Properties

Label 1248.2.a.l.1.2
Level $1248$
Weight $2$
Character 1248.1
Self dual yes
Analytic conductor $9.965$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1248,2,Mod(1,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, 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("1248.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.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: \(9.96533017226\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1248.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} -3.23607 q^{15} -4.47214 q^{17} -0.763932 q^{19} +3.23607 q^{21} -6.47214 q^{23} +5.47214 q^{25} -1.00000 q^{27} -4.47214 q^{29} +5.70820 q^{31} +2.00000 q^{33} -10.4721 q^{35} -8.47214 q^{37} +1.00000 q^{39} -3.23607 q^{41} +2.47214 q^{43} +3.23607 q^{45} +10.9443 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} -6.47214 q^{55} +0.763932 q^{57} +0.472136 q^{59} -3.52786 q^{61} -3.23607 q^{63} -3.23607 q^{65} -11.2361 q^{67} +6.47214 q^{69} +4.47214 q^{71} -8.47214 q^{73} -5.47214 q^{75} +6.47214 q^{77} -8.94427 q^{79} +1.00000 q^{81} -16.4721 q^{83} -14.4721 q^{85} +4.47214 q^{87} +12.1803 q^{89} +3.23607 q^{91} -5.70820 q^{93} -2.47214 q^{95} +4.47214 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 4 q^{33} - 12 q^{35} - 8 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} - 2 q^{49} + 8 q^{53} - 4 q^{55} + 6 q^{57} - 8 q^{59} - 16 q^{61} - 2 q^{63} - 2 q^{65} - 18 q^{67} + 4 q^{69} - 8 q^{73} - 2 q^{75} + 4 q^{77} + 2 q^{81} - 24 q^{83} - 20 q^{85} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 4 q^{95} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −3.23607 −0.835549
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −0.763932 −0.175258 −0.0876290 0.996153i \(-0.527929\pi\)
−0.0876290 + 0.996153i \(0.527929\pi\)
\(20\) 0 0
\(21\) 3.23607 0.706168
\(22\) 0 0
\(23\) −6.47214 −1.34953 −0.674767 0.738031i \(-0.735756\pi\)
−0.674767 + 0.738031i \(0.735756\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 5.70820 1.02522 0.512612 0.858620i \(-0.328678\pi\)
0.512612 + 0.858620i \(0.328678\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −10.4721 −1.77011
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −3.23607 −0.505389 −0.252694 0.967546i \(-0.581317\pi\)
−0.252694 + 0.967546i \(0.581317\pi\)
\(42\) 0 0
\(43\) 2.47214 0.376997 0.188499 0.982073i \(-0.439638\pi\)
0.188499 + 0.982073i \(0.439638\pi\)
\(44\) 0 0
\(45\) 3.23607 0.482405
\(46\) 0 0
\(47\) 10.9443 1.59639 0.798193 0.602402i \(-0.205789\pi\)
0.798193 + 0.602402i \(0.205789\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 0 0
\(57\) 0.763932 0.101185
\(58\) 0 0
\(59\) 0.472136 0.0614669 0.0307334 0.999528i \(-0.490216\pi\)
0.0307334 + 0.999528i \(0.490216\pi\)
\(60\) 0 0
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 0 0
\(63\) −3.23607 −0.407706
\(64\) 0 0
\(65\) −3.23607 −0.401385
\(66\) 0 0
\(67\) −11.2361 −1.37270 −0.686352 0.727269i \(-0.740789\pi\)
−0.686352 + 0.727269i \(0.740789\pi\)
\(68\) 0 0
\(69\) 6.47214 0.779154
\(70\) 0 0
\(71\) 4.47214 0.530745 0.265372 0.964146i \(-0.414505\pi\)
0.265372 + 0.964146i \(0.414505\pi\)
\(72\) 0 0
\(73\) −8.47214 −0.991589 −0.495794 0.868440i \(-0.665123\pi\)
−0.495794 + 0.868440i \(0.665123\pi\)
\(74\) 0 0
\(75\) −5.47214 −0.631868
\(76\) 0 0
\(77\) 6.47214 0.737568
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.4721 −1.80805 −0.904026 0.427478i \(-0.859402\pi\)
−0.904026 + 0.427478i \(0.859402\pi\)
\(84\) 0 0
\(85\) −14.4721 −1.56972
\(86\) 0 0
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) 12.1803 1.29111 0.645557 0.763712i \(-0.276625\pi\)
0.645557 + 0.763712i \(0.276625\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 0 0
\(93\) −5.70820 −0.591913
\(94\) 0 0
\(95\) −2.47214 −0.253636
\(96\) 0 0
\(97\) 4.47214 0.454077 0.227038 0.973886i \(-0.427096\pi\)
