Properties

Label 2175.2.a.r.1.1
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
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 \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.79129 q^{2} -1.00000 q^{3} +1.20871 q^{4} +1.79129 q^{6} -1.00000 q^{7} +1.41742 q^{8} +1.00000 q^{9} +5.00000 q^{11} -1.20871 q^{12} +4.58258 q^{13} +1.79129 q^{14} -4.95644 q^{16} +3.00000 q^{17} -1.79129 q^{18} +3.58258 q^{19} +1.00000 q^{21} -8.95644 q^{22} +4.00000 q^{23} -1.41742 q^{24} -8.20871 q^{26} -1.00000 q^{27} -1.20871 q^{28} +1.00000 q^{29} +4.00000 q^{31} +6.04356 q^{32} -5.00000 q^{33} -5.37386 q^{34} +1.20871 q^{36} +4.00000 q^{37} -6.41742 q^{38} -4.58258 q^{39} -9.16515 q^{41} -1.79129 q^{42} +9.58258 q^{43} +6.04356 q^{44} -7.16515 q^{46} -10.5826 q^{47} +4.95644 q^{48} -6.00000 q^{49} -3.00000 q^{51} +5.53901 q^{52} -0.417424 q^{53} +1.79129 q^{54} -1.41742 q^{56} -3.58258 q^{57} -1.79129 q^{58} -7.58258 q^{59} +12.7477 q^{61} -7.16515 q^{62} -1.00000 q^{63} -0.912878 q^{64} +8.95644 q^{66} +4.16515 q^{67} +3.62614 q^{68} -4.00000 q^{69} -9.58258 q^{71} +1.41742 q^{72} -4.00000 q^{73} -7.16515 q^{74} +4.33030 q^{76} -5.00000 q^{77} +8.20871 q^{78} +7.58258 q^{79} +1.00000 q^{81} +16.4174 q^{82} +11.5826 q^{83} +1.20871 q^{84} -17.1652 q^{86} -1.00000 q^{87} +7.08712 q^{88} +1.41742 q^{89} -4.58258 q^{91} +4.83485 q^{92} -4.00000 q^{93} +18.9564 q^{94} -6.04356 q^{96} -11.5826 q^{97} +10.7477 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26} - 2 q^{27} - 7 q^{28} + 2 q^{29} + 8 q^{31} + 35 q^{32} - 10 q^{33} + 3 q^{34} + 7 q^{36} + 8 q^{37} - 22 q^{38} + q^{42} + 10 q^{43} + 35 q^{44} + 4 q^{46} - 12 q^{47} - 13 q^{48} - 12 q^{49} - 6 q^{51} - 21 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} - 2 q^{61} + 4 q^{62} - 2 q^{63} + 44 q^{64} - 5 q^{66} - 10 q^{67} + 21 q^{68} - 8 q^{69} - 10 q^{71} + 12 q^{72} - 8 q^{73} + 4 q^{74} - 28 q^{76} - 10 q^{77} + 21 q^{78} + 6 q^{79} + 2 q^{81} + 42 q^{82} + 14 q^{83} + 7 q^{84} - 16 q^{86} - 2 q^{87} + 60 q^{88} + 12 q^{89} + 28 q^{92} - 8 q^{93} + 15 q^{94} - 35 q^{96} - 14 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79129 −1.26663 −0.633316 0.773893i \(-0.718307\pi\)
−0.633316 + 0.773893i \(0.718307\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.20871 0.604356
\(5\) 0 0
\(6\) 1.79129 0.731290
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.41742 0.501135
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −1.20871 −0.348925
\(13\) 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 \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.79129 −0.422211
\(19\) 3.58258 0.821899 0.410950 0.911658i \(-0.365197\pi\)
0.410950 + 0.911658i \(0.365197\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −8.95644 −1.90952
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.41742 −0.289331
\(25\) 0 0
\(26\) −8.20871 −1.60986
\(27\) −1.00000 −0.192450
\(28\) −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
\(33\) −5.00000 −0.870388
\(34\) −5.37386 −0.921610
\(35\) 0 0
\(36\) 1.20871 0.201452
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −6.41742 −1.04104
\(39\) −4.58258 −0.733799
\(40\) 0 0
