\pi\)
\(42\) 0 0
\(43\) −48.1238 −1.11916 −0.559579 0.828777i \(-0.689037\pi\)
−0.559579 + 0.828777i \(0.689037\pi\)
\(44\) 1.45017 0.0329583
\(45\) 0 0
\(46\) 62.5840i 1.36052i
\(47\) 12.4743 0.265410 0.132705 0.991156i \(-0.457634\pi\)
0.132705 + 0.991156i \(0.457634\pi\)
\(48\) 0 0
\(49\) 101.674 2.07497
\(50\) 43.7774i 0.875549i
\(51\) 0 0
\(52\) 68.6750i 1.32067i
\(53\) 25.8081i 0.486946i −0.969908 0.243473i \(-0.921713\pi\)
0.969908 0.243473i \(-0.0782867\pi\)
\(54\) 0 0
\(55\) −1.72508 −0.0313651
\(56\) 47.6602i 0.851074i
\(57\) 0 0
\(58\) 79.2990 1.36722
\(59\) 0.230175i 0.00390128i 0.999998 + 0.00195064i \(0.000620908\pi\)
−0.999998 + 0.00195064i \(0.999379\pi\)
\(60\) 0 0
\(61\) −28.8248 −0.472537 −0.236268 0.971688i \(-0.575924\pi\)
−0.236268 + 0.971688i \(0.575924\pi\)
\(62\) 71.6495 1.15564
\(63\) 0 0
\(64\) 96.2234 1.50349
\(65\) 81.6941i 1.25683i
\(66\) 0 0
\(67\) 102.939i 1.53640i 0.640208 + 0.768202i \(0.278848\pi\)
−0.640208 + 0.768202i \(0.721152\pi\)
\(68\) 91.6495 1.34779
\(69\) 0 0
\(70\) 234.575i 3.35107i
\(71\) 107.732i 1.51736i 0.651465 + 0.758679i \(0.274155\pi\)
−0.651465 + 0.758679i \(0.725845\pi\)
\(72\) 0 0
\(73\) 11.1752 0.153086 0.0765428 0.997066i \(-0.475612\pi\)
0.0765428 + 0.997066i \(0.475612\pi\)
\(74\) −203.347 −2.74793
\(75\) 0 0
\(76\) −39.8248 + 91.9713i −0.524010 + 1.21015i
\(77\) −3.37459 −0.0438258
\(78\) 0 0
\(79\) 26.6454i 0.337283i 0.985677 + 0.168642i \(0.0539381\pi\)
−0.985677 + 0.168642i \(0.946062\pi\)
\(80\) −58.1993 −0.727492
\(81\) 0 0
\(82\) 10.9003 0.132931
\(83\) 89.6977 1.08069 0.540347 0.841442i \(-0.318293\pi\)
0.540347 + 0.841442i \(0.318293\pi\)
\(84\) 0 0
\(85\) −109.024 −1.28264
\(86\) 146.560i 1.70418i
\(87\) 0 0
\(88\) 1.06743i 0.0121299i
\(89\) 139.255i 1.56466i −0.622865 0.782329i \(-0.714032\pi\)
0.622865 0.782329i \(-0.285968\pi\)
\(90\) 0 0
\(91\) 159.809i 1.75614i
\(92\) 108.399 1.17825
\(93\) 0 0
\(94\) 37.9900i 0.404149i
\(95\) 47.3746 109.407i 0.498680 1.15165i
\(96\) 0 0
\(97\) 41.3390i 0.426176i 0.977033 + 0.213088i \(0.0683521\pi\)
−0.977033 + 0.213088i \(0.931648\pi\)
\(98\) 309.644i 3.15964i
\(99\) 0 0
\(100\) −75.8248 −0.758248
\(101\) −150.749 −1.49257 −0.746283 0.665629i \(-0.768163\pi\)
−0.746283 + 0.665629i \(0.768163\pi\)
\(102\) 0 0
\(103\) 22.8360i 0.221709i −0.993837 0.110854i \(-0.964641\pi\)
0.993837 0.110854i \(-0.0353587\pi\)
\(104\) −50.5498 −0.486056
\(105\) 0 0
\(106\) −78.5980 −0.741491
\(107\) 165.063i 1.54264i 0.636446 + 0.771321i \(0.280404\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(108\) 0 0
\(109\) 46.4460i 0.426110i −0.977040 0.213055i \(-0.931659\pi\)
0.977040 0.213055i \(-0.0683414\pi\)
\(110\) 5.25370i 0.0477609i
\(111\) 0 0
\(112\) −113.849 −1.01651
\(113\) 4.56317i 0.0403820i 0.999796 + 0.0201910i \(0.00642744\pi\)
−0.999796 + 0.0201910i \(0.993573\pi\)
\(114\) 0 0
\(115\) −128.949 −1.12129
\(116\) 137.350i 1.18405i
\(117\) 0 0
\(118\) 0.700993 0.00594062
\(119\) −213.272 −1.79220
\(120\) 0 0
\(121\) −120.924 −0.999375
\(122\) 87.7851i 0.719550i
\(123\) 0 0
\(124\) 124.101i 1.00081i
\(125\) −66.6736 −0.533389
\(126\) 0 0
\(127\) 146.790i 1.15583i −0.816098 0.577913i \(-0.803867\pi\)
0.816098 0.577913i \(-0.196133\pi\)
\(128\) 117.936i 0.921377i
\(129\) 0 0
\(130\) 248.797 1.91383
\(131\) 188.921 1.44215 0.721073 0.692859i \(-0.243649\pi\)
0.721073 + 0.692859i \(0.243649\pi\)
\(132\) 0 0
\(133\) 92.6736 214.020i 0.696794 1.60918i
\(134\) 313.498 2.33954
\(135\) 0 0
\(136\) 67.4608i 0.496035i
\(137\) 71.3264 0.520631 0.260315 0.965524i \(-0.416173\pi\)
0.260315 + 0.965524i \(0.416173\pi\)
\(138\) 0 0
\(139\) 43.5739 0.313481 0.156741 0.987640i \(-0.449901\pi\)
0.156741 + 0.987640i \(0.449901\pi\)
\(140\) −406.296 −2.90211
\(141\) 0 0
\(142\) 328.096 2.31054
\(143\) 3.57919i 0.0250293i
\(144\) 0 0
\(145\) 163.388i 1.12682i
\(146\) 34.0339i 0.233109i
\(147\) 0 0
\(148\) 352.208i 2.37978i
\(149\) 71.6769 0.481053 0.240527 0.970643i \(-0.422680\pi\)
0.240527 + 0.970643i \(0.422680\pi\)
\(150\) 0 0
\(151\) 68.9051i 0.456325i −0.973623 0.228163i \(-0.926728\pi\)
0.973623 0.228163i \(-0.0732718\pi\)
\(152\) 67.6977 + 29.3140i 0.445379 + 0.192855i
\(153\) 0 0
\(154\) 10.2772i 0.0667352i