0.227038 + 0.973886i \(0.427096\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 14.9443 1.48701 0.743505 0.668730i \(-0.233162\pi\)
0.743505 + 0.668730i \(0.233162\pi\)
\(102\) 0 0
\(103\) 10.4721 1.03185 0.515925 0.856634i \(-0.327448\pi\)
0.515925 + 0.856634i \(0.327448\pi\)
\(104\) 0 0
\(105\) 10.4721 1.02198
\(106\) 0 0
\(107\) −5.52786 −0.534399 −0.267199 0.963641i \(-0.586098\pi\)
−0.267199 + 0.963641i \(0.586098\pi\)
\(108\) 0 0
\(109\) −18.9443 −1.81453 −0.907266 0.420557i \(-0.861835\pi\)
−0.907266 + 0.420557i \(0.861835\pi\)
\(110\) 0 0
\(111\) 8.47214 0.804140
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −20.9443 −1.95306
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 14.4721 1.32666
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 3.23607 0.291786
\(124\) 0 0
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) 2.47214 0.219367 0.109683 0.993967i \(-0.465016\pi\)
0.109683 + 0.993967i \(0.465016\pi\)
\(128\) 0 0
\(129\) −2.47214 −0.217659
\(130\) 0 0
\(131\) −15.4164 −1.34694 −0.673469 0.739216i \(-0.735196\pi\)
−0.673469 + 0.739216i \(0.735196\pi\)
\(132\) 0 0
\(133\) 2.47214 0.214361
\(134\) 0 0
\(135\) −3.23607 −0.278516
\(136\) 0 0
\(137\) −11.2361 −0.959962 −0.479981 0.877279i \(-0.659356\pi\)
−0.479981 + 0.877279i \(0.659356\pi\)
\(138\) 0 0
\(139\) −21.8885 −1.85656 −0.928281 0.371879i \(-0.878713\pi\)
−0.928281 + 0.371879i \(0.878713\pi\)
\(140\) 0 0
\(141\) −10.9443 −0.921674
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −14.4721 −1.20185
\(146\) 0 0
\(147\) −3.47214 −0.286377
\(148\) 0 0
\(149\) 21.7082 1.77841 0.889203 0.457513i \(-0.151260\pi\)
0.889203 + 0.457513i \(0.151260\pi\)
\(150\) 0 0
\(151\) 23.2361 1.89092 0.945462 0.325732i \(-0.105611\pi\)
0.945462 + 0.325732i \(0.105611\pi\)
\(152\) 0 0
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 18.4721 1.48372
\(156\) 0 0
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) 0 0
\(159\) 0.472136 0.0374428
\(160\) 0 0
\(161\) 20.9443 1.65064
\(162\) 0 0
\(163\) −7.23607 −0.566773 −0.283386 0.959006i \(-0.591458\pi\)
−0.283386 + 0.959006i \(0.591458\pi\)
\(164\) 0 0
\(165\) 6.47214 0.503855
\(166\) 0 0
\(167\) −19.8885 −1.53902 −0.769511 0.638634i \(-0.779500\pi\)
−0.769511 + 0.638634i \(0.779500\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.763932 −0.0584193
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −17.7082 −1.33861
\(176\) 0 0
\(177\) −0.472136 −0.0354879
\(178\) 0 0
\(179\) 19.4164 1.45125 0.725625 0.688090i \(-0.241551\pi\)
0.725625 + 0.688090i \(0.241551\pi\)
\(180\) 0 0
\(181\) −17.4164 −1.29455 −0.647276 0.762256i \(-0.724092\pi\)
−0.647276 + 0.762256i \(0.724092\pi\)
\(182\) 0 0
\(183\) 3.52786 0.260787
\(184\) 0 0
\(185\) −27.4164 −2.01569
\(186\) 0 0
\(187\) 8.94427 0.654070
\(188\) 0 0
\(189\) 3.23607 0.235389
\(190\) 0 0
\(191\) 10.4721 0.757737 0.378869 0.925450i \(-0.376313\pi\)
0.378869 + 0.925450i \(0.376313\pi\)
\(192\) 0 0
\(193\) −15.8885 −1.14368 −0.571841 0.820364i \(-0.693771\pi\)
−0.571841 + 0.820364i \(0.693771\pi\)
\(194\) 0 0
\(195\) 3.23607 0.231740
\(196\) 0 0
\(197\) 27.2361 1.94049 0.970245 0.242126i \(-0.0778448\pi\)
0.970245 + 0.242126i \(0.0778448\pi\)
\(198\) 0 0
\(199\) 18.4721 1.30945 0.654727 0.755865i \(-0.272783\pi\)
0.654727 + 0.755865i \(0.272783\pi\)
\(200\) 0 0
\(201\) 11.2361 0.792531
\(202\) 0 0
\(203\) 14.4721 1.01574
\(204\) 0 0
\(205\) −10.4721 −0.731406
\(206\) 0 0
\(207\) −6.47214 −0.449845
\(208\) 0 0
\(209\) 1.52786 0.105685
\(210\) 0 0
\(211\) 0.944272 0.0650064 0.0325032 0.999472i \(-0.489652\pi\)
0.0325032 + 0.999472i \(0.489652\pi\)
\(212\) 0 0
\(213\) −4.47214 −0.306426
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −18.4721 −1.25397
\(218\) 0 0
\(219\) 8.47214 0.572494
\(220\) 0 0
\(221\) 4.47214 0.300828
\(222\) 0 0
\(223\) 26.6525 1.78478 0.892391 0.451263i \(-0.149026\pi\)
0.892391 + 0.451263i \(0.149026\pi\)
