Properties

Label 912.3.o.b.721.3
Level $912$
Weight $3$
Character 912.721
Analytic conductor $24.850$
Analytic rank $0$
Dimension $4$
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] = [912,3,Mod(721,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("912.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (of order \(2\), degree \(1\), not 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: \(24.8502001097\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.3
Root \(2.13746 - 0.656712i\) of defining polynomial
Character \(\chi\) \(=\) 912.721
Dual form 912.3.o.b.721.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} -6.27492 q^{5} -12.2749 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} -6.27492 q^{5} -12.2749 q^{7} -3.00000 q^{9} -0.274917 q^{11} -13.0192i q^{13} -10.8685i q^{15} +17.3746 q^{17} +(-7.54983 - 17.4356i) q^{19} -21.2608i q^{21} -20.5498 q^{23} +14.3746 q^{25} -5.19615i q^{27} +26.0383i q^{29} +23.5265i q^{31} -0.476171i q^{33} +77.0241 q^{35} +66.7703i q^{37} +22.5498 q^{39} +3.57919i q^{41} +48.1238 q^{43} +18.8248 q^{45} +12.4743 q^{47} +101.674 q^{49} +30.0937i q^{51} -25.8081i q^{53} +1.72508 q^{55} +(30.1993 - 13.0767i) q^{57} -0.230175i q^{59} -28.8248 q^{61} +36.8248 q^{63} +81.6941i q^{65} +102.939i q^{67} -35.5934i q^{69} -107.732i q^{71} +11.1752 q^{73} +24.8975i q^{75} +3.37459 q^{77} +26.6454i q^{79} +9.00000 q^{81} +89.6977 q^{83} -109.024 q^{85} -45.0997 q^{87} -139.255i q^{89} +159.809i q^{91} -40.7492 q^{93} +(47.3746 + 109.407i) q^{95} -41.3390i q^{97} +0.824752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9} + 14 q^{11} - 6 q^{17} - 52 q^{23} - 18 q^{25} + 142 q^{35} + 60 q^{39} - 34 q^{43} + 30 q^{45} - 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 102 q^{63} + 90 q^{73} - 62 q^{77} + 36 q^{81} - 64 q^{83} - 270 q^{85} - 120 q^{87} - 12 q^{93} + 114 q^{95} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) −6.27492 −1.25498 −0.627492 0.778623i \(-0.715918\pi\)
−0.627492 + 0.778623i \(0.715918\pi\)
\(6\) 0 0
\(7\) −12.2749 −1.75356 −0.876780 0.480892i \(-0.840313\pi\)
−0.876780 + 0.480892i \(0.840313\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −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 \(-0.833063\pi\)
0.865600 0.500737i \(-0.166937\pi\)
\(14\) 0 0
\(15\) 10.8685i 0.724565i
\(16\) 0 0
\(17\) 17.3746 1.02203 0.511017 0.859570i \(-0.329269\pi\)
0.511017 + 0.859570i \(0.329269\pi\)
\(18\) 0 0
\(19\) −7.54983 17.4356i −0.397360 0.917663i
\(20\) 0 0
\(21\) 21.2608i 1.01242i
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 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 \(0.876110\pi\)
\(32\) 0 0
\(33\) 0.476171i 0.0144294i
\(34\) 0 0
\(35\) 77.0241 2.20069
\(36\) 0 0
\(37\) 66.7703i 1.80460i 0.431107 + 0.902301i \(0.358123\pi\)
−0.431107 + 0.902301i \(0.641877\pi\)
\(38\) 0 0
\(39\) 22.5498 0.578201
\(40\) 0 0
\(41\) 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 \(-0.310963\pi\)
0.559579 + 0.828777i \(0.310963\pi\)
\(44\) 0 0
\(45\) 18.8248 0.418328
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 30.0937i 0.590072i
\(52\) 0 0
\(53\) 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
\(56\) 0 0
\(57\) 30.1993 13.0767i 0.529813 0.229416i
\(58\) 0 0
\(59\) 0.230175i 0.00390128i −0.999998 0.00195064i \(-0.999379\pi\)
0.999998 0.00195064i \(-0.000620908\pi\)
\(60\) 0 0
\(61\) −28.8248 −0.472537 −0.236268 0.971688i \(-0.575924\pi\)
−0.236268 + 0.971688i \(0.575924\pi\)
\(62\) 0 0
\(63\) 36.8248 0.584520
\(64\) 0 0
\(65\) 81.6941i 1.25683i
\(66\) 0 0
\(67\) 102.939i 1.53640i 0.640208 + 0.768202i \(0.278848\pi\)
−0.640208 + 0.768202i \(0.721152\pi\)
\(68\) 0 0
\(69\) 35.5934i 0.515846i
\(70\) 0 0