\(155\) 147.627i 0.952433i
\(156\) 0 0
\(157\) 157.698 1.00444 0.502222 0.864739i \(-0.332516\pi\)
0.502222 + 0.864739i \(0.332516\pi\)
\(158\) 81.1478 0.513594
\(159\) 0 0
\(160\) 274.700i 1.71687i
\(161\) −252.248 −1.56675
\(162\) 0 0
\(163\) −42.3987 −0.260115 −0.130057 0.991506i \(-0.541516\pi\)
−0.130057 + 0.991506i \(0.541516\pi\)
\(164\) 18.8799i 0.115122i
\(165\) 0 0
\(166\) 273.172i 1.64561i
\(167\) 101.558i 0.608132i −0.952651 0.304066i \(-0.901656\pi\)
0.952651 0.304066i \(-0.0983443\pi\)
\(168\) 0 0
\(169\) −0.498344 −0.00294878
\(170\) 332.030i 1.95312i
\(171\) 0 0
\(172\) 253.849 1.47587
\(173\) 24.3638i 0.140831i −0.997518 0.0704156i \(-0.977567\pi\)
0.997518 0.0704156i \(-0.0224325\pi\)
\(174\) 0 0
\(175\) 176.447 1.00827
\(176\) 2.54983 0.0144877
\(177\) 0 0
\(178\) −424.096 −2.38256
\(179\) 74.7659i 0.417687i 0.977949 + 0.208843i \(0.0669699\pi\)
−0.977949 + 0.208843i \(0.933030\pi\)
\(180\) 0 0
\(181\) 28.3199i 0.156463i −0.996935 0.0782317i \(-0.975073\pi\)
0.996935 0.0782317i \(-0.0249274\pi\)
\(182\) 486.694 2.67414
\(183\) 0 0
\(184\) 79.7894i 0.433638i
\(185\) 418.978i 2.26475i
\(186\) 0 0
\(187\) 4.77657 0.0255432
\(188\) −65.8007 −0.350004
\(189\) 0 0
\(190\) −333.196 144.278i −1.75366 0.759358i
\(191\) 237.973 1.24593 0.622965 0.782250i \(-0.285928\pi\)
0.622965 + 0.782250i \(0.285928\pi\)
\(192\) 0 0
\(193\) 249.876i 1.29469i −0.762196 0.647346i \(-0.775879\pi\)
0.762196 0.647346i \(-0.224121\pi\)
\(194\) 125.897 0.648954
\(195\) 0 0
\(196\) −536.320 −2.73633
\(197\) −253.251 −1.28554 −0.642769 0.766060i \(-0.722214\pi\)
−0.642769 + 0.766060i \(0.722214\pi\)
\(198\) 0 0
\(199\) −48.0274 −0.241344 −0.120672 0.992692i \(-0.538505\pi\)
−0.120672 + 0.992692i \(0.538505\pi\)
\(200\) 55.8126i 0.279063i
\(201\) 0 0
\(202\) 459.103i 2.27279i
\(203\) 319.618i 1.57447i
\(204\) 0 0
\(205\) 22.4591i 0.109557i
\(206\) −69.5465 −0.337604
\(207\) 0 0
\(208\) 120.752i 0.580536i
\(209\) −2.07558 + 4.79335i −0.00993100 + 0.0229347i
\(210\) 0 0
\(211\) 278.279i 1.31886i 0.751767 + 0.659429i \(0.229202\pi\)
−0.751767 + 0.659429i \(0.770798\pi\)
\(212\) 136.136i 0.642150i
\(213\) 0 0
\(214\) 502.694 2.34904
\(215\) −301.973 −1.40452
\(216\) 0 0
\(217\) 288.786i 1.33081i
\(218\) −141.450 −0.648854
\(219\) 0 0
\(220\) 9.09967 0.0413621
\(221\) 226.202i 1.02354i
\(222\) 0 0
\(223\) 67.6910i 0.303547i −0.988415 0.151773i \(-0.951502\pi\)
0.988415 0.151773i \(-0.0484984\pi\)
\(224\) 537.364i 2.39895i
\(225\) 0 0
\(226\) 13.8970 0.0614912
\(227\) 409.224i 1.80275i −0.433039 0.901375i \(-0.642559\pi\)
0.433039 0.901375i \(-0.357441\pi\)
\(228\) 0 0
\(229\) −246.419 −1.07607 −0.538034 0.842923i \(-0.680833\pi\)
−0.538034 + 0.842923i \(0.680833\pi\)
\(230\) 392.709i 1.70743i
\(231\) 0 0
\(232\) −101.100 −0.435774
\(233\) 226.323 0.971344 0.485672 0.874141i \(-0.338575\pi\)
0.485672 + 0.874141i \(0.338575\pi\)
\(234\) 0 0
\(235\) 78.2749 0.333085
\(236\) 1.21416i 0.00514473i
\(237\) 0 0
\(238\) 649.513i 2.72905i
\(239\) −69.1204 −0.289207 −0.144603 0.989490i \(-0.546191\pi\)
−0.144603 + 0.989490i \(0.546191\pi\)
\(240\) 0 0
\(241\) 108.486i 0.450150i 0.974341 + 0.225075i \(0.0722628\pi\)
−0.974341 + 0.225075i \(0.927737\pi\)
\(242\) 368.272i 1.52179i
\(243\) 0 0
\(244\) 152.048 0.623148
\(245\) 637.993 2.60405
\(246\) 0 0
\(247\) 226.997 + 98.2924i 0.919015 + 0.397945i
\(248\) −91.3472 −0.368335
\(249\) 0 0
\(250\) 203.053i 0.812211i
\(251\) −46.7766 −0.186361 −0.0931804 0.995649i \(-0.529703\pi\)
−0.0931804 + 0.995649i \(0.529703\pi\)
\(252\) 0 0
\(253\) 5.64950 0.0223301
\(254\) −447.045 −1.76002
\(255\) 0 0
\(256\) 25.7218 0.100476
\(257\) 217.139i 0.844900i 0.906386 + 0.422450i \(0.138830\pi\)
−0.906386 + 0.422450i \(0.861170\pi\)
\(258\) 0 0
\(259\) 819.600i 3.16448i
\(260\) 430.930i 1.65742i
\(261\) 0 0
\(262\) 575.354i 2.19601i
\(263\) 34.8796 0.132622 0.0663109 0.997799i \(-0.478877\pi\)
0.0663109 + 0.997799i \(0.478877\pi\)
\(264\) 0 0
\(265\) 161.944i 0.611109i
\(266\) −651.794 282.235i −2.45035 1.06103i
\(267\) 0 0
\(268\) 542.995i 2.02610i
\(269\) 90.2335i 0.335441i −0.985835 0.167720i \(-0.946359\pi\)
0.985835 0.167720i \(-0.0536406\pi\)
\(270\) 0 0
\(271\) −144.997 −0.535043 −0.267522 0.963552i \(-0.586205\pi\)