\(224\) 0 0
\(225\) 5.47214 0.364809
\(226\) 0 0
\(227\) 6.94427 0.460908 0.230454 0.973083i \(-0.425979\pi\)
0.230454 + 0.973083i \(0.425979\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −6.47214 −0.425835
\(232\) 0 0
\(233\) 9.41641 0.616889 0.308445 0.951242i \(-0.400192\pi\)
0.308445 + 0.951242i \(0.400192\pi\)
\(234\) 0 0
\(235\) 35.4164 2.31031
\(236\) 0 0
\(237\) 8.94427 0.580993
\(238\) 0 0
\(239\) −3.52786 −0.228199 −0.114099 0.993469i \(-0.536398\pi\)
−0.114099 + 0.993469i \(0.536398\pi\)
\(240\) 0 0
\(241\) 22.3607 1.44038 0.720189 0.693778i \(-0.244055\pi\)
0.720189 + 0.693778i \(0.244055\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 11.2361 0.717846
\(246\) 0 0
\(247\) 0.763932 0.0486078
\(248\) 0 0
\(249\) 16.4721 1.04388
\(250\) 0 0
\(251\) 11.0557 0.697831 0.348916 0.937154i \(-0.386550\pi\)
0.348916 + 0.937154i \(0.386550\pi\)
\(252\) 0 0
\(253\) 12.9443 0.813799
\(254\) 0 0
\(255\) 14.4721 0.906280
\(256\) 0 0
\(257\) −3.52786 −0.220062 −0.110031 0.993928i \(-0.535095\pi\)
−0.110031 + 0.993928i \(0.535095\pi\)
\(258\) 0 0
\(259\) 27.4164 1.70357
\(260\) 0 0
\(261\) −4.47214 −0.276818
\(262\) 0 0
\(263\) −8.94427 −0.551527 −0.275764 0.961225i \(-0.588931\pi\)
−0.275764 + 0.961225i \(0.588931\pi\)
\(264\) 0 0
\(265\) −1.52786 −0.0938559
\(266\) 0 0
\(267\) −12.1803 −0.745425
\(268\) 0 0
\(269\) 11.5279 0.702866 0.351433 0.936213i \(-0.385694\pi\)
0.351433 + 0.936213i \(0.385694\pi\)
\(270\) 0 0
\(271\) −21.1246 −1.28323 −0.641614 0.767027i \(-0.721735\pi\)
−0.641614 + 0.767027i \(0.721735\pi\)
\(272\) 0 0
\(273\) −3.23607 −0.195856
\(274\) 0 0
\(275\) −10.9443 −0.659964
\(276\) 0 0
\(277\) 2.94427 0.176904 0.0884521 0.996080i \(-0.471808\pi\)
0.0884521 + 0.996080i \(0.471808\pi\)
\(278\) 0 0
\(279\) 5.70820 0.341741
\(280\) 0 0
\(281\) 9.70820 0.579143 0.289571 0.957156i \(-0.406487\pi\)
0.289571 + 0.957156i \(0.406487\pi\)
\(282\) 0 0
\(283\) 13.5279 0.804148 0.402074 0.915607i \(-0.368289\pi\)
0.402074 + 0.915607i \(0.368289\pi\)
\(284\) 0 0
\(285\) 2.47214 0.146437
\(286\) 0 0
\(287\) 10.4721 0.618151
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) −4.47214 −0.262161
\(292\) 0 0
\(293\) −12.1803 −0.711583 −0.355792 0.934565i \(-0.615789\pi\)
−0.355792 + 0.934565i \(0.615789\pi\)
\(294\) 0 0
\(295\) 1.52786 0.0889557
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 6.47214 0.374293
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −14.9443 −0.858526
\(304\) 0 0
\(305\) −11.4164 −0.653702
\(306\) 0 0
\(307\) 11.2361 0.641276 0.320638 0.947202i \(-0.396103\pi\)
0.320638 + 0.947202i \(0.396103\pi\)
\(308\) 0 0
\(309\) −10.4721 −0.595739
\(310\) 0 0
\(311\) 27.4164 1.55464 0.777321 0.629104i \(-0.216578\pi\)
0.777321 + 0.629104i \(0.216578\pi\)
\(312\) 0 0
\(313\) −3.88854 −0.219793 −0.109897 0.993943i \(-0.535052\pi\)
−0.109897 + 0.993943i \(0.535052\pi\)
\(314\) 0 0
\(315\) −10.4721 −0.590038
\(316\) 0 0
\(317\) −4.76393 −0.267569 −0.133785 0.991010i \(-0.542713\pi\)
−0.133785 + 0.991010i \(0.542713\pi\)
\(318\) 0 0
\(319\) 8.94427 0.500783
\(320\) 0 0
\(321\) 5.52786 0.308535
\(322\) 0 0
\(323\) 3.41641 0.190094
\(324\) 0 0
\(325\) −5.47214 −0.303539
\(326\) 0 0
\(327\) 18.9443 1.04762
\(328\) 0 0
\(329\) −35.4164 −1.95257
\(330\) 0 0
\(331\) −8.18034 −0.449632 −0.224816 0.974401i \(-0.572178\pi\)
−0.224816 + 0.974401i \(0.572178\pi\)
\(332\) 0 0
\(333\) −8.47214 −0.464270
\(334\) 0 0
\(335\) −36.3607 −1.98660
\(336\) 0 0
\(337\) 10.9443 0.596172 0.298086 0.954539i \(-0.403652\pi\)
0.298086 + 0.954539i \(0.403652\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −11.4164 −0.618233
\(342\) 0 0
\(343\) 11.4164 0.616428
\(344\) 0 0
\(345\) 20.9443 1.12760
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 30.3607 1.62517 0.812585 0.582843i \(-0.198060\pi\)