\(71\) 107.732i 1.51736i −0.651465 0.758679i \(-0.725845\pi\)
0.651465 0.758679i \(-0.274155\pi\)
\(72\) 0 0
\(73\) 11.1752 0.153086 0.0765428 0.997066i \(-0.475612\pi\)
0.0765428 + 0.997066i \(0.475612\pi\)
\(74\) 0 0
\(75\) 24.8975i 0.331967i
\(76\) 0 0
\(77\) 3.37459 0.0438258
\(78\) 0 0
\(79\) 26.6454i 0.337283i 0.985677 + 0.168642i \(0.0539381\pi\)
−0.985677 + 0.168642i \(0.946062\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) −45.0997 −0.518387
\(88\) 0 0
\(89\) 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
\(92\) 0 0
\(93\) −40.7492 −0.438163
\(94\) 0 0
\(95\) 47.3746 + 109.407i 0.498680 + 1.15165i
\(96\) 0 0
\(97\) 41.3390i 0.426176i −0.977033 0.213088i \(-0.931648\pi\)
0.977033 0.213088i \(-0.0683521\pi\)
\(98\) 0 0
\(99\) 0.824752 0.00833082
\(100\) 0 0
\(101\) 150.749 1.49257 0.746283 0.665629i \(-0.231837\pi\)
0.746283 + 0.665629i \(0.231837\pi\)
\(102\) 0 0
\(103\) 22.8360i 0.221709i −0.993837 0.110854i \(-0.964641\pi\)
0.993837 0.110854i \(-0.0353587\pi\)
\(104\) 0 0
\(105\) 133.410i 1.27057i
\(106\) 0 0
\(107\) 165.063i 1.54264i −0.636446 0.771321i \(-0.719596\pi\)
0.636446 0.771321i \(-0.280404\pi\)
\(108\) 0 0
\(109\) 46.4460i 0.426110i 0.977040 + 0.213055i \(0.0683414\pi\)
−0.977040 + 0.213055i \(0.931659\pi\)
\(110\) 0 0
\(111\) −115.650 −1.04189
\(112\) 0 0
\(113\) 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
\(116\) 0 0
\(117\) 39.0575i 0.333824i
\(118\) 0 0
\(119\) −213.272 −1.79220
\(120\) 0 0
\(121\) −120.924 −0.999375
\(122\) 0 0
\(123\) −6.19934 −0.0504011
\(124\) 0 0
\(125\) 66.6736 0.533389
\(126\) 0 0
\(127\) 146.790i 1.15583i −0.816098 0.577913i \(-0.803867\pi\)
0.816098 0.577913i \(-0.196133\pi\)
\(128\) 0 0
\(129\) 83.3528i 0.646146i
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) 32.6054i 0.241522i
\(136\) 0 0
\(137\) −71.3264 −0.520631 −0.260315 0.965524i \(-0.583827\pi\)
−0.260315 + 0.965524i \(0.583827\pi\)
\(138\) 0 0
\(139\) −43.5739 −0.313481 −0.156741 0.987640i \(-0.550099\pi\)
−0.156741 + 0.987640i \(0.550099\pi\)
\(140\) 0 0
\(141\) 21.6060i 0.153234i
\(142\) 0 0
\(143\) 3.57919i 0.0250293i
\(144\) 0 0
\(145\) 163.388i 1.12682i
\(146\) 0 0
\(147\) 176.104i 1.19799i
\(148\) 0 0
\(149\) −71.6769 −0.481053 −0.240527 0.970643i \(-0.577320\pi\)
−0.240527 + 0.970643i \(0.577320\pi\)
\(150\) 0 0
\(151\) 68.9051i 0.456325i −0.973623 0.228163i \(-0.926728\pi\)
0.973623 0.228163i \(-0.0732718\pi\)
\(152\) 0 0
\(153\) −52.1238 −0.340678
\(154\) 0 0
\(155\) 147.627i 0.952433i
\(156\) 0 0
\(157\) 157.698 1.00444 0.502222 0.864739i \(-0.332516\pi\)
0.502222 + 0.864739i \(0.332516\pi\)
\(158\) 0 0
\(159\) 44.7010 0.281138
\(160\) 0 0
\(161\) 252.248 1.56675
\(162\) 0 0
\(163\) 42.3987 0.260115 0.130057 0.991506i \(-0.458484\pi\)
0.130057 + 0.991506i \(0.458484\pi\)
\(164\) 0 0
\(165\) 2.98793i 0.0181087i
\(166\) 0 0
\(167\) 101.558i 0.608132i 0.952651 + 0.304066i \(0.0983443\pi\)
−0.952651 + 0.304066i \(0.901656\pi\)
\(168\) 0 0
\(169\) −0.498344 −0.00294878
\(170\) 0 0
\(171\) 22.6495 + 52.3068i 0.132453 + 0.305888i
\(172\) 0 0
\(173\) 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
\(176\) 0 0
\(177\) 0.398675 0.00225240
\(178\) 0 0
\(179\) 74.7659i 0.417687i −0.977949 0.208843i \(-0.933030\pi\)
0.977949 0.208843i \(-0.0669699\pi\)
\(180\) 0 0
\(181\) 28.3199i 0.156463i 0.996935 + 0.0782317i \(0.0249274\pi\)
−0.996935 + 0.0782317i \(0.975073\pi\)
\(182\) 0 0
\(183\) 49.9259i 0.272819i
\(184\) 0 0
\(185\) 418.978i 2.26475i
\(186\) 0 0
\(187\) −4.77657 −0.0255432
\(188\) 0 0
\(189\) 63.7823i 0.337473i
\(190\) 0 0
\(191\) 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 \(0.224121\pi\)
−0.762196 + 0.647346i \(0.775879\pi\)
\(194\) 0 0
