Properties

Label 6525.2.a.t.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{7} -1.41742 q^{8} +O(q^{10})\) \(q+1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{7} -1.41742 q^{8} -5.00000 q^{11} +4.58258 q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000 q^{17} +3.58258 q^{19} -8.95644 q^{22} -4.00000 q^{23} +8.20871 q^{26} -1.20871 q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356 q^{32} -5.37386 q^{34} +4.00000 q^{37} +6.41742 q^{38} +9.16515 q^{41} +9.58258 q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826 q^{47} -6.00000 q^{49} +5.53901 q^{52} +0.417424 q^{53} +1.41742 q^{56} -1.79129 q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515 q^{62} -0.912878 q^{64} +4.16515 q^{67} -3.62614 q^{68} +9.58258 q^{71} -4.00000 q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000 q^{77} +7.58258 q^{79} +16.4174 q^{82} -11.5826 q^{83} +17.1652 q^{86} +7.08712 q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485 q^{92} +18.9564 q^{94} -11.5826 q^{97} -10.7477 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8} - 10 q^{11} + q^{14} + 13 q^{16} - 6 q^{17} - 2 q^{19} + 5 q^{22} - 8 q^{23} + 21 q^{26} - 7 q^{28} - 2 q^{29} + 8 q^{31} - 35 q^{32} + 3 q^{34} + 8 q^{37} + 22 q^{38} + 10 q^{43} - 35 q^{44} + 4 q^{46} + 12 q^{47} - 12 q^{49} - 21 q^{52} + 10 q^{53} + 12 q^{56} + q^{58} + 6 q^{59} - 2 q^{61} - 4 q^{62} + 44 q^{64} - 10 q^{67} - 21 q^{68} + 10 q^{71} - 8 q^{73} - 4 q^{74} - 28 q^{76} + 10 q^{77} + 6 q^{79} + 42 q^{82} - 14 q^{83} + 16 q^{86} + 60 q^{88} - 12 q^{89} - 28 q^{92} + 15 q^{94} - 14 q^{97} + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.79129 1.26663 0.633316 0.773893i \(-0.281693\pi\)
0.633316 + 0.773893i \(0.281693\pi\)
\(3\) 0 0
\(4\) 1.20871 0.604356
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.41742 −0.501135
\(9\) 0 0
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 4.58258 1.27098 0.635489 0.772110i \(-0.280799\pi\)
0.635489 + 0.772110i \(0.280799\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) −4.95644 −1.23911
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 3.58258 0.821899 0.410950 0.911658i \(-0.365197\pi\)
0.410950 + 0.911658i \(0.365197\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.95644 −1.90952
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.20871 1.60986
\(27\) 0 0
\(28\) −1.20871 −0.228425
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −6.04356 −1.06836
\(33\) 0 0
\(34\) −5.37386 −0.921610
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 6.41742 1.04104
\(39\) 0 0
\(40\) 0 0
\(41\) 9.16515 1.43136 0.715678 0.698430i \(-0.246118\pi\)
0.715678 + 0.698430i \(0.246118\pi\)
\(42\) 0 0
\(43\) 9.58258 1.46133 0.730665 0.682737i \(-0.239210\pi\)
0.730665 + 0.682737i \(0.239210\pi\)
\(44\) −6.04356 −0.911101
\(45\) 0 0
\(46\) −7.16515 −1.05644
\(47\) 10.5826 1.54363 0.771814 0.635849i \(-0.219350\pi\)
0.771814 + 0.635849i \(0.219350\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 5.53901 0.768123
\(53\) 0.417424 0.0573376 0.0286688 0.999589i \(-0.490873\pi\)
0.0286688 + 0.999589i \(0.490873\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.41742 0.189411
\(57\) 0 0
\(58\) −1.79129 −0.235208
\(59\) 7.58258 0.987167 0.493584 0.869698i \(-0.335687\pi\)
0.493584 + 0.869698i \(0.335687\pi\)
\(60\) 0 0
\(61\) 12.7477 1.63218 0.816090 0.577925i \(-0.196138\pi\)
0.816090 + 0.577925i \(0.196138\pi\)
\(62\) 7.16515 0.909975
\(63\) 0 0
\(64\) −0.912878 −0.114110
\(65\) 0 0
\(66\) 0 0
\(67\) 4.16515 0.508854 0.254427 0.967092i \(-0.418113\pi\)
0.254427 + 0.967092i \(0.418113\pi\)
\(68\) −3.62614 −0.439734
\(69\) 0 0
\(70\) 0 0
\(71\) 9.58258 1.13724 0.568621 0.822599i \(-0.307477\pi\)
0.568621 + 0.822599i \(0.307477\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 7.16515 0.832932
\(75\) 0 0
\(76\) 4.33030 0.496720
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 7.58258 0.853106 0.426553 0.904462i \(-0.359728\pi\)
0.426553 + 0.904462i \(0.359728\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 16.4174 1.81300
