Properties

Label 3344.2.a.m.1.2
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{3} -1.00000 q^{5} -1.41421 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q+2.41421 q^{3} -1.00000 q^{5} -1.41421 q^{7} +2.82843 q^{9} -1.00000 q^{11} -1.41421 q^{13} -2.41421 q^{15} +1.41421 q^{17} +1.00000 q^{19} -3.41421 q^{21} -4.65685 q^{23} -4.00000 q^{25} -0.414214 q^{27} -8.24264 q^{29} -3.24264 q^{31} -2.41421 q^{33} +1.41421 q^{35} -8.41421 q^{37} -3.41421 q^{39} -4.00000 q^{41} +6.00000 q^{43} -2.82843 q^{45} +6.48528 q^{47} -5.00000 q^{49} +3.41421 q^{51} +1.17157 q^{53} +1.00000 q^{55} +2.41421 q^{57} +7.24264 q^{59} -3.41421 q^{61} -4.00000 q^{63} +1.41421 q^{65} +9.72792 q^{67} -11.2426 q^{69} -1.24264 q^{71} +0.485281 q^{73} -9.65685 q^{75} +1.41421 q^{77} +10.7279 q^{79} -9.48528 q^{81} -15.5563 q^{83} -1.41421 q^{85} -19.8995 q^{87} +2.41421 q^{89} +2.00000 q^{91} -7.82843 q^{93} -1.00000 q^{95} +1.58579 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{11} - 2 q^{15} + 2 q^{19} - 4 q^{21} + 2 q^{23} - 8 q^{25} + 2 q^{27} - 8 q^{29} + 2 q^{31} - 2 q^{33} - 14 q^{37} - 4 q^{39} - 8 q^{41} + 12 q^{43} - 4 q^{47} - 10 q^{49} + 4 q^{51} + 8 q^{53} + 2 q^{55} + 2 q^{57} + 6 q^{59} - 4 q^{61} - 8 q^{63} - 6 q^{67} - 14 q^{69} + 6 q^{71} - 16 q^{73} - 8 q^{75} - 4 q^{79} - 2 q^{81} - 20 q^{87} + 2 q^{89} + 4 q^{91} - 10 q^{93} - 2 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41421 1.39385 0.696923 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 0 0
\(23\) −4.65685 −0.971021 −0.485511 0.874231i \(-0.661366\pi\)
−0.485511 + 0.874231i \(0.661366\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) −8.24264 −1.53062 −0.765310 0.643662i \(-0.777414\pi\)
−0.765310 + 0.643662i \(0.777414\pi\)
\(30\) 0 0
\(31\) −3.24264 −0.582395 −0.291198 0.956663i \(-0.594054\pi\)
−0.291198 + 0.956663i \(0.594054\pi\)
\(32\) 0 0
\(33\) −2.41421 −0.420261
\(34\) 0 0
\(35\) 1.41421 0.239046
\(36\) 0 0
\(37\) −8.41421 −1.38329 −0.691644 0.722238i \(-0.743113\pi\)
−0.691644 + 0.722238i \(0.743113\pi\)
\(38\) 0 0
\(39\) −3.41421 −0.546712
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 6.48528 0.945976 0.472988 0.881069i \(-0.343175\pi\)
0.472988 + 0.881069i \(0.343175\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 3.41421 0.478086
\(52\) 0 0
\(53\) 1.17157 0.160928 0.0804640 0.996758i \(-0.474360\pi\)
0.0804640 + 0.996758i \(0.474360\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 2.41421 0.319770
\(58\) 0 0
\(59\) 7.24264 0.942912 0.471456 0.881890i \(-0.343729\pi\)
0.471456 + 0.881890i \(0.343729\pi\)
\(60\) 0 0
\(61\) −3.41421 −0.437145 −0.218573 0.975821i \(-0.570140\pi\)
−0.218573 + 0.975821i \(0.570140\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 1.41421 0.175412
\(66\) 0 0
\(67\) 9.72792 1.18845 0.594227 0.804297i \(-0.297458\pi\)
0.594227 + 0.804297i \(0.297458\pi\)
\(68\) 0 0
\(69\) −11.2426 −1.35345
\(70\) 0 0
\(71\) −1.24264 −0.147474 −0.0737372 0.997278i \(-0.523493\pi\)
−0.0737372 + 0.997278i \(0.523493\pi\)
\(72\) 0 0
\(73\) 0.485281 0.0567979 0.0283989 0.999597i \(-0.490959\pi\)
0.0283989 + 0.999597i \(0.490959\pi\)
\(74\) 0 0
\(75\) −9.65685 −1.11508
\(76\) 0 0
\(77\) 1.41421 0.161165
\(78\) 0 0
\(79\) 10.7279 1.20699 0.603493 0.797368i \(-0.293775\pi\)
0.603493 + 0.797368i \(0.293775\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −15.5563 −1.70753 −0.853766 0.520658i \(-0.825687\pi\)
−0.853766 + 0.520658i \(0.825687\pi\)
\(84\) 0 0
\(85\) −1.41421 −0.153393
\(86\) 0 0
\(87\) −19.8995 −2.13345
\(88\) 0 0
\(89\) 2.41421 0.255906 0.127953 0.991780i \(-0.459159\pi\)
0.127953 + 0.991780i \(0.459159\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −7.82843 −0.811770
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 1.58579 0.161012 0.0805061 0.996754i \(-0.474346\pi\)
0.0805061 + 0.996754i \(0.474346\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
