Properties

Label 1200.3.l.x.401.8
Level $1200$
Weight $3$
Character 1200.401
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
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] = [1200,3,Mod(401,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.681615360000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.8
Root \(1.54294 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1200.401
Dual form 1200.3.l.x.401.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.98254 + 0.323191i) q^{3} +4.72640 q^{7} +(8.79110 + 1.92786i) q^{9} +O(q^{10})\) \(q+(2.98254 + 0.323191i) q^{3} +4.72640 q^{7} +(8.79110 + 1.92786i) q^{9} +4.76442i q^{11} +1.06692 q^{13} -26.7847i q^{17} +8.12938 q^{19} +(14.0967 + 1.52753i) q^{21} -40.0468i q^{23} +(25.5967 + 8.59112i) q^{27} +20.8744i q^{29} +33.7860 q^{31} +(-1.53982 + 14.2101i) q^{33} +60.4351 q^{37} +(3.18213 + 0.344819i) q^{39} +59.2611i q^{41} -56.4424 q^{43} -9.68942i q^{47} -26.6611 q^{49} +(8.65657 - 79.8864i) q^{51} +93.1378i q^{53} +(24.2462 + 2.62734i) q^{57} -17.4907i q^{59} +57.7400 q^{61} +(41.5503 + 9.11184i) q^{63} +101.531 q^{67} +(12.9428 - 119.441i) q^{69} -90.1745i q^{71} -40.0700 q^{73} +22.5186i q^{77} -65.3727 q^{79} +(73.5667 + 33.8960i) q^{81} -117.888i q^{83} +(-6.74641 + 62.2587i) q^{87} +119.679i q^{89} +5.04269 q^{91} +(100.768 + 10.9193i) q^{93} +15.2522 q^{97} +(-9.18513 + 41.8845i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} + 8 q^{13} + 8 q^{19} + 28 q^{21} + 20 q^{27} - 120 q^{31} + 112 q^{33} - 8 q^{37} + 72 q^{39} - 328 q^{43} + 64 q^{49} - 64 q^{51} - 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} - 32 q^{73} - 88 q^{79} + 224 q^{81} - 152 q^{87} - 560 q^{91} + 368 q^{93} - 144 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.98254 + 0.323191i 0.994180 + 0.107730i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.72640 0.675201 0.337600 0.941290i \(-0.390385\pi\)
0.337600 + 0.941290i \(0.390385\pi\)
\(8\) 0 0
\(9\) 8.79110 + 1.92786i 0.976788 + 0.214207i
\(10\) 0 0
\(11\) 4.76442i 0.433129i 0.976268 + 0.216565i \(0.0694852\pi\)
−0.976268 + 0.216565i \(0.930515\pi\)
\(12\) 0 0
\(13\) 1.06692 0.0820707 0.0410354 0.999158i \(-0.486934\pi\)
0.0410354 + 0.999158i \(0.486934\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 26.7847i 1.57557i −0.615950 0.787785i \(-0.711228\pi\)
0.615950 0.787785i \(-0.288772\pi\)
\(18\) 0 0
\(19\) 8.12938 0.427862 0.213931 0.976849i \(-0.431373\pi\)
0.213931 + 0.976849i \(0.431373\pi\)
\(20\) 0 0
\(21\) 14.0967 + 1.52753i 0.671271 + 0.0727395i
\(22\) 0 0
\(23\) 40.0468i 1.74117i −0.492021 0.870583i \(-0.663742\pi\)
0.492021 0.870583i \(-0.336258\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 25.5967 + 8.59112i 0.948027 + 0.318190i
\(28\) 0 0
\(29\) 20.8744i 0.719807i 0.932990 + 0.359903i \(0.117190\pi\)
−0.932990 + 0.359903i \(0.882810\pi\)
\(30\) 0 0
\(31\) 33.7860 1.08987 0.544936 0.838478i \(-0.316554\pi\)
0.544936 + 0.838478i \(0.316554\pi\)
\(32\) 0 0
\(33\) −1.53982 + 14.2101i −0.0466611 + 0.430608i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 60.4351 1.63338 0.816691 0.577075i \(-0.195806\pi\)
0.816691 + 0.577075i \(0.195806\pi\)
\(38\) 0 0
\(39\) 3.18213 + 0.344819i 0.0815931 + 0.00884150i
\(40\) 0 0
\(41\) 59.2611i 1.44539i 0.691166 + 0.722696i \(0.257097\pi\)
−0.691166 + 0.722696i \(0.742903\pi\)
\(42\) 0 0
\(43\) −56.4424 −1.31261 −0.656307 0.754494i \(-0.727883\pi\)
−0.656307 + 0.754494i \(0.727883\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.68942i 0.206158i −0.994673 0.103079i \(-0.967131\pi\)
0.994673 0.103079i \(-0.0328694\pi\)
\(48\) 0 0
\(49\) −26.6611 −0.544104
\(50\) 0 0
\(51\) 8.65657 79.8864i 0.169737 1.56640i
\(52\) 0 0
\(53\) 93.1378i 1.75732i 0.477451 + 0.878659i \(0.341561\pi\)
−0.477451 + 0.878659i \(0.658439\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.2462 + 2.62734i 0.425372 + 0.0460937i
\(58\) 0 0
\(59\) 17.4907i 0.296453i −0.988953 0.148227i \(-0.952643\pi\)
0.988953 0.148227i \(-0.0473565\pi\)
\(60\) 0 0
\(61\) 57.7400 0.946558 0.473279 0.880913i \(-0.343070\pi\)
0.473279 + 0.880913i \(0.343070\pi\)
\(62\) 0 0
\(63\) 41.5503 + 9.11184i 0.659528 + 0.144632i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 101.531 1.51539 0.757695 0.652609i \(-0.226325\pi\)
0.757695 + 0.652609i \(0.226325\pi\)
\(68\) 0 0
\(69\) 12.9428 119.441i 0.187576 1.73103i
\(70\) 0 0
\(71\) 90.1745i 1.27006i −0.772486 0.635032i \(-0.780987\pi\)
0.772486 0.635032i \(-0.219013\pi\)
\(72\) 0 0
\(73\) −40.0700 −0.548904 −0.274452 0.961601i \(-0.588496\pi\)