\(83\) −11.5826 −1.27135 −0.635676 0.771956i \(-0.719279\pi\)
−0.635676 + 0.771956i \(0.719279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 17.1652 1.85097
\(87\) 0 0
\(88\) 7.08712 0.755490
\(89\) −1.41742 −0.150247 −0.0751233 0.997174i \(-0.523935\pi\)
−0.0751233 + 0.997174i \(0.523935\pi\)
\(90\) 0 0
\(91\) −4.58258 −0.480384
\(92\) −4.83485 −0.504068
\(93\) 0 0
\(94\) 18.9564 1.95521
\(95\) 0 0
\(96\) 0 0
\(97\) −11.5826 −1.17603 −0.588016 0.808849i \(-0.700091\pi\)
−0.588016 + 0.808849i \(0.700091\pi\)
\(98\) −10.7477 −1.08568
\(99\) 0 0
\(100\) 0 0
\(101\) 0.582576 0.0579684 0.0289842 0.999580i \(-0.490773\pi\)
0.0289842 + 0.999580i \(0.490773\pi\)
\(102\) 0 0
\(103\) −15.1652 −1.49427 −0.747133 0.664674i \(-0.768570\pi\)
−0.747133 + 0.664674i \(0.768570\pi\)
\(104\) −6.49545 −0.636932
\(105\) 0 0
\(106\) 0.747727 0.0726257
\(107\) 5.16515 0.499334 0.249667 0.968332i \(-0.419679\pi\)
0.249667 + 0.968332i \(0.419679\pi\)
\(108\) 0 0
\(109\) 14.1652 1.35678 0.678388 0.734704i \(-0.262679\pi\)
0.678388 + 0.734704i \(0.262679\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.95644 0.468339
\(113\) −14.1652 −1.33255 −0.666273 0.745708i \(-0.732111\pi\)
−0.666273 + 0.745708i \(0.732111\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.20871 −0.112226
\(117\) 0 0
\(118\) 13.5826 1.25038
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 22.8348 2.06737
\(123\) 0 0
\(124\) 4.83485 0.434182
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 10.4519 0.923826
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −3.58258 −0.310649
\(134\) 7.46099 0.644531
\(135\) 0 0
\(136\) 4.25227 0.364629
\(137\) 16.3303 1.39519 0.697596 0.716491i \(-0.254253\pi\)
0.697596 + 0.716491i \(0.254253\pi\)
\(138\) 0 0
\(139\) −9.41742 −0.798776 −0.399388 0.916782i \(-0.630777\pi\)
−0.399388 + 0.916782i \(0.630777\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 17.1652 1.44047
\(143\) −22.9129 −1.91607
\(144\) 0 0
\(145\) 0 0
\(146\) −7.16515 −0.592992
\(147\) 0 0
\(148\) 4.83485 0.397422
\(149\) −16.7477 −1.37203 −0.686014 0.727589i \(-0.740641\pi\)
−0.686014 + 0.727589i \(0.740641\pi\)
\(150\) 0 0
\(151\) 7.16515 0.583092 0.291546 0.956557i \(-0.405830\pi\)
0.291546 + 0.956557i \(0.405830\pi\)
\(152\) −5.07803 −0.411883
\(153\) 0 0
\(154\) 8.95644 0.721730
\(155\) 0 0
\(156\) 0 0
\(157\) −10.7477 −0.857762 −0.428881 0.903361i \(-0.641092\pi\)
−0.428881 + 0.903361i \(0.641092\pi\)
\(158\) 13.5826 1.08057
\(159\) 0 0
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −7.58258 −0.593913 −0.296957 0.954891i \(-0.595972\pi\)
−0.296957 + 0.954891i \(0.595972\pi\)
\(164\) 11.0780 0.865049
\(165\) 0 0
\(166\) −20.7477 −1.61034
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) 0 0
\(172\) 11.5826 0.883163
\(173\) −3.16515 −0.240642 −0.120321 0.992735i \(-0.538392\pi\)
−0.120321 + 0.992735i \(0.538392\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 24.7822 1.86803
\(177\) 0 0
\(178\) −2.53901 −0.190307
\(179\) −4.74773 −0.354862 −0.177431 0.984133i \(-0.556779\pi\)
−0.177431 + 0.984133i \(0.556779\pi\)
\(180\) 0 0
\(181\) 16.1652 1.20155 0.600773 0.799420i \(-0.294860\pi\)
0.600773 + 0.799420i \(0.294860\pi\)
\(182\) −8.20871 −0.608470
\(183\) 0 0
\(184\) 5.66970 0.417976
\(185\) 0 0
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 12.7913 0.932901
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 20.3303 1.46341 0.731704 0.681623i \(-0.238726\pi\)
0.731704 + 0.681623i \(0.238726\pi\)
\(194\) −20.7477 −1.48960
\(195\) 0 0
\(196\) −7.25227 −0.518019
\(197\) −16.3303 −1.16349 −0.581743 0.813373i \(-0.697629\pi\)
−0.581743 + 0.813373i \(0.697629\pi\)
\(198\) 0 0
\(199\) 13.4174 0.951136 0.475568 0.879679i \(-0.342243\pi\)
0.475568 + 0.879679i \(0.342243\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.04356 0.0734247
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) 0 0
\(206\) −27.1652 −1.89269
\(207\) 0 0
\(208\) −22.7133 −1.57488
\(209\) −17.9129 −1.23906
\(210\) 0 0