−0.267522 + 0.963552i \(0.586205\pi\)
\(272\) 161.148 0.592455
\(273\) 0 0
\(274\) 217.223i 0.792784i
\(275\) −3.95182 −0.0143703
\(276\) 0 0
\(277\) 496.371 1.79195 0.895977 0.444100i \(-0.146477\pi\)
0.895977 + 0.444100i \(0.146477\pi\)
\(278\) 132.703i 0.477350i
\(279\) 0 0
\(280\) 299.064i 1.06808i
\(281\) 187.459i 0.667112i −0.942730 0.333556i \(-0.891751\pi\)
0.942730 0.333556i \(-0.108249\pi\)
\(282\) 0 0
\(283\) 51.5739 0.182240 0.0911200 0.995840i \(-0.470955\pi\)
0.0911200 + 0.995840i \(0.470955\pi\)
\(284\) 568.280i 2.00098i
\(285\) 0 0
\(286\) −10.9003 −0.0381130
\(287\) 43.9343i 0.153081i
\(288\) 0 0
\(289\) 12.8762 0.0445545
\(290\) 497.595 1.71584
\(291\) 0 0
\(292\) −58.9485 −0.201878
\(293\) 82.6781i 0.282178i 0.989997 + 0.141089i \(0.0450603\pi\)
−0.989997 + 0.141089i \(0.954940\pi\)
\(294\) 0 0
\(295\) 1.44433i 0.00489604i
\(296\) 259.251 0.875847
\(297\) 0 0
\(298\) 218.290i 0.732517i
\(299\) 267.541i 0.894787i
\(300\) 0 0
\(301\) −590.715 −1.96251
\(302\) −209.849 −0.694864
\(303\) 0 0
\(304\) −70.0241 + 161.714i −0.230342 + 0.531953i
\(305\) −180.873 −0.593026
\(306\) 0 0
\(307\) 83.3053i 0.271353i 0.990753 + 0.135676i \(0.0433208\pi\)
−0.990753 + 0.135676i \(0.956679\pi\)
\(308\) 17.8007 0.0577944
\(309\) 0 0
\(310\) 449.595 1.45031
\(311\) −4.22674 −0.0135908 −0.00679540 0.999977i \(-0.502163\pi\)
−0.00679540 + 0.999977i \(0.502163\pi\)
\(312\) 0 0
\(313\) −158.894 −0.507648 −0.253824 0.967250i \(-0.581688\pi\)
−0.253824 + 0.967250i \(0.581688\pi\)
\(314\) 480.264i 1.52950i
\(315\) 0 0
\(316\) 140.552i 0.444785i
\(317\) 278.572i 0.878777i −0.898297 0.439389i \(-0.855195\pi\)
0.898297 0.439389i \(-0.144805\pi\)
\(318\) 0 0
\(319\) 7.15838i 0.0224401i
\(320\) 603.794 1.88686
\(321\) 0 0
\(322\) 768.213i 2.38576i
\(323\) −131.175 + 302.936i −0.406115 + 0.937883i
\(324\) 0 0
\(325\) 187.145i 0.575831i
\(326\) 129.124i 0.396086i
\(327\) 0 0
\(328\) −13.8970 −0.0423690
\(329\) 153.120 0.465412
\(330\) 0 0
\(331\) 133.080i 0.402055i 0.979586 + 0.201027i \(0.0644281\pi\)
−0.979586 + 0.201027i \(0.935572\pi\)
\(332\) −473.148 −1.42514
\(333\) 0 0
\(334\) −309.292 −0.926025
\(335\) 645.934i 1.92816i
\(336\) 0 0
\(337\) 351.831i 1.04401i −0.852943 0.522004i \(-0.825185\pi\)
0.852943 0.522004i \(-0.174815\pi\)
\(338\) 1.51770i 0.00449022i
\(339\) 0 0
\(340\) 575.093 1.69145
\(341\) 6.46785i 0.0189673i
\(342\) 0 0
\(343\) 646.564 1.88503
\(344\) 186.851i 0.543173i
\(345\) 0 0
\(346\) −74.1993 −0.214449
\(347\) 348.124 1.00324 0.501619 0.865089i \(-0.332738\pi\)
0.501619 + 0.865089i \(0.332738\pi\)
\(348\) 0 0
\(349\) −614.323 −1.76024 −0.880119 0.474753i \(-0.842537\pi\)
−0.880119 + 0.474753i \(0.842537\pi\)
\(350\) 537.364i 1.53533i
\(351\) 0 0
\(352\) 12.0352i 0.0341908i
\(353\) −151.292 −0.428590 −0.214295 0.976769i \(-0.568745\pi\)
−0.214295 + 0.976769i \(0.568745\pi\)
\(354\) 0 0
\(355\) 676.012i 1.90426i
\(356\) 734.556i 2.06336i
\(357\) 0 0
\(358\) 227.698 0.636027
\(359\) −497.519 −1.38585 −0.692924 0.721011i \(-0.743678\pi\)
−0.692924 + 0.721011i \(0.743678\pi\)
\(360\) 0 0
\(361\) −247.000 263.272i −0.684211 0.729285i
\(362\) −86.2475 −0.238253
\(363\) 0 0
\(364\) 842.979i 2.31588i
\(365\) 70.1238 0.192120
\(366\) 0 0
\(367\) −267.003 −0.727529 −0.363765 0.931491i \(-0.618509\pi\)
−0.363765 + 0.931491i \(0.618509\pi\)
\(368\) 190.598 0.517929
\(369\) 0 0
\(370\) −1275.99 −3.44861
\(371\) 316.793i 0.853889i
\(372\) 0 0
\(373\) 51.4695i 0.137988i −0.997617 0.0689940i \(-0.978021\pi\)
0.997617 0.0689940i \(-0.0219789\pi\)
\(374\) 14.5469i 0.0388955i
\(375\) 0 0
\(376\) 48.4341i 0.128814i
\(377\) −338.997 −0.899195
\(378\) 0 0
\(379\) 447.214i 1.17999i 0.807409 + 0.589993i \(0.200869\pi\)
−0.807409 + 0.589993i \(0.799131\pi\)
\(380\) −249.897 + 577.112i −0.657624 + 1.51872i
\(381\) 0 0
\(382\) 724.740i 1.89722i
\(383\) 201.839i 0.526994i 0.964660 + 0.263497i \(0.0848759\pi\)
−0.964660 + 0.263497i \(0.915124\pi\)
\(384\) 0 0
\(385\) −21.1752 −0.0550006
\(386\) −760.990 −1.97148
\(387\) 0 0
\(388\) 218.060i 0.562010i
\(389\) 26.7700 0.0688174 0.0344087 0.999408i \(-0.489045\pi\)
0.0344087 + 0.999408i \(0.489045\pi\)
\(390\) 0 0
\(391\) 357.045 0.913158