0.812585 + 0.582843i \(0.198060\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 27.5967 1.46883 0.734413 0.678702i \(-0.237457\pi\)
0.734413 + 0.678702i \(0.237457\pi\)
\(354\) 0 0
\(355\) 14.4721 0.768101
\(356\) 0 0
\(357\) −14.4721 −0.765947
\(358\) 0 0
\(359\) −16.8328 −0.888402 −0.444201 0.895927i \(-0.646512\pi\)
−0.444201 + 0.895927i \(0.646512\pi\)
\(360\) 0 0
\(361\) −18.4164 −0.969285
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −27.4164 −1.43504
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −3.23607 −0.168463
\(370\) 0 0
\(371\) 1.52786 0.0793227
\(372\) 0 0
\(373\) −34.9443 −1.80935 −0.904673 0.426107i \(-0.859885\pi\)
−0.904673 + 0.426107i \(0.859885\pi\)
\(374\) 0 0
\(375\) −1.52786 −0.0788986
\(376\) 0 0
\(377\) 4.47214 0.230327
\(378\) 0 0
\(379\) −23.5967 −1.21208 −0.606042 0.795433i \(-0.707244\pi\)
−0.606042 + 0.795433i \(0.707244\pi\)
\(380\) 0 0
\(381\) −2.47214 −0.126651
\(382\) 0 0
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 20.9443 1.06742
\(386\) 0 0
\(387\) 2.47214 0.125666
\(388\) 0 0
\(389\) −11.8885 −0.602773 −0.301387 0.953502i \(-0.597449\pi\)
−0.301387 + 0.953502i \(0.597449\pi\)
\(390\) 0 0
\(391\) 28.9443 1.46377
\(392\) 0 0
\(393\) 15.4164 0.777655
\(394\) 0 0
\(395\) −28.9443 −1.45634
\(396\) 0 0
\(397\) −16.4721 −0.826713 −0.413356 0.910569i \(-0.635644\pi\)
−0.413356 + 0.910569i \(0.635644\pi\)
\(398\) 0 0
\(399\) −2.47214 −0.123762
\(400\) 0 0
\(401\) 4.18034 0.208756 0.104378 0.994538i \(-0.466715\pi\)
0.104378 + 0.994538i \(0.466715\pi\)
\(402\) 0 0
\(403\) −5.70820 −0.284346
\(404\) 0 0
\(405\) 3.23607 0.160802
\(406\) 0 0
\(407\) 16.9443 0.839896
\(408\) 0 0
\(409\) −10.3607 −0.512303 −0.256151 0.966637i \(-0.582455\pi\)
−0.256151 + 0.966637i \(0.582455\pi\)
\(410\) 0 0
\(411\) 11.2361 0.554234
\(412\) 0 0
\(413\) −1.52786 −0.0751813
\(414\) 0 0
\(415\) −53.3050 −2.61664
\(416\) 0 0
\(417\) 21.8885 1.07189
\(418\) 0 0
\(419\) −29.5279 −1.44253 −0.721265 0.692659i \(-0.756439\pi\)
−0.721265 + 0.692659i \(0.756439\pi\)
\(420\) 0 0
\(421\) −9.05573 −0.441349 −0.220675 0.975347i \(-0.570826\pi\)
−0.220675 + 0.975347i \(0.570826\pi\)
\(422\) 0 0
\(423\) 10.9443 0.532129
\(424\) 0 0
\(425\) −24.4721 −1.18707
\(426\) 0 0
\(427\) 11.4164 0.552479
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 0 0
\(433\) 0.472136 0.0226894 0.0113447 0.999936i \(-0.496389\pi\)
0.0113447 + 0.999936i \(0.496389\pi\)
\(434\) 0 0
\(435\) 14.4721 0.693886
\(436\) 0 0
\(437\) 4.94427 0.236517
\(438\) 0 0
\(439\) −13.5279 −0.645650 −0.322825 0.946459i \(-0.604632\pi\)
−0.322825 + 0.946459i \(0.604632\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) 0 0
\(443\) 21.5279 1.02282 0.511410 0.859337i \(-0.329123\pi\)
0.511410 + 0.859337i \(0.329123\pi\)
\(444\) 0 0
\(445\) 39.4164 1.86852
\(446\) 0 0
\(447\) −21.7082 −1.02676
\(448\) 0 0
\(449\) −36.5410 −1.72448 −0.862239 0.506502i \(-0.830938\pi\)
−0.862239 + 0.506502i \(0.830938\pi\)
\(450\) 0 0
\(451\) 6.47214 0.304761
\(452\) 0 0
\(453\) −23.2361 −1.09173
\(454\) 0 0
\(455\) 10.4721 0.490941
\(456\) 0 0
\(457\) −34.3607 −1.60732 −0.803662 0.595085i \(-0.797118\pi\)
−0.803662 + 0.595085i \(0.797118\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) −22.6525 −1.05503 −0.527515 0.849545i \(-0.676876\pi\)
−0.527515 + 0.849545i \(0.676876\pi\)
\(462\) 0 0
\(463\) 17.7082 0.822970 0.411485 0.911417i \(-0.365010\pi\)
0.411485 + 0.911417i \(0.365010\pi\)
\(464\) 0 0
\(465\) −18.4721 −0.856625
\(466\) 0 0
\(467\) −1.88854 −0.0873914 −0.0436957 0.999045i \(-0.513913\pi\)
−0.0436957 + 0.999045i \(0.513913\pi\)
\(468\) 0 0
\(469\) 36.3607 1.67898
\(470\) 0 0
\(471\) −13.4164 −0.618195
\(472\) 0 0
\(473\) −4.94427 −0.227338
\(474\) 0 0
\(475\) −4.18034 −0.191807
\(476\) 0 0
\(477\) −0.472136 −0.0216176
\(478\) 0 0
\(479\) 23.3050 1.06483 0.532415 0.846483i \(-0.321285\pi\)
0.532415 + 0.846483i \(0.321285\pi\)
\(480\) 0 0
\(481\) 8.47214 0.386296