\(100\) 0 0
\(101\) 13.8995 1.38305 0.691526 0.722352i \(-0.256939\pi\)
0.691526 + 0.722352i \(0.256939\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 3.41421 0.333193
\(106\) 0 0
\(107\) 5.31371 0.513696 0.256848 0.966452i \(-0.417316\pi\)
0.256848 + 0.966452i \(0.417316\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −20.3137 −1.92809
\(112\) 0 0
\(113\) −0.414214 −0.0389659 −0.0194830 0.999810i \(-0.506202\pi\)
−0.0194830 + 0.999810i \(0.506202\pi\)
\(114\) 0 0
\(115\) 4.65685 0.434254
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −9.65685 −0.870729
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −7.07107 −0.627456 −0.313728 0.949513i \(-0.601578\pi\)
−0.313728 + 0.949513i \(0.601578\pi\)
\(128\) 0 0
\(129\) 14.4853 1.27536
\(130\) 0 0
\(131\) −3.17157 −0.277102 −0.138551 0.990355i \(-0.544244\pi\)
−0.138551 + 0.990355i \(0.544244\pi\)
\(132\) 0 0
\(133\) −1.41421 −0.122628
\(134\) 0 0
\(135\) 0.414214 0.0356498
\(136\) 0 0
\(137\) −14.6569 −1.25222 −0.626110 0.779735i \(-0.715354\pi\)
−0.626110 + 0.779735i \(0.715354\pi\)
\(138\) 0 0
\(139\) 3.75736 0.318695 0.159348 0.987223i \(-0.449061\pi\)
0.159348 + 0.987223i \(0.449061\pi\)
\(140\) 0 0
\(141\) 15.6569 1.31854
\(142\) 0 0
\(143\) 1.41421 0.118262
\(144\) 0 0
\(145\) 8.24264 0.684514
\(146\) 0 0
\(147\) −12.0711 −0.995605
\(148\) 0 0
\(149\) −1.31371 −0.107623 −0.0538116 0.998551i \(-0.517137\pi\)
−0.0538116 + 0.998551i \(0.517137\pi\)
\(150\) 0 0
\(151\) 10.4853 0.853280 0.426640 0.904422i \(-0.359697\pi\)
0.426640 + 0.904422i \(0.359697\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 3.24264 0.260455
\(156\) 0 0
\(157\) −12.6569 −1.01013 −0.505063 0.863082i \(-0.668531\pi\)
−0.505063 + 0.863082i \(0.668531\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) 6.58579 0.519033
\(162\) 0 0
\(163\) −22.4853 −1.76118 −0.880592 0.473876i \(-0.842854\pi\)
−0.880592 + 0.473876i \(0.842854\pi\)
\(164\) 0 0
\(165\) 2.41421 0.187946
\(166\) 0 0
\(167\) −15.7574 −1.21934 −0.609671 0.792655i \(-0.708698\pi\)
−0.609671 + 0.792655i \(0.708698\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −1.75736 −0.133610 −0.0668048 0.997766i \(-0.521280\pi\)
−0.0668048 + 0.997766i \(0.521280\pi\)
\(174\) 0 0
\(175\) 5.65685 0.427618
\(176\) 0 0
\(177\) 17.4853 1.31427
\(178\) 0 0
\(179\) 5.58579 0.417501 0.208751 0.977969i \(-0.433060\pi\)
0.208751 + 0.977969i \(0.433060\pi\)
\(180\) 0 0
\(181\) −13.5858 −1.00982 −0.504912 0.863171i \(-0.668475\pi\)
−0.504912 + 0.863171i \(0.668475\pi\)
\(182\) 0 0
\(183\) −8.24264 −0.609314
\(184\) 0 0
\(185\) 8.41421 0.618625
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) 0.585786 0.0426097
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 0 0
\(193\) 14.1421 1.01797 0.508987 0.860774i \(-0.330020\pi\)
0.508987 + 0.860774i \(0.330020\pi\)
\(194\) 0 0
\(195\) 3.41421 0.244497
\(196\) 0 0
\(197\) −9.41421 −0.670735 −0.335367 0.942087i \(-0.608860\pi\)
−0.335367 + 0.942087i \(0.608860\pi\)
\(198\) 0 0
\(199\) −1.51472 −0.107376 −0.0536878 0.998558i \(-0.517098\pi\)
−0.0536878 + 0.998558i \(0.517098\pi\)
\(200\) 0 0
\(201\) 23.4853 1.65652
\(202\) 0 0
\(203\) 11.6569 0.818151
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) −13.1716 −0.915488
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 3.75736 0.258667 0.129334 0.991601i \(-0.458716\pi\)
0.129334 + 0.991601i \(0.458716\pi\)
\(212\) 0 0
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 4.58579 0.311303
\(218\) 0 0
\(219\) 1.17157 0.0791676
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −1.24264 −0.0832134 −0.0416067 0.999134i \(-0.513248\pi\)
−0.0416067 + 0.999134i \(0.513248\pi\)
\(224\) 0 0
\(225\) −11.3137 −0.754247
\(226\) 0 0
\(227\) 12.5858 0.835348 0.417674 0.908597i \(-0.362845\pi\)
0.417674 + 0.908597i \(0.362845\pi\)
\(228\) 0 0
\(229\) −0.656854 −0.0434062 −0.0217031 0.999764i \(-0.506909\pi\)