\(211\) 20.3303 1.39960 0.699798 0.714341i \(-0.253273\pi\)
0.699798 + 0.714341i \(0.253273\pi\)
\(212\) 0.504546 0.0346523
\(213\) 0 0
\(214\) 9.25227 0.632472
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 25.3739 1.71853
\(219\) 0 0
\(220\) 0 0
\(221\) −13.7477 −0.924772
\(222\) 0 0
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) 6.04356 0.403802
\(225\) 0 0
\(226\) −25.3739 −1.68784
\(227\) −7.58258 −0.503273 −0.251637 0.967822i \(-0.580969\pi\)
−0.251637 + 0.967822i \(0.580969\pi\)
\(228\) 0 0
\(229\) −17.1652 −1.13431 −0.567153 0.823613i \(-0.691955\pi\)
−0.567153 + 0.823613i \(0.691955\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.41742 0.0930585
\(233\) −13.1652 −0.862478 −0.431239 0.902238i \(-0.641923\pi\)
−0.431239 + 0.902238i \(0.641923\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.16515 0.596601
\(237\) 0 0
\(238\) 5.37386 0.348336
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 29.3303 1.88933 0.944665 0.328035i \(-0.106387\pi\)
0.944665 + 0.328035i \(0.106387\pi\)
\(242\) 25.0780 1.61208
\(243\) 0 0
\(244\) 15.4083 0.986417
\(245\) 0 0
\(246\) 0 0
\(247\) 16.4174 1.04462
\(248\) −5.66970 −0.360026
\(249\) 0 0
\(250\) 0 0
\(251\) −16.1652 −1.02034 −0.510168 0.860075i \(-0.670417\pi\)
−0.510168 + 0.860075i \(0.670417\pi\)
\(252\) 0 0
\(253\) 20.0000 1.25739
\(254\) −3.58258 −0.224791
\(255\) 0 0
\(256\) 20.5481 1.28426
\(257\) 12.7477 0.795181 0.397591 0.917563i \(-0.369846\pi\)
0.397591 + 0.917563i \(0.369846\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 26.8693 1.65999
\(263\) 30.3303 1.87025 0.935123 0.354322i \(-0.115288\pi\)
0.935123 + 0.354322i \(0.115288\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.41742 −0.393478
\(267\) 0 0
\(268\) 5.03447 0.307529
\(269\) −22.5826 −1.37688 −0.688442 0.725291i \(-0.741705\pi\)
−0.688442 + 0.725291i \(0.741705\pi\)
\(270\) 0 0
\(271\) 1.16515 0.0707779 0.0353890 0.999374i \(-0.488733\pi\)
0.0353890 + 0.999374i \(0.488733\pi\)
\(272\) 14.8693 0.901585
\(273\) 0 0
\(274\) 29.2523 1.76719
\(275\) 0 0
\(276\) 0 0
\(277\) 24.9129 1.49687 0.748435 0.663208i \(-0.230806\pi\)
0.748435 + 0.663208i \(0.230806\pi\)
\(278\) −16.8693 −1.01175
\(279\) 0 0
\(280\) 0 0
\(281\) 0.417424 0.0249014 0.0124507 0.999922i \(-0.496037\pi\)
0.0124507 + 0.999922i \(0.496037\pi\)
\(282\) 0 0
\(283\) 0.834849 0.0496266 0.0248133 0.999692i \(-0.492101\pi\)
0.0248133 + 0.999692i \(0.492101\pi\)
\(284\) 11.5826 0.687299
\(285\) 0 0
\(286\) −41.0436 −2.42696
\(287\) −9.16515 −0.541002
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) −4.83485 −0.282938
\(293\) 30.1652 1.76227 0.881133 0.472868i \(-0.156781\pi\)
0.881133 + 0.472868i \(0.156781\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.66970 −0.329544
\(297\) 0 0
\(298\) −30.0000 −1.73785
\(299\) −18.3303 −1.06007
\(300\) 0 0
\(301\) −9.58258 −0.552330
\(302\) 12.8348 0.738563
\(303\) 0 0
\(304\) −17.7568 −1.01842
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 6.04356 0.344364
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) −10.5826 −0.598163 −0.299081 0.954228i \(-0.596680\pi\)
−0.299081 + 0.954228i \(0.596680\pi\)
\(314\) −19.2523 −1.08647
\(315\) 0 0
\(316\) 9.16515 0.515580
\(317\) −25.0000 −1.40414 −0.702070 0.712108i \(-0.747741\pi\)
−0.702070 + 0.712108i \(0.747741\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) 0 0
\(322\) 7.16515 0.399298
\(323\) −10.7477 −0.598020
\(324\) 0 0
\(325\) 0 0
\(326\) −13.5826 −0.752269
\(327\) 0 0
\(328\) −12.9909 −0.717303
\(329\) −10.5826 −0.583436
\(330\) 0 0
\(331\) −8.33030 −0.457875 −0.228937 0.973441i \(-0.573525\pi\)
−0.228937 + 0.973441i \(0.573525\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 21.1652 1.15294 0.576470 0.817119i \(-0.304430\pi\)
0.576470 + 0.817119i \(0.304430\pi\)
\(338\) 14.3303 0.779466
\(339\) 0 0
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −13.5826 −0.732323
\(345\) 0 0
\(346\) −5.66970 −0.304805
\(347\) 7.16515 0.384645 0.192323 0.981332i \(-0.438398\pi\)
0.192323 + 0.981332i \(0.438398\pi\)