−0.274452 + 0.961601i \(0.588496\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 22.5186i 0.292449i
\(78\) 0 0
\(79\) −65.3727 −0.827502 −0.413751 0.910390i \(-0.635782\pi\)
−0.413751 + 0.910390i \(0.635782\pi\)
\(80\) 0 0
\(81\) 73.5667 + 33.8960i 0.908231 + 0.418469i
\(82\) 0 0
\(83\) 117.888i 1.42033i −0.704034 0.710166i \(-0.748620\pi\)
0.704034 0.710166i \(-0.251380\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.74641 + 62.2587i −0.0775450 + 0.715618i
\(88\) 0 0
\(89\) 119.679i 1.34471i 0.740228 + 0.672356i \(0.234718\pi\)
−0.740228 + 0.672356i \(0.765282\pi\)
\(90\) 0 0
\(91\) 5.04269 0.0554142
\(92\) 0 0
\(93\) 100.768 + 10.9193i 1.08353 + 0.117412i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.2522 0.157239 0.0786196 0.996905i \(-0.474949\pi\)
0.0786196 + 0.996905i \(0.474949\pi\)
\(98\) 0 0
\(99\) −9.18513 + 41.8845i −0.0927791 + 0.423075i
\(100\) 0 0
\(101\) 72.0047i 0.712918i −0.934311 0.356459i \(-0.883984\pi\)
0.934311 0.356459i \(-0.116016\pi\)
\(102\) 0 0
\(103\) 110.950 1.07718 0.538591 0.842567i \(-0.318957\pi\)
0.538591 + 0.842567i \(0.318957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 30.6020i 0.286000i −0.989723 0.143000i \(-0.954325\pi\)
0.989723 0.143000i \(-0.0456748\pi\)
\(108\) 0 0
\(109\) −17.3694 −0.159353 −0.0796763 0.996821i \(-0.525389\pi\)
−0.0796763 + 0.996821i \(0.525389\pi\)
\(110\) 0 0
\(111\) 180.250 + 19.5321i 1.62388 + 0.175965i
\(112\) 0 0
\(113\) 4.71526i 0.0417280i −0.999782 0.0208640i \(-0.993358\pi\)
0.999782 0.0208640i \(-0.00664170\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.37939 + 2.05687i 0.0801657 + 0.0175801i
\(118\) 0 0
\(119\) 126.595i 1.06383i
\(120\) 0 0
\(121\) 98.3003 0.812399
\(122\) 0 0
\(123\) −19.1526 + 176.749i −0.155712 + 1.43698i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.39528 −0.0109865 −0.00549325 0.999985i \(-0.501749\pi\)
−0.00549325 + 0.999985i \(0.501749\pi\)
\(128\) 0 0
\(129\) −168.342 18.2417i −1.30498 0.141408i
\(130\) 0 0
\(131\) 226.220i 1.72687i 0.504460 + 0.863435i \(0.331692\pi\)
−0.504460 + 0.863435i \(0.668308\pi\)
\(132\) 0 0
\(133\) 38.4227 0.288893
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 76.2589i 0.556634i −0.960489 0.278317i \(-0.910223\pi\)
0.960489 0.278317i \(-0.0897766\pi\)
\(138\) 0 0
\(139\) −127.660 −0.918417 −0.459208 0.888329i \(-0.651867\pi\)
−0.459208 + 0.888329i \(0.651867\pi\)
\(140\) 0 0
\(141\) 3.13153 28.8991i 0.0222094 0.204958i
\(142\) 0 0
\(143\) 5.08325i 0.0355472i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −79.5178 8.61662i −0.540938 0.0586165i
\(148\) 0 0
\(149\) 63.3790i 0.425362i 0.977122 + 0.212681i \(0.0682195\pi\)
−0.977122 + 0.212681i \(0.931780\pi\)
\(150\) 0 0
\(151\) 115.233 0.763134 0.381567 0.924341i \(-0.375385\pi\)
0.381567 + 0.924341i \(0.375385\pi\)
\(152\) 0 0
\(153\) 51.6371 235.467i 0.337498 1.53900i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.3695 0.117003 0.0585017 0.998287i \(-0.481368\pi\)
0.0585017 + 0.998287i \(0.481368\pi\)
\(158\) 0 0
\(159\) −30.1013 + 277.787i −0.189316 + 1.74709i
\(160\) 0 0
\(161\) 189.278i 1.17564i
\(162\) 0 0
\(163\) −163.693 −1.00425 −0.502126 0.864794i \(-0.667449\pi\)
−0.502126 + 0.864794i \(0.667449\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 48.7025i 0.291632i 0.989312 + 0.145816i \(0.0465807\pi\)
−0.989312 + 0.145816i \(0.953419\pi\)
\(168\) 0 0
\(169\) −167.862 −0.993264
\(170\) 0 0
\(171\) 71.4662 + 15.6723i 0.417931 + 0.0916509i
\(172\) 0 0
\(173\) 140.785i 0.813787i −0.913476 0.406894i \(-0.866612\pi\)
0.913476 0.406894i \(-0.133388\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.65285 52.1669i 0.0319370 0.294728i
\(178\) 0 0
\(179\) 11.1350i 0.0622065i −0.999516 0.0311033i \(-0.990098\pi\)
0.999516 0.0311033i \(-0.00990207\pi\)
\(180\) 0 0
\(181\) −150.235 −0.830028 −0.415014 0.909815i \(-0.636223\pi\)
−0.415014 + 0.909815i \(0.636223\pi\)
\(182\) 0 0
\(183\) 172.212 + 18.6610i 0.941049 + 0.101973i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 127.614 0.682425
\(188\) 0 0
\(189\) 120.981 + 40.6051i 0.640108 + 0.214842i
\(190\) 0 0
\(191\) 168.060i 0.879895i 0.898023 + 0.439948i \(0.145003\pi\)
−0.898023 + 0.439948i \(0.854997\pi\)
\(192\) 0 0
\(193\) −312.926 −1.62138 −0.810688 0.585479i \(-0.800907\pi\)
−0.810688 + 0.585479i \(0.800907\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 166.150i 0.843400i 0.906735 + 0.421700i \(0.138566\pi\)
−0.906735 + 0.421700i \(0.861434\pi\)
\(198\) 0 0
\(199\) 105.535 0.530327 0.265163 0.964204i \(-0.414574\pi\)
0.265163 + 0.964204i \(0.414574\pi\)
\(200\) 0 0
\(201\) 302.821 + 32.8139i 1.50657 + 0.163253i
\(202\) 0 0