−0.0217031 + 0.999764i \(0.506909\pi\)
\(230\) 0 0
\(231\) 3.41421 0.224639
\(232\) 0 0
\(233\) 14.7279 0.964858 0.482429 0.875935i \(-0.339755\pi\)
0.482429 + 0.875935i \(0.339755\pi\)
\(234\) 0 0
\(235\) −6.48528 −0.423053
\(236\) 0 0
\(237\) 25.8995 1.68235
\(238\) 0 0
\(239\) 3.65685 0.236542 0.118271 0.992981i \(-0.462265\pi\)
0.118271 + 0.992981i \(0.462265\pi\)
\(240\) 0 0
\(241\) −19.3137 −1.24411 −0.622053 0.782975i \(-0.713701\pi\)
−0.622053 + 0.782975i \(0.713701\pi\)
\(242\) 0 0
\(243\) −21.6569 −1.38929
\(244\) 0 0
\(245\) 5.00000 0.319438
\(246\) 0 0
\(247\) −1.41421 −0.0899843
\(248\) 0 0
\(249\) −37.5563 −2.38004
\(250\) 0 0
\(251\) 10.3137 0.650996 0.325498 0.945543i \(-0.394468\pi\)
0.325498 + 0.945543i \(0.394468\pi\)
\(252\) 0 0
\(253\) 4.65685 0.292774
\(254\) 0 0
\(255\) −3.41421 −0.213806
\(256\) 0 0
\(257\) 5.51472 0.343999 0.171999 0.985097i \(-0.444977\pi\)
0.171999 + 0.985097i \(0.444977\pi\)
\(258\) 0 0
\(259\) 11.8995 0.739399
\(260\) 0 0
\(261\) −23.3137 −1.44308
\(262\) 0 0
\(263\) 16.1421 0.995367 0.497683 0.867359i \(-0.334184\pi\)
0.497683 + 0.867359i \(0.334184\pi\)
\(264\) 0 0
\(265\) −1.17157 −0.0719691
\(266\) 0 0
\(267\) 5.82843 0.356694
\(268\) 0 0
\(269\) −24.6274 −1.50156 −0.750780 0.660552i \(-0.770322\pi\)
−0.750780 + 0.660552i \(0.770322\pi\)
\(270\) 0 0
\(271\) 20.4853 1.24439 0.622196 0.782861i \(-0.286241\pi\)
0.622196 + 0.782861i \(0.286241\pi\)
\(272\) 0 0
\(273\) 4.82843 0.292230
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 24.1421 1.45056 0.725280 0.688454i \(-0.241710\pi\)
0.725280 + 0.688454i \(0.241710\pi\)
\(278\) 0 0
\(279\) −9.17157 −0.549088
\(280\) 0 0
\(281\) 18.9706 1.13169 0.565844 0.824512i \(-0.308550\pi\)
0.565844 + 0.824512i \(0.308550\pi\)
\(282\) 0 0
\(283\) 2.24264 0.133311 0.0666556 0.997776i \(-0.478767\pi\)
0.0666556 + 0.997776i \(0.478767\pi\)
\(284\) 0 0
\(285\) −2.41421 −0.143006
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 3.82843 0.224426
\(292\) 0 0
\(293\) 15.5563 0.908812 0.454406 0.890795i \(-0.349852\pi\)
0.454406 + 0.890795i \(0.349852\pi\)
\(294\) 0 0
\(295\) −7.24264 −0.421683
\(296\) 0 0
\(297\) 0.414214 0.0240351
\(298\) 0 0
\(299\) 6.58579 0.380866
\(300\) 0 0
\(301\) −8.48528 −0.489083
\(302\) 0 0
\(303\) 33.5563 1.92776
\(304\) 0 0
\(305\) 3.41421 0.195497
\(306\) 0 0
\(307\) 3.07107 0.175275 0.0876375 0.996152i \(-0.472068\pi\)
0.0876375 + 0.996152i \(0.472068\pi\)
\(308\) 0 0
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) −21.3137 −1.20859 −0.604295 0.796761i \(-0.706545\pi\)
−0.604295 + 0.796761i \(0.706545\pi\)
\(312\) 0 0
\(313\) 25.6274 1.44855 0.724274 0.689513i \(-0.242175\pi\)
0.724274 + 0.689513i \(0.242175\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 2.75736 0.154869 0.0774344 0.996997i \(-0.475327\pi\)
0.0774344 + 0.996997i \(0.475327\pi\)
\(318\) 0 0
\(319\) 8.24264 0.461499
\(320\) 0 0
\(321\) 12.8284 0.716013
\(322\) 0 0
\(323\) 1.41421 0.0786889
\(324\) 0 0
\(325\) 5.65685 0.313786
\(326\) 0 0
\(327\) −33.7990 −1.86909
\(328\) 0 0
\(329\) −9.17157 −0.505645
\(330\) 0 0
\(331\) 1.10051 0.0604892 0.0302446 0.999543i \(-0.490371\pi\)
0.0302446 + 0.999543i \(0.490371\pi\)
\(332\) 0 0
\(333\) −23.7990 −1.30418
\(334\) 0 0
\(335\) −9.72792 −0.531493
\(336\) 0 0
\(337\) 4.38478 0.238854 0.119427 0.992843i \(-0.461894\pi\)
0.119427 + 0.992843i \(0.461894\pi\)
\(338\) 0 0
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 3.24264 0.175599
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 11.2426 0.605283
\(346\) 0 0
\(347\) 27.4142 1.47167 0.735836 0.677160i \(-0.236789\pi\)
0.735836 + 0.677160i \(0.236789\pi\)
\(348\) 0 0
\(349\) −4.44365 −0.237863 −0.118932 0.992902i \(-0.537947\pi\)
−0.118932 + 0.992902i \(0.537947\pi\)
\(350\) 0 0
\(351\) 0.585786 0.0312670
\(352\) 0 0
\(353\) 7.68629 0.409100 0.204550 0.978856i \(-0.434427\pi\)
0.204550 + 0.978856i \(0.434427\pi\)