\(348\) 0 0
\(349\) −4.33030 −0.231796 −0.115898 0.993261i \(-0.536975\pi\)
−0.115898 + 0.993261i \(0.536975\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 30.2178 1.61061
\(353\) −14.8348 −0.789579 −0.394790 0.918772i \(-0.629183\pi\)
−0.394790 + 0.918772i \(0.629183\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.71326 −0.0908025
\(357\) 0 0
\(358\) −8.50455 −0.449479
\(359\) 27.1652 1.43372 0.716861 0.697216i \(-0.245578\pi\)
0.716861 + 0.697216i \(0.245578\pi\)
\(360\) 0 0
\(361\) −6.16515 −0.324482
\(362\) 28.9564 1.52192
\(363\) 0 0
\(364\) −5.53901 −0.290323
\(365\) 0 0
\(366\) 0 0
\(367\) 37.4955 1.95725 0.978623 0.205661i \(-0.0659343\pi\)
0.978623 + 0.205661i \(0.0659343\pi\)
\(368\) 19.8258 1.03349
\(369\) 0 0
\(370\) 0 0
\(371\) −0.417424 −0.0216716
\(372\) 0 0
\(373\) −28.3303 −1.46689 −0.733444 0.679750i \(-0.762088\pi\)
−0.733444 + 0.679750i \(0.762088\pi\)
\(374\) 26.8693 1.38938
\(375\) 0 0
\(376\) −15.0000 −0.773566
\(377\) −4.58258 −0.236015
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.16515 0.366601
\(383\) 3.58258 0.183061 0.0915305 0.995802i \(-0.470824\pi\)
0.0915305 + 0.995802i \(0.470824\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.4174 1.85360
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 6.58258 0.333750 0.166875 0.985978i \(-0.446632\pi\)
0.166875 + 0.985978i \(0.446632\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 8.50455 0.429544
\(393\) 0 0
\(394\) −29.2523 −1.47371
\(395\) 0 0
\(396\) 0 0
\(397\) −10.8348 −0.543785 −0.271893 0.962328i \(-0.587650\pi\)
−0.271893 + 0.962328i \(0.587650\pi\)
\(398\) 24.0345 1.20474
\(399\) 0 0
\(400\) 0 0
\(401\) −12.4174 −0.620097 −0.310048 0.950721i \(-0.600345\pi\)
−0.310048 + 0.950721i \(0.600345\pi\)
\(402\) 0 0
\(403\) 18.3303 0.913097
\(404\) 0.704166 0.0350336
\(405\) 0 0
\(406\) 1.79129 0.0889001
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) −2.74773 −0.135866 −0.0679332 0.997690i \(-0.521640\pi\)
−0.0679332 + 0.997690i \(0.521640\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −18.3303 −0.903069
\(413\) −7.58258 −0.373114
\(414\) 0 0
\(415\) 0 0
\(416\) −27.6951 −1.35786
\(417\) 0 0
\(418\) −32.0871 −1.56943
\(419\) 37.1652 1.81564 0.907818 0.419364i \(-0.137747\pi\)
0.907818 + 0.419364i \(0.137747\pi\)
\(420\) 0 0
\(421\) 4.41742 0.215292 0.107646 0.994189i \(-0.465669\pi\)
0.107646 + 0.994189i \(0.465669\pi\)
\(422\) 36.4174 1.77277
\(423\) 0 0
\(424\) −0.591667 −0.0287339
\(425\) 0 0
\(426\) 0 0
\(427\) −12.7477 −0.616906
\(428\) 6.24318 0.301776
\(429\) 0 0
\(430\) 0 0
\(431\) 39.1652 1.88652 0.943259 0.332057i \(-0.107742\pi\)
0.943259 + 0.332057i \(0.107742\pi\)
\(432\) 0 0
\(433\) −25.0780 −1.20517 −0.602587 0.798053i \(-0.705863\pi\)
−0.602587 + 0.798053i \(0.705863\pi\)
\(434\) −7.16515 −0.343938
\(435\) 0 0
\(436\) 17.1216 0.819975
\(437\) −14.3303 −0.685511
\(438\) 0 0
\(439\) −16.9129 −0.807208 −0.403604 0.914934i \(-0.632243\pi\)
−0.403604 + 0.914934i \(0.632243\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.6261 −1.17135
\(443\) 0.582576 0.0276790 0.0138395 0.999904i \(-0.495595\pi\)
0.0138395 + 0.999904i \(0.495595\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −12.5390 −0.593740
\(447\) 0 0
\(448\) 0.912878 0.0431295
\(449\) 30.0780 1.41947 0.709735 0.704469i \(-0.248815\pi\)
0.709735 + 0.704469i \(0.248815\pi\)
\(450\) 0 0
\(451\) −45.8258 −2.15785
\(452\) −17.1216 −0.805332
\(453\) 0 0
\(454\) −13.5826 −0.637462
\(455\) 0 0
\(456\) 0 0
\(457\) 3.74773 0.175311 0.0876556 0.996151i \(-0.472062\pi\)
0.0876556 + 0.996151i \(0.472062\pi\)
\(458\) −30.7477 −1.43675
\(459\) 0 0
\(460\) 0 0
\(461\) 9.16515 0.426864 0.213432 0.976958i \(-0.431536\pi\)
0.213432 + 0.976958i \(0.431536\pi\)
\(462\) 0 0
\(463\) 0.165151 0.00767524 0.00383762 0.999993i \(-0.498778\pi\)
0.00383762 + 0.999993i \(0.498778\pi\)
\(464\) 4.95644 0.230097
\(465\) 0 0
\(466\) −23.5826 −1.09244
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −4.16515 −0.192329
\(470\) 0 0
\(471\) 0 0