\(203\) 98.6608i 0.486014i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 77.2046 352.056i 0.372969 1.70075i
\(208\) 0 0
\(209\) 38.7318i 0.185320i
\(210\) 0 0
\(211\) −283.373 −1.34300 −0.671500 0.741004i \(-0.734350\pi\)
−0.671500 + 0.741004i \(0.734350\pi\)
\(212\) 0 0
\(213\) 29.1436 268.949i 0.136824 1.26267i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 159.686 0.735882
\(218\) 0 0
\(219\) −119.510 12.9503i −0.545710 0.0591336i
\(220\) 0 0
\(221\) 28.5771i 0.129308i
\(222\) 0 0
\(223\) −100.108 −0.448915 −0.224458 0.974484i \(-0.572061\pi\)
−0.224458 + 0.974484i \(0.572061\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 239.794i 1.05636i −0.849132 0.528181i \(-0.822874\pi\)
0.849132 0.528181i \(-0.177126\pi\)
\(228\) 0 0
\(229\) 393.036 1.71632 0.858158 0.513386i \(-0.171609\pi\)
0.858158 + 0.513386i \(0.171609\pi\)
\(230\) 0 0
\(231\) −7.27779 + 67.1626i −0.0315056 + 0.290747i
\(232\) 0 0
\(233\) 278.691i 1.19610i 0.801459 + 0.598050i \(0.204058\pi\)
−0.801459 + 0.598050i \(0.795942\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −194.977 21.1278i −0.822686 0.0891470i
\(238\) 0 0
\(239\) 196.594i 0.822570i −0.911507 0.411285i \(-0.865080\pi\)
0.911507 0.411285i \(-0.134920\pi\)
\(240\) 0 0
\(241\) −231.153 −0.959139 −0.479570 0.877504i \(-0.659207\pi\)
−0.479570 + 0.877504i \(0.659207\pi\)
\(242\) 0 0
\(243\) 208.461 + 124.872i 0.857864 + 0.513877i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.67340 0.0351150
\(248\) 0 0
\(249\) 38.1002 351.604i 0.153013 1.41207i
\(250\) 0 0
\(251\) 243.442i 0.969887i −0.874545 0.484944i \(-0.838840\pi\)
0.874545 0.484944i \(-0.161160\pi\)
\(252\) 0 0
\(253\) 190.800 0.754150
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 358.482i 1.39487i 0.716646 + 0.697437i \(0.245676\pi\)
−0.716646 + 0.697437i \(0.754324\pi\)
\(258\) 0 0
\(259\) 285.641 1.10286
\(260\) 0 0
\(261\) −40.2429 + 183.509i −0.154187 + 0.703099i
\(262\) 0 0
\(263\) 330.913i 1.25823i 0.777314 + 0.629113i \(0.216582\pi\)
−0.777314 + 0.629113i \(0.783418\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −38.6793 + 356.949i −0.144866 + 1.33689i
\(268\) 0 0
\(269\) 97.0354i 0.360727i −0.983600 0.180363i \(-0.942273\pi\)
0.983600 0.180363i \(-0.0577273\pi\)
\(270\) 0 0
\(271\) −67.1851 −0.247915 −0.123958 0.992288i \(-0.539559\pi\)
−0.123958 + 0.992288i \(0.539559\pi\)
\(272\) 0 0
\(273\) 15.0400 + 1.62975i 0.0550917 + 0.00596979i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 361.801 1.30614 0.653070 0.757298i \(-0.273481\pi\)
0.653070 + 0.757298i \(0.273481\pi\)
\(278\) 0 0
\(279\) 297.016 + 65.1347i 1.06457 + 0.233458i
\(280\) 0 0
\(281\) 288.193i 1.02560i −0.858509 0.512798i \(-0.828609\pi\)
0.858509 0.512798i \(-0.171391\pi\)
\(282\) 0 0
\(283\) −272.474 −0.962805 −0.481402 0.876500i \(-0.659872\pi\)
−0.481402 + 0.876500i \(0.659872\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 280.092i 0.975930i
\(288\) 0 0
\(289\) −428.420 −1.48242
\(290\) 0 0
\(291\) 45.4903 + 4.92937i 0.156324 + 0.0169394i
\(292\) 0 0
\(293\) 70.5674i 0.240845i 0.992723 + 0.120422i \(0.0384248\pi\)
−0.992723 + 0.120422i \(0.961575\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −40.9317 + 121.954i −0.137817 + 0.410618i
\(298\) 0 0
\(299\) 42.7267i 0.142899i
\(300\) 0 0
\(301\) −266.770 −0.886278
\(302\) 0 0
\(303\) 23.2713 214.757i 0.0768028 0.708769i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −124.274 −0.404801 −0.202401 0.979303i \(-0.564874\pi\)
−0.202401 + 0.979303i \(0.564874\pi\)
\(308\) 0 0
\(309\) 330.912 + 35.8580i 1.07091 + 0.116045i
\(310\) 0 0
\(311\) 229.006i 0.736353i −0.929756 0.368176i \(-0.879982\pi\)
0.929756 0.368176i \(-0.120018\pi\)
\(312\) 0 0
\(313\) −465.490 −1.48719 −0.743594 0.668631i \(-0.766880\pi\)
−0.743594 + 0.668631i \(0.766880\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 405.458i 1.27905i −0.768771 0.639524i \(-0.779131\pi\)
0.768771 0.639524i \(-0.220869\pi\)
\(318\) 0 0
\(319\) −99.4544 −0.311769
\(320\) 0 0
\(321\) 9.89027 91.2716i 0.0308108 0.284335i
\(322\) 0 0
\(323\) 217.743i 0.674127i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −51.8051 5.61364i −0.158425 0.0171671i
\(328\) 0 0
\(329\) 45.7961i 0.139198i
\(330\) 0 0
\(331\) −45.9271 −0.138753 −0.0693763 0.997591i \(-0.522101\pi\)
−0.0693763 + 0.997591i \(0.522101\pi\)
\(332\) 0 0
\(333\) 531.291 + 116.510i 1.59547 + 0.349881i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −516.924 −1.53390 −0.766950 0.641707i \(-0.778226\pi\)
−0.766950 + 0.641707i \(0.778226\pi\)
\(338\) 0 0
\(339\) 1.52393 14.0635i 0.00449537 0.0414852i