\(354\) 0 0
\(355\) 1.24264 0.0659525
\(356\) 0 0
\(357\) −4.82843 −0.255547
\(358\) 0 0
\(359\) −2.14214 −0.113058 −0.0565288 0.998401i \(-0.518003\pi\)
−0.0565288 + 0.998401i \(0.518003\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.41421 0.126713
\(364\) 0 0
\(365\) −0.485281 −0.0254008
\(366\) 0 0
\(367\) 7.14214 0.372816 0.186408 0.982472i \(-0.440315\pi\)
0.186408 + 0.982472i \(0.440315\pi\)
\(368\) 0 0
\(369\) −11.3137 −0.588968
\(370\) 0 0
\(371\) −1.65685 −0.0860196
\(372\) 0 0
\(373\) −25.3137 −1.31069 −0.655347 0.755328i \(-0.727478\pi\)
−0.655347 + 0.755328i \(0.727478\pi\)
\(374\) 0 0
\(375\) 21.7279 1.12203
\(376\) 0 0
\(377\) 11.6569 0.600359
\(378\) 0 0
\(379\) 34.4142 1.76774 0.883870 0.467733i \(-0.154929\pi\)
0.883870 + 0.467733i \(0.154929\pi\)
\(380\) 0 0
\(381\) −17.0711 −0.874577
\(382\) 0 0
\(383\) −17.2426 −0.881058 −0.440529 0.897738i \(-0.645209\pi\)
−0.440529 + 0.897738i \(0.645209\pi\)
\(384\) 0 0
\(385\) −1.41421 −0.0720750
\(386\) 0 0
\(387\) 16.9706 0.862662
\(388\) 0 0
\(389\) 31.9706 1.62097 0.810486 0.585758i \(-0.199203\pi\)
0.810486 + 0.585758i \(0.199203\pi\)
\(390\) 0 0
\(391\) −6.58579 −0.333058
\(392\) 0 0
\(393\) −7.65685 −0.386237
\(394\) 0 0
\(395\) −10.7279 −0.539780
\(396\) 0 0
\(397\) 5.31371 0.266687 0.133344 0.991070i \(-0.457429\pi\)
0.133344 + 0.991070i \(0.457429\pi\)
\(398\) 0 0
\(399\) −3.41421 −0.170924
\(400\) 0 0
\(401\) 18.2843 0.913073 0.456536 0.889705i \(-0.349090\pi\)
0.456536 + 0.889705i \(0.349090\pi\)
\(402\) 0 0
\(403\) 4.58579 0.228434
\(404\) 0 0
\(405\) 9.48528 0.471327
\(406\) 0 0
\(407\) 8.41421 0.417077
\(408\) 0 0
\(409\) 9.07107 0.448535 0.224268 0.974528i \(-0.428001\pi\)
0.224268 + 0.974528i \(0.428001\pi\)
\(410\) 0 0
\(411\) −35.3848 −1.74540
\(412\) 0 0
\(413\) −10.2426 −0.504007
\(414\) 0 0
\(415\) 15.5563 0.763631
\(416\) 0 0
\(417\) 9.07107 0.444212
\(418\) 0 0
\(419\) −33.1127 −1.61766 −0.808831 0.588042i \(-0.799899\pi\)
−0.808831 + 0.588042i \(0.799899\pi\)
\(420\) 0 0
\(421\) 29.1127 1.41887 0.709433 0.704773i \(-0.248951\pi\)
0.709433 + 0.704773i \(0.248951\pi\)
\(422\) 0 0
\(423\) 18.3431 0.891874
\(424\) 0 0
\(425\) −5.65685 −0.274398
\(426\) 0 0
\(427\) 4.82843 0.233664
\(428\) 0 0
\(429\) 3.41421 0.164840
\(430\) 0 0
\(431\) 37.7990 1.82071 0.910357 0.413825i \(-0.135807\pi\)
0.910357 + 0.413825i \(0.135807\pi\)
\(432\) 0 0
\(433\) −34.6985 −1.66750 −0.833751 0.552140i \(-0.813811\pi\)
−0.833751 + 0.552140i \(0.813811\pi\)
\(434\) 0 0
\(435\) 19.8995 0.954108
\(436\) 0 0
\(437\) −4.65685 −0.222768
\(438\) 0 0
\(439\) −22.8701 −1.09153 −0.545764 0.837939i \(-0.683761\pi\)
−0.545764 + 0.837939i \(0.683761\pi\)
\(440\) 0 0
\(441\) −14.1421 −0.673435
\(442\) 0 0
\(443\) 12.3137 0.585042 0.292521 0.956259i \(-0.405506\pi\)
0.292521 + 0.956259i \(0.405506\pi\)
\(444\) 0 0
\(445\) −2.41421 −0.114445
\(446\) 0 0
\(447\) −3.17157 −0.150010
\(448\) 0 0
\(449\) −11.8701 −0.560183 −0.280091 0.959973i \(-0.590365\pi\)
−0.280091 + 0.959973i \(0.590365\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 25.3137 1.18934
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −0.384776 −0.0179991 −0.00899954 0.999960i \(-0.502865\pi\)
−0.00899954 + 0.999960i \(0.502865\pi\)
\(458\) 0 0
\(459\) −0.585786 −0.0273422
\(460\) 0 0
\(461\) 12.9706 0.604099 0.302050 0.953292i \(-0.402329\pi\)
0.302050 + 0.953292i \(0.402329\pi\)
\(462\) 0 0
\(463\) −24.7990 −1.15251 −0.576253 0.817271i \(-0.695486\pi\)
−0.576253 + 0.817271i \(0.695486\pi\)
\(464\) 0 0
\(465\) 7.82843 0.363035
\(466\) 0 0
\(467\) −37.6274 −1.74119 −0.870595 0.492001i \(-0.836266\pi\)
−0.870595 + 0.492001i \(0.836266\pi\)
\(468\) 0 0
\(469\) −13.7574 −0.635256
\(470\) 0 0
\(471\) −30.5563 −1.40796
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 3.31371 0.151724
\(478\) 0 0
\(479\) 17.7574 0.811354 0.405677 0.914016i \(-0.367036\pi\)
0.405677 + 0.914016i \(0.367036\pi\)