\(472\) −10.7477 −0.494704
\(473\) −47.9129 −2.20304
\(474\) 0 0
\(475\) 0 0
\(476\) 3.62614 0.166204
\(477\) 0 0
\(478\) 46.5735 2.13022
\(479\) 19.1652 0.875678 0.437839 0.899053i \(-0.355744\pi\)
0.437839 + 0.899053i \(0.355744\pi\)
\(480\) 0 0
\(481\) 18.3303 0.835790
\(482\) 52.5390 2.39309
\(483\) 0 0
\(484\) 16.9220 0.769180
\(485\) 0 0
\(486\) 0 0
\(487\) −2.33030 −0.105596 −0.0527980 0.998605i \(-0.516814\pi\)
−0.0527980 + 0.998605i \(0.516814\pi\)
\(488\) −18.0689 −0.817942
\(489\) 0 0
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 29.4083 1.32314
\(495\) 0 0
\(496\) −19.8258 −0.890203
\(497\) −9.58258 −0.429837
\(498\) 0 0
\(499\) −13.4174 −0.600646 −0.300323 0.953837i \(-0.597095\pi\)
−0.300323 + 0.953837i \(0.597095\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −28.9564 −1.29239
\(503\) −22.9129 −1.02163 −0.510817 0.859689i \(-0.670657\pi\)
−0.510817 + 0.859689i \(0.670657\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 35.8258 1.59265
\(507\) 0 0
\(508\) −2.41742 −0.107256
\(509\) −26.7477 −1.18557 −0.592786 0.805360i \(-0.701972\pi\)
−0.592786 + 0.805360i \(0.701972\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 15.9038 0.702855
\(513\) 0 0
\(514\) 22.8348 1.00720
\(515\) 0 0
\(516\) 0 0
\(517\) −52.9129 −2.32711
\(518\) −7.16515 −0.314819
\(519\) 0 0
\(520\) 0 0
\(521\) −43.0780 −1.88728 −0.943641 0.330970i \(-0.892624\pi\)
−0.943641 + 0.330970i \(0.892624\pi\)
\(522\) 0 0
\(523\) 33.3303 1.45743 0.728716 0.684816i \(-0.240117\pi\)
0.728716 + 0.684816i \(0.240117\pi\)
\(524\) 18.1307 0.792043
\(525\) 0 0
\(526\) 54.3303 2.36891
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −4.33030 −0.187742
\(533\) 42.0000 1.81922
\(534\) 0 0
\(535\) 0 0
\(536\) −5.90379 −0.255005
\(537\) 0 0
\(538\) −40.4519 −1.74400
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −31.4955 −1.35410 −0.677048 0.735939i \(-0.736741\pi\)
−0.677048 + 0.735939i \(0.736741\pi\)
\(542\) 2.08712 0.0896495
\(543\) 0 0
\(544\) 18.1307 0.777347
\(545\) 0 0
\(546\) 0 0
\(547\) −4.16515 −0.178089 −0.0890445 0.996028i \(-0.528381\pi\)
−0.0890445 + 0.996028i \(0.528381\pi\)
\(548\) 19.7386 0.843193
\(549\) 0 0
\(550\) 0 0
\(551\) −3.58258 −0.152623
\(552\) 0 0
\(553\) −7.58258 −0.322444
\(554\) 44.6261 1.89598
\(555\) 0 0
\(556\) −11.3830 −0.482745
\(557\) −22.7477 −0.963852 −0.481926 0.876212i \(-0.660063\pi\)
−0.481926 + 0.876212i \(0.660063\pi\)
\(558\) 0 0
\(559\) 43.9129 1.85732
\(560\) 0 0
\(561\) 0 0
\(562\) 0.747727 0.0315410
\(563\) −14.5826 −0.614582 −0.307291 0.951616i \(-0.599423\pi\)
−0.307291 + 0.951616i \(0.599423\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.49545 0.0628586
\(567\) 0 0
\(568\) −13.5826 −0.569912
\(569\) −28.5826 −1.19824 −0.599122 0.800658i \(-0.704484\pi\)
−0.599122 + 0.800658i \(0.704484\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −27.6951 −1.15799
\(573\) 0 0
\(574\) −16.4174 −0.685250
\(575\) 0 0
\(576\) 0 0
\(577\) 19.1652 0.797856 0.398928 0.916982i \(-0.369382\pi\)
0.398928 + 0.916982i \(0.369382\pi\)
\(578\) −14.3303 −0.596062
\(579\) 0 0
\(580\) 0 0
\(581\) 11.5826 0.480526
\(582\) 0 0
\(583\) −2.08712 −0.0864397
\(584\) 5.66970 0.234614
\(585\) 0 0
\(586\) 54.0345 2.23214
\(587\) −11.5826 −0.478064 −0.239032 0.971012i \(-0.576830\pi\)
−0.239032 + 0.971012i \(0.576830\pi\)
\(588\) 0 0
\(589\) 14.3303 0.590470
\(590\) 0 0
\(591\) 0 0
\(592\) −19.8258 −0.814834
\(593\) 8.41742 0.345662 0.172831 0.984951i \(-0.444709\pi\)
0.172831 + 0.984951i \(0.444709\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.2432 −0.829193
\(597\) 0 0
\(598\) −32.8348 −1.34272
\(599\) −0.165151 −0.00674790 −0.00337395 0.999994i \(-0.501074\pi\)
−0.00337395 + 0.999994i \(0.501074\pi\)
\(600\) 0 0
\(601\) −32.3303 −1.31878 −0.659390 0.751801i \(-0.729185\pi\)
−0.659390 + 0.751801i \(0.729185\pi\)
\(602\) −17.1652 −0.699599
\(603\) 0 0
\(604\) 8.66061 0.352395
\(605\) 0 0
\(606\) 0 0
\(607\) −3.58258 −0.145412 −0.0727061 0.997353i \(-0.523164\pi\)
−0.0727061 + 0.997353i \(0.523164\pi\)