\(392\) 394.771i 1.00707i
\(393\) 0 0
\(394\) 771.269i 1.95754i
\(395\) 167.198i 0.423285i
\(396\) 0 0
\(397\) 268.069 0.675237 0.337618 0.941283i \(-0.390379\pi\)
0.337618 + 0.941283i \(0.390379\pi\)
\(398\) 146.266i 0.367503i
\(399\) 0 0
\(400\) −133.323 −0.333308
\(401\) 445.540i 1.11107i −0.831492 0.555536i \(-0.812513\pi\)
0.831492 0.555536i \(-0.187487\pi\)
\(402\) 0 0
\(403\) −306.296 −0.760039
\(404\) 795.189 1.96829
\(405\) 0 0
\(406\) 973.389 2.39751
\(407\) 18.3563i 0.0451015i
\(408\) 0 0
\(409\) 325.039i 0.794716i −0.917664 0.397358i \(-0.869927\pi\)
0.917664 0.397358i \(-0.130073\pi\)
\(410\) 68.3987 0.166826
\(411\) 0 0
\(412\) 120.458i 0.292374i
\(413\) 2.82538i 0.00684112i
\(414\) 0 0
\(415\) 562.846 1.35625
\(416\) −569.945 −1.37006
\(417\) 0 0
\(418\) 14.5980 + 6.32113i 0.0349235 + 0.0151223i
\(419\) 51.0033 0.121726 0.0608631 0.998146i \(-0.480615\pi\)
0.0608631 + 0.998146i \(0.480615\pi\)
\(420\) 0 0
\(421\) 214.794i 0.510201i 0.966915 + 0.255100i \(0.0821085\pi\)
−0.966915 + 0.255100i \(0.917892\pi\)
\(422\) 847.492 2.00827
\(423\) 0 0
\(424\) 100.206 0.236335
\(425\) −249.752 −0.587653
\(426\) 0 0
\(427\) −353.821 −0.828622
\(428\) 870.692i 2.03433i
\(429\) 0 0
\(430\) 919.650i 2.13872i
\(431\) 469.547i 1.08944i 0.838619 + 0.544718i \(0.183363\pi\)
−0.838619 + 0.544718i \(0.816637\pi\)
\(432\) 0 0
\(433\) 175.110i 0.404411i 0.979343 + 0.202205i \(0.0648108\pi\)
−0.979343 + 0.202205i \(0.935189\pi\)
\(434\) 879.492 2.02648
\(435\) 0 0
\(436\) 244.999i 0.561924i
\(437\) −155.148 + 358.299i −0.355029 + 0.819905i
\(438\) 0 0
\(439\) 542.848i 1.23656i 0.785959 + 0.618278i \(0.212170\pi\)
−0.785959 + 0.618278i \(0.787830\pi\)
\(440\) 6.69803i 0.0152228i
\(441\) 0 0
\(442\) −688.894 −1.55858
\(443\) −134.680 −0.304019 −0.152009 0.988379i \(-0.548574\pi\)
−0.152009 + 0.988379i \(0.548574\pi\)
\(444\) 0 0
\(445\) 873.811i 1.96362i
\(446\) −206.151 −0.462222
\(447\) 0 0
\(448\) 1181.13 2.63646
\(449\) 722.541i 1.60922i −0.593801 0.804612i \(-0.702373\pi\)
0.593801 0.804612i \(-0.297627\pi\)
\(450\) 0 0
\(451\) 0.983981i 0.00218178i
\(452\) 24.0703i 0.0532530i
\(453\) 0 0
\(454\) −1246.28 −2.74512
\(455\) 1002.79i 2.20393i
\(456\) 0 0
\(457\) −42.7218 −0.0934831 −0.0467415 0.998907i \(-0.514884\pi\)
−0.0467415 + 0.998907i \(0.514884\pi\)
\(458\) 750.464i 1.63857i
\(459\) 0 0
\(460\) 680.193 1.47868
\(461\) −39.6221 −0.0859482 −0.0429741 0.999076i \(-0.513683\pi\)
−0.0429741 + 0.999076i \(0.513683\pi\)
\(462\) 0 0
\(463\) −158.625 −0.342603 −0.171302 0.985219i \(-0.554797\pi\)
−0.171302 + 0.985219i \(0.554797\pi\)
\(464\) 241.503i 0.520481i
\(465\) 0 0
\(466\) 689.261i 1.47910i
\(467\) 672.619 1.44030 0.720149 0.693820i \(-0.244074\pi\)
0.720149 + 0.693820i \(0.244074\pi\)
\(468\) 0 0
\(469\) 1263.57i 2.69418i
\(470\) 238.384i 0.507201i
\(471\) 0 0
\(472\) −0.893709 −0.00189345
\(473\) 13.2300 0.0279705
\(474\) 0 0
\(475\) 108.526 250.629i 0.228475 0.527641i
\(476\) 1124.99 2.36342
\(477\) 0 0
\(478\) 210.505i 0.440386i
\(479\) −870.846 −1.81805 −0.909025 0.416743i \(-0.863172\pi\)
−0.909025 + 0.416743i \(0.863172\pi\)
\(480\) 0 0
\(481\) 869.292 1.80726
\(482\) 330.392 0.685461
\(483\) 0 0
\(484\) 637.866 1.31791
\(485\) 259.399i 0.534843i
\(486\) 0 0
\(487\) 274.616i 0.563894i −0.959430 0.281947i \(-0.909020\pi\)
0.959430 0.281947i \(-0.0909802\pi\)
\(488\) 111.919i 0.229342i
\(489\) 0 0
\(490\) 1942.99i 3.96529i
\(491\) 781.588 1.59183 0.795915 0.605409i \(-0.206990\pi\)
0.795915 + 0.605409i \(0.206990\pi\)
\(492\) 0 0
\(493\) 452.405i 0.917657i
\(494\) 299.347 691.313i 0.605966 1.39942i
\(495\) 0 0
\(496\) 218.207i 0.439933i
\(497\) 1322.41i 2.66078i
\(498\) 0 0
\(499\) −387.918 −0.777390 −0.388695 0.921366i \(-0.627074\pi\)
−0.388695 + 0.921366i \(0.627074\pi\)
\(500\) 351.698 0.703395
\(501\) 0 0
\(502\) 142.457i 0.283779i
\(503\) 914.846 1.81878 0.909389 0.415946i \(-0.136550\pi\)
0.909389 + 0.415946i \(0.136550\pi\)
\(504\) 0 0
\(505\) −945.939 −1.87315
\(506\) 17.2054i 0.0340028i
\(507\) 0 0
\(508\) 774.304i 1.52422i
\(509\) 185.554i 0.364546i 0.983248 + 0.182273i \(0.0583455\pi\)
−0.983248 + 0.182273i \(0.941655\pi\)
\(510\) 0 0
\(511\) 137.175 0.268445
\(512\) 550.080i 1.07438i