\(480\) 0 0
\(481\) 11.8995 0.542570
\(482\) 0 0
\(483\) 15.8995 0.723452
\(484\) 0 0
\(485\) −1.58579 −0.0720069
\(486\) 0 0
\(487\) −18.8995 −0.856418 −0.428209 0.903680i \(-0.640855\pi\)
−0.428209 + 0.903680i \(0.640855\pi\)
\(488\) 0 0
\(489\) −54.2843 −2.45482
\(490\) 0 0
\(491\) 18.7279 0.845179 0.422590 0.906321i \(-0.361121\pi\)
0.422590 + 0.906321i \(0.361121\pi\)
\(492\) 0 0
\(493\) −11.6569 −0.524998
\(494\) 0 0
\(495\) 2.82843 0.127128
\(496\) 0 0
\(497\) 1.75736 0.0788283
\(498\) 0 0
\(499\) 12.3431 0.552555 0.276278 0.961078i \(-0.410899\pi\)
0.276278 + 0.961078i \(0.410899\pi\)
\(500\) 0 0
\(501\) −38.0416 −1.69957
\(502\) 0 0
\(503\) 6.48528 0.289164 0.144582 0.989493i \(-0.453816\pi\)
0.144582 + 0.989493i \(0.453816\pi\)
\(504\) 0 0
\(505\) −13.8995 −0.618519
\(506\) 0 0
\(507\) −26.5563 −1.17941
\(508\) 0 0
\(509\) 27.2426 1.20751 0.603754 0.797170i \(-0.293671\pi\)
0.603754 + 0.797170i \(0.293671\pi\)
\(510\) 0 0
\(511\) −0.686292 −0.0303597
\(512\) 0 0
\(513\) −0.414214 −0.0182880
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −6.48528 −0.285222
\(518\) 0 0
\(519\) −4.24264 −0.186231
\(520\) 0 0
\(521\) −1.92893 −0.0845081 −0.0422540 0.999107i \(-0.513454\pi\)
−0.0422540 + 0.999107i \(0.513454\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 13.6569 0.596034
\(526\) 0 0
\(527\) −4.58579 −0.199760
\(528\) 0 0
\(529\) −1.31371 −0.0571178
\(530\) 0 0
\(531\) 20.4853 0.888985
\(532\) 0 0
\(533\) 5.65685 0.245026
\(534\) 0 0
\(535\) −5.31371 −0.229732
\(536\) 0 0
\(537\) 13.4853 0.581933
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) −4.38478 −0.188516 −0.0942581 0.995548i \(-0.530048\pi\)
−0.0942581 + 0.995548i \(0.530048\pi\)
\(542\) 0 0
\(543\) −32.7990 −1.40754
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −29.6985 −1.26982 −0.634908 0.772588i \(-0.718962\pi\)
−0.634908 + 0.772588i \(0.718962\pi\)
\(548\) 0 0
\(549\) −9.65685 −0.412144
\(550\) 0 0
\(551\) −8.24264 −0.351148
\(552\) 0 0
\(553\) −15.1716 −0.645161
\(554\) 0 0
\(555\) 20.3137 0.862269
\(556\) 0 0
\(557\) −13.3137 −0.564120 −0.282060 0.959397i \(-0.591018\pi\)
−0.282060 + 0.959397i \(0.591018\pi\)
\(558\) 0 0
\(559\) −8.48528 −0.358889
\(560\) 0 0
\(561\) −3.41421 −0.144148
\(562\) 0 0
\(563\) 7.75736 0.326934 0.163467 0.986549i \(-0.447732\pi\)
0.163467 + 0.986549i \(0.447732\pi\)
\(564\) 0 0
\(565\) 0.414214 0.0174261
\(566\) 0 0
\(567\) 13.4142 0.563344
\(568\) 0 0
\(569\) −28.2426 −1.18399 −0.591997 0.805941i \(-0.701660\pi\)
−0.591997 + 0.805941i \(0.701660\pi\)
\(570\) 0 0
\(571\) −15.8995 −0.665373 −0.332687 0.943037i \(-0.607955\pi\)
−0.332687 + 0.943037i \(0.607955\pi\)
\(572\) 0 0
\(573\) −26.5563 −1.10941
\(574\) 0 0
\(575\) 18.6274 0.776817
\(576\) 0 0
\(577\) −27.0000 −1.12402 −0.562012 0.827129i \(-0.689973\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(578\) 0 0
\(579\) 34.1421 1.41890
\(580\) 0 0
\(581\) 22.0000 0.912714
\(582\) 0 0
\(583\) −1.17157 −0.0485216
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 4.62742 0.190994 0.0954970 0.995430i \(-0.469556\pi\)
0.0954970 + 0.995430i \(0.469556\pi\)
\(588\) 0 0
\(589\) −3.24264 −0.133611
\(590\) 0 0
\(591\) −22.7279 −0.934902
\(592\) 0 0
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 0 0
\(597\) −3.65685 −0.149665
\(598\) 0 0
\(599\) −37.3137 −1.52460 −0.762298 0.647226i \(-0.775929\pi\)
−0.762298 + 0.647226i \(0.775929\pi\)
\(600\) 0 0
\(601\) −47.6985 −1.94566 −0.972831 0.231517i \(-0.925631\pi\)
−0.972831 + 0.231517i \(0.925631\pi\)
\(602\) 0 0
\(603\) 27.5147 1.12049
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 12.5858 0.510841 0.255421 0.966830i \(-0.417786\pi\)
0.255421 + 0.966830i \(0.417786\pi\)
\(608\) 0 0
\(609\) 28.1421 1.14038
\(610\) 0 0
\(611\) −9.17157 −0.371042
\(612\) 0 0
\(613\) −20.3848 −0.823333 −0.411667 0.911334i \(-0.635053\pi\)
−0.411667 + 0.911334i \(0.635053\pi\)