\(195\) −141.498 −0.725633
\(196\) 0 0
\(197\) 253.251 1.28554 0.642769 0.766060i \(-0.277786\pi\)
0.642769 + 0.766060i \(0.277786\pi\)
\(198\) 0 0
\(199\) 48.0274 0.241344 0.120672 0.992692i \(-0.461495\pi\)
0.120672 + 0.992692i \(0.461495\pi\)
\(200\) 0 0
\(201\) −178.296 −0.887043
\(202\) 0 0
\(203\) 319.618i 1.57447i
\(204\) 0 0
\(205\) 22.4591i 0.109557i
\(206\) 0 0
\(207\) 61.6495 0.297824
\(208\) 0 0
\(209\) 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 \(0.770798\pi\)
\(212\) 0 0
\(213\) 186.598 0.876047
\(214\) 0 0
\(215\) −301.973 −1.40452
\(216\) 0 0
\(217\) 288.786i 1.33081i
\(218\) 0 0
\(219\) 19.3561i 0.0883840i
\(220\) 0 0
\(221\) 226.202i 1.02354i
\(222\) 0 0
\(223\) 67.6910i 0.303547i −0.988415 0.151773i \(-0.951502\pi\)
0.988415 0.151773i \(-0.0484984\pi\)
\(224\) 0 0
\(225\) −43.1238 −0.191661
\(226\) 0 0
\(227\) 409.224i 1.80275i 0.433039 + 0.901375i \(0.357441\pi\)
−0.433039 + 0.901375i \(0.642559\pi\)
\(228\) 0 0
\(229\) −246.419 −1.07607 −0.538034 0.842923i \(-0.680833\pi\)
−0.538034 + 0.842923i \(0.680833\pi\)
\(230\) 0 0
\(231\) 5.84495i 0.0253028i
\(232\) 0 0
\(233\) −226.323 −0.971344 −0.485672 0.874141i \(-0.661425\pi\)
−0.485672 + 0.874141i \(0.661425\pi\)
\(234\) 0 0
\(235\) −78.2749 −0.333085
\(236\) 0 0
\(237\) −46.1512 −0.194731
\(238\) 0 0
\(239\) −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 \(-0.927737\pi\)
0.974341 0.225075i \(-0.0722628\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) −637.993 −2.60405
\(246\) 0 0
\(247\) −226.997 + 98.2924i −0.919015 + 0.397945i
\(248\) 0 0
\(249\) 155.361i 0.623940i
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) 188.835i 0.740530i
\(256\) 0 0
\(257\) 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
\(260\) 0 0
\(261\) 78.1149i 0.299291i
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) 241.196 0.903356
\(268\) 0 0
\(269\) 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 \(-0.413795\pi\)
0.267522 + 0.963552i \(0.413795\pi\)
\(272\) 0 0
\(273\) −276.797 −1.01391
\(274\) 0 0
\(275\) −3.95182 −0.0143703
\(276\) 0 0
\(277\) 496.371 1.79195 0.895977 0.444100i \(-0.146477\pi\)
0.895977 + 0.444100i \(0.146477\pi\)
\(278\) 0 0
\(279\) 70.5796i 0.252974i
\(280\) 0 0
\(281\) 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 \(-0.529045\pi\)
−0.0911200 + 0.995840i \(0.529045\pi\)
\(284\) 0 0
\(285\) −189.498 + 82.0552i −0.664906 + 0.287913i
\(286\) 0 0
\(287\) 43.9343i 0.153081i
\(288\) 0 0
\(289\) 12.8762 0.0445545
\(290\) 0 0
\(291\) 71.6013 0.246053
\(292\) 0 0
\(293\) 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
\(296\) 0 0
\(297\) 1.42851i 0.00480980i
\(298\) 0 0
\(299\) 267.541i 0.894787i
\(300\) 0 0
\(301\) −590.715 −1.96251
\(302\) 0 0
\(303\) 261.105i 0.861733i
\(304\) 0 0
\(305\) 180.873 0.593026
\(306\) 0 0
\(307\) 83.3053i 0.271353i 0.990753 + 0.135676i \(0.0433208\pi\)
−0.990753 + 0.135676i \(0.956679\pi\)
\(308\) 0 0
\(309\) 39.5531 0.128004
\(310\) 0 0
\(311\) −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 \(0.581688\pi\)
\(314\) 0 0
\(315\) −231.072 −0.733563
\(316\) 0 0
\(317\) 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
\(320\) 0 0
\(321\) 285.897 0.890645
\(322\) 0 0
\(323\) −131.175 302.936i −0.406115 0.937883i
\(324\) 0 0
\(325\) 187.145i 0.575831i
\(326\) 0 0
\(327\) −80.4469 −0.246015
\(328\) 0 0
\(329\) −153.120 −0.465412
\(330\) 0 0
\(331\) 133.080i 0.402055i 0.979586 + 0.201027i \(0.0644281\pi\)
−0.979586 + 0.201027i \(0.935572\pi\)
\(332\) 0 0
\(333\) 200.311i 0.601534i
\(334\) 0 0
\(335\) 645.934i 1.92816i
\(336\) 0 0
\(337\) 351.831i 1.04401i 0.852943 + 0.522004i \(0.174815\pi\)
−0.852943 + 0.522004i \(0.825185\pi\)
\(338\) 0 0
\(339\) −7.90364 −0.0233146
