Properties

Label 1305.2.a.m.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
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 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +O(q^{10})\) \(q-1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +1.79129 q^{10} -5.00000 q^{11} -4.58258 q^{13} -1.79129 q^{14} -4.95644 q^{16} +3.00000 q^{17} +3.58258 q^{19} -1.20871 q^{20} +8.95644 q^{22} +4.00000 q^{23} +1.00000 q^{25} +8.20871 q^{26} +1.20871 q^{28} -1.00000 q^{29} +4.00000 q^{31} +6.04356 q^{32} -5.37386 q^{34} -1.00000 q^{35} -4.00000 q^{37} -6.41742 q^{38} -1.41742 q^{40} +9.16515 q^{41} -9.58258 q^{43} -6.04356 q^{44} -7.16515 q^{46} -10.5826 q^{47} -6.00000 q^{49} -1.79129 q^{50} -5.53901 q^{52} -0.417424 q^{53} +5.00000 q^{55} +1.41742 q^{56} +1.79129 q^{58} +7.58258 q^{59} +12.7477 q^{61} -7.16515 q^{62} -0.912878 q^{64} +4.58258 q^{65} -4.16515 q^{67} +3.62614 q^{68} +1.79129 q^{70} +9.58258 q^{71} +4.00000 q^{73} +7.16515 q^{74} +4.33030 q^{76} -5.00000 q^{77} +7.58258 q^{79} +4.95644 q^{80} -16.4174 q^{82} +11.5826 q^{83} -3.00000 q^{85} +17.1652 q^{86} -7.08712 q^{88} -1.41742 q^{89} -4.58258 q^{91} +4.83485 q^{92} +18.9564 q^{94} -3.58258 q^{95} +11.5826 q^{97} +10.7477 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 7 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 7 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8} - q^{10} - 10 q^{11} + q^{14} + 13 q^{16} + 6 q^{17} - 2 q^{19} - 7 q^{20} - 5 q^{22} + 8 q^{23} + 2 q^{25} + 21 q^{26} + 7 q^{28} - 2 q^{29} + 8 q^{31} + 35 q^{32} + 3 q^{34} - 2 q^{35} - 8 q^{37} - 22 q^{38} - 12 q^{40} - 10 q^{43} - 35 q^{44} + 4 q^{46} - 12 q^{47} - 12 q^{49} + q^{50} + 21 q^{52} - 10 q^{53} + 10 q^{55} + 12 q^{56} - q^{58} + 6 q^{59} - 2 q^{61} + 4 q^{62} + 44 q^{64} + 10 q^{67} + 21 q^{68} - q^{70} + 10 q^{71} + 8 q^{73} - 4 q^{74} - 28 q^{76} - 10 q^{77} + 6 q^{79} - 13 q^{80} - 42 q^{82} + 14 q^{83} - 6 q^{85} + 16 q^{86} - 60 q^{88} - 12 q^{89} + 28 q^{92} + 15 q^{94} + 2 q^{95} + 14 q^{97} - 6 q^{98}+O(q^{100}) \) 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 \(-0.718307\pi\)
−0.633316 + 0.773893i \(0.718307\pi\)
\(3\) 0 0
\(4\) 1.20871 0.604356
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.41742 0.501135
\(9\) 0 0
\(10\) 1.79129 0.566455
\(11\) −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 \(-0.719201\pi\)
−0.635489 + 0.772110i \(0.719201\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) −4.95644 −1.23911
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 3.58258 0.821899 0.410950 0.911658i \(-0.365197\pi\)
0.410950 + 0.911658i \(0.365197\pi\)
\(20\) −1.20871 −0.270276
\(21\) 0 0
\(22\) 8.95644 1.90952
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 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
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −6.41742 −1.04104
\(39\) 0 0
\(40\) −1.41742 −0.224114
\(41\) 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 \(-0.760790\pi\)
−0.730665 + 0.682737i \(0.760790\pi\)
\(44\) −6.04356 −0.911101
\(45\) 0 0
\(46\) −7.16515 −1.05644
\(47\) −10.5826 −1.54363 −0.771814 0.635849i \(-0.780650\pi\)
−0.771814 + 0.635849i \(0.780650\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −1.79129 −0.253326
\(51\) 0 0
\(52\) −5.53901 −0.768123
\(53\) −0.417424 −0.0573376 −0.0286688 0.999589i \(-0.509127\pi\)