\(614\) 0 0
\(615\) 9.65685 0.389402
\(616\) 0 0
\(617\) 28.4853 1.14677 0.573387 0.819285i \(-0.305629\pi\)
0.573387 + 0.819285i \(0.305629\pi\)
\(618\) 0 0
\(619\) 5.14214 0.206680 0.103340 0.994646i \(-0.467047\pi\)
0.103340 + 0.994646i \(0.467047\pi\)
\(620\) 0 0
\(621\) 1.92893 0.0774054
\(622\) 0 0
\(623\) −3.41421 −0.136788
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −2.41421 −0.0964144
\(628\) 0 0
\(629\) −11.8995 −0.474464
\(630\) 0 0
\(631\) −28.3137 −1.12715 −0.563575 0.826065i \(-0.690575\pi\)
−0.563575 + 0.826065i \(0.690575\pi\)
\(632\) 0 0
\(633\) 9.07107 0.360543
\(634\) 0 0
\(635\) 7.07107 0.280607
\(636\) 0 0
\(637\) 7.07107 0.280166
\(638\) 0 0
\(639\) −3.51472 −0.139040
\(640\) 0 0
\(641\) 5.04163 0.199132 0.0995662 0.995031i \(-0.468254\pi\)
0.0995662 + 0.995031i \(0.468254\pi\)
\(642\) 0 0
\(643\) −38.9411 −1.53569 −0.767844 0.640637i \(-0.778670\pi\)
−0.767844 + 0.640637i \(0.778670\pi\)
\(644\) 0 0
\(645\) −14.4853 −0.570357
\(646\) 0 0
\(647\) 21.6274 0.850261 0.425131 0.905132i \(-0.360228\pi\)
0.425131 + 0.905132i \(0.360228\pi\)
\(648\) 0 0
\(649\) −7.24264 −0.284299
\(650\) 0 0
\(651\) 11.0711 0.433909
\(652\) 0 0
\(653\) 20.1716 0.789375 0.394687 0.918815i \(-0.370853\pi\)
0.394687 + 0.918815i \(0.370853\pi\)
\(654\) 0 0
\(655\) 3.17157 0.123924
\(656\) 0 0
\(657\) 1.37258 0.0535496
\(658\) 0 0
\(659\) 21.5147 0.838094 0.419047 0.907964i \(-0.362364\pi\)
0.419047 + 0.907964i \(0.362364\pi\)
\(660\) 0 0
\(661\) −17.0416 −0.662843 −0.331421 0.943483i \(-0.607528\pi\)
−0.331421 + 0.943483i \(0.607528\pi\)
\(662\) 0 0
\(663\) −4.82843 −0.187521
\(664\) 0 0
\(665\) 1.41421 0.0548408
\(666\) 0 0
\(667\) 38.3848 1.48626
\(668\) 0 0
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) 3.41421 0.131804
\(672\) 0 0
\(673\) 12.8284 0.494500 0.247250 0.968952i \(-0.420473\pi\)
0.247250 + 0.968952i \(0.420473\pi\)
\(674\) 0 0
\(675\) 1.65685 0.0637723
\(676\) 0 0
\(677\) 40.6274 1.56144 0.780719 0.624882i \(-0.214853\pi\)
0.780719 + 0.624882i \(0.214853\pi\)
\(678\) 0 0
\(679\) −2.24264 −0.0860647
\(680\) 0 0
\(681\) 30.3848 1.16435
\(682\) 0 0
\(683\) 14.8284 0.567394 0.283697 0.958914i \(-0.408439\pi\)
0.283697 + 0.958914i \(0.408439\pi\)
\(684\) 0 0
\(685\) 14.6569 0.560010
\(686\) 0 0
\(687\) −1.58579 −0.0605015
\(688\) 0 0
\(689\) −1.65685 −0.0631211
\(690\) 0 0
\(691\) 44.5980 1.69659 0.848294 0.529526i \(-0.177630\pi\)
0.848294 + 0.529526i \(0.177630\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) −3.75736 −0.142525
\(696\) 0 0
\(697\) −5.65685 −0.214269
\(698\) 0 0
\(699\) 35.5563 1.34486
\(700\) 0 0
\(701\) 32.6274 1.23232 0.616160 0.787621i \(-0.288687\pi\)
0.616160 + 0.787621i \(0.288687\pi\)
\(702\) 0 0
\(703\) −8.41421 −0.317348
\(704\) 0 0
\(705\) −15.6569 −0.589671
\(706\) 0 0
\(707\) −19.6569 −0.739272
\(708\) 0 0
\(709\) −5.68629 −0.213553 −0.106777 0.994283i \(-0.534053\pi\)
−0.106777 + 0.994283i \(0.534053\pi\)
\(710\) 0 0
\(711\) 30.3431 1.13796
\(712\) 0 0
\(713\) 15.1005 0.565518
\(714\) 0 0
\(715\) −1.41421 −0.0528886
\(716\) 0 0
\(717\) 8.82843 0.329704
\(718\) 0 0
\(719\) −27.6274 −1.03033 −0.515164 0.857091i \(-0.672269\pi\)
−0.515164 + 0.857091i \(0.672269\pi\)
\(720\) 0 0
\(721\) 11.3137 0.421345
\(722\) 0 0
\(723\) −46.6274 −1.73409
\(724\) 0 0
\(725\) 32.9706 1.22450
\(726\) 0 0
\(727\) −19.4853 −0.722669 −0.361335 0.932436i \(-0.617679\pi\)
−0.361335 + 0.932436i \(0.617679\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 8.48528 0.313839
\(732\) 0 0
\(733\) 29.7574 1.09911 0.549557 0.835457i \(-0.314797\pi\)
0.549557 + 0.835457i \(0.314797\pi\)
\(734\) 0 0
\(735\) 12.0711 0.445248
\(736\) 0 0
\(737\) −9.72792 −0.358333
\(738\) 0 0
\(739\) −42.3848 −1.55915 −0.779575 0.626309i \(-0.784565\pi\)
−0.779575 + 0.626309i \(0.784565\pi\)
\(740\) 0 0
\(741\) −3.41421 −0.125424
\(742\) 0 0
\(743\) −0.928932 −0.0340792 −0.0170396 0.999855i \(-0.505424\pi\)