\(513\) 0 0
\(514\) 661.292 1.28656
\(515\) 143.294i 0.278241i
\(516\) 0 0
\(517\) −3.42939 −0.00663324
\(518\) −2496.07 −4.81867
\(519\) 0 0
\(520\) −317.196 −0.609992
\(521\) 212.283i 0.407452i −0.979028 0.203726i \(-0.934695\pi\)
0.979028 0.203726i \(-0.0653052\pi\)
\(522\) 0 0
\(523\) 209.227i 0.400052i 0.979791 + 0.200026i \(0.0641026\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(524\) −996.543 −1.90180
\(525\) 0 0
\(526\) 106.225i 0.201948i
\(527\) 408.764i 0.775643i
\(528\) 0 0
\(529\) −106.704 −0.201709
\(530\) −493.196 −0.930559
\(531\) 0 0
\(532\) −488.846 + 1128.94i −0.918883 + 2.12207i
\(533\) −46.5980 −0.0874259
\(534\) 0 0
\(535\) 1035.75i 1.93599i
\(536\) −399.684 −0.745680
\(537\) 0 0
\(538\) −274.804 −0.510788
\(539\) −27.9518 −0.0518587
\(540\) 0 0
\(541\) 989.664 1.82932 0.914661 0.404221i \(-0.132457\pi\)
0.914661 + 0.404221i \(0.132457\pi\)
\(542\) 441.584i 0.814730i
\(543\) 0 0
\(544\) 760.615i 1.39819i
\(545\) 291.445i 0.534761i
\(546\) 0 0
\(547\) 851.749i 1.55713i −0.627565 0.778564i \(-0.715948\pi\)
0.627565 0.778564i \(-0.284052\pi\)
\(548\) −376.241 −0.686571
\(549\) 0 0
\(550\) 12.0352i 0.0218821i
\(551\) 453.993 + 196.585i 0.823944 + 0.356778i
\(552\) 0 0
\(553\) 327.070i 0.591446i
\(554\) 1511.69i 2.72868i
\(555\) 0 0
\(556\) −229.849 −0.413397
\(557\) −125.423 −0.225176 −0.112588 0.993642i \(-0.535914\pi\)
−0.112588 + 0.993642i \(0.535914\pi\)
\(558\) 0 0
\(559\) 626.531i 1.12081i
\(560\) −714.392 −1.27570
\(561\) 0 0
\(562\) −570.900 −1.01584
\(563\) 762.666i 1.35465i −0.735685 0.677324i \(-0.763140\pi\)
0.735685 0.677324i \(-0.236860\pi\)
\(564\) 0 0
\(565\) 28.6335i 0.0506788i
\(566\) 157.067i 0.277504i
\(567\) 0 0
\(568\) −418.296 −0.736436
\(569\) 598.818i 1.05240i −0.850360 0.526202i \(-0.823616\pi\)
0.850360 0.526202i \(-0.176384\pi\)
\(570\) 0 0
\(571\) −165.492 −0.289828 −0.144914 0.989444i \(-0.546291\pi\)
−0.144914 + 0.989444i \(0.546291\pi\)
\(572\) 18.8799i 0.0330069i
\(573\) 0 0
\(574\) 133.801 0.233102
\(575\) −295.395 −0.513731
\(576\) 0 0
\(577\) 852.262 1.47706 0.738528 0.674222i \(-0.235521\pi\)
0.738528 + 0.674222i \(0.235521\pi\)
\(578\) 39.2143i 0.0678448i
\(579\) 0 0
\(580\) 861.859i 1.48596i
\(581\) 1101.03 1.89506
\(582\) 0 0
\(583\) 7.09510i 0.0121700i
\(584\) 43.3905i 0.0742987i
\(585\) 0 0
\(586\) 251.794 0.429683
\(587\) −429.375 −0.731473 −0.365736 0.930718i \(-0.619183\pi\)
−0.365736 + 0.930718i \(0.619183\pi\)
\(588\) 0 0
\(589\) 410.199 + 177.622i 0.696434 + 0.301565i
\(590\) 4.39868 0.00745538
\(591\) 0 0
\(592\) 619.289i 1.04610i
\(593\) 243.100 0.409949 0.204974 0.978767i \(-0.434289\pi\)
0.204974 + 0.978767i \(0.434289\pi\)
\(594\) 0 0
\(595\) −1338.26 −2.24918
\(596\) −378.090 −0.634379
\(597\) 0 0
\(598\) −814.791 −1.36253
\(599\) 725.096i 1.21051i 0.796031 + 0.605256i \(0.206929\pi\)
−0.796031 + 0.605256i \(0.793071\pi\)
\(600\) 0 0
\(601\) 595.929i 0.991563i −0.868447 0.495781i \(-0.834882\pi\)
0.868447 0.495781i \(-0.165118\pi\)
\(602\) 1799.01i 2.98839i
\(603\) 0 0
\(604\) 363.469i 0.601770i
\(605\) −758.791 −1.25420
\(606\) 0 0
\(607\) 724.886i 1.19421i −0.802163 0.597106i \(-0.796317\pi\)
0.802163 0.597106i \(-0.203683\pi\)
\(608\) 763.286 + 330.512i 1.25540 + 0.543606i
\(609\) 0 0
\(610\) 550.844i 0.903023i
\(611\) 162.404i 0.265801i
\(612\) 0 0
\(613\) 499.368 0.814630 0.407315 0.913288i \(-0.366465\pi\)
0.407315 + 0.913288i \(0.366465\pi\)
\(614\) 253.704 0.413199
\(615\) 0 0
\(616\) 13.1026i 0.0212705i
\(617\) 87.2300 0.141378 0.0706889 0.997498i \(-0.477480\pi\)
0.0706889 + 0.997498i \(0.477480\pi\)
\(618\) 0 0
\(619\) −846.482 −1.36750 −0.683749 0.729717i \(-0.739652\pi\)
−0.683749 + 0.729717i \(0.739652\pi\)
\(620\) 778.721i 1.25600i
\(621\) 0 0
\(622\) 12.8724i 0.0206952i
\(623\) 1709.34i 2.74372i
\(624\) 0 0
\(625\) −777.736 −1.24438
\(626\) 483.907i 0.773014i
\(627\) 0 0
\(628\) −831.842 −1.32459
\(629\) 1160.11i 1.84437i
\(630\) 0 0
\(631\) −800.509 −1.26864 −0.634318 0.773072i \(-0.718719\pi\)
−0.634318 + 0.773072i \(0.718719\pi\)
\(632\) −103.457 −0.163697
\(633\) 0 0
\(634\) −848.385 −1.33815
\(635\) 921.094i 1.45054i
\(636\) 0 0
\(637\) 1323.70i 2.07803i
\(638\) −21.8007 −0.0341703
\(639\) 0 0