\(608\) −21.6515 −0.878085
\(609\) 0 0
\(610\) 0 0
\(611\) 48.4955 1.96192
\(612\) 0 0
\(613\) 43.7477 1.76695 0.883477 0.468474i \(-0.155196\pi\)
0.883477 + 0.468474i \(0.155196\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −7.08712 −0.285548
\(617\) 15.4955 0.623823 0.311912 0.950111i \(-0.399031\pi\)
0.311912 + 0.950111i \(0.399031\pi\)
\(618\) 0 0
\(619\) 13.1652 0.529152 0.264576 0.964365i \(-0.414768\pi\)
0.264576 + 0.964365i \(0.414768\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.37386 0.215472
\(623\) 1.41742 0.0567879
\(624\) 0 0
\(625\) 0 0
\(626\) −18.9564 −0.757652
\(627\) 0 0
\(628\) −12.9909 −0.518394
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 10.9129 0.434435 0.217217 0.976123i \(-0.430302\pi\)
0.217217 + 0.976123i \(0.430302\pi\)
\(632\) −10.7477 −0.427522
\(633\) 0 0
\(634\) −44.7822 −1.77853
\(635\) 0 0
\(636\) 0 0
\(637\) −27.4955 −1.08941
\(638\) 8.95644 0.354589
\(639\) 0 0
\(640\) 0 0
\(641\) 6.58258 0.259996 0.129998 0.991514i \(-0.458503\pi\)
0.129998 + 0.991514i \(0.458503\pi\)
\(642\) 0 0
\(643\) −43.6606 −1.72181 −0.860903 0.508769i \(-0.830101\pi\)
−0.860903 + 0.508769i \(0.830101\pi\)
\(644\) 4.83485 0.190520
\(645\) 0 0
\(646\) −19.2523 −0.757471
\(647\) 24.7477 0.972934 0.486467 0.873699i \(-0.338285\pi\)
0.486467 + 0.873699i \(0.338285\pi\)
\(648\) 0 0
\(649\) −37.9129 −1.48821
\(650\) 0 0
\(651\) 0 0
\(652\) −9.16515 −0.358935
\(653\) 26.1652 1.02392 0.511961 0.859009i \(-0.328919\pi\)
0.511961 + 0.859009i \(0.328919\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −45.4265 −1.77361
\(657\) 0 0
\(658\) −18.9564 −0.738999
\(659\) −18.1652 −0.707614 −0.353807 0.935318i \(-0.615113\pi\)
−0.353807 + 0.935318i \(0.615113\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) −14.9220 −0.579959
\(663\) 0 0
\(664\) 16.4174 0.637120
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −63.7386 −2.46060
\(672\) 0 0
\(673\) −1.74773 −0.0673699 −0.0336850 0.999433i \(-0.510724\pi\)
−0.0336850 + 0.999433i \(0.510724\pi\)
\(674\) 37.9129 1.46035
\(675\) 0 0
\(676\) 9.66970 0.371911
\(677\) 44.8258 1.72279 0.861397 0.507932i \(-0.169590\pi\)
0.861397 + 0.507932i \(0.169590\pi\)
\(678\) 0 0
\(679\) 11.5826 0.444498
\(680\) 0 0
\(681\) 0 0
\(682\) −35.8258 −1.37184
\(683\) 7.16515 0.274167 0.137083 0.990560i \(-0.456227\pi\)
0.137083 + 0.990560i \(0.456227\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 23.2867 0.889092
\(687\) 0 0
\(688\) −47.4955 −1.81075
\(689\) 1.91288 0.0728749
\(690\) 0 0
\(691\) 42.9129 1.63248 0.816241 0.577711i \(-0.196054\pi\)
0.816241 + 0.577711i \(0.196054\pi\)
\(692\) −3.82576 −0.145433
\(693\) 0 0
\(694\) 12.8348 0.487204
\(695\) 0 0
\(696\) 0 0
\(697\) −27.4955 −1.04146
\(698\) −7.75682 −0.293600
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0780 0.871645 0.435823 0.900033i \(-0.356458\pi\)
0.435823 + 0.900033i \(0.356458\pi\)
\(702\) 0 0
\(703\) 14.3303 0.540478
\(704\) 4.56439 0.172027
\(705\) 0 0
\(706\) −26.5735 −1.00011
\(707\) −0.582576 −0.0219100
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.00909 0.0752939
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −5.73864 −0.214463
\(717\) 0 0
\(718\) 48.6606 1.81600
\(719\) −7.91288 −0.295101 −0.147550 0.989055i \(-0.547139\pi\)
−0.147550 + 0.989055i \(0.547139\pi\)
\(720\) 0 0
\(721\) 15.1652 0.564780
\(722\) −11.0436 −0.410999
\(723\) 0 0
\(724\) 19.5390 0.726162
\(725\) 0 0
\(726\) 0 0
\(727\) −9.66970 −0.358629 −0.179315 0.983792i \(-0.557388\pi\)
−0.179315 + 0.983792i \(0.557388\pi\)
\(728\) 6.49545 0.240738
\(729\) 0 0
\(730\) 0 0
\(731\) −28.7477 −1.06327
\(732\) 0 0
\(733\) 38.4174 1.41898 0.709490 0.704716i \(-0.248925\pi\)
0.709490 + 0.704716i \(0.248925\pi\)
\(734\) 67.1652 2.47911
\(735\) 0 0
\(736\) 24.1742 0.891074
\(737\) −20.8258 −0.767127
\(738\) 0 0
\(739\) 19.2523 0.708206 0.354103 0.935206i \(-0.384786\pi\)
0.354103 + 0.935206i \(0.384786\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.747727 −0.0274499
\(743\) 29.7477 1.09134 0.545669 0.838001i \(-0.316276\pi\)