−0.0286688 + 0.999589i \(0.509127\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 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
\(65\) 4.58258 0.568399
\(66\) 0 0
\(67\) −4.16515 −0.508854 −0.254427 0.967092i \(-0.581887\pi\)
−0.254427 + 0.967092i \(0.581887\pi\)
\(68\) 3.62614 0.439734
\(69\) 0 0
\(70\) 1.79129 0.214100
\(71\) 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 \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 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 \(0.359728\pi\)
\(80\) 4.95644 0.554147
\(81\) 0 0
\(82\) −16.4174 −1.81300
\(83\) 11.5826 1.27135 0.635676 0.771956i \(-0.280721\pi\)
0.635676 + 0.771956i \(0.280721\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 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
\(95\) −3.58258 −0.367565
\(96\) 0 0
\(97\) 11.5826 1.17603 0.588016 0.808849i \(-0.299909\pi\)
0.588016 + 0.808849i \(0.299909\pi\)
\(98\) 10.7477 1.08568
\(99\) 0 0
\(100\) 1.20871 0.120871
\(101\) 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 \(-0.231430\pi\)
0.747133 + 0.664674i \(0.231430\pi\)
\(104\) −6.49545 −0.636932
\(105\) 0 0
\(106\) 0.747727 0.0726257
\(107\) −5.16515 −0.499334 −0.249667 0.968332i \(-0.580321\pi\)
−0.249667 + 0.968332i \(0.580321\pi\)
\(108\) 0 0
\(109\) 14.1652 1.35678 0.678388 0.734704i \(-0.262679\pi\)
0.678388 + 0.734704i \(0.262679\pi\)
\(110\) −8.95644 −0.853963
\(111\) 0 0
\(112\) −4.95644 −0.468339
\(113\) 14.1652 1.33255 0.666273 0.745708i \(-0.267889\pi\)
0.666273 + 0.745708i \(0.267889\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −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
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −10.4519 −0.923826
\(129\) 0 0
\(130\) −8.20871 −0.719952
\(131\) 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 \(-0.745747\pi\)
−0.697596 + 0.716491i \(0.745747\pi\)
\(138\) 0 0
\(139\) −9.41742 −0.798776 −0.399388 0.916782i \(-0.630777\pi\)
−0.399388 + 0.916782i \(0.630777\pi\)
\(140\) −1.20871 −0.102155
\(141\) 0 0
\(142\) −17.1652 −1.44047
\(143\) 22.9129 1.91607
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −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
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 10.7477 0.857762 0.428881 0.903361i \(-0.358908\pi\)
0.428881 + 0.903361i \(0.358908\pi\)
\(158\) −13.5826 −1.08057
\(159\) 0 0
\(160\) −6.04356 −0.477785
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 7.58258 0.593913 0.296957 0.954891i \(-0.404028\pi\)
0.296957 + 0.954891i \(0.404028\pi\)
\(164\) 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
\(170\) 5.37386 0.412157
\(171\) 0 0
\(172\) −11.5826 −0.883163
\(173\) 3.16515 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 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
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) −12.7913 −0.932901
\(189\) 0 0
\(190\) 6.41742 0.465569
\(191\) 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 \(-0.761274\pi\)
−0.731704 + 0.681623i \(0.761274\pi\)
\(194\) −20.7477 −1.48960
\(195\) 0 0
\(196\) −7.25227 −0.518019
\(197\) 16.3303 1.16349 0.581743 0.813373i \(-0.302371\pi\)
0.581743 + 0.813373i \(0.302371\pi\)
\(198\) 0 0
\(199\) 13.4174 0.951136 0.475568 0.879679i \(-0.342243\pi\)
0.475568 + 0.879679i \(0.342243\pi\)
\(200\) 1.41742 0.100227
\(201\) 0 0
\(202\) −1.04356 −0.0734247
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) −9.16515 −0.640122