\(640\) 740.040i 1.15631i
\(641\) 492.363i 0.768117i 0.923309 + 0.384058i \(0.125474\pi\)
−0.923309 + 0.384058i \(0.874526\pi\)
\(642\) 0 0
\(643\) 1212.76 1.88609 0.943046 0.332663i \(-0.107947\pi\)
0.943046 + 0.332663i \(0.107947\pi\)
\(644\) 1330.58 2.06613
\(645\) 0 0
\(646\) 922.585 + 399.491i 1.42815 + 0.618407i
\(647\) 157.368 0.243227 0.121614 0.992578i \(-0.461193\pi\)
0.121614 + 0.992578i \(0.461193\pi\)
\(648\) 0 0
\(649\) 0.0632792i 9.75026e-5i
\(650\) 569.945 0.876839
\(651\) 0 0
\(652\) 223.650 0.343021
\(653\) −726.220 −1.11213 −0.556064 0.831139i \(-0.687689\pi\)
−0.556064 + 0.831139i \(0.687689\pi\)
\(654\) 0 0
\(655\) 1185.46 1.80987
\(656\) 33.1967i 0.0506047i
\(657\) 0 0
\(658\) 466.324i 0.708700i
\(659\) 259.756i 0.394167i 0.980387 + 0.197083i \(0.0631469\pi\)
−0.980387 + 0.197083i \(0.936853\pi\)
\(660\) 0 0
\(661\) 874.021i 1.32227i 0.750267 + 0.661135i \(0.229925\pi\)
−0.750267 + 0.661135i \(0.770075\pi\)
\(662\) 405.292 0.612224
\(663\) 0 0
\(664\) 348.272i 0.524506i
\(665\) 581.519 1342.96i 0.874465 2.01949i
\(666\) 0 0
\(667\) 535.083i 0.802223i
\(668\) 535.710i 0.801961i
\(669\) 0 0
\(670\) 1967.18 2.93608
\(671\) 7.92442 0.0118099
\(672\) 0 0
\(673\) 1035.99i 1.53935i −0.638434 0.769677i \(-0.720417\pi\)
0.638434 0.769677i \(-0.279583\pi\)
\(674\) −1071.49 −1.58975
\(675\) 0 0
\(676\) 2.62873 0.00388865
\(677\) 973.505i 1.43797i 0.695026 + 0.718984i \(0.255393\pi\)
−0.695026 + 0.718984i \(0.744607\pi\)
\(678\) 0 0
\(679\) 507.433i 0.747325i
\(680\) 423.311i 0.622516i
\(681\) 0 0
\(682\) −19.6977 −0.0288822
\(683\) 1186.44i 1.73710i −0.495604 0.868549i \(-0.665053\pi\)
0.495604 0.868549i \(-0.334947\pi\)
\(684\) 0 0
\(685\) 447.567 0.653383
\(686\) 1969.09i 2.87040i
\(687\) 0 0
\(688\) 446.344 0.648756
\(689\) 336.000 0.487663
\(690\) 0 0
\(691\) 563.670 0.815731 0.407866 0.913042i \(-0.366273\pi\)
0.407866 + 0.913042i \(0.366273\pi\)
\(692\) 128.517i 0.185718i
\(693\) 0 0
\(694\) 1060.20i 1.52767i
\(695\) 273.423 0.393414
\(696\) 0 0
\(697\) 62.1869i 0.0892209i
\(698\) 1870.91i 2.68038i
\(699\) 0 0
\(700\) −930.743 −1.32963
\(701\) 163.958 0.233892 0.116946 0.993138i \(-0.462690\pi\)
0.116946 + 0.993138i \(0.462690\pi\)
\(702\) 0 0
\(703\) −1164.18 504.104i −1.65602 0.717076i
\(704\) −26.4535 −0.0375760
\(705\) 0 0
\(706\) 460.757i 0.652631i
\(707\) −1850.43 −2.61730
\(708\) 0 0
\(709\) −315.993 −0.445689 −0.222844 0.974854i \(-0.571534\pi\)
−0.222844 + 0.974854i \(0.571534\pi\)
\(710\) 2058.78 2.89969
\(711\) 0 0
\(712\) 540.688 0.759393
\(713\) 483.467i 0.678074i
\(714\) 0 0
\(715\) 22.4591i 0.0314114i
\(716\) 394.384i 0.550816i
\(717\) 0 0
\(718\) 1515.18i 2.11028i
\(719\) 171.072 0.237931 0.118965 0.992898i \(-0.462042\pi\)
0.118965 + 0.992898i \(0.462042\pi\)
\(720\) 0 0
\(721\) 280.310i 0.388780i
\(722\) −801.787 + 752.232i −1.11051 + 1.04187i
\(723\) 0 0
\(724\) 149.385i 0.206333i
\(725\) 374.290i 0.516262i
\(726\) 0 0
\(727\) 261.169 0.359242 0.179621 0.983736i \(-0.442513\pi\)
0.179621 + 0.983736i \(0.442513\pi\)
\(728\) −620.495 −0.852328
\(729\) 0 0
\(730\) 213.560i 0.292548i
\(731\) 836.130 1.14382
\(732\) 0 0
\(733\) 1185.78 1.61771 0.808855 0.588009i \(-0.200088\pi\)
0.808855 + 0.588009i \(0.200088\pi\)
\(734\) 813.152i 1.10784i
\(735\) 0 0
\(736\) 899.619i 1.22231i
\(737\) 28.2997i 0.0383985i
\(738\) 0 0
\(739\) 738.557 0.999401 0.499701 0.866198i \(-0.333443\pi\)
0.499701 + 0.866198i \(0.333443\pi\)
\(740\) 2210.07i 2.98659i
\(741\) 0 0
\(742\) −964.784 −1.30025
\(743\) 809.783i 1.08988i −0.838474 0.544941i \(-0.816552\pi\)
0.838474 0.544941i \(-0.183448\pi\)
\(744\) 0 0
\(745\) 449.767 0.603714
\(746\) −156.749 −0.210120
\(747\) 0 0
\(748\) −25.1960 −0.0336845
\(749\) 2026.13i 2.70512i
\(750\) 0 0
\(751\) 1167.89i 1.55512i −0.628809 0.777559i \(-0.716457\pi\)
0.628809 0.777559i \(-0.283543\pi\)
\(752\) −115.698 −0.153853
\(753\) 0 0
\(754\) 1032.41i 1.36924i
\(755\) 432.374i 0.572681i
\(756\) 0 0
\(757\) −161.828 −0.213776 −0.106888 0.994271i \(-0.534089\pi\)
−0.106888 + 0.994271i \(0.534089\pi\)
\(758\) 1361.98 1.79681
\(759\) 0 0
\(760\) 424.797 + 183.943i 0.558944 + 0.242030i
\(761\) −1148.47 −1.50916 −0.754578 0.656210i \(-0.772158\pi\)