\(340\) 0 0
\(341\) 160.971i 0.472055i
\(342\) 0 0
\(343\) −357.605 −1.04258
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.39280i 0.0155412i −0.999970 0.00777061i \(-0.997527\pi\)
0.999970 0.00777061i \(-0.00247349\pi\)
\(348\) 0 0
\(349\) 284.894 0.816315 0.408157 0.912912i \(-0.366171\pi\)
0.408157 + 0.912912i \(0.366171\pi\)
\(350\) 0 0
\(351\) 27.3097 + 9.16603i 0.0778053 + 0.0261141i
\(352\) 0 0
\(353\) 73.2882i 0.207615i 0.994597 + 0.103808i \(0.0331026\pi\)
−0.994597 + 0.103808i \(0.966897\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 40.9144 377.576i 0.114606 1.05763i
\(358\) 0 0
\(359\) 361.674i 1.00745i −0.863865 0.503724i \(-0.831963\pi\)
0.863865 0.503724i \(-0.168037\pi\)
\(360\) 0 0
\(361\) −294.913 −0.816934
\(362\) 0 0
\(363\) 293.185 + 31.7697i 0.807671 + 0.0875200i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −131.432 −0.358124 −0.179062 0.983838i \(-0.557306\pi\)
−0.179062 + 0.983838i \(0.557306\pi\)
\(368\) 0 0
\(369\) −114.247 + 520.970i −0.309612 + 1.41184i
\(370\) 0 0
\(371\) 440.207i 1.18654i
\(372\) 0 0
\(373\) −211.216 −0.566262 −0.283131 0.959081i \(-0.591373\pi\)
−0.283131 + 0.959081i \(0.591373\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.2713i 0.0590751i
\(378\) 0 0
\(379\) −47.4989 −0.125327 −0.0626635 0.998035i \(-0.519959\pi\)
−0.0626635 + 0.998035i \(0.519959\pi\)
\(380\) 0 0
\(381\) −4.16149 0.450943i −0.0109226 0.00118358i
\(382\) 0 0
\(383\) 517.991i 1.35246i −0.736692 0.676228i \(-0.763613\pi\)
0.736692 0.676228i \(-0.236387\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −496.191 108.813i −1.28215 0.281171i
\(388\) 0 0
\(389\) 253.951i 0.652830i 0.945227 + 0.326415i \(0.105841\pi\)
−0.945227 + 0.326415i \(0.894159\pi\)
\(390\) 0 0
\(391\) −1072.64 −2.74333
\(392\) 0 0
\(393\) −73.1122 + 674.710i −0.186036 + 1.71682i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −527.905 −1.32974 −0.664868 0.746961i \(-0.731512\pi\)
−0.664868 + 0.746961i \(0.731512\pi\)
\(398\) 0 0
\(399\) 114.597 + 12.4179i 0.287212 + 0.0311225i
\(400\) 0 0
\(401\) 664.097i 1.65610i 0.560653 + 0.828051i \(0.310550\pi\)
−0.560653 + 0.828051i \(0.689450\pi\)
\(402\) 0 0
\(403\) 36.0470 0.0894466
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 287.938i 0.707465i
\(408\) 0 0
\(409\) 79.2471 0.193758 0.0968791 0.995296i \(-0.469114\pi\)
0.0968791 + 0.995296i \(0.469114\pi\)
\(410\) 0 0
\(411\) 24.6462 227.445i 0.0599664 0.553395i
\(412\) 0 0
\(413\) 82.6683i 0.200165i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −380.751 41.2585i −0.913072 0.0989413i
\(418\) 0 0
\(419\) 666.530i 1.59076i 0.606109 + 0.795381i \(0.292729\pi\)
−0.606109 + 0.795381i \(0.707271\pi\)
\(420\) 0 0
\(421\) −306.220 −0.727364 −0.363682 0.931523i \(-0.618480\pi\)
−0.363682 + 0.931523i \(0.618480\pi\)
\(422\) 0 0
\(423\) 18.6798 85.1806i 0.0441604 0.201373i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 272.903 0.639116
\(428\) 0 0
\(429\) −1.64286 + 15.1610i −0.00382951 + 0.0353403i
\(430\) 0 0
\(431\) 254.551i 0.590605i −0.955404 0.295303i \(-0.904579\pi\)
0.955404 0.295303i \(-0.0954205\pi\)
\(432\) 0 0
\(433\) −442.391 −1.02169 −0.510845 0.859673i \(-0.670667\pi\)
−0.510845 + 0.859673i \(0.670667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 325.556i 0.744979i
\(438\) 0 0
\(439\) −561.568 −1.27920 −0.639599 0.768709i \(-0.720899\pi\)
−0.639599 + 0.768709i \(0.720899\pi\)
\(440\) 0 0
\(441\) −234.380 51.3988i −0.531475 0.116551i
\(442\) 0 0
\(443\) 564.841i 1.27504i −0.770436 0.637518i \(-0.779961\pi\)
0.770436 0.637518i \(-0.220039\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −20.4835 + 189.030i −0.0458244 + 0.422887i
\(448\) 0 0
\(449\) 512.215i 1.14079i −0.821370 0.570396i \(-0.806790\pi\)
0.821370 0.570396i \(-0.193210\pi\)
\(450\) 0 0
\(451\) −282.345 −0.626041
\(452\) 0 0
\(453\) 343.688 + 37.2423i 0.758692 + 0.0822126i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −52.5358 −0.114958 −0.0574790 0.998347i \(-0.518306\pi\)
−0.0574790 + 0.998347i \(0.518306\pi\)
\(458\) 0 0
\(459\) 230.110 685.601i 0.501330 1.49368i
\(460\) 0 0
\(461\) 625.737i 1.35735i 0.734441 + 0.678673i \(0.237445\pi\)
−0.734441 + 0.678673i \(0.762555\pi\)
\(462\) 0 0
\(463\) −49.8782 −0.107728 −0.0538641 0.998548i \(-0.517154\pi\)
−0.0538641 + 0.998548i \(0.517154\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 64.6312i 0.138397i 0.997603 + 0.0691983i \(0.0220441\pi\)
−0.997603 + 0.0691983i \(0.977956\pi\)
\(468\) 0 0
\(469\) 479.877 1.02319
\(470\) 0 0
\(471\) 54.7879 + 5.93686i 0.116322 + 0.0126048i
\(472\) 0 0