\(340\) 0 0
\(341\) 6.46785i 0.0189673i
\(342\) 0 0
\(343\) −646.564 −1.88503
\(344\) 0 0
\(345\) 223.345i 0.647378i
\(346\) 0 0
\(347\) 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 \(0.842537\pi\)
\(350\) 0 0
\(351\) −67.6495 −0.192734
\(352\) 0 0
\(353\) 151.292 0.428590 0.214295 0.976769i \(-0.431255\pi\)
0.214295 + 0.976769i \(0.431255\pi\)
\(354\) 0 0
\(355\) 676.012i 1.90426i
\(356\) 0 0
\(357\) 369.397i 1.03473i
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 209.447i 0.576990i
\(364\) 0 0
\(365\) −70.1238 −0.192120
\(366\) 0 0
\(367\) 267.003 0.727529 0.363765 0.931491i \(-0.381491\pi\)
0.363765 + 0.931491i \(0.381491\pi\)
\(368\) 0 0
\(369\) 10.7376i 0.0290991i
\(370\) 0 0
\(371\) 316.793i 0.853889i
\(372\) 0 0
\(373\) 51.4695i 0.137988i 0.997617 + 0.0689940i \(0.0219789\pi\)
−0.997617 + 0.0689940i \(0.978021\pi\)
\(374\) 0 0
\(375\) 115.482i 0.307952i
\(376\) 0 0
\(377\) 338.997 0.899195
\(378\) 0 0
\(379\) 447.214i 1.17999i 0.807409 + 0.589993i \(0.200869\pi\)
−0.807409 + 0.589993i \(0.799131\pi\)
\(380\) 0 0
\(381\) 254.248 0.667316
\(382\) 0 0
\(383\) 201.839i 0.526994i −0.964660 0.263497i \(-0.915124\pi\)
0.964660 0.263497i \(-0.0848759\pi\)
\(384\) 0 0
\(385\) −21.1752 −0.0550006
\(386\) 0 0
\(387\) −144.371 −0.373052
\(388\) 0 0
\(389\) −26.7700 −0.0688174 −0.0344087 0.999408i \(-0.510955\pi\)
−0.0344087 + 0.999408i \(0.510955\pi\)
\(390\) 0 0
\(391\) −357.045 −0.913158
\(392\) 0 0
\(393\) 327.221i 0.832623i
\(394\) 0 0
\(395\) 167.198i 0.423285i
\(396\) 0 0
\(397\) 268.069 0.675237 0.337618 0.941283i \(-0.390379\pi\)
0.337618 + 0.941283i \(0.390379\pi\)
\(398\) 0 0
\(399\) −370.694 + 160.515i −0.929059 + 0.402294i
\(400\) 0 0
\(401\) 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
\(404\) 0 0
\(405\) −56.4743 −0.139443
\(406\) 0 0
\(407\) 18.3563i 0.0451015i
\(408\) 0 0
\(409\) 325.039i 0.794716i 0.917664 + 0.397358i \(0.130073\pi\)
−0.917664 + 0.397358i \(0.869927\pi\)
\(410\) 0 0
\(411\) 123.541i 0.300586i
\(412\) 0 0
\(413\) 2.82538i 0.00684112i
\(414\) 0 0
\(415\) −562.846 −1.35625
\(416\) 0 0
\(417\) 75.4722i 0.180989i
\(418\) 0 0
\(419\) 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 \(-0.917892\pi\)
0.966915 0.255100i \(-0.0821085\pi\)
\(422\) 0 0
\(423\) −37.4228 −0.0884699
\(424\) 0 0
\(425\) 249.752 0.587653
\(426\) 0 0
\(427\) 353.821 0.828622
\(428\) 0 0
\(429\) −6.19934 −0.0144507
\(430\) 0 0
\(431\) 469.547i 1.08944i −0.838619 0.544718i \(-0.816637\pi\)
0.838619 0.544718i \(-0.183363\pi\)
\(432\) 0 0
\(433\) 175.110i 0.404411i −0.979343 0.202205i \(-0.935189\pi\)
0.979343 0.202205i \(-0.0648108\pi\)
\(434\) 0 0
\(435\) 282.997 0.650567
\(436\) 0 0
\(437\) 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 \(0.787830\pi\)
\(440\) 0 0
\(441\) −305.021 −0.691657
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 124.148i 0.277736i
\(448\) 0 0
\(449\) 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
\(452\) 0 0
\(453\) 119.347 0.263460
\(454\) 0 0
\(455\) 1002.79i 2.20393i
\(456\) 0 0
\(457\) −42.7218 −0.0934831 −0.0467415 0.998907i \(-0.514884\pi\)
−0.0467415 + 0.998907i \(0.514884\pi\)
\(458\) 0 0
\(459\) 90.2810i 0.196691i
\(460\) 0 0
\(461\) 39.6221 0.0859482 0.0429741 0.999076i \(-0.486317\pi\)
0.0429741 + 0.999076i \(0.486317\pi\)
\(462\) 0 0
\(463\) 158.625 0.342603 0.171302 0.985219i \(-0.445203\pi\)
0.171302 + 0.985219i \(0.445203\pi\)
\(464\) 0 0
\(465\) 255.698 0.549887
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) 273.140i 0.579916i
\(472\) 0 0
\(473\) −13.2300 −0.0279705
\(474\) 0 0
\(475\) −108.526 250.629i −0.228475 0.527641i
\(476\) 0 0
\(477\) 77.4244i 0.162315i
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) 436.906i 0.904566i