−0.0170396 + 0.999855i \(0.505424\pi\)
\(744\) 0 0
\(745\) 1.31371 0.0481306
\(746\) 0 0
\(747\) −44.0000 −1.60988
\(748\) 0 0
\(749\) −7.51472 −0.274582
\(750\) 0 0
\(751\) −16.4142 −0.598963 −0.299482 0.954102i \(-0.596814\pi\)
−0.299482 + 0.954102i \(0.596814\pi\)
\(752\) 0 0
\(753\) 24.8995 0.907388
\(754\) 0 0
\(755\) −10.4853 −0.381598
\(756\) 0 0
\(757\) −1.65685 −0.0602194 −0.0301097 0.999547i \(-0.509586\pi\)
−0.0301097 + 0.999547i \(0.509586\pi\)
\(758\) 0 0
\(759\) 11.2426 0.408082
\(760\) 0 0
\(761\) 10.8284 0.392530 0.196265 0.980551i \(-0.437119\pi\)
0.196265 + 0.980551i \(0.437119\pi\)
\(762\) 0 0
\(763\) 19.7990 0.716772
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) −10.2426 −0.369840
\(768\) 0 0
\(769\) −13.4558 −0.485230 −0.242615 0.970123i \(-0.578005\pi\)
−0.242615 + 0.970123i \(0.578005\pi\)
\(770\) 0 0
\(771\) 13.3137 0.479481
\(772\) 0 0
\(773\) 50.2843 1.80860 0.904300 0.426898i \(-0.140394\pi\)
0.904300 + 0.426898i \(0.140394\pi\)
\(774\) 0 0
\(775\) 12.9706 0.465916
\(776\) 0 0
\(777\) 28.7279 1.03061
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 1.24264 0.0444652
\(782\) 0 0
\(783\) 3.41421 0.122014
\(784\) 0 0
\(785\) 12.6569 0.451742
\(786\) 0 0
\(787\) 20.9706 0.747520 0.373760 0.927525i \(-0.378068\pi\)
0.373760 + 0.927525i \(0.378068\pi\)
\(788\) 0 0
\(789\) 38.9706 1.38739
\(790\) 0 0
\(791\) 0.585786 0.0208282
\(792\) 0 0
\(793\) 4.82843 0.171462
\(794\) 0 0
\(795\) −2.82843 −0.100314
\(796\) 0 0
\(797\) 18.8995 0.669454 0.334727 0.942315i \(-0.391356\pi\)
0.334727 + 0.942315i \(0.391356\pi\)
\(798\) 0 0
\(799\) 9.17157 0.324467
\(800\) 0 0
\(801\) 6.82843 0.241271
\(802\) 0 0
\(803\) −0.485281 −0.0171252
\(804\) 0 0
\(805\) −6.58579 −0.232118
\(806\) 0 0
\(807\) −59.4558 −2.09294
\(808\) 0 0
\(809\) −43.7990 −1.53989 −0.769945 0.638110i \(-0.779717\pi\)
−0.769945 + 0.638110i \(0.779717\pi\)
\(810\) 0 0
\(811\) −31.2132 −1.09604 −0.548022 0.836464i \(-0.684619\pi\)
−0.548022 + 0.836464i \(0.684619\pi\)
\(812\) 0 0
\(813\) 49.4558 1.73449
\(814\) 0 0
\(815\) 22.4853 0.787625
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 5.65685 0.197666
\(820\) 0 0
\(821\) 18.3848 0.641633 0.320817 0.947141i \(-0.396043\pi\)
0.320817 + 0.947141i \(0.396043\pi\)
\(822\) 0 0
\(823\) 24.3137 0.847523 0.423761 0.905774i \(-0.360710\pi\)
0.423761 + 0.905774i \(0.360710\pi\)
\(824\) 0 0
\(825\) 9.65685 0.336209
\(826\) 0 0
\(827\) −40.0416 −1.39238 −0.696192 0.717856i \(-0.745124\pi\)
−0.696192 + 0.717856i \(0.745124\pi\)
\(828\) 0 0
\(829\) 20.3553 0.706970 0.353485 0.935440i \(-0.384997\pi\)
0.353485 + 0.935440i \(0.384997\pi\)
\(830\) 0 0
\(831\) 58.2843 2.02186
\(832\) 0 0
\(833\) −7.07107 −0.244998
\(834\) 0 0
\(835\) 15.7574 0.545306
\(836\) 0 0
\(837\) 1.34315 0.0464259
\(838\) 0 0
\(839\) −40.0711 −1.38341 −0.691703 0.722182i \(-0.743139\pi\)
−0.691703 + 0.722182i \(0.743139\pi\)
\(840\) 0 0
\(841\) 38.9411 1.34280
\(842\) 0 0
\(843\) 45.7990 1.57740
\(844\) 0 0
\(845\) 11.0000 0.378412
\(846\) 0 0
\(847\) −1.41421 −0.0485930
\(848\) 0 0
\(849\) 5.41421 0.185815
\(850\) 0 0
\(851\) 39.1838 1.34320
\(852\) 0 0
\(853\) 31.6569 1.08391 0.541955 0.840407i \(-0.317684\pi\)
0.541955 + 0.840407i \(0.317684\pi\)
\(854\) 0 0
\(855\) −2.82843 −0.0967302
\(856\) 0 0
\(857\) −50.6690 −1.73082 −0.865411 0.501063i \(-0.832943\pi\)
−0.865411 + 0.501063i \(0.832943\pi\)
\(858\) 0 0
\(859\) −46.6569 −1.59191 −0.795956 0.605355i \(-0.793031\pi\)
−0.795956 + 0.605355i \(0.793031\pi\)
\(860\) 0 0
\(861\) 13.6569 0.465424
\(862\) 0 0
\(863\) −27.3137 −0.929769 −0.464885 0.885371i \(-0.653904\pi\)
−0.464885 + 0.885371i \(0.653904\pi\)
\(864\) 0 0
\(865\) 1.75736 0.0597520
\(866\) 0 0
\(867\) −36.2132 −1.22986
\(868\) 0 0
\(869\) −10.7279 −0.363920
\(870\) 0 0
\(871\) −13.7574 −0.466150
\(872\) 0 0
\(873\) 4.48528 0.151804
\(874\) 0 0
\(875\) −12.7279 −0.430282