\(473\) 268.915i 0.568532i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −179.557 + 818.783i −0.376429 + 1.71653i
\(478\) 0 0
\(479\) 872.673i 1.82186i 0.412556 + 0.910932i \(0.364636\pi\)
−0.412556 + 0.910932i \(0.635364\pi\)
\(480\) 0 0
\(481\) 64.4794 0.134053
\(482\) 0 0
\(483\) 61.1727 564.528i 0.126652 1.16879i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −883.613 −1.81440 −0.907201 0.420698i \(-0.861785\pi\)
−0.907201 + 0.420698i \(0.861785\pi\)
\(488\) 0 0
\(489\) −488.221 52.9041i −0.998408 0.108188i
\(490\) 0 0
\(491\) 596.247i 1.21435i −0.794567 0.607177i \(-0.792302\pi\)
0.794567 0.607177i \(-0.207698\pi\)
\(492\) 0 0
\(493\) 559.114 1.13411
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 426.201i 0.857548i
\(498\) 0 0
\(499\) 560.109 1.12246 0.561231 0.827659i \(-0.310328\pi\)
0.561231 + 0.827659i \(0.310328\pi\)
\(500\) 0 0
\(501\) −15.7402 + 145.257i −0.0314175 + 0.289934i
\(502\) 0 0
\(503\) 505.038i 1.00405i 0.864853 + 0.502026i \(0.167412\pi\)
−0.864853 + 0.502026i \(0.832588\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −500.654 54.2513i −0.987484 0.107005i
\(508\) 0 0
\(509\) 13.3027i 0.0261350i 0.999915 + 0.0130675i \(0.00415963\pi\)
−0.999915 + 0.0130675i \(0.995840\pi\)
\(510\) 0 0
\(511\) −189.387 −0.370620
\(512\) 0 0
\(513\) 208.086 + 69.8405i 0.405625 + 0.136141i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 46.1645 0.0892930
\(518\) 0 0
\(519\) 45.5005 419.898i 0.0876695 0.809051i
\(520\) 0 0
\(521\) 267.898i 0.514200i 0.966385 + 0.257100i \(0.0827670\pi\)
−0.966385 + 0.257100i \(0.917233\pi\)
\(522\) 0 0
\(523\) 176.493 0.337463 0.168731 0.985662i \(-0.446033\pi\)
0.168731 + 0.985662i \(0.446033\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 904.949i 1.71717i
\(528\) 0 0
\(529\) −1074.75 −2.03166
\(530\) 0 0
\(531\) 33.7197 153.763i 0.0635022 0.289572i
\(532\) 0 0
\(533\) 63.2268i 0.118624i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.59872 33.2105i 0.00670152 0.0618445i
\(538\) 0 0
\(539\) 127.025i 0.235667i
\(540\) 0 0
\(541\) −790.757 −1.46166 −0.730829 0.682561i \(-0.760866\pi\)
−0.730829 + 0.682561i \(0.760866\pi\)
\(542\) 0 0
\(543\) −448.082 48.5546i −0.825198 0.0894191i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −44.1668 −0.0807436 −0.0403718 0.999185i \(-0.512854\pi\)
−0.0403718 + 0.999185i \(0.512854\pi\)
\(548\) 0 0
\(549\) 507.598 + 111.315i 0.924587 + 0.202759i
\(550\) 0 0
\(551\) 169.696i 0.307978i
\(552\) 0 0
\(553\) −308.978 −0.558730
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 753.685i 1.35312i 0.736390 + 0.676558i \(0.236529\pi\)
−0.736390 + 0.676558i \(0.763471\pi\)
\(558\) 0 0
\(559\) −60.2195 −0.107727
\(560\) 0 0
\(561\) 380.613 + 41.2435i 0.678454 + 0.0735179i
\(562\) 0 0
\(563\) 609.590i 1.08275i 0.840780 + 0.541376i \(0.182097\pi\)
−0.840780 + 0.541376i \(0.817903\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 347.706 + 160.206i 0.613238 + 0.282550i
\(568\) 0 0
\(569\) 528.082i 0.928088i 0.885812 + 0.464044i \(0.153602\pi\)
−0.885812 + 0.464044i \(0.846398\pi\)
\(570\) 0 0
\(571\) 708.097 1.24010 0.620050 0.784562i \(-0.287112\pi\)
0.620050 + 0.784562i \(0.287112\pi\)
\(572\) 0 0
\(573\) −54.3154 + 501.246i −0.0947913 + 0.874774i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 55.0939 0.0954834 0.0477417 0.998860i \(-0.484798\pi\)
0.0477417 + 0.998860i \(0.484798\pi\)
\(578\) 0 0
\(579\) −933.313 101.135i −1.61194 0.174671i
\(580\) 0 0
\(581\) 557.184i 0.959009i
\(582\) 0 0
\(583\) −443.748 −0.761145
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 596.377i 1.01598i 0.861364 + 0.507988i \(0.169610\pi\)
−0.861364 + 0.507988i \(0.830390\pi\)
\(588\) 0 0
\(589\) 274.660 0.466315
\(590\) 0 0
\(591\) −53.6981 + 495.549i −0.0908597 + 0.838492i
\(592\) 0 0
\(593\) 269.915i 0.455169i −0.973758 0.227584i \(-0.926917\pi\)
0.973758 0.227584i \(-0.0730828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 314.762 + 34.1079i 0.527240 + 0.0571322i
\(598\) 0 0
\(599\) 622.634i 1.03946i 0.854332 + 0.519728i \(0.173967\pi\)
−0.854332 + 0.519728i \(0.826033\pi\)
\(600\) 0 0
\(601\) 865.760 1.44053 0.720266 0.693698i \(-0.244020\pi\)
0.720266 + 0.693698i \(0.244020\pi\)
\(602\) 0 0
\(603\) 892.570 + 195.738i 1.48022 + 0.324606i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 390.780 0.643790 0.321895 0.946775i \(-0.395680\pi\)
0.321895 + 0.946775i \(0.395680\pi\)
\(608\) 0 0
\(609\) −31.8863 + 294.260i −0.0523584 + 0.483185i
\(610\) 0 0
\(611\) 10.3378i 0.0169195i
\(612\) 0 0
\(613\) 398.441 0.649985 0.324993 0.945717i \(-0.394638\pi\)