\(484\) 0 0
\(485\) 259.399i 0.534843i
\(486\) 0 0
\(487\) 274.616i 0.563894i −0.959430 0.281947i \(-0.909020\pi\)
0.959430 0.281947i \(-0.0909802\pi\)
\(488\) 0 0
\(489\) 73.4367i 0.150177i
\(490\) 0 0
\(491\) 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
\(494\) 0 0
\(495\) −5.17525 −0.0104550
\(496\) 0 0
\(497\) 1322.41i 2.66078i
\(498\) 0 0
\(499\) 387.918 0.777390 0.388695 0.921366i \(-0.372926\pi\)
0.388695 + 0.921366i \(0.372926\pi\)
\(500\) 0 0
\(501\) −175.904 −0.351105
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 0.863158i 0.00170248i
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) −90.5980 + 39.2301i −0.176604 + 0.0764719i
\(514\) 0 0
\(515\) 143.294i 0.278241i
\(516\) 0 0
\(517\) −3.42939 −0.00663324
\(518\) 0 0
\(519\) 42.1993 0.0813089
\(520\) 0 0
\(521\) 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 \(0.935897\pi\)
\(524\) 0 0
\(525\) 305.615i 0.582124i
\(526\) 0 0
\(527\) 408.764i 0.775643i
\(528\) 0 0
\(529\) −106.704 −0.201709
\(530\) 0 0
\(531\) 0.690526i 0.00130043i
\(532\) 0 0
\(533\) 46.5980 0.0874259
\(534\) 0 0
\(535\) 1035.75i 1.93599i
\(536\) 0 0
\(537\) 129.498 0.241151
\(538\) 0 0
\(539\) −27.9518 −0.0518587
\(540\) 0 0
\(541\) 989.664 1.82932 0.914661 0.404221i \(-0.132457\pi\)
0.914661 + 0.404221i \(0.132457\pi\)
\(542\) 0 0
\(543\) −49.0515 −0.0903342
\(544\) 0 0
\(545\) 291.445i 0.534761i
\(546\) 0 0
\(547\) 851.749i 1.55713i −0.627565 0.778564i \(-0.715948\pi\)
0.627565 0.778564i \(-0.284052\pi\)
\(548\) 0 0
\(549\) 86.4743 0.157512
\(550\) 0 0
\(551\) 453.993 196.585i 0.823944 0.356778i
\(552\) 0 0
\(553\) 327.070i 0.591446i
\(554\) 0 0
\(555\) 725.691 1.30755
\(556\) 0 0
\(557\) 125.423 0.225176 0.112588 0.993642i \(-0.464086\pi\)
0.112588 + 0.993642i \(0.464086\pi\)
\(558\) 0 0
\(559\) 626.531i 1.12081i
\(560\) 0 0
\(561\) 8.27327i 0.0147474i
\(562\) 0 0
\(563\) 762.666i 1.35465i 0.735685 + 0.677324i \(0.236860\pi\)
−0.735685 + 0.677324i \(0.763140\pi\)
\(564\) 0 0
\(565\) 28.6335i 0.0506788i
\(566\) 0 0
\(567\) −110.474 −0.194840
\(568\) 0 0
\(569\) 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 \(-0.453709\pi\)
0.144914 + 0.989444i \(0.453709\pi\)
\(572\) 0 0
\(573\) 412.181i 0.719338i
\(574\) 0 0
\(575\) −295.395 −0.513731
\(576\) 0 0
\(577\) 852.262 1.47706 0.738528 0.674222i \(-0.235521\pi\)
0.738528 + 0.674222i \(0.235521\pi\)
\(578\) 0 0
\(579\) −432.797 −0.747491
\(580\) 0 0
\(581\) −1101.03 −1.89506
\(582\) 0 0
\(583\) 7.09510i 0.0121700i
\(584\) 0 0
\(585\) 245.082i 0.418944i
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) 438.643i 0.742205i
\(592\) 0 0
\(593\) −243.100 −0.409949 −0.204974 0.978767i \(-0.565711\pi\)
−0.204974 + 0.978767i \(0.565711\pi\)
\(594\) 0 0
\(595\) 1338.26 2.24918
\(596\) 0 0
\(597\) 83.1859i 0.139340i
\(598\) 0 0
\(599\) 725.096i 1.21051i −0.796031 0.605256i \(-0.793071\pi\)
0.796031 0.605256i \(-0.206929\pi\)
\(600\) 0 0
\(601\) 595.929i 0.991563i 0.868447 + 0.495781i \(0.165118\pi\)
−0.868447 + 0.495781i \(0.834882\pi\)
\(602\) 0 0
\(603\) 308.817i 0.512135i
\(604\) 0 0
\(605\) 758.791 1.25420
\(606\) 0 0
\(607\) 724.886i 1.19421i −0.802163 0.597106i \(-0.796317\pi\)
0.802163 0.597106i \(-0.203683\pi\)
\(608\) 0 0
\(609\) 553.595 0.909022
\(610\) 0 0
\(611\) 162.404i 0.265801i
\(612\) 0 0
\(613\) 499.368 0.814630 0.407315 0.913288i \(-0.366465\pi\)
0.407315 + 0.913288i \(0.366465\pi\)
\(614\) 0 0
\(615\) 38.9003 0.0632526
\(616\) 0 0
\(617\) −87.2300 −0.141378 −0.0706889 0.997498i \(-0.522520\pi\)
−0.0706889 + 0.997498i \(0.522520\pi\)
\(618\) 0 0
\(619\) 846.482 1.36750 0.683749 0.729717i \(-0.260348\pi\)
0.683749 + 0.729717i \(0.260348\pi\)
\(620\) 0 0
\(621\) 106.780i 0.171949i
\(622\) 0 0