−0.754578 + 0.656210i \(0.772158\pi\)
\(762\) 0 0
\(763\) 570.121i 0.747210i
\(764\) −1255.29 −1.64304
\(765\) 0 0
\(766\) 614.694 0.802473
\(767\) −2.99669 −0.00390703
\(768\) 0 0
\(769\) 77.5673 0.100868 0.0504339 0.998727i \(-0.483940\pi\)
0.0504339 + 0.998727i \(0.483940\pi\)
\(770\) 64.4887i 0.0837515i
\(771\) 0 0
\(772\) 1318.07i 1.70735i
\(773\) 1165.42i 1.50766i 0.657068 + 0.753831i \(0.271796\pi\)
−0.657068 + 0.753831i \(0.728204\pi\)
\(774\) 0 0
\(775\) 338.184i 0.436367i
\(776\) −160.508 −0.206841
\(777\) 0 0
\(778\) 81.5272i 0.104791i
\(779\) 62.4053 + 27.0223i 0.0801095 + 0.0346884i
\(780\) 0 0
\(781\) 29.6175i 0.0379225i
\(782\) 1087.37i 1.39050i
\(783\) 0 0
\(784\) −943.014 −1.20282
\(785\) 989.540 1.26056
\(786\) 0 0
\(787\) 1141.88i 1.45092i 0.688263 + 0.725461i \(0.258374\pi\)
−0.688263 + 0.725461i \(0.741626\pi\)
\(788\) 1335.88 1.69528
\(789\) 0 0
\(790\) 509.196 0.644552
\(791\) 56.0125i 0.0708123i
\(792\) 0 0
\(793\) 375.274i 0.473233i
\(794\) 816.397i 1.02821i
\(795\) 0 0
\(796\) 253.341 0.318267
\(797\) 798.061i 1.00133i 0.865641 + 0.500666i \(0.166911\pi\)
−0.865641 + 0.500666i \(0.833089\pi\)
\(798\) 0 0
\(799\) −216.735 −0.271258
\(800\) 629.283i 0.786603i
\(801\) 0 0
\(802\) −1356.88 −1.69187
\(803\) −3.07227 −0.00382599
\(804\) 0 0
\(805\) −1582.83 −1.96625
\(806\) 932.816i 1.15734i
\(807\) 0 0
\(808\) 585.318i 0.724404i
\(809\) −879.175 −1.08674 −0.543372 0.839492i \(-0.682853\pi\)
−0.543372 + 0.839492i \(0.682853\pi\)
\(810\) 0 0
\(811\) 654.537i 0.807074i 0.914963 + 0.403537i \(0.132219\pi\)
−0.914963 + 0.403537i \(0.867781\pi\)
\(812\) 1685.96i 2.07630i
\(813\) 0 0
\(814\) 55.9036 0.0686777
\(815\) −266.048 −0.326439
\(816\) 0 0
\(817\) −363.326 + 839.066i −0.444708 + 1.02701i
\(818\) −989.897 −1.21014
\(819\) 0 0
\(820\) 118.470i 0.144476i
\(821\) 1419.35 1.72880 0.864402 0.502801i \(-0.167697\pi\)
0.864402 + 0.502801i \(0.167697\pi\)
\(822\) 0 0
\(823\) −671.107 −0.815440 −0.407720 0.913107i \(-0.633676\pi\)
−0.407720 + 0.913107i \(0.633676\pi\)
\(824\) 88.6661 0.107604
\(825\) 0 0
\(826\) 8.60464 0.0104172
\(827\) 186.204i 0.225156i 0.993643 + 0.112578i \(0.0359108\pi\)
−0.993643 + 0.112578i \(0.964089\pi\)
\(828\) 0 0
\(829\) 924.130i 1.11475i 0.830260 + 0.557376i \(0.188192\pi\)
−0.830260 + 0.557376i \(0.811808\pi\)
\(830\) 1714.13i 2.06522i
\(831\) 0 0
\(832\) 1252.75i 1.50571i
\(833\) −1766.54 −2.12069
\(834\) 0 0
\(835\) 637.268i 0.763195i
\(836\) 10.9485 25.2845i 0.0130963 0.0302446i
\(837\) 0 0
\(838\) 155.329i 0.185357i
\(839\) 934.157i 1.11342i −0.830708 0.556708i \(-0.812064\pi\)
0.830708 0.556708i \(-0.187936\pi\)
\(840\) 0 0
\(841\) 163.007 0.193825
\(842\) 654.151 0.776902
\(843\) 0 0
\(844\) 1467.90i 1.73922i
\(845\) −3.12707 −0.00370067
\(846\) 0 0
\(847\) −1484.34 −1.75246
\(848\) 239.368i 0.282274i
\(849\) 0 0
\(850\) 760.615i 0.894841i
\(851\) 1372.12i 1.61236i
\(852\) 0 0
\(853\) 344.405 0.403758 0.201879 0.979411i \(-0.435295\pi\)
0.201879 + 0.979411i \(0.435295\pi\)
\(854\) 1077.55i 1.26177i
\(855\) 0 0
\(856\) −640.894 −0.748708
\(857\) 931.245i 1.08663i −0.839528 0.543317i \(-0.817168\pi\)
0.839528 0.543317i \(-0.182832\pi\)
\(858\) 0 0
\(859\) −1072.73 −1.24881 −0.624405 0.781100i \(-0.714659\pi\)
−0.624405 + 0.781100i \(0.714659\pi\)
\(860\) 1592.88 1.85219
\(861\) 0 0
\(862\) 1429.99 1.65893
\(863\) 1240.65i 1.43760i −0.695217 0.718800i \(-0.744692\pi\)
0.695217 0.718800i \(-0.255308\pi\)
\(864\) 0 0
\(865\) 152.881i 0.176741i
\(866\) 533.292 0.615811
\(867\) 0 0
\(868\) 1523.32i 1.75498i
\(869\) 7.32527i 0.00842955i
\(870\) 0 0
\(871\) −1340.18 −1.53867
\(872\) 180.337 0.206809
\(873\) 0 0
\(874\) 1091.19 + 472.499i 1.24850 + 0.540617i
\(875\) −818.413 −0.935329
\(876\) 0 0
\(877\) 526.564i 0.600415i 0.953874 + 0.300207i \(0.0970559\pi\)
−0.953874 + 0.300207i \(0.902944\pi\)
\(878\) 1653.23 1.88295
\(879\) 0 0
\(880\) 16.0000 0.0181818
\(881\) −1444.25 −1.63933 −0.819664 0.572844i \(-0.805840\pi\)
−0.819664 + 0.572844i \(0.805840\pi\)
\(882\) 0 0
\(883\) 593.086 0.671671 0.335836 0.941921i \(-0.390981\pi\)
0.335836 + 0.941921i \(0.390981\pi\)
\(884\) 1193.20i 1.34977i
\(885\) 0 0
\(886\) 410.165i 0.462940i