\(482\) 0 0
\(483\) −20.9443 −0.952997
\(484\) 0 0
\(485\) 14.4721 0.657146
\(486\) 0 0
\(487\) −26.2918 −1.19140 −0.595698 0.803209i \(-0.703124\pi\)
−0.595698 + 0.803209i \(0.703124\pi\)
\(488\) 0 0
\(489\) 7.23607 0.327226
\(490\) 0 0
\(491\) 11.0557 0.498938 0.249469 0.968383i \(-0.419744\pi\)
0.249469 + 0.968383i \(0.419744\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −6.47214 −0.290901
\(496\) 0 0
\(497\) −14.4721 −0.649164
\(498\) 0 0
\(499\) −40.1803 −1.79872 −0.899360 0.437210i \(-0.855967\pi\)
−0.899360 + 0.437210i \(0.855967\pi\)
\(500\) 0 0
\(501\) 19.8885 0.888555
\(502\) 0 0
\(503\) 39.7771 1.77357 0.886786 0.462180i \(-0.152932\pi\)
0.886786 + 0.462180i \(0.152932\pi\)
\(504\) 0 0
\(505\) 48.3607 2.15202
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 16.1803 0.717181 0.358590 0.933495i \(-0.383257\pi\)
0.358590 + 0.933495i \(0.383257\pi\)
\(510\) 0 0
\(511\) 27.4164 1.21283
\(512\) 0 0
\(513\) 0.763932 0.0337284
\(514\) 0 0
\(515\) 33.8885 1.49331
\(516\) 0 0
\(517\) −21.8885 −0.962657
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 18.9443 0.829964 0.414982 0.909830i \(-0.363788\pi\)
0.414982 + 0.909830i \(0.363788\pi\)
\(522\) 0 0
\(523\) −23.0557 −1.00816 −0.504078 0.863658i \(-0.668168\pi\)
−0.504078 + 0.863658i \(0.668168\pi\)
\(524\) 0 0
\(525\) 17.7082 0.772849
\(526\) 0 0
\(527\) −25.5279 −1.11201
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 0 0
\(531\) 0.472136 0.0204890
\(532\) 0 0
\(533\) 3.23607 0.140170
\(534\) 0 0
\(535\) −17.8885 −0.773389
\(536\) 0 0
\(537\) −19.4164 −0.837880
\(538\) 0 0
\(539\) −6.94427 −0.299111
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 17.4164 0.747410
\(544\) 0 0
\(545\) −61.3050 −2.62602
\(546\) 0 0
\(547\) −0.944272 −0.0403742 −0.0201871 0.999796i \(-0.506426\pi\)
−0.0201871 + 0.999796i \(0.506426\pi\)
\(548\) 0 0
\(549\) −3.52786 −0.150566
\(550\) 0 0
\(551\) 3.41641 0.145544
\(552\) 0 0
\(553\) 28.9443 1.23084
\(554\) 0 0
\(555\) 27.4164 1.16376
\(556\) 0 0
\(557\) 13.7082 0.580835 0.290418 0.956900i \(-0.406206\pi\)
0.290418 + 0.956900i \(0.406206\pi\)
\(558\) 0 0
\(559\) −2.47214 −0.104560
\(560\) 0 0
\(561\) −8.94427 −0.377627
\(562\) 0 0
\(563\) −1.52786 −0.0643918 −0.0321959 0.999482i \(-0.510250\pi\)
−0.0321959 + 0.999482i \(0.510250\pi\)
\(564\) 0 0
\(565\) −6.47214 −0.272285
\(566\) 0 0
\(567\) −3.23607 −0.135902
\(568\) 0 0
\(569\) 30.3607 1.27279 0.636393 0.771365i \(-0.280426\pi\)
0.636393 + 0.771365i \(0.280426\pi\)
\(570\) 0 0
\(571\) 34.4721 1.44261 0.721307 0.692615i \(-0.243542\pi\)
0.721307 + 0.692615i \(0.243542\pi\)
\(572\) 0 0
\(573\) −10.4721 −0.437480
\(574\) 0 0
\(575\) −35.4164 −1.47697
\(576\) 0 0
\(577\) −18.9443 −0.788660 −0.394330 0.918969i \(-0.629023\pi\)
−0.394330 + 0.918969i \(0.629023\pi\)
\(578\) 0 0
\(579\) 15.8885 0.660305
\(580\) 0 0
\(581\) 53.3050 2.21146
\(582\) 0 0
\(583\) 0.944272 0.0391077
\(584\) 0 0
\(585\) −3.23607 −0.133795
\(586\) 0 0
\(587\) −6.58359 −0.271734 −0.135867 0.990727i \(-0.543382\pi\)
−0.135867 + 0.990727i \(0.543382\pi\)
\(588\) 0 0
\(589\) −4.36068 −0.179679
\(590\) 0 0
\(591\) −27.2361 −1.12034
\(592\) 0 0
\(593\) 15.2361 0.625670 0.312835 0.949807i \(-0.398721\pi\)
0.312835 + 0.949807i \(0.398721\pi\)
\(594\) 0 0
\(595\) 46.8328 1.91996
\(596\) 0 0
\(597\) −18.4721 −0.756014
\(598\) 0 0
\(599\) 24.9443 1.01920 0.509598 0.860413i \(-0.329794\pi\)
0.509598 + 0.860413i \(0.329794\pi\)
\(600\) 0 0
\(601\) 47.8885 1.95341 0.976707 0.214576i \(-0.0688371\pi\)
0.976707 + 0.214576i \(0.0688371\pi\)
\(602\) 0 0
\(603\) −11.2361 −0.457568
\(604\) 0 0
\(605\) −22.6525 −0.920954
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) −14.4721 −0.586441
\(610\) 0 0
\(611\) −10.9443 −0.442758
\(612\) 0 0
\(613\) 30.3607 1.22626 0.613128 0.789983i \(-0.289911\pi\)
0.613128 + 0.789983i \(0.289911\pi\)
\(614\) 0 0