\(623\) 1709.34i 2.74372i
\(624\) 0 0
\(625\) −777.736 −1.24438
\(626\) 0 0
\(627\) −8.30232 + 3.59501i −0.0132413 + 0.00573367i
\(628\) 0 0
\(629\) 1160.11i 1.84437i
\(630\) 0 0
\(631\) 800.509 1.26864 0.634318 0.773072i \(-0.281281\pi\)
0.634318 + 0.773072i \(0.281281\pi\)
\(632\) 0 0
\(633\) −481.993 −0.761443
\(634\) 0 0
\(635\) 921.094i 1.45054i
\(636\) 0 0
\(637\) 1323.70i 2.07803i
\(638\) 0 0
\(639\) 323.197i 0.505786i
\(640\) 0 0
\(641\) 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 \(-0.892053\pi\)
−0.943046 + 0.332663i \(0.892053\pi\)
\(644\) 0 0
\(645\) 523.032i 0.810902i
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) 500.193 0.768345
\(652\) 0 0
\(653\) 726.220 1.11213 0.556064 0.831139i \(-0.312311\pi\)
0.556064 + 0.831139i \(0.312311\pi\)
\(654\) 0 0
\(655\) −1185.46 −1.80987
\(656\) 0 0
\(657\) −33.5257 −0.0510285
\(658\) 0 0
\(659\) 259.756i 0.394167i −0.980387 0.197083i \(-0.936853\pi\)
0.980387 0.197083i \(-0.0631469\pi\)
\(660\) 0 0
\(661\) 874.021i 1.32227i −0.750267 0.661135i \(-0.770075\pi\)
0.750267 0.661135i \(-0.229925\pi\)
\(662\) 0 0
\(663\) 391.794 0.590941
\(664\) 0 0
\(665\) −581.519 1342.96i −0.874465 2.01949i
\(666\) 0 0
\(667\) 535.083i 0.802223i
\(668\) 0 0
\(669\) 117.244 0.175253
\(670\) 0 0
\(671\) 7.92442 0.0118099
\(672\) 0 0
\(673\) 1035.99i 1.53935i 0.638434 + 0.769677i \(0.279583\pi\)
−0.638434 + 0.769677i \(0.720417\pi\)
\(674\) 0 0
\(675\) 74.6925i 0.110656i
\(676\) 0 0
\(677\) 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
\(680\) 0 0
\(681\) −708.797 −1.04082
\(682\) 0 0
\(683\) 1186.44i 1.73710i 0.495604 + 0.868549i \(0.334947\pi\)
−0.495604 + 0.868549i \(0.665053\pi\)
\(684\) 0 0
\(685\) 447.567 0.653383
\(686\) 0 0
\(687\) 426.811i 0.621268i
\(688\) 0 0
\(689\) −336.000 −0.487663
\(690\) 0 0
\(691\) −563.670 −0.815731 −0.407866 0.913042i \(-0.633727\pi\)
−0.407866 + 0.913042i \(0.633727\pi\)
\(692\) 0 0
\(693\) −10.1238 −0.0146086
\(694\) 0 0
\(695\) 273.423 0.393414
\(696\) 0 0
\(697\) 62.1869i 0.0892209i
\(698\) 0 0
\(699\) 392.003i 0.560806i
\(700\) 0 0
\(701\) −163.958 −0.233892 −0.116946 0.993138i \(-0.537310\pi\)
−0.116946 + 0.993138i \(0.537310\pi\)
\(702\) 0 0
\(703\) 1164.18 504.104i 1.65602 0.717076i
\(704\) 0 0
\(705\) 135.576i 0.192307i
\(706\) 0 0
\(707\) −1850.43 −2.61730
\(708\) 0 0
\(709\) −315.993 −0.445689 −0.222844 0.974854i \(-0.571534\pi\)
−0.222844 + 0.974854i \(0.571534\pi\)
\(710\) 0 0
\(711\) 79.9362i 0.112428i
\(712\) 0 0
\(713\) 483.467i 0.678074i
\(714\) 0 0
\(715\) 22.4591i 0.0314114i
\(716\) 0 0
\(717\) 119.720i 0.166974i
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 187.904 0.259894
\(724\) 0 0
\(725\) 374.290i 0.516262i
\(726\) 0 0
\(727\) −261.169 −0.359242 −0.179621 0.983736i \(-0.557487\pi\)
−0.179621 + 0.983736i \(0.557487\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 836.130 1.14382
\(732\) 0 0
\(733\) 1185.78 1.61771 0.808855 0.588009i \(-0.200088\pi\)
0.808855 + 0.588009i \(0.200088\pi\)
\(734\) 0 0
\(735\) 1105.04i 1.50345i
\(736\) 0 0
\(737\) 28.2997i 0.0383985i
\(738\) 0 0
\(739\) −738.557 −0.999401 −0.499701 0.866198i \(-0.666557\pi\)
−0.499701 + 0.866198i \(0.666557\pi\)
\(740\) 0 0
\(741\) −170.248 393.170i −0.229754 0.530594i
\(742\) 0 0
\(743\) 809.783i 1.08988i 0.838474 + 0.544941i \(0.183448\pi\)
−0.838474 + 0.544941i \(0.816552\pi\)
\(744\) 0 0
\(745\) 449.767 0.603714
\(746\) 0 0
\(747\) −269.093 −0.360232
\(748\) 0 0
\(749\) 2026.13i 2.70512i
\(750\) 0 0
\(751\) 1167.89i 1.55512i −0.628809 0.777559i \(-0.716457\pi\)
0.628809 0.777559i \(-0.283543\pi\)
\(752\) 0 0
\(753\) 81.0194i 0.107595i
\(754\) 0 0
\(755\) 432.374i 0.572681i
\(756\) 0 0