\(206\) −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
\(215\) 9.58258 0.653526
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −25.3739 −1.71853
\(219\) 0 0
\(220\) 6.04356 0.407457
\(221\) −13.7477 −0.924772
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 6.04356 0.403802
\(225\) 0 0
\(226\) −25.3739 −1.68784
\(227\) 7.58258 0.503273 0.251637 0.967822i \(-0.419031\pi\)
0.251637 + 0.967822i \(0.419031\pi\)
\(228\) 0 0
\(229\) −17.1652 −1.13431 −0.567153 0.823613i \(-0.691955\pi\)
−0.567153 + 0.823613i \(0.691955\pi\)
\(230\) 7.16515 0.472456
\(231\) 0 0
\(232\) −1.41742 −0.0930585
\(233\) 13.1652 0.862478 0.431239 0.902238i \(-0.358077\pi\)
0.431239 + 0.902238i \(0.358077\pi\)
\(234\) 0 0
\(235\) 10.5826 0.690331
\(236\) 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
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −16.4174 −1.04462
\(248\) 5.66970 0.360026
\(249\) 0 0
\(250\) 1.79129 0.113291
\(251\) −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 \(-0.630154\pi\)
−0.397591 + 0.917563i \(0.630154\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 5.53901 0.343515
\(261\) 0 0
\(262\) −26.8693 −1.65999
\(263\) −30.3303 −1.87025 −0.935123 0.354322i \(-0.884712\pi\)
−0.935123 + 0.354322i \(0.884712\pi\)
\(264\) 0 0
\(265\) 0.417424 0.0256422
\(266\) −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
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −24.9129 −1.49687 −0.748435 0.663208i \(-0.769194\pi\)
−0.748435 + 0.663208i \(0.769194\pi\)
\(278\) 16.8693 1.01175
\(279\) 0 0
\(280\) −1.41742 −0.0847073
\(281\) 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 \(-0.507899\pi\)
−0.0248133 + 0.999692i \(0.507899\pi\)
\(284\) 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
\(290\) −1.79129 −0.105188
\(291\) 0 0
\(292\) 4.83485 0.282938
\(293\) −30.1652 −1.76227 −0.881133 0.472868i \(-0.843219\pi\)
−0.881133 + 0.472868i \(0.843219\pi\)
\(294\) 0 0
\(295\) −7.58258 −0.441475
\(296\) −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
\(305\) −12.7477 −0.729933
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −6.04356 −0.344364
\(309\) 0 0
\(310\) 7.16515 0.406953
\(311\) 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 \(-0.403320\pi\)
0.299081 + 0.954228i \(0.403320\pi\)
\(314\) −19.2523 −1.08647
\(315\) 0 0
\(316\) 9.16515 0.515580
\(317\) 25.0000 1.40414 0.702070 0.712108i \(-0.252259\pi\)
0.702070 + 0.712108i \(0.252259\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0.912878 0.0510315
\(321\) 0 0
\(322\) −7.16515 −0.399298
\(323\) 10.7477 0.598020
\(324\) 0 0
\(325\) −4.58258 −0.254196
\(326\) −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
\(335\) 4.16515 0.227567
\(336\) 0 0
\(337\) −21.1652 −1.15294 −0.576470 0.817119i \(-0.695570\pi\)
−0.576470 + 0.817119i \(0.695570\pi\)
\(338\) −14.3303 −0.779466
\(339\) 0 0
\(340\) −3.62614 −0.196655
\(341\) −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 \(-0.561602\pi\)
−0.192323 + 0.981332i \(0.561602\pi\)
\(348\) 0 0
\(349\) −4.33030 −0.231796 −0.115898 0.993261i \(-0.536975\pi\)
−0.115898 + 0.993261i \(0.536975\pi\)
\(350\) −1.79129 −0.0957484
\(351\) 0 0
\(352\) −30.2178 −1.61061
\(353\) 14.8348 0.789579 0.394790 0.918772i \(-0.370817\pi\)
0.394790 + 0.918772i \(0.370817\pi\)
\(354\) 0 0
\(355\) −9.58258 −0.508590
\(356\) −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