\(615\) 10.4721 0.422277
\(616\) 0 0
\(617\) 3.59675 0.144800 0.0723998 0.997376i \(-0.476934\pi\)
0.0723998 + 0.997376i \(0.476934\pi\)
\(618\) 0 0
\(619\) −38.6525 −1.55357 −0.776787 0.629763i \(-0.783152\pi\)
−0.776787 + 0.629763i \(0.783152\pi\)
\(620\) 0 0
\(621\) 6.47214 0.259718
\(622\) 0 0
\(623\) −39.4164 −1.57919
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) −1.52786 −0.0610170
\(628\) 0 0
\(629\) 37.8885 1.51072
\(630\) 0 0
\(631\) −18.6525 −0.742543 −0.371272 0.928524i \(-0.621078\pi\)
−0.371272 + 0.928524i \(0.621078\pi\)
\(632\) 0 0
\(633\) −0.944272 −0.0375314
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −3.47214 −0.137571
\(638\) 0 0
\(639\) 4.47214 0.176915
\(640\) 0 0
\(641\) 21.4164 0.845897 0.422949 0.906154i \(-0.360995\pi\)
0.422949 + 0.906154i \(0.360995\pi\)
\(642\) 0 0
\(643\) −29.1246 −1.14856 −0.574281 0.818658i \(-0.694718\pi\)
−0.574281 + 0.818658i \(0.694718\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −0.944272 −0.0370659
\(650\) 0 0
\(651\) 18.4721 0.723980
\(652\) 0 0
\(653\) 10.3607 0.405445 0.202722 0.979236i \(-0.435021\pi\)
0.202722 + 0.979236i \(0.435021\pi\)
\(654\) 0 0
\(655\) −49.8885 −1.94931
\(656\) 0 0
\(657\) −8.47214 −0.330530
\(658\) 0 0
\(659\) −34.8328 −1.35689 −0.678447 0.734649i \(-0.737347\pi\)
−0.678447 + 0.734649i \(0.737347\pi\)
\(660\) 0 0
\(661\) −36.2492 −1.40993 −0.704966 0.709241i \(-0.749038\pi\)
−0.704966 + 0.709241i \(0.749038\pi\)
\(662\) 0 0
\(663\) −4.47214 −0.173683
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 28.9443 1.12073
\(668\) 0 0
\(669\) −26.6525 −1.03044
\(670\) 0 0
\(671\) 7.05573 0.272383
\(672\) 0 0
\(673\) 18.5836 0.716345 0.358172 0.933655i \(-0.383400\pi\)
0.358172 + 0.933655i \(0.383400\pi\)
\(674\) 0 0
\(675\) −5.47214 −0.210623
\(676\) 0 0
\(677\) −34.3607 −1.32059 −0.660294 0.751007i \(-0.729568\pi\)
−0.660294 + 0.751007i \(0.729568\pi\)
\(678\) 0 0
\(679\) −14.4721 −0.555390
\(680\) 0 0
\(681\) −6.94427 −0.266105
\(682\) 0 0
\(683\) −10.0000 −0.382639 −0.191320 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(684\) 0 0
\(685\) −36.3607 −1.38927
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) 0.472136 0.0179869
\(690\) 0 0
\(691\) 27.5967 1.04983 0.524915 0.851155i \(-0.324097\pi\)
0.524915 + 0.851155i \(0.324097\pi\)
\(692\) 0 0
\(693\) 6.47214 0.245856
\(694\) 0 0
\(695\) −70.8328 −2.68684
\(696\) 0 0
\(697\) 14.4721 0.548171
\(698\) 0 0
\(699\) −9.41641 −0.356161
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 6.47214 0.244101
\(704\) 0 0
\(705\) −35.4164 −1.33386
\(706\) 0 0
\(707\) −48.3607 −1.81879
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −8.94427 −0.335436
\(712\) 0 0
\(713\) −36.9443 −1.38357
\(714\) 0 0
\(715\) 6.47214 0.242044
\(716\) 0 0
\(717\) 3.52786 0.131750
\(718\) 0 0
\(719\) −6.47214 −0.241370 −0.120685 0.992691i \(-0.538509\pi\)
−0.120685 + 0.992691i \(0.538509\pi\)
\(720\) 0 0
\(721\) −33.8885 −1.26208
\(722\) 0 0
\(723\) −22.3607 −0.831603
\(724\) 0 0
\(725\) −24.4721 −0.908872
\(726\) 0 0
\(727\) −10.4721 −0.388390 −0.194195 0.980963i \(-0.562209\pi\)
−0.194195 + 0.980963i \(0.562209\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.0557 −0.408911
\(732\) 0 0
\(733\) −7.88854 −0.291370 −0.145685 0.989331i \(-0.546539\pi\)
−0.145685 + 0.989331i \(0.546539\pi\)
\(734\) 0 0
\(735\) −11.2361 −0.414449
\(736\) 0 0
\(737\) 22.4721 0.827772
\(738\) 0 0
\(739\) −23.8197 −0.876220 −0.438110 0.898921i \(-0.644352\pi\)
−0.438110 + 0.898921i \(0.644352\pi\)
\(740\) 0 0
\(741\) −0.763932 −0.0280637
\(742\) 0 0
\(743\) 7.52786 0.276171 0.138085 0.990420i \(-0.455905\pi\)
0.138085 + 0.990420i \(0.455905\pi\)
\(744\) 0 0
\(745\) 70.2492 2.57373
\(746\) 0 0
\(747\) −16.4721 −0.602684
\(748\) 0 0
\(749\) 17.8885 0.653633
\(750\) 0 0
\(751\) 12.3607 0.451048 0.225524 0.974238i \(-0.427591\pi\)
0.225524 + 0.974238i \(0.427591\pi\)