0.324993 + 0.945717i \(0.394638\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 178.765i 0.289732i −0.989451 0.144866i \(-0.953725\pi\)
0.989451 0.144866i \(-0.0462752\pi\)
\(618\) 0 0
\(619\) 224.867 0.363274 0.181637 0.983366i \(-0.441860\pi\)
0.181637 + 0.983366i \(0.441860\pi\)
\(620\) 0 0
\(621\) 344.047 1025.07i 0.554021 1.65067i
\(622\) 0 0
\(623\) 565.653i 0.907950i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −12.5178 + 115.519i −0.0199645 + 0.184241i
\(628\) 0 0
\(629\) 1618.74i 2.57351i
\(630\) 0 0
\(631\) 9.36019 0.0148339 0.00741695 0.999972i \(-0.497639\pi\)
0.00741695 + 0.999972i \(0.497639\pi\)
\(632\) 0 0
\(633\) −845.172 91.5836i −1.33518 0.144682i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −28.4453 −0.0446550
\(638\) 0 0
\(639\) 173.844 792.733i 0.272056 1.24058i
\(640\) 0 0
\(641\) 785.281i 1.22509i −0.790437 0.612543i \(-0.790146\pi\)
0.790437 0.612543i \(-0.209854\pi\)
\(642\) 0 0
\(643\) 131.320 0.204230 0.102115 0.994773i \(-0.467439\pi\)
0.102115 + 0.994773i \(0.467439\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 155.225i 0.239915i 0.992779 + 0.119958i \(0.0382759\pi\)
−0.992779 + 0.119958i \(0.961724\pi\)
\(648\) 0 0
\(649\) 83.3333 0.128403
\(650\) 0 0
\(651\) 476.271 + 51.6092i 0.731599 + 0.0792768i
\(652\) 0 0
\(653\) 668.464i 1.02368i −0.859080 0.511841i \(-0.828964\pi\)
0.859080 0.511841i \(-0.171036\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −352.259 77.2493i −0.536163 0.117579i
\(658\) 0 0
\(659\) 579.510i 0.879377i −0.898150 0.439689i \(-0.855089\pi\)
0.898150 0.439689i \(-0.144911\pi\)
\(660\) 0 0
\(661\) −307.070 −0.464553 −0.232277 0.972650i \(-0.574617\pi\)
−0.232277 + 0.972650i \(0.574617\pi\)
\(662\) 0 0
\(663\) 9.23586 85.2324i 0.0139304 0.128556i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 835.953 1.25330
\(668\) 0 0
\(669\) −298.576 32.3540i −0.446302 0.0483617i
\(670\) 0 0
\(671\) 275.098i 0.409982i
\(672\) 0 0
\(673\) −7.02018 −0.0104312 −0.00521559 0.999986i \(-0.501660\pi\)
−0.00521559 + 0.999986i \(0.501660\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.21742i 0.00179825i 1.00000 0.000899127i \(0.000286201\pi\)
−1.00000 0.000899127i \(0.999714\pi\)
\(678\) 0 0
\(679\) 72.0881 0.106168
\(680\) 0 0
\(681\) 77.4993 715.196i 0.113802 1.05021i
\(682\) 0 0
\(683\) 1261.23i 1.84661i 0.384067 + 0.923305i \(0.374523\pi\)
−0.384067 + 0.923305i \(0.625477\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1172.25 + 127.026i 1.70633 + 0.184899i
\(688\) 0 0
\(689\) 99.3706i 0.144224i
\(690\) 0 0
\(691\) 158.177 0.228910 0.114455 0.993428i \(-0.463488\pi\)
0.114455 + 0.993428i \(0.463488\pi\)
\(692\) 0 0
\(693\) −43.4126 + 197.963i −0.0626445 + 0.285661i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1587.29 2.27732
\(698\) 0 0
\(699\) −90.0705 + 831.208i −0.128856 + 1.18914i
\(700\) 0 0
\(701\) 331.746i 0.473247i −0.971601 0.236623i \(-0.923959\pi\)
0.971601 0.236623i \(-0.0760408\pi\)
\(702\) 0 0
\(703\) 491.300 0.698863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 340.323i 0.481363i
\(708\) 0 0
\(709\) 854.366 1.20503 0.602515 0.798108i \(-0.294165\pi\)
0.602515 + 0.798108i \(0.294165\pi\)
\(710\) 0 0
\(711\) −574.697 126.029i −0.808295 0.177256i
\(712\) 0 0
\(713\) 1353.02i 1.89765i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 63.5374 586.350i 0.0886156 0.817782i
\(718\) 0 0
\(719\) 218.882i 0.304426i −0.988348 0.152213i \(-0.951360\pi\)
0.988348 0.152213i \(-0.0486400\pi\)
\(720\) 0 0
\(721\) 524.394 0.727314
\(722\) 0 0
\(723\) −689.422 74.7064i −0.953557 0.103328i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.2420 0.0168391 0.00841954 0.999965i \(-0.497320\pi\)
0.00841954 + 0.999965i \(0.497320\pi\)
\(728\) 0 0
\(729\) 581.385 + 439.809i 0.797511 + 0.603305i
\(730\) 0 0
\(731\) 1511.79i 2.06812i
\(732\) 0 0
\(733\) −86.8331 −0.118463 −0.0592313 0.998244i \(-0.518865\pi\)
−0.0592313 + 0.998244i \(0.518865\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 483.737i 0.656360i
\(738\) 0 0
\(739\) 187.725 0.254026 0.127013 0.991901i \(-0.459461\pi\)
0.127013 + 0.991901i \(0.459461\pi\)
\(740\) 0 0
\(741\) 25.8688 + 2.80316i 0.0349106 + 0.00378294i
\(742\) 0 0
\(743\) 178.264i 0.239925i −0.992778 0.119963i \(-0.961723\pi\)
0.992778 0.119963i \(-0.0382775\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 227.271 1036.36i 0.304244 1.38736i
\(748\) 0 0
\(749\) 144.637i 0.193107i
\(750\) 0 0
\(751\) 1328.32 1.76874 0.884370 0.466786i \(-0.154588\pi\)
0.884370 + 0.466786i \(0.154588\pi\)
\(752\) 0 0
\(753\) 78.6781 726.075i 0.104486 0.964243i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 545.957 0.721211 0.360606 0.932718i \(-0.382570\pi\)