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −37.4955 −1.95725 −0.978623 0.205661i \(-0.934066\pi\)
−0.978623 + 0.205661i \(0.934066\pi\)
\(368\) −19.8258 −1.03349
\(369\) 0 0
\(370\) −7.16515 −0.372498
\(371\) −0.417424 −0.0216716
\(372\) 0 0
\(373\) 28.3303 1.46689 0.733444 0.679750i \(-0.237912\pi\)
0.733444 + 0.679750i \(0.237912\pi\)
\(374\) 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 \(0.732749\pi\)
\(380\) −4.33030 −0.222140
\(381\) 0 0
\(382\) −7.16515 −0.366601
\(383\) −3.58258 −0.183061 −0.0915305 0.995802i \(-0.529176\pi\)
−0.0915305 + 0.995802i \(0.529176\pi\)
\(384\) 0 0
\(385\) 5.00000 0.254824
\(386\) 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
\(395\) −7.58258 −0.381521
\(396\) 0 0
\(397\) 10.8348 0.543785 0.271893 0.962328i \(-0.412350\pi\)
0.271893 + 0.962328i \(0.412350\pi\)
\(398\) −24.0345 −1.20474
\(399\) 0 0
\(400\) −4.95644 −0.247822
\(401\) −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 \(0.521640\pi\)
\(410\) 16.4174 0.810799
\(411\) 0 0
\(412\) 18.3303 0.903069
\(413\) 7.58258 0.373114
\(414\) 0 0
\(415\) −11.5826 −0.568566
\(416\) −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
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 12.7477 0.616906
\(428\) −6.24318 −0.301776
\(429\) 0 0
\(430\) −17.1652 −0.827777
\(431\) 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 \(-0.294137\pi\)
0.602587 + 0.798053i \(0.294137\pi\)
\(434\) −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 \(0.632243\pi\)
\(440\) 7.08712 0.337865
\(441\) 0 0
\(442\) 24.6261 1.17135
\(443\) −0.582576 −0.0276790 −0.0138395 0.999904i \(-0.504405\pi\)
−0.0138395 + 0.999904i \(0.504405\pi\)
\(444\) 0 0
\(445\) 1.41742 0.0671924
\(446\) −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
\(455\) 4.58258 0.214834
\(456\) 0 0
\(457\) −3.74773 −0.175311 −0.0876556 0.996151i \(-0.527938\pi\)
−0.0876556 + 0.996151i \(0.527938\pi\)
\(458\) 30.7477 1.43675
\(459\) 0 0
\(460\) −4.83485 −0.225426
\(461\) 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 \(-0.501222\pi\)
−0.00383762 + 0.999993i \(0.501222\pi\)
\(464\) 4.95644 0.230097
\(465\) 0 0
\(466\) −23.5826 −1.09244
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −4.16515 −0.192329
\(470\) −18.9564 −0.874395
\(471\) 0 0
\(472\) 10.7477 0.494704
\(473\) 47.9129 2.20304
\(474\) 0 0
\(475\) 3.58258 0.164380
\(476\) 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
\(485\) −11.5826 −0.525938
\(486\) 0 0
\(487\) 2.33030 0.105596 0.0527980 0.998605i \(-0.483186\pi\)
0.0527980 + 0.998605i \(0.483186\pi\)
\(488\) 18.0689 0.817942
\(489\) 0 0
\(490\) −10.7477 −0.485533
\(491\) −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 \(0.597095\pi\)
\(500\) −1.20871 −0.0540553
\(501\) 0 0
\(502\) 28.9564 1.29239
\(503\) 22.9129 1.02163 0.510817 0.859689i \(-0.329343\pi\)
0.510817 + 0.859689i \(0.329343\pi\)
\(504\) 0 0
\(505\) −0.582576 −0.0259243
\(506\) 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
\(515\) −15.1652 −0.668256
\(516\) 0 0
\(517\) 52.9129 2.32711
\(518\) 7.16515 0.314819
\(519\) 0 0
\(520\) 6.49545 0.284845
\(521\) −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 \(-0.759883\pi\)
−0.728716 + 0.684816i \(0.759883\pi\)
\(524\) 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
\(530\) −0.747727 −0.0324792
\(531\) 0 0
\(532\) 4.33030 0.187742
\(533\) −42.0000 −1.81922
\(534\) 0 0
\(535\) 5.16515 0.223309