0.545669 + 0.838001i \(0.316276\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −50.7477 −1.85801
\(747\) 0 0
\(748\) 18.1307 0.662923
\(749\) −5.16515 −0.188731
\(750\) 0 0
\(751\) −17.4955 −0.638418 −0.319209 0.947684i \(-0.603417\pi\)
−0.319209 + 0.947684i \(0.603417\pi\)
\(752\) −52.4519 −1.91272
\(753\) 0 0
\(754\) −8.20871 −0.298944
\(755\) 0 0
\(756\) 0 0
\(757\) 2.33030 0.0846963 0.0423481 0.999103i \(-0.486516\pi\)
0.0423481 + 0.999103i \(0.486516\pi\)
\(758\) −46.5735 −1.69163
\(759\) 0 0
\(760\) 0 0
\(761\) −36.4174 −1.32013 −0.660065 0.751208i \(-0.729471\pi\)
−0.660065 + 0.751208i \(0.729471\pi\)
\(762\) 0 0
\(763\) −14.1652 −0.512813
\(764\) 4.83485 0.174919
\(765\) 0 0
\(766\) 6.41742 0.231871
\(767\) 34.7477 1.25467
\(768\) 0 0
\(769\) 25.5826 0.922531 0.461266 0.887262i \(-0.347396\pi\)
0.461266 + 0.887262i \(0.347396\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 24.5735 0.884419
\(773\) −5.16515 −0.185778 −0.0928888 0.995676i \(-0.529610\pi\)
−0.0928888 + 0.995676i \(0.529610\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.4174 0.589351
\(777\) 0 0
\(778\) 11.7913 0.422738
\(779\) 32.8348 1.17643
\(780\) 0 0
\(781\) −47.9129 −1.71446
\(782\) 21.4955 0.768676
\(783\) 0 0
\(784\) 29.7386 1.06209
\(785\) 0 0
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) −19.7386 −0.703160
\(789\) 0 0
\(790\) 0 0
\(791\) 14.1652 0.503655
\(792\) 0 0
\(793\) 58.4174 2.07446
\(794\) −19.4083 −0.688776
\(795\) 0 0
\(796\) 16.2178 0.574825
\(797\) 32.3303 1.14520 0.572599 0.819836i \(-0.305935\pi\)
0.572599 + 0.819836i \(0.305935\pi\)
\(798\) 0 0
\(799\) −31.7477 −1.12315
\(800\) 0 0
\(801\) 0 0
\(802\) −22.2432 −0.785434
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 0 0
\(806\) 32.8348 1.15656
\(807\) 0 0
\(808\) −0.825757 −0.0290500
\(809\) −44.0780 −1.54970 −0.774851 0.632145i \(-0.782175\pi\)
−0.774851 + 0.632145i \(0.782175\pi\)
\(810\) 0 0
\(811\) 32.5826 1.14413 0.572064 0.820209i \(-0.306143\pi\)
0.572064 + 0.820209i \(0.306143\pi\)
\(812\) 1.20871 0.0424175
\(813\) 0 0
\(814\) −35.8258 −1.25569
\(815\) 0 0
\(816\) 0 0
\(817\) 34.3303 1.20107
\(818\) −4.92197 −0.172093
\(819\) 0 0
\(820\) 0 0
\(821\) 31.4955 1.09920 0.549599 0.835428i \(-0.314780\pi\)
0.549599 + 0.835428i \(0.314780\pi\)
\(822\) 0 0
\(823\) −51.0780 −1.78047 −0.890234 0.455503i \(-0.849459\pi\)
−0.890234 + 0.455503i \(0.849459\pi\)
\(824\) 21.4955 0.748830
\(825\) 0 0
\(826\) −13.5826 −0.472598
\(827\) −43.8258 −1.52397 −0.761985 0.647594i \(-0.775775\pi\)
−0.761985 + 0.647594i \(0.775775\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.18333 −0.145031
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) −21.6515 −0.748833
\(837\) 0 0
\(838\) 66.5735 2.29974
\(839\) 48.4955 1.67425 0.837125 0.547012i \(-0.184235\pi\)
0.837125 + 0.547012i \(0.184235\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.91288 0.272696
\(843\) 0 0
\(844\) 24.5735 0.845854
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −2.06894 −0.0710476
\(849\) 0 0
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −20.7477 −0.710389 −0.355194 0.934792i \(-0.615585\pi\)
−0.355194 + 0.934792i \(0.615585\pi\)
\(854\) −22.8348 −0.781392
\(855\) 0 0
\(856\) −7.32121 −0.250234
\(857\) −46.6606 −1.59390 −0.796948 0.604048i \(-0.793554\pi\)
−0.796948 + 0.604048i \(0.793554\pi\)
\(858\) 0 0
\(859\) −47.5826 −1.62350 −0.811748 0.584007i \(-0.801484\pi\)
−0.811748 + 0.584007i \(0.801484\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 70.1561 2.38952
\(863\) −6.41742 −0.218452 −0.109226 0.994017i \(-0.534837\pi\)
−0.109226 + 0.994017i \(0.534837\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −44.9220 −1.52651
\(867\) 0 0
\(868\) −4.83485 −0.164105
\(869\) −37.9129 −1.28611
\(870\) 0 0
\(871\) 19.0871 0.646742
\(872\) −20.0780 −0.679928
\(873\) 0 0
\(874\) −25.6697 −0.868290
\(875\) 0 0
\(876\) 0 0
\(877\) −19.6697 −0.664198 −0.332099 0.943244i \(-0.607757\pi\)
−0.332099 + 0.943244i \(0.607757\pi\)