0.360606 + 0.932718i \(0.382570\pi\)
\(758\) 0 0
\(759\) 569.068 + 61.6648i 0.749761 + 0.0812448i
\(760\) 0 0
\(761\) 828.655i 1.08890i 0.838793 + 0.544451i \(0.183262\pi\)
−0.838793 + 0.544451i \(0.816738\pi\)
\(762\) 0 0
\(763\) −82.0950 −0.107595
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.6612i 0.0243301i
\(768\) 0 0
\(769\) −303.757 −0.395002 −0.197501 0.980303i \(-0.563283\pi\)
−0.197501 + 0.980303i \(0.563283\pi\)
\(770\) 0 0
\(771\) −115.858 + 1069.19i −0.150270 + 1.38676i
\(772\) 0 0
\(773\) 59.2137i 0.0766025i −0.999266 0.0383013i \(-0.987805\pi\)
0.999266 0.0383013i \(-0.0121947\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 851.936 + 92.3165i 1.09644 + 0.118811i
\(778\) 0 0
\(779\) 481.756i 0.618429i
\(780\) 0 0
\(781\) 429.629 0.550102
\(782\) 0 0
\(783\) −179.334 + 534.316i −0.229035 + 0.682396i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 214.281 0.272276 0.136138 0.990690i \(-0.456531\pi\)
0.136138 + 0.990690i \(0.456531\pi\)
\(788\) 0 0
\(789\) −106.948 + 986.963i −0.135549 + 1.25090i
\(790\) 0 0
\(791\) 22.2862i 0.0281748i
\(792\) 0 0
\(793\) 61.6040 0.0776847
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 124.487i 0.156195i −0.996946 0.0780974i \(-0.975115\pi\)
0.996946 0.0780974i \(-0.0248845\pi\)
\(798\) 0 0
\(799\) −259.528 −0.324816
\(800\) 0 0
\(801\) −230.725 + 1052.11i −0.288046 + 1.31350i
\(802\) 0 0
\(803\) 190.910i 0.237746i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 31.3610 289.412i 0.0388612 0.358627i
\(808\) 0 0
\(809\) 44.3423i 0.0548113i 0.999624 + 0.0274056i \(0.00872458\pi\)
−0.999624 + 0.0274056i \(0.991275\pi\)
\(810\) 0 0
\(811\) 686.962 0.847056 0.423528 0.905883i \(-0.360792\pi\)
0.423528 + 0.905883i \(0.360792\pi\)
\(812\) 0 0
\(813\) −200.382 21.7136i −0.246473 0.0267080i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −458.842 −0.561618
\(818\) 0 0
\(819\) 44.3308 + 9.72160i 0.0541280 + 0.0118701i
\(820\) 0 0
\(821\) 865.772i 1.05453i −0.849700 0.527267i \(-0.823217\pi\)
0.849700 0.527267i \(-0.176783\pi\)
\(822\) 0 0
\(823\) −240.142 −0.291789 −0.145895 0.989300i \(-0.546606\pi\)
−0.145895 + 0.989300i \(0.546606\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 311.704i 0.376910i −0.982082 0.188455i \(-0.939652\pi\)
0.982082 0.188455i \(-0.0603479\pi\)
\(828\) 0 0
\(829\) −1125.26 −1.35737 −0.678684 0.734430i \(-0.737449\pi\)
−0.678684 + 0.734430i \(0.737449\pi\)
\(830\) 0 0
\(831\) 1079.08 + 116.931i 1.29854 + 0.140711i
\(832\) 0 0
\(833\) 714.110i 0.857274i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 864.812 + 290.260i 1.03323 + 0.346786i
\(838\) 0 0
\(839\) 40.3794i 0.0481280i 0.999710 + 0.0240640i \(0.00766055\pi\)
−0.999710 + 0.0240640i \(0.992339\pi\)
\(840\) 0 0
\(841\) 405.259 0.481878
\(842\) 0 0
\(843\) 93.1412 859.546i 0.110488 1.01963i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 464.607 0.548532
\(848\) 0 0
\(849\) −812.664 88.0610i −0.957201 0.103723i
\(850\) 0 0
\(851\) 2420.24i 2.84399i
\(852\) 0 0
\(853\) 290.487 0.340547 0.170274 0.985397i \(-0.445535\pi\)
0.170274 + 0.985397i \(0.445535\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1111.79i 1.29731i −0.761083 0.648655i \(-0.775332\pi\)
0.761083 0.648655i \(-0.224668\pi\)
\(858\) 0 0
\(859\) 159.069 0.185179 0.0925894 0.995704i \(-0.470486\pi\)
0.0925894 + 0.995704i \(0.470486\pi\)
\(860\) 0 0
\(861\) −90.5231 + 835.385i −0.105137 + 0.970250i
\(862\) 0 0
\(863\) 75.4630i 0.0874426i 0.999044 + 0.0437213i \(0.0139214\pi\)
−0.999044 + 0.0437213i \(0.986079\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1277.78 138.461i −1.47379 0.159702i
\(868\) 0 0
\(869\) 311.463i 0.358415i
\(870\) 0 0
\(871\) 108.326 0.124369
\(872\) 0 0
\(873\) 134.084 + 29.4041i 0.153589 + 0.0336817i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 511.799 0.583579 0.291790 0.956483i \(-0.405749\pi\)
0.291790 + 0.956483i \(0.405749\pi\)
\(878\) 0 0
\(879\) −22.8067 + 210.470i −0.0259462 + 0.239443i
\(880\) 0 0
\(881\) 855.549i 0.971111i −0.874206 0.485556i \(-0.838617\pi\)
0.874206 0.485556i \(-0.161383\pi\)
\(882\) 0 0
\(883\) 11.2299 0.0127179 0.00635894 0.999980i \(-0.497976\pi\)
0.00635894 + 0.999980i \(0.497976\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 733.251i 0.826664i −0.910580 0.413332i \(-0.864365\pi\)
0.910580 0.413332i \(-0.135635\pi\)
\(888\) 0 0
\(889\) −6.59468 −0.00741808
\(890\) 0 0
\(891\) −161.495 + 350.503i −0.181251 + 0.393381i
\(892\) 0 0
\(893\) 78.7690i 0.0882072i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 13.8089 127.434i 0.0153945 0.142067i