\(752\) 0 0
\(753\) −11.0557 −0.402893
\(754\) 0 0
\(755\) 75.1935 2.73657
\(756\) 0 0
\(757\) 10.5836 0.384667 0.192334 0.981330i \(-0.438394\pi\)
0.192334 + 0.981330i \(0.438394\pi\)
\(758\) 0 0
\(759\) −12.9443 −0.469847
\(760\) 0 0
\(761\) −32.1803 −1.16654 −0.583268 0.812280i \(-0.698226\pi\)
−0.583268 + 0.812280i \(0.698226\pi\)
\(762\) 0 0
\(763\) 61.3050 2.21939
\(764\) 0 0
\(765\) −14.4721 −0.523241
\(766\) 0 0
\(767\) −0.472136 −0.0170478
\(768\) 0 0
\(769\) 0.832816 0.0300321 0.0150161 0.999887i \(-0.495220\pi\)
0.0150161 + 0.999887i \(0.495220\pi\)
\(770\) 0 0
\(771\) 3.52786 0.127053
\(772\) 0 0
\(773\) 21.7082 0.780790 0.390395 0.920647i \(-0.372338\pi\)
0.390395 + 0.920647i \(0.372338\pi\)
\(774\) 0 0
\(775\) 31.2361 1.12203
\(776\) 0 0
\(777\) −27.4164 −0.983558
\(778\) 0 0
\(779\) 2.47214 0.0885735
\(780\) 0 0
\(781\) −8.94427 −0.320051
\(782\) 0 0
\(783\) 4.47214 0.159821
\(784\) 0 0
\(785\) 43.4164 1.54960
\(786\) 0 0
\(787\) −17.7082 −0.631229 −0.315615 0.948887i \(-0.602211\pi\)
−0.315615 + 0.948887i \(0.602211\pi\)
\(788\) 0 0
\(789\) 8.94427 0.318425
\(790\) 0 0
\(791\) 6.47214 0.230123
\(792\) 0 0
\(793\) 3.52786 0.125278
\(794\) 0 0
\(795\) 1.52786 0.0541878
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −48.9443 −1.73152
\(800\) 0 0
\(801\) 12.1803 0.430371
\(802\) 0 0
\(803\) 16.9443 0.597950
\(804\) 0 0
\(805\) 67.7771 2.38883
\(806\) 0 0
\(807\) −11.5279 −0.405800
\(808\) 0 0
\(809\) −13.0557 −0.459015 −0.229507 0.973307i \(-0.573712\pi\)
−0.229507 + 0.973307i \(0.573712\pi\)
\(810\) 0 0
\(811\) −3.81966 −0.134126 −0.0670632 0.997749i \(-0.521363\pi\)
−0.0670632 + 0.997749i \(0.521363\pi\)
\(812\) 0 0
\(813\) 21.1246 0.740872
\(814\) 0 0
\(815\) −23.4164 −0.820241
\(816\) 0 0
\(817\) −1.88854 −0.0660718
\(818\) 0 0
\(819\) 3.23607 0.113077
\(820\) 0 0
\(821\) −9.12461 −0.318451 −0.159226 0.987242i \(-0.550900\pi\)
−0.159226 + 0.987242i \(0.550900\pi\)
\(822\) 0 0
\(823\) −17.3050 −0.603213 −0.301606 0.953433i \(-0.597523\pi\)
−0.301606 + 0.953433i \(0.597523\pi\)
\(824\) 0 0
\(825\) 10.9443 0.381031
\(826\) 0 0
\(827\) −37.7771 −1.31364 −0.656819 0.754048i \(-0.728098\pi\)
−0.656819 + 0.754048i \(0.728098\pi\)
\(828\) 0 0
\(829\) 39.5279 1.37286 0.686430 0.727196i \(-0.259177\pi\)
0.686430 + 0.727196i \(0.259177\pi\)
\(830\) 0 0
\(831\) −2.94427 −0.102136
\(832\) 0 0
\(833\) −15.5279 −0.538009
\(834\) 0 0
\(835\) −64.3607 −2.22729
\(836\) 0 0
\(837\) −5.70820 −0.197304
\(838\) 0 0
\(839\) −14.3607 −0.495786 −0.247893 0.968787i \(-0.579738\pi\)
−0.247893 + 0.968787i \(0.579738\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) −9.70820 −0.334368
\(844\) 0 0
\(845\) 3.23607 0.111324
\(846\) 0 0
\(847\) 22.6525 0.778348
\(848\) 0 0
\(849\) −13.5279 −0.464275
\(850\) 0 0
\(851\) 54.8328 1.87964
\(852\) 0 0
\(853\) −11.5279 −0.394707 −0.197353 0.980332i \(-0.563235\pi\)
−0.197353 + 0.980332i \(0.563235\pi\)
\(854\) 0 0
\(855\) −2.47214 −0.0845453
\(856\) 0 0
\(857\) −37.4164 −1.27812 −0.639060 0.769157i \(-0.720676\pi\)
−0.639060 + 0.769157i \(0.720676\pi\)
\(858\) 0 0
\(859\) 32.9443 1.12404 0.562022 0.827122i \(-0.310024\pi\)
0.562022 + 0.827122i \(0.310024\pi\)
\(860\) 0 0
\(861\) −10.4721 −0.356889
\(862\) 0 0
\(863\) −47.8885 −1.63014 −0.815072 0.579359i \(-0.803303\pi\)
−0.815072 + 0.579359i \(0.803303\pi\)
\(864\) 0 0
\(865\) 6.47214 0.220059
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 17.8885 0.606827
\(870\) 0 0
\(871\) 11.2361 0.380720
\(872\) 0 0
\(873\) 4.47214 0.151359
\(874\) 0 0
\(875\) −4.94427 −0.167147
\(876\) 0 0
\(877\) 27.3050 0.922023 0.461011 0.887394i \(-0.347487\pi\)
0.461011 + 0.887394i \(0.347487\pi\)
\(878\) 0 0
\(879\) 12.1803 0.410833
\(880\) 0 0
\(881\) 5.63932 0.189994 0.0949968 0.995478i \(-0.469716\pi\)
0.0949968 + 0.995478i \(0.469716\pi\)
\(882\) 0 0
\(883\) 3.63932 0.122473 0.0612364 0.998123i \(-0.480496\pi\)