\(536\) −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
\(545\) −14.1652 −0.606768
\(546\) 0 0
\(547\) 4.16515 0.178089 0.0890445 0.996028i \(-0.471619\pi\)
0.0890445 + 0.996028i \(0.471619\pi\)
\(548\) −19.7386 −0.843193
\(549\) 0 0
\(550\) 8.95644 0.381904
\(551\) −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 \(-0.339937\pi\)
0.481926 + 0.876212i \(0.339937\pi\)
\(558\) 0 0
\(559\) 43.9129 1.85732
\(560\) 4.95644 0.209448
\(561\) 0 0
\(562\) −0.747727 −0.0315410
\(563\) 14.5826 0.614582 0.307291 0.951616i \(-0.400577\pi\)
0.307291 + 0.951616i \(0.400577\pi\)
\(564\) 0 0
\(565\) −14.1652 −0.595932
\(566\) 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
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −19.1652 −0.797856 −0.398928 0.916982i \(-0.630618\pi\)
−0.398928 + 0.916982i \(0.630618\pi\)
\(578\) 14.3303 0.596062
\(579\) 0 0
\(580\) 1.20871 0.0501890
\(581\) 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 \(-0.423170\pi\)
0.239032 + 0.971012i \(0.423170\pi\)
\(588\) 0 0
\(589\) 14.3303 0.590470
\(590\) 13.5826 0.559186
\(591\) 0 0
\(592\) 19.8258 0.814834
\(593\) −8.41742 −0.345662 −0.172831 0.984951i \(-0.555291\pi\)
−0.172831 + 0.984951i \(0.555291\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −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
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 3.58258 0.145412 0.0727061 0.997353i \(-0.476836\pi\)
0.0727061 + 0.997353i \(0.476836\pi\)
\(608\) 21.6515 0.878085
\(609\) 0 0
\(610\) 22.8348 0.924556
\(611\) 48.4955 1.96192
\(612\) 0 0
\(613\) −43.7477 −1.76695 −0.883477 0.468474i \(-0.844804\pi\)
−0.883477 + 0.468474i \(0.844804\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −7.08712 −0.285548
\(617\) −15.4955 −0.623823 −0.311912 0.950111i \(-0.600969\pi\)
−0.311912 + 0.950111i \(0.600969\pi\)
\(618\) 0 0
\(619\) 13.1652 0.529152 0.264576 0.964365i \(-0.414768\pi\)
0.264576 + 0.964365i \(0.414768\pi\)
\(620\) −4.83485 −0.194172
\(621\) 0 0
\(622\) −5.37386 −0.215472
\(623\) −1.41742 −0.0567879
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −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
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 27.4955 1.08941
\(638\) −8.95644 −0.354589
\(639\) 0 0
\(640\) 10.4519 0.413147
\(641\) 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 \(-0.169899\pi\)
0.860903 + 0.508769i \(0.169899\pi\)
\(644\) 4.83485 0.190520
\(645\) 0 0
\(646\) −19.2523 −0.757471
\(647\) −24.7477 −0.972934 −0.486467 0.873699i \(-0.661715\pi\)
−0.486467 + 0.873699i \(0.661715\pi\)
\(648\) 0 0
\(649\) −37.9129 −1.48821
\(650\) 8.20871 0.321972
\(651\) 0 0
\(652\) 9.16515 0.358935
\(653\) −26.1652 −1.02392 −0.511961 0.859009i \(-0.671081\pi\)
−0.511961 + 0.859009i \(0.671081\pi\)
\(654\) 0 0
\(655\) −15.0000 −0.586098
\(656\) −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
\(665\) −3.58258 −0.138926
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 0 0
\(669\) 0 0
\(670\) −7.46099 −0.288243
\(671\) −63.7386 −2.46060
\(672\) 0 0
\(673\) 1.74773 0.0673699 0.0336850 0.999433i \(-0.489276\pi\)
0.0336850 + 0.999433i \(0.489276\pi\)
\(674\) 37.9129 1.46035
\(675\) 0 0
\(676\) 9.66970 0.371911
\(677\) −44.8258 −1.72279 −0.861397 0.507932i \(-0.830410\pi\)
−0.861397 + 0.507932i \(0.830410\pi\)
\(678\) 0 0
\(679\) 11.5826 0.444498
\(680\) −4.25227 −0.163067
\(681\) 0 0