\(876\) 0 0
\(877\) 25.1716 0.849984 0.424992 0.905197i \(-0.360277\pi\)
0.424992 + 0.905197i \(0.360277\pi\)
\(878\) 0 0
\(879\) 37.5563 1.26674
\(880\) 0 0
\(881\) 20.5147 0.691158 0.345579 0.938390i \(-0.387682\pi\)
0.345579 + 0.938390i \(0.387682\pi\)
\(882\) 0 0
\(883\) 43.4558 1.46241 0.731203 0.682160i \(-0.238959\pi\)
0.731203 + 0.682160i \(0.238959\pi\)
\(884\) 0 0
\(885\) −17.4853 −0.587761
\(886\) 0 0
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 9.48528 0.317769
\(892\) 0 0
\(893\) 6.48528 0.217022
\(894\) 0 0
\(895\) −5.58579 −0.186712
\(896\) 0 0
\(897\) 15.8995 0.530869
\(898\) 0 0
\(899\) 26.7279 0.891426
\(900\) 0 0
\(901\) 1.65685 0.0551978
\(902\) 0 0
\(903\) −20.4853 −0.681707
\(904\) 0 0
\(905\) 13.5858 0.451607
\(906\) 0 0
\(907\) −25.4558 −0.845247 −0.422624 0.906305i \(-0.638891\pi\)
−0.422624 + 0.906305i \(0.638891\pi\)
\(908\) 0 0
\(909\) 39.3137 1.30395
\(910\) 0 0
\(911\) 12.2010 0.404237 0.202119 0.979361i \(-0.435217\pi\)
0.202119 + 0.979361i \(0.435217\pi\)
\(912\) 0 0
\(913\) 15.5563 0.514840
\(914\) 0 0
\(915\) 8.24264 0.272493
\(916\) 0 0
\(917\) 4.48528 0.148117
\(918\) 0 0
\(919\) −37.0122 −1.22092 −0.610460 0.792047i \(-0.709015\pi\)
−0.610460 + 0.792047i \(0.709015\pi\)
\(920\) 0 0
\(921\) 7.41421 0.244307
\(922\) 0 0
\(923\) 1.75736 0.0578442
\(924\) 0 0
\(925\) 33.6569 1.10663
\(926\) 0 0
\(927\) −22.6274 −0.743182
\(928\) 0 0
\(929\) −14.6274 −0.479910 −0.239955 0.970784i \(-0.577133\pi\)
−0.239955 + 0.970784i \(0.577133\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) −51.4558 −1.68459
\(934\) 0 0
\(935\) 1.41421 0.0462497
\(936\) 0 0
\(937\) 5.07107 0.165665 0.0828323 0.996564i \(-0.473603\pi\)
0.0828323 + 0.996564i \(0.473603\pi\)
\(938\) 0 0
\(939\) 61.8701 2.01905
\(940\) 0 0
\(941\) −17.3553 −0.565768 −0.282884 0.959154i \(-0.591291\pi\)
−0.282884 + 0.959154i \(0.591291\pi\)
\(942\) 0 0
\(943\) 18.6274 0.606592
\(944\) 0 0
\(945\) −0.585786 −0.0190556
\(946\) 0 0
\(947\) 12.3137 0.400142 0.200071 0.979781i \(-0.435883\pi\)
0.200071 + 0.979781i \(0.435883\pi\)
\(948\) 0 0
\(949\) −0.686292 −0.0222780
\(950\) 0 0
\(951\) 6.65685 0.215863
\(952\) 0 0
\(953\) −7.61522 −0.246681 −0.123341 0.992364i \(-0.539361\pi\)
−0.123341 + 0.992364i \(0.539361\pi\)
\(954\) 0 0
\(955\) 11.0000 0.355952
\(956\) 0 0
\(957\) 19.8995 0.643259
\(958\) 0 0
\(959\) 20.7279 0.669340
\(960\) 0 0
\(961\) −20.4853 −0.660816
\(962\) 0 0
\(963\) 15.0294 0.484317
\(964\) 0 0
\(965\) −14.1421 −0.455251
\(966\) 0 0
\(967\) −28.3848 −0.912793 −0.456396 0.889777i \(-0.650860\pi\)
−0.456396 + 0.889777i \(0.650860\pi\)
\(968\) 0 0
\(969\) 3.41421 0.109680
\(970\) 0 0
\(971\) −12.5563 −0.402952 −0.201476 0.979493i \(-0.564574\pi\)
−0.201476 + 0.979493i \(0.564574\pi\)
\(972\) 0 0
\(973\) −5.31371 −0.170350
\(974\) 0 0
\(975\) 13.6569 0.437369
\(976\) 0 0
\(977\) 3.78680 0.121150 0.0605752 0.998164i \(-0.480707\pi\)
0.0605752 + 0.998164i \(0.480707\pi\)
\(978\) 0 0
\(979\) −2.41421 −0.0771586
\(980\) 0 0
\(981\) −39.5980 −1.26427
\(982\) 0 0
\(983\) 55.3848 1.76650 0.883250 0.468902i \(-0.155350\pi\)
0.883250 + 0.468902i \(0.155350\pi\)
\(984\) 0 0
\(985\) 9.41421 0.299962
\(986\) 0 0
\(987\) −22.1421 −0.704792
\(988\) 0 0
\(989\) −27.9411 −0.888476
\(990\) 0 0
\(991\) −14.3431 −0.455625 −0.227813 0.973705i \(-0.573157\pi\)
−0.227813 + 0.973705i \(0.573157\pi\)
\(992\) 0 0
\(993\) 2.65685 0.0843127
\(994\) 0 0
\(995\) 1.51472 0.0480198
\(996\) 0 0
\(997\) 8.24264 0.261047 0.130524 0.991445i \(-0.458334\pi\)
0.130524 + 0.991445i \(0.458334\pi\)
\(998\) 0 0
\(999\) 3.48528 0.110269
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 3344.2.a.m.1.2 2
4.3 odd 2 836.2.a.c.1.1 2
12.11 even 2 7524.2.a.h.1.2 2
44.43 even 2 9196.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.c.1.1 2 4.3 odd 2
3344.2.a.m.1.2 2 1.1 even 1 trivial
7524.2.a.h.1.2 2 12.11 even 2
9196.2.a.h.1.1 2 44.43 even 2