\(878\) −30.2958 −1.02243
\(879\) 0 0
\(880\) 0 0
\(881\) 32.0780 1.08074 0.540368 0.841429i \(-0.318285\pi\)
0.540368 + 0.841429i \(0.318285\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −16.6170 −0.558892
\(885\) 0 0
\(886\) 1.04356 0.0350591
\(887\) −44.9129 −1.50803 −0.754013 0.656859i \(-0.771885\pi\)
−0.754013 + 0.656859i \(0.771885\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) −8.46099 −0.283295
\(893\) 37.9129 1.26871
\(894\) 0 0
\(895\) 0 0
\(896\) −10.4519 −0.349173
\(897\) 0 0
\(898\) 53.8784 1.79795
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −1.25227 −0.0417193
\(902\) −82.0871 −2.73320
\(903\) 0 0
\(904\) 20.0780 0.667785
\(905\) 0 0
\(906\) 0 0
\(907\) −23.5826 −0.783047 −0.391523 0.920168i \(-0.628052\pi\)
−0.391523 + 0.920168i \(0.628052\pi\)
\(908\) −9.16515 −0.304156
\(909\) 0 0
\(910\) 0 0
\(911\) 50.8258 1.68393 0.841966 0.539530i \(-0.181398\pi\)
0.841966 + 0.539530i \(0.181398\pi\)
\(912\) 0 0
\(913\) 57.9129 1.91664
\(914\) 6.71326 0.222055
\(915\) 0 0
\(916\) −20.7477 −0.685524
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) 18.9129 0.623878 0.311939 0.950102i \(-0.399021\pi\)
0.311939 + 0.950102i \(0.399021\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 16.4174 0.540679
\(923\) 43.9129 1.44541
\(924\) 0 0
\(925\) 0 0
\(926\) 0.295834 0.00972170
\(927\) 0 0
\(928\) 6.04356 0.198390
\(929\) −15.6697 −0.514106 −0.257053 0.966397i \(-0.582752\pi\)
−0.257053 + 0.966397i \(0.582752\pi\)
\(930\) 0 0
\(931\) −21.4955 −0.704485
\(932\) −15.9129 −0.521244
\(933\) 0 0
\(934\) −14.3303 −0.468902
\(935\) 0 0
\(936\) 0 0
\(937\) 18.5826 0.607066 0.303533 0.952821i \(-0.401834\pi\)
0.303533 + 0.952821i \(0.401834\pi\)
\(938\) −7.46099 −0.243610
\(939\) 0 0
\(940\) 0 0
\(941\) −19.9129 −0.649141 −0.324571 0.945861i \(-0.605220\pi\)
−0.324571 + 0.945861i \(0.605220\pi\)
\(942\) 0 0
\(943\) −36.6606 −1.19383
\(944\) −37.5826 −1.22321
\(945\) 0 0
\(946\) −85.8258 −2.79044
\(947\) −54.9129 −1.78443 −0.892214 0.451612i \(-0.850849\pi\)
−0.892214 + 0.451612i \(0.850849\pi\)
\(948\) 0 0
\(949\) −18.3303 −0.595027
\(950\) 0 0
\(951\) 0 0
\(952\) −4.25227 −0.137817
\(953\) −37.5826 −1.21742 −0.608710 0.793393i \(-0.708313\pi\)
−0.608710 + 0.793393i \(0.708313\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 31.4265 1.01641
\(957\) 0 0
\(958\) 34.3303 1.10916
\(959\) −16.3303 −0.527333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 32.8348 1.05864
\(963\) 0 0
\(964\) 35.4519 1.14183
\(965\) 0 0
\(966\) 0 0
\(967\) −9.25227 −0.297533 −0.148767 0.988872i \(-0.547530\pi\)
−0.148767 + 0.988872i \(0.547530\pi\)
\(968\) −19.8439 −0.637808
\(969\) 0 0
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 9.41742 0.301909
\(974\) −4.17424 −0.133751
\(975\) 0 0
\(976\) −63.1833 −2.02245
\(977\) −2.50455 −0.0801275 −0.0400638 0.999197i \(-0.512756\pi\)
−0.0400638 + 0.999197i \(0.512756\pi\)
\(978\) 0 0
\(979\) 7.08712 0.226505
\(980\) 0 0
\(981\) 0 0
\(982\) −28.6606 −0.914597
\(983\) 55.1652 1.75950 0.879748 0.475441i \(-0.157712\pi\)
0.879748 + 0.475441i \(0.157712\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 5.37386 0.171139
\(987\) 0 0
\(988\) 19.8439 0.631320
\(989\) −38.3303 −1.21883
\(990\) 0 0
\(991\) 16.0780 0.510735 0.255368 0.966844i \(-0.417803\pi\)
0.255368 + 0.966844i \(0.417803\pi\)
\(992\) −24.1742 −0.767533
\(993\) 0 0
\(994\) −17.1652 −0.544446
\(995\) 0 0
\(996\) 0 0
\(997\) 0.834849 0.0264399 0.0132200 0.999913i \(-0.495792\pi\)
0.0132200 + 0.999913i \(0.495792\pi\)
\(998\) −24.0345 −0.760798
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6525.2.a.t.1.2 2
3.2 odd 2 2175.2.a.r.1.1 2
5.4 even 2 1305.2.a.m.1.1 2
15.2 even 4 2175.2.c.f.349.2 4
15.8 even 4 2175.2.c.f.349.3 4
15.14 odd 2 435.2.a.f.1.2 2
60.59 even 2 6960.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 15.14 odd 2
1305.2.a.m.1.1 2 5.4 even 2
2175.2.a.r.1.1 2 3.2 odd 2
2175.2.c.f.349.2 4 15.2 even 4
2175.2.c.f.349.3 4 15.8 even 4
6525.2.a.t.1.2 2 1.1 even 1 trivial
6960.2.a.bw.1.1 2 60.59 even 2