\(682\) 35.8258 1.37184
\(683\) −7.16515 −0.274167 −0.137083 0.990560i \(-0.543773\pi\)
−0.137083 + 0.990560i \(0.543773\pi\)
\(684\) 0 0
\(685\) 16.3303 0.623949
\(686\) 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
\(695\) 9.41742 0.357223
\(696\) 0 0
\(697\) 27.4955 1.04146
\(698\) 7.75682 0.293600
\(699\) 0 0
\(700\) 1.20871 0.0456850
\(701\) 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 \(0.439871\pi\)
\(710\) 17.1652 0.644197
\(711\) 0 0
\(712\) −2.00909 −0.0752939
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −22.9129 −0.856893
\(716\) −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
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 9.66970 0.358629 0.179315 0.983792i \(-0.442612\pi\)
0.179315 + 0.983792i \(0.442612\pi\)
\(728\) −6.49545 −0.240738
\(729\) 0 0
\(730\) 7.16515 0.265194
\(731\) −28.7477 −1.06327
\(732\) 0 0
\(733\) −38.4174 −1.41898 −0.709490 0.704716i \(-0.751075\pi\)
−0.709490 + 0.704716i \(0.751075\pi\)
\(734\) 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 \(0.384786\pi\)
\(740\) 4.83485 0.177733
\(741\) 0 0
\(742\) 0.747727 0.0274499
\(743\) −29.7477 −1.09134 −0.545669 0.838001i \(-0.683724\pi\)
−0.545669 + 0.838001i \(0.683724\pi\)
\(744\) 0 0
\(745\) 16.7477 0.613589
\(746\) −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
\(755\) −7.16515 −0.260767
\(756\) 0 0
\(757\) −2.33030 −0.0846963 −0.0423481 0.999103i \(-0.513484\pi\)
−0.0423481 + 0.999103i \(0.513484\pi\)
\(758\) 46.5735 1.69163
\(759\) 0 0
\(760\) −5.07803 −0.184200
\(761\) −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 \(0.347396\pi\)
\(770\) −8.95644 −0.322768
\(771\) 0 0
\(772\) −24.5735 −0.884419
\(773\) 5.16515 0.185778 0.0928888 0.995676i \(-0.470390\pi\)
0.0928888 + 0.995676i \(0.470390\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 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
\(785\) −10.7477 −0.383603
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) 19.7386 0.703160
\(789\) 0 0
\(790\) 13.5826 0.483246
\(791\) 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 \(-0.694065\pi\)
−0.572599 + 0.819836i \(0.694065\pi\)
\(798\) 0 0
\(799\) −31.7477 −1.12315
\(800\) 6.04356 0.213672
\(801\) 0 0
\(802\) 22.2432 0.785434
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 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
\(815\) −7.58258 −0.265606
\(816\) 0 0
\(817\) −34.3303 −1.20107
\(818\) 4.92197 0.172093
\(819\) 0 0
\(820\) −11.0780 −0.386862
\(821\) 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 \(-0.150541\pi\)
0.890234 + 0.455503i \(0.150541\pi\)
\(824\) 21.4955 0.748830
\(825\) 0 0
\(826\) −13.5826 −0.472598
\(827\) 43.8258 1.52397 0.761985 0.647594i \(-0.224225\pi\)
0.761985 + 0.647594i \(0.224225\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 20.7477 0.720164
\(831\) 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
\(845\) −8.00000 −0.275208
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 2.06894 0.0710476
\(849\) 0 0
\(850\) −5.37386 −0.184322
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 20.7477 0.710389 0.355194 0.934792i \(-0.384415\pi\)
0.355194 + 0.934792i \(0.384415\pi\)
\(854\) −22.8348 −0.781392
\(855\) 0 0
\(856\) −7.32121 −0.250234
\(857\) 46.6606 1.59390 0.796948 0.604048i \(-0.206446\pi\)
0.796948 + 0.604048i \(0.206446\pi\)
\(858\) 0 0
\(859\) −47.5826 −1.62350 −0.811748 0.584007i \(-0.801484\pi\)