\(757\) −161.828 −0.213776 −0.106888 0.994271i \(-0.534089\pi\)
−0.106888 + 0.994271i \(0.534089\pi\)
\(758\) 0 0
\(759\) 9.78523i 0.0128923i
\(760\) 0 0
\(761\) 1148.47 1.50916 0.754578 0.656210i \(-0.227842\pi\)
0.754578 + 0.656210i \(0.227842\pi\)
\(762\) 0 0
\(763\) 570.121i 0.747210i
\(764\) 0 0
\(765\) 327.072 0.427545
\(766\) 0 0
\(767\) −2.99669 −0.00390703
\(768\) 0 0
\(769\) 77.5673 0.100868 0.0504339 0.998727i \(-0.483940\pi\)
0.0504339 + 0.998727i \(0.483940\pi\)
\(770\) 0 0
\(771\) −376.096 −0.487803
\(772\) 0 0
\(773\) 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
\(776\) 0 0
\(777\) 1419.59 1.82701
\(778\) 0 0
\(779\) 62.4053 27.0223i 0.0801095 0.0346884i
\(780\) 0 0
\(781\) 29.6175i 0.0379225i
\(782\) 0 0
\(783\) 135.299 0.172796
\(784\) 0 0
\(785\) −989.540 −1.26056
\(786\) 0 0
\(787\) 1141.88i 1.45092i 0.688263 + 0.725461i \(0.258374\pi\)
−0.688263 + 0.725461i \(0.741626\pi\)
\(788\) 0 0
\(789\) 60.4132i 0.0765693i
\(790\) 0 0
\(791\) 56.0125i 0.0708123i
\(792\) 0 0
\(793\) 375.274i 0.473233i
\(794\) 0 0
\(795\) −280.495 −0.352824
\(796\) 0 0
\(797\) 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
\(800\) 0 0
\(801\) 417.764i 0.521553i
\(802\) 0 0
\(803\) −3.07227 −0.00382599
\(804\) 0 0
\(805\) −1582.83 −1.96625
\(806\) 0 0
\(807\) 156.289 0.193667
\(808\) 0 0
\(809\) 879.175 1.08674 0.543372 0.839492i \(-0.317147\pi\)
0.543372 + 0.839492i \(0.317147\pi\)
\(810\) 0 0
\(811\) 654.537i 0.807074i 0.914963 + 0.403537i \(0.132219\pi\)
−0.914963 + 0.403537i \(0.867781\pi\)
\(812\) 0 0
\(813\) 251.142i 0.308907i
\(814\) 0 0
\(815\) −266.048 −0.326439
\(816\) 0 0
\(817\) −363.326 839.066i −0.444708 1.02701i
\(818\) 0 0
\(819\) 479.427i 0.585381i
\(820\) 0 0
\(821\) −1419.35 −1.72880 −0.864402 0.502801i \(-0.832303\pi\)
−0.864402 + 0.502801i \(0.832303\pi\)
\(822\) 0 0
\(823\) 671.107 0.815440 0.407720 0.913107i \(-0.366324\pi\)
0.407720 + 0.913107i \(0.366324\pi\)
\(824\) 0 0
\(825\) 6.84476i 0.00829667i
\(826\) 0 0
\(827\) 186.204i 0.225156i −0.993643 0.112578i \(-0.964089\pi\)
0.993643 0.112578i \(-0.0359108\pi\)
\(828\) 0 0
\(829\) 924.130i 1.11475i −0.830260 0.557376i \(-0.811808\pi\)
0.830260 0.557376i \(-0.188192\pi\)
\(830\) 0 0
\(831\) 859.740i 1.03459i
\(832\) 0 0
\(833\) 1766.54 2.12069
\(834\) 0 0
\(835\) 637.268i 0.763195i
\(836\) 0 0
\(837\) 122.248 0.146054
\(838\) 0 0
\(839\) 934.157i 1.11342i 0.830708 + 0.556708i \(0.187936\pi\)
−0.830708 + 0.556708i \(0.812064\pi\)
\(840\) 0 0
\(841\) 163.007 0.193825
\(842\) 0 0
\(843\) 324.688 0.385157
\(844\) 0 0
\(845\) 3.12707 0.00370067
\(846\) 0 0
\(847\) 1484.34 1.75246
\(848\) 0 0
\(849\) 89.3287i 0.105216i
\(850\) 0 0
\(851\) 1372.12i 1.61236i
\(852\) 0 0
\(853\) 344.405 0.403758 0.201879 0.979411i \(-0.435295\pi\)
0.201879 + 0.979411i \(0.435295\pi\)
\(854\) 0 0
\(855\) −142.124 328.221i −0.166227 0.383884i
\(856\) 0 0
\(857\) 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 \(-0.285341\pi\)
0.624405 + 0.781100i \(0.285341\pi\)
\(860\) 0 0
\(861\) 76.0964 0.0883814
\(862\) 0 0
\(863\) 1240.65i 1.43760i 0.695217 + 0.718800i \(0.255308\pi\)
−0.695217 + 0.718800i \(0.744692\pi\)
\(864\) 0 0
\(865\) 152.881i 0.176741i
\(866\) 0 0
\(867\) 22.3023i 0.0257235i
\(868\) 0 0
\(869\) 7.32527i 0.00842955i
\(870\) 0 0
\(871\) 1340.18 1.53867
\(872\) 0 0
\(873\) 124.017i 0.142059i
\(874\) 0 0
\(875\) −818.413 −0.935329
\(876\) 0 0
\(877\) 526.564i 0.600415i −0.953874 0.300207i \(-0.902944\pi\)
0.953874 0.300207i \(-0.0970559\pi\)
\(878\) 0 0
\(879\) −143.203 −0.162915
\(880\) 0 0
\(881\) 1444.25 1.63933 0.819664 0.572844i \(-0.194160\pi\)
0.819664 + 0.572844i \(0.194160\pi\)
\(882\) 0 0
\(883\) −593.086 −0.671671 −0.335836 0.941921i \(-0.609019\pi\)