\(898\) 0 0
\(899\) 705.263i 0.784497i
\(900\) 0 0
\(901\) 2494.67 2.76878
\(902\) 0 0
\(903\) −795.652 86.2175i −0.881120 0.0954790i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1141.39 1.25843 0.629214 0.777232i \(-0.283377\pi\)
0.629214 + 0.777232i \(0.283377\pi\)
\(908\) 0 0
\(909\) 138.815 633.000i 0.152712 0.696370i
\(910\) 0 0
\(911\) 157.145i 0.172497i −0.996274 0.0862487i \(-0.972512\pi\)
0.996274 0.0862487i \(-0.0274880\pi\)
\(912\) 0 0
\(913\) 561.666 0.615187
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1069.21i 1.16598i
\(918\) 0 0
\(919\) −1542.94 −1.67893 −0.839466 0.543412i \(-0.817132\pi\)
−0.839466 + 0.543412i \(0.817132\pi\)
\(920\) 0 0
\(921\) −370.652 40.1642i −0.402446 0.0436094i
\(922\) 0 0
\(923\) 96.2090i 0.104235i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 975.371 + 213.896i 1.05218 + 0.230740i
\(928\) 0 0
\(929\) 165.617i 0.178274i −0.996019 0.0891372i \(-0.971589\pi\)
0.996019 0.0891372i \(-0.0284110\pi\)
\(930\) 0 0
\(931\) −216.738 −0.232802
\(932\) 0 0
\(933\) 74.0125 683.019i 0.0793275 0.732067i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1418.05 −1.51339 −0.756695 0.653768i \(-0.773187\pi\)
−0.756695 + 0.653768i \(0.773187\pi\)
\(938\) 0 0
\(939\) −1388.34 150.442i −1.47853 0.160215i
\(940\) 0 0
\(941\) 1274.33i 1.35423i 0.735878 + 0.677114i \(0.236769\pi\)
−0.735878 + 0.677114i \(0.763231\pi\)
\(942\) 0 0
\(943\) 2373.22 2.51667
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1406.16i 1.48486i 0.669926 + 0.742428i \(0.266326\pi\)
−0.669926 + 0.742428i \(0.733674\pi\)
\(948\) 0 0
\(949\) −42.7515 −0.0450490
\(950\) 0 0
\(951\) 131.040 1209.30i 0.137792 1.27160i
\(952\) 0 0
\(953\) 1415.36i 1.48516i 0.669759 + 0.742579i \(0.266398\pi\)
−0.669759 + 0.742579i \(0.733602\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −296.627 32.1427i −0.309955 0.0335870i
\(958\) 0 0
\(959\) 360.430i 0.375840i
\(960\) 0 0
\(961\) 180.496 0.187821
\(962\) 0 0
\(963\) 58.9963 269.025i 0.0612630 0.279361i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1662.07 −1.71879 −0.859394 0.511315i \(-0.829159\pi\)
−0.859394 + 0.511315i \(0.829159\pi\)
\(968\) 0 0
\(969\) 70.3725 649.427i 0.0726239 0.670204i
\(970\) 0 0
\(971\) 868.901i 0.894852i −0.894321 0.447426i \(-0.852341\pi\)
0.894321 0.447426i \(-0.147659\pi\)
\(972\) 0 0
\(973\) −603.373 −0.620116
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1336.81i 1.36828i 0.729350 + 0.684140i \(0.239822\pi\)
−0.729350 + 0.684140i \(0.760178\pi\)
\(978\) 0 0
\(979\) −570.203 −0.582434
\(980\) 0 0
\(981\) −152.696 33.4858i −0.155654 0.0341344i
\(982\) 0 0
\(983\) 1193.86i 1.21451i −0.794508 0.607254i \(-0.792271\pi\)
0.794508 0.607254i \(-0.207729\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14.8009 136.589i 0.0149958 0.138388i
\(988\) 0 0
\(989\) 2260.34i 2.28548i
\(990\) 0 0
\(991\) 460.690 0.464874 0.232437 0.972611i \(-0.425330\pi\)
0.232437 + 0.972611i \(0.425330\pi\)
\(992\) 0 0
\(993\) −136.980 14.8432i −0.137945 0.0149479i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −673.924 −0.675952 −0.337976 0.941155i \(-0.609742\pi\)
−0.337976 + 0.941155i \(0.609742\pi\)
\(998\) 0 0
\(999\) 1546.94 + 519.205i 1.54849 + 0.519725i
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 1200.3.l.x.401.8 8
3.2 odd 2 inner 1200.3.l.x.401.7 8
4.3 odd 2 600.3.l.f.401.1 8
5.2 odd 4 1200.3.c.m.449.9 16
5.3 odd 4 1200.3.c.m.449.8 16
5.4 even 2 240.3.l.d.161.1 8
12.11 even 2 600.3.l.f.401.2 8
15.2 even 4 1200.3.c.m.449.7 16
15.8 even 4 1200.3.c.m.449.10 16
15.14 odd 2 240.3.l.d.161.2 8
20.3 even 4 600.3.c.d.449.9 16
20.7 even 4 600.3.c.d.449.8 16
20.19 odd 2 120.3.l.a.41.8 yes 8
40.19 odd 2 960.3.l.h.641.1 8
40.29 even 2 960.3.l.g.641.8 8
60.23 odd 4 600.3.c.d.449.7 16
60.47 odd 4 600.3.c.d.449.10 16
60.59 even 2 120.3.l.a.41.7 8
120.29 odd 2 960.3.l.g.641.7 8
120.59 even 2 960.3.l.h.641.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.7 8 60.59 even 2
120.3.l.a.41.8 yes 8 20.19 odd 2
240.3.l.d.161.1 8 5.4 even 2
240.3.l.d.161.2 8 15.14 odd 2
600.3.c.d.449.7 16 60.23 odd 4
600.3.c.d.449.8 16 20.7 even 4
600.3.c.d.449.9 16 20.3 even 4
600.3.c.d.449.10 16 60.47 odd 4
600.3.l.f.401.1 8 4.3 odd 2
600.3.l.f.401.2 8 12.11 even 2
960.3.l.g.641.7 8 120.29 odd 2
960.3.l.g.641.8 8 40.29 even 2
960.3.l.h.641.1 8 40.19 odd 2
960.3.l.h.641.2 8 120.59 even 2
1200.3.c.m.449.7 16 15.2 even 4
1200.3.c.m.449.8 16 5.3 odd 4
1200.3.c.m.449.9 16 5.2 odd 4
1200.3.c.m.449.10 16 15.8 even 4
1200.3.l.x.401.7 8 3.2 odd 2 inner
1200.3.l.x.401.8 8 1.1 even 1 trivial