\(887\) 589.691i 0.664816i −0.943136 0.332408i \(-0.892139\pi\)
0.943136 0.332408i \(-0.107861\pi\)
\(888\) 0 0
\(889\) 1801.83i 2.02681i
\(890\) −2661.17 −2.99008
\(891\) 0 0
\(892\) 357.064i 0.400296i
\(893\) 94.1786 217.496i 0.105463 0.243557i
\(894\) 0 0
\(895\) 469.150i 0.524190i
\(896\) 1447.66i 1.61569i
\(897\) 0 0
\(898\) −2200.48 −2.45043
\(899\) −612.591 −0.681414
\(900\) 0 0
\(901\) 448.406i 0.497675i
\(902\) −2.99669 −0.00332227
\(903\) 0 0
\(904\) −17.7175 −0.0195991
\(905\) 177.705i 0.196359i
\(906\) 0 0
\(907\) 164.499i 0.181366i 0.995880 + 0.0906829i \(0.0289050\pi\)
−0.995880 + 0.0906829i \(0.971095\pi\)
\(908\) 2158.62i 2.37734i
\(909\) 0 0
\(910\) 3053.97 3.35601
\(911\) 474.507i 0.520864i −0.965492 0.260432i \(-0.916135\pi\)
0.965492 0.260432i \(-0.0838650\pi\)
\(912\) 0 0
\(913\) −24.6594 −0.0270092
\(914\) 130.108i 0.142350i
\(915\) 0 0
\(916\) 1299.84 1.41904
\(917\) 2318.99 2.52889
\(918\) 0 0
\(919\) 708.488 0.770934 0.385467 0.922722i \(-0.374040\pi\)
0.385467 + 0.922722i \(0.374040\pi\)
\(920\) 500.672i 0.544209i
\(921\) 0 0
\(922\) 120.668i 0.130876i
\(923\) −1402.58 −1.51959
\(924\) 0 0
\(925\) 959.795i 1.03762i
\(926\) 483.090i 0.521695i
\(927\) 0 0
\(928\) −1139.89 −1.22833
\(929\) −154.482 −0.166288 −0.0831441 0.996538i \(-0.526496\pi\)
−0.0831441 + 0.996538i \(0.526496\pi\)
\(930\) 0 0
\(931\) 767.619 1772.74i 0.824510 1.90412i
\(932\) −1193.84 −1.28094
\(933\) 0 0
\(934\) 2048.44i 2.19319i
\(935\) 29.9726 0.0320563
\(936\) 0 0
\(937\) −289.223 −0.308670 −0.154335 0.988019i \(-0.549323\pi\)
−0.154335 + 0.988019i \(0.549323\pi\)
\(938\) 3848.17 4.10252
\(939\) 0 0
\(940\) −412.894 −0.439249
\(941\) 1484.31i 1.57738i −0.614794 0.788688i \(-0.710761\pi\)
0.614794 0.788688i \(-0.289239\pi\)
\(942\) 0 0
\(943\) 73.5517i 0.0779976i
\(944\) 2.13486i 0.00226150i
\(945\) 0 0
\(946\) 40.2918i 0.0425917i
\(947\) −222.206 −0.234642 −0.117321 0.993094i \(-0.537431\pi\)
−0.117321 + 0.993094i \(0.537431\pi\)
\(948\) 0 0
\(949\) 145.492i 0.153311i
\(950\) −763.286 330.512i −0.803459 0.347908i
\(951\) 0 0
\(952\) 828.076i 0.869827i
\(953\) 1149.54i 1.20623i −0.797655 0.603114i \(-0.793926\pi\)
0.797655 0.603114i \(-0.206074\pi\)
\(954\) 0 0
\(955\) 1493.26 1.56362
\(956\) 364.605 0.381386
\(957\) 0 0
\(958\) 2652.14i 2.76841i
\(959\) 875.526 0.912957
\(960\) 0 0
\(961\) 407.502 0.424039
\(962\) 2647.41i 2.75198i
\(963\) 0 0
\(964\) 572.256i 0.593626i
\(965\) 1567.95i 1.62482i
\(966\) 0 0
\(967\) 980.296 1.01375 0.506875 0.862020i \(-0.330801\pi\)
0.506875 + 0.862020i \(0.330801\pi\)
\(968\) 469.517i 0.485038i
\(969\) 0 0
\(970\) 789.993 0.814426
\(971\) 1699.98i 1.75075i 0.483441 + 0.875377i \(0.339387\pi\)
−0.483441 + 0.875377i \(0.660613\pi\)
\(972\) 0 0
\(973\) 534.866 0.549708
\(974\) −836.337 −0.858662
\(975\) 0 0
\(976\) 267.347 0.273921
\(977\) 73.3216i 0.0750477i −0.999296 0.0375238i \(-0.988053\pi\)
0.999296 0.0375238i \(-0.0119470\pi\)
\(978\) 0 0
\(979\) 38.2835i 0.0391047i
\(980\) −3365.36 −3.43404
\(981\) 0 0
\(982\) 2380.31i 2.42394i
\(983\) 541.654i 0.551022i −0.961298 0.275511i \(-0.911153\pi\)
0.961298 0.275511i \(-0.0888470\pi\)
\(984\) 0 0
\(985\) −1589.13 −1.61333
\(986\) −1377.79 −1.39735
\(987\) 0 0
\(988\) −1197.39 518.485i −1.21193 0.524782i
\(989\) 988.935 0.999935
\(990\) 0 0
\(991\) 1546.81i 1.56086i 0.625244 + 0.780429i \(0.284999\pi\)
−0.625244 + 0.780429i \(0.715001\pi\)
\(992\) −1029.93 −1.03824
\(993\) 0 0
\(994\) 4027.36 4.05167
\(995\) −301.368 −0.302882
\(996\) 0 0
\(997\) −1366.20 −1.37031 −0.685156 0.728397i \(-0.740266\pi\)
−0.685156 + 0.728397i \(0.740266\pi\)
\(998\) 1181.39i 1.18376i
\(999\) 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 171.3.c.f.37.1 4
3.2 odd 2 57.3.c.b.37.4 yes 4
4.3 odd 2 2736.3.o.l.721.4 4
12.11 even 2 912.3.o.b.721.1 4
19.18 odd 2 inner 171.3.c.f.37.4 4
57.56 even 2 57.3.c.b.37.1 4
76.75 even 2 2736.3.o.l.721.3 4
228.227 odd 2 912.3.o.b.721.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.3.c.b.37.1 4 57.56 even 2
57.3.c.b.37.4 yes 4 3.2 odd 2
171.3.c.f.37.1 4 1.1 even 1 trivial
171.3.c.f.37.4 4 19.18 odd 2 inner
912.3.o.b.721.1 4 12.11 even 2
912.3.o.b.721.3 4 228.227 odd 2
2736.3.o.l.721.3 4 76.75 even 2
2736.3.o.l.721.4 4 4.3 odd 2