−0.335836 + 0.941921i \(0.609019\pi\)
\(884\) 0 0
\(885\) −2.50166 −0.00282673
\(886\) 0 0
\(887\) 589.691i 0.664816i 0.943136 + 0.332408i \(0.107861\pi\)
−0.943136 + 0.332408i \(0.892139\pi\)
\(888\) 0 0
\(889\) 1801.83i 2.02681i
\(890\) 0 0
\(891\) −2.47425 −0.00277694
\(892\) 0 0
\(893\) −94.1786 217.496i −0.105463 0.243557i
\(894\) 0 0
\(895\) 469.150i 0.524190i
\(896\) 0 0
\(897\) −463.395 −0.516606
\(898\) 0 0
\(899\) −612.591 −0.681414
\(900\) 0 0
\(901\) 448.406i 0.497675i
\(902\) 0 0
\(903\) 1023.15i 1.13305i
\(904\) 0 0
\(905\) 177.705i 0.196359i
\(906\) 0 0
\(907\) 164.499i 0.181366i 0.995880 + 0.0906829i \(0.0289050\pi\)
−0.995880 + 0.0906829i \(0.971095\pi\)
\(908\) 0 0
\(909\) −452.248 −0.497522
\(910\) 0 0
\(911\) 474.507i 0.520864i 0.965492 + 0.260432i \(0.0838650\pi\)
−0.965492 + 0.260432i \(0.916135\pi\)
\(912\) 0 0
\(913\) −24.6594 −0.0270092
\(914\) 0 0
\(915\) 313.281i 0.342384i
\(916\) 0 0
\(917\) −2318.99 −2.52889
\(918\) 0 0
\(919\) −708.488 −0.770934 −0.385467 0.922722i \(-0.625960\pi\)
−0.385467 + 0.922722i \(0.625960\pi\)
\(920\) 0 0
\(921\) −144.289 −0.156666
\(922\) 0 0
\(923\) −1402.58 −1.51959
\(924\) 0 0
\(925\) 959.795i 1.03762i
\(926\) 0 0
\(927\) 68.5081i 0.0739030i
\(928\) 0 0
\(929\) 154.482 0.166288 0.0831441 0.996538i \(-0.473504\pi\)
0.0831441 + 0.996538i \(0.473504\pi\)
\(930\) 0 0
\(931\) −767.619 1772.74i −0.824510 1.90412i
\(932\) 0 0
\(933\) 7.32093i 0.00784665i
\(934\) 0 0
\(935\) 29.9726 0.0320563
\(936\) 0 0
\(937\) −289.223 −0.308670 −0.154335 0.988019i \(-0.549323\pi\)
−0.154335 + 0.988019i \(0.549323\pi\)
\(938\) 0 0
\(939\) 275.212i 0.293090i
\(940\) 0 0
\(941\) 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
\(944\) 0 0
\(945\) 400.229i 0.423523i
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) 482.502 0.507362
\(952\) 0 0
\(953\) 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
\(956\) 0 0
\(957\) 12.3987 0.0129558
\(958\) 0 0
\(959\) 875.526 0.912957
\(960\) 0 0
\(961\) 407.502 0.424039
\(962\) 0 0
\(963\) 495.188i 0.514214i
\(964\) 0 0
\(965\) 1567.95i 1.62482i
\(966\) 0 0
\(967\) −980.296 −1.01375 −0.506875 0.862020i \(-0.669199\pi\)
−0.506875 + 0.862020i \(0.669199\pi\)
\(968\) 0 0
\(969\) 524.701 227.202i 0.541487 0.234471i
\(970\) 0 0
\(971\) 1699.98i 1.75075i −0.483441 0.875377i \(-0.660613\pi\)
0.483441 0.875377i \(-0.339387\pi\)
\(972\) 0 0
\(973\) 534.866 0.549708
\(974\) 0 0
\(975\) 324.145 0.332456
\(976\) 0 0
\(977\) 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
\(980\) 0 0
\(981\) 139.338i 0.142037i
\(982\) 0 0
\(983\) 541.654i 0.551022i 0.961298 + 0.275511i \(0.0888470\pi\)
−0.961298 + 0.275511i \(0.911153\pi\)
\(984\) 0 0
\(985\) −1589.13 −1.61333
\(986\) 0 0
\(987\) 265.212i 0.268706i
\(988\) 0 0
\(989\) −988.935 −0.999935
\(990\) 0 0
\(991\) 1546.81i 1.56086i 0.625244 + 0.780429i \(0.284999\pi\)
−0.625244 + 0.780429i \(0.715001\pi\)
\(992\) 0 0
\(993\) −230.502 −0.232127
\(994\) 0 0
\(995\) −301.368 −0.302882
\(996\) 0 0
\(997\) −1366.20 −1.37031 −0.685156 0.728397i \(-0.740266\pi\)
−0.685156 + 0.728397i \(0.740266\pi\)
\(998\) 0 0
\(999\) 346.949 0.347296
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 912.3.o.b.721.3 4
3.2 odd 2 2736.3.o.l.721.3 4
4.3 odd 2 57.3.c.b.37.1 4
12.11 even 2 171.3.c.f.37.4 4
19.18 odd 2 inner 912.3.o.b.721.1 4
57.56 even 2 2736.3.o.l.721.4 4
76.75 even 2 57.3.c.b.37.4 yes 4
228.227 odd 2 171.3.c.f.37.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.3.c.b.37.1 4 4.3 odd 2
57.3.c.b.37.4 yes 4 76.75 even 2
171.3.c.f.37.1 4 228.227 odd 2
171.3.c.f.37.4 4 12.11 even 2
912.3.o.b.721.1 4 19.18 odd 2 inner
912.3.o.b.721.3 4 1.1 even 1 trivial
2736.3.o.l.721.3 4 3.2 odd 2
2736.3.o.l.721.4 4 57.56 even 2