−0.811748 + 0.584007i \(0.801484\pi\)
\(860\) 11.5826 0.394963
\(861\) 0 0
\(862\) −70.1561 −2.38952
\(863\) 6.41742 0.218452 0.109226 0.994017i \(-0.465163\pi\)
0.109226 + 0.994017i \(0.465163\pi\)
\(864\) 0 0
\(865\) −3.16515 −0.107618
\(866\) −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
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 19.6697 0.664198 0.332099 0.943244i \(-0.392243\pi\)
0.332099 + 0.943244i \(0.392243\pi\)
\(878\) 30.2958 1.02243
\(879\) 0 0
\(880\) −24.7822 −0.835408
\(881\) 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 \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) −16.6170 −0.558892
\(885\) 0 0
\(886\) 1.04356 0.0350591
\(887\) 44.9129 1.50803 0.754013 0.656859i \(-0.228115\pi\)
0.754013 + 0.656859i \(0.228115\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) −2.53901 −0.0851080
\(891\) 0 0
\(892\) 8.46099 0.283295
\(893\) −37.9129 −1.26871
\(894\) 0 0
\(895\) 4.74773 0.158699
\(896\) −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
\(905\) −16.1652 −0.537348
\(906\) 0 0
\(907\) 23.5826 0.783047 0.391523 0.920168i \(-0.371948\pi\)
0.391523 + 0.920168i \(0.371948\pi\)
\(908\) 9.16515 0.304156
\(909\) 0 0
\(910\) −8.20871 −0.272116
\(911\) 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 \(0.399021\pi\)
\(920\) −5.66970 −0.186924
\(921\) 0 0
\(922\) −16.4174 −0.540679
\(923\) −43.9129 −1.44541
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 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
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −18.5826 −0.607066 −0.303533 0.952821i \(-0.598166\pi\)
−0.303533 + 0.952821i \(0.598166\pi\)
\(938\) 7.46099 0.243610
\(939\) 0 0
\(940\) 12.7913 0.417206
\(941\) −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 \(-0.149151\pi\)
0.892214 + 0.451612i \(0.149151\pi\)
\(948\) 0 0
\(949\) −18.3303 −0.595027
\(950\) −6.41742 −0.208209
\(951\) 0 0
\(952\) 4.25227 0.137817
\(953\) 37.5826 1.21742 0.608710 0.793393i \(-0.291687\pi\)
0.608710 + 0.793393i \(0.291687\pi\)
\(954\) 0 0
\(955\) −4.00000 −0.129437
\(956\) 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
\(965\) 20.3303 0.654456
\(966\) 0 0
\(967\) 9.25227 0.297533 0.148767 0.988872i \(-0.452470\pi\)
0.148767 + 0.988872i \(0.452470\pi\)
\(968\) 19.8439 0.637808
\(969\) 0 0
\(970\) 20.7477 0.666169
\(971\) −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 \(-0.487244\pi\)
0.0400638 + 0.999197i \(0.487244\pi\)
\(978\) 0 0
\(979\) 7.08712 0.226505
\(980\) 7.25227 0.231665
\(981\) 0 0
\(982\) 28.6606 0.914597
\(983\) −55.1652 −1.75950 −0.879748 0.475441i \(-0.842288\pi\)
−0.879748 + 0.475441i \(0.842288\pi\)
\(984\) 0 0
\(985\) −16.3303 −0.520327
\(986\) 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
\(995\) −13.4174 −0.425361
\(996\) 0 0
\(997\) −0.834849 −0.0264399 −0.0132200 0.999913i \(-0.504208\pi\)
−0.0132200 + 0.999913i \(0.504208\pi\)
\(998\) 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 1305.2.a.m.1.1 2
3.2 odd 2 435.2.a.f.1.2 2
5.4 even 2 6525.2.a.t.1.2 2
12.11 even 2 6960.2.a.bw.1.1 2
15.2 even 4 2175.2.c.f.349.3 4
15.8 even 4 2175.2.c.f.349.2 4
15.14 odd 2 2175.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 3.2 odd 2
1305.2.a.m.1.1 2 1.1 even 1 trivial
2175.2.a.r.1.1 2 15.14 odd 2
2175.2.c.f.349.2 4 15.8 even 4
2175.2.c.f.349.3 4 15.2 even 4
6525.2.a.t.1.2 2 5.4 even 2
6960.2.a.bw.1.1 2 12.11 even 2