0.0612364 + 0.998123i \(0.480496\pi\)
\(884\) 0 0
\(885\) −1.52786 −0.0513586
\(886\) 0 0
\(887\) 17.3050 0.581043 0.290522 0.956868i \(-0.406171\pi\)
0.290522 + 0.956868i \(0.406171\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) −8.36068 −0.279779
\(894\) 0 0
\(895\) 62.8328 2.10027
\(896\) 0 0
\(897\) −6.47214 −0.216098
\(898\) 0 0
\(899\) −25.5279 −0.851402
\(900\) 0 0
\(901\) 2.11146 0.0703428
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) −56.3607 −1.87349
\(906\) 0 0
\(907\) −20.3607 −0.676065 −0.338033 0.941134i \(-0.609761\pi\)
−0.338033 + 0.941134i \(0.609761\pi\)
\(908\) 0 0
\(909\) 14.9443 0.495670
\(910\) 0 0
\(911\) 45.5279 1.50841 0.754203 0.656642i \(-0.228024\pi\)
0.754203 + 0.656642i \(0.228024\pi\)
\(912\) 0 0
\(913\) 32.9443 1.09030
\(914\) 0 0
\(915\) 11.4164 0.377415
\(916\) 0 0
\(917\) 49.8885 1.64746
\(918\) 0 0
\(919\) 13.8885 0.458141 0.229070 0.973410i \(-0.426431\pi\)
0.229070 + 0.973410i \(0.426431\pi\)
\(920\) 0 0
\(921\) −11.2361 −0.370241
\(922\) 0 0
\(923\) −4.47214 −0.147202
\(924\) 0 0
\(925\) −46.3607 −1.52433
\(926\) 0 0
\(927\) 10.4721 0.343950
\(928\) 0 0
\(929\) 8.54102 0.280222 0.140111 0.990136i \(-0.455254\pi\)
0.140111 + 0.990136i \(0.455254\pi\)
\(930\) 0 0
\(931\) −2.65248 −0.0869314
\(932\) 0 0
\(933\) −27.4164 −0.897573
\(934\) 0 0
\(935\) 28.9443 0.946579
\(936\) 0 0
\(937\) 31.3050 1.02269 0.511344 0.859376i \(-0.329148\pi\)
0.511344 + 0.859376i \(0.329148\pi\)
\(938\) 0 0
\(939\) 3.88854 0.126898
\(940\) 0 0
\(941\) −31.2361 −1.01827 −0.509133 0.860688i \(-0.670034\pi\)
−0.509133 + 0.860688i \(0.670034\pi\)
\(942\) 0 0
\(943\) 20.9443 0.682039
\(944\) 0 0
\(945\) 10.4721 0.340659
\(946\) 0 0
\(947\) −41.4164 −1.34585 −0.672926 0.739710i \(-0.734963\pi\)
−0.672926 + 0.739710i \(0.734963\pi\)
\(948\) 0 0
\(949\) 8.47214 0.275017
\(950\) 0 0
\(951\) 4.76393 0.154481
\(952\) 0 0
\(953\) 31.5279 1.02129 0.510644 0.859792i \(-0.329407\pi\)
0.510644 + 0.859792i \(0.329407\pi\)
\(954\) 0 0
\(955\) 33.8885 1.09661
\(956\) 0 0
\(957\) −8.94427 −0.289127
\(958\) 0 0
\(959\) 36.3607 1.17415
\(960\) 0 0
\(961\) 1.58359 0.0510836
\(962\) 0 0
\(963\) −5.52786 −0.178133
\(964\) 0 0
\(965\) −51.4164 −1.65515
\(966\) 0 0
\(967\) −15.2361 −0.489959 −0.244979 0.969528i \(-0.578781\pi\)
−0.244979 + 0.969528i \(0.578781\pi\)
\(968\) 0 0
\(969\) −3.41641 −0.109751
\(970\) 0 0
\(971\) 50.8328 1.63130 0.815651 0.578544i \(-0.196379\pi\)
0.815651 + 0.578544i \(0.196379\pi\)
\(972\) 0 0
\(973\) 70.8328 2.27080
\(974\) 0 0
\(975\) 5.47214 0.175249
\(976\) 0 0
\(977\) −13.7082 −0.438564 −0.219282 0.975661i \(-0.570372\pi\)
−0.219282 + 0.975661i \(0.570372\pi\)
\(978\) 0 0
\(979\) −24.3607 −0.778571
\(980\) 0 0
\(981\) −18.9443 −0.604844
\(982\) 0 0
\(983\) 22.5836 0.720305 0.360152 0.932893i \(-0.382725\pi\)
0.360152 + 0.932893i \(0.382725\pi\)
\(984\) 0 0
\(985\) 88.1378 2.80830
\(986\) 0 0
\(987\) 35.4164 1.12732
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 40.9443 1.30064 0.650319 0.759661i \(-0.274635\pi\)
0.650319 + 0.759661i \(0.274635\pi\)
\(992\) 0 0
\(993\) 8.18034 0.259595
\(994\) 0 0
\(995\) 59.7771 1.89506
\(996\) 0 0
\(997\) −59.8885 −1.89669 −0.948345 0.317242i \(-0.897243\pi\)
−0.948345 + 0.317242i \(0.897243\pi\)
\(998\) 0 0
\(999\) 8.47214 0.268047
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 1248.2.a.l.1.2 2
3.2 odd 2 3744.2.a.r.1.1 2
4.3 odd 2 1248.2.a.n.1.2 yes 2
8.3 odd 2 2496.2.a.be.1.1 2
8.5 even 2 2496.2.a.bh.1.1 2
12.11 even 2 3744.2.a.s.1.1 2
24.5 odd 2 7488.2.a.cs.1.2 2
24.11 even 2 7488.2.a.ct.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.l.1.2 2 1.1 even 1 trivial
1248.2.a.n.1.2 yes 2 4.3 odd 2
2496.2.a.be.1.1 2 8.3 odd 2
2496.2.a.bh.1.1 2 8.5 even 2
3744.2.a.r.1.1 2 3.2 odd 2
3744.2.a.s.1.1 2 12.11 even 2
7488.2.a.cs.1.2 2 24.5 odd 2
7488.2.a.ct.1.2 2 24.11 even 2