\(41\) −9.16515 −1.43136 −0.715678 0.698430i \(-0.753882\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) −1.79129 −0.276402
\(43\) 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 \(-0.780650\pi\)
−0.771814 + 0.635849i \(0.780650\pi\)
\(48\) 4.95644 0.715400
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 5.53901 0.768123
\(53\) −0.417424 −0.0573376 −0.0286688 0.999589i \(-0.509127\pi\)
−0.0286688 + 0.999589i \(0.509127\pi\)
\(54\) 1.79129 0.243763
\(55\) 0 0
\(56\) −1.41742 −0.189411
\(57\) −3.58258 −0.474524
\(58\) −1.79129 −0.235208
\(59\) −7.58258 −0.987167 −0.493584 0.869698i \(-0.664313\pi\)
−0.493584 + 0.869698i \(0.664313\pi\)
\(60\) 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
\(63\) −1.00000 −0.125988
\(64\) −0.912878 −0.114110
\(65\) 0 0
\(66\) 8.95644 1.10246
\(67\) 4.16515 0.508854 0.254427 0.967092i \(-0.418113\pi\)
0.254427 + 0.967092i \(0.418113\pi\)
\(68\) 3.62614 0.439734
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −9.58258 −1.13724 −0.568621 0.822599i \(-0.692523\pi\)
−0.568621 + 0.822599i \(0.692523\pi\)
\(72\) 1.41742 0.167045
\(73\) −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
\(78\) 8.20871 0.929454
\(79\) 7.58258 0.853106 0.426553 0.904462i \(-0.359728\pi\)
0.426553 + 0.904462i \(0.359728\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 16.4174 1.81300
\(83\) 11.5826 1.27135 0.635676 0.771956i \(-0.280721\pi\)
0.635676 + 0.771956i \(0.280721\pi\)
\(84\) 1.20871 0.131881
\(85\) 0 0
\(86\) −17.1652 −1.85097
\(87\) −1.00000 −0.107211
\(88\) 7.08712 0.755490
\(89\) 1.41742 0.150247 0.0751233 0.997174i \(-0.476065\pi\)
0.0751233 + 0.997174i \(0.476065\pi\)
\(90\) 0 0
\(91\) −4.58258 −0.480384
\(92\) 4.83485 0.504068
\(93\) −4.00000 −0.414781
\(94\) 18.9564 1.95521
\(95\) 0 0
\(96\) −6.04356 −0.616818
\(97\) −11.5826 −1.17603 −0.588016 0.808849i \(-0.700091\pi\)
−0.588016 + 0.808849i \(0.700091\pi\)
\(98\) 10.7477 1.08568
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) −0.582576 −0.0579684 −0.0289842 0.999580i \(-0.509227\pi\)
−0.0289842 + 0.999580i \(0.509227\pi\)
\(102\) 5.37386 0.532092
\(103\) −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 \(-0.580321\pi\)
−0.249667 + 0.968332i \(0.580321\pi\)
\(108\) −1.20871 −0.116308
\(109\) 14.1652 1.35678 0.678388 0.734704i \(-0.262679\pi\)
0.678388 + 0.734704i \(0.262679\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 4.95644 0.468339
\(113\) 14.1652 1.33255 0.666273 0.745708i \(-0.267889\pi\)
0.666273 + 0.745708i \(0.267889\pi\)
\(114\) 6.41742 0.601047
\(115\) 0 0
\(116\) 1.20871 0.112226
\(117\) 4.58258 0.423659
\(118\) 13.5826 1.25038
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −22.8348 −2.06737
\(123\) 9.16515 0.826394
\(124\) 4.83485 0.434182
\(125\) 0 0
\(126\) 1.79129 0.159581
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −10.4519 −0.923826
\(129\) −9.58258 −0.843699
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) −6.04356 −0.526024
\(133\) −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 \(0.745747\pi\)
\(138\) 7.16515 0.609938
\(139\) −9.41742 −0.798776 −0.399388 0.916782i \(-0.630777\pi\)
−0.399388 + 0.916782i \(0.630777\pi\)
\(140\) 0 0
\(141\) 10.5826 0.891214
\(142\) 17.1652 1.44047
\(143\) 22.9129 1.91607
\(144\) −4.95644 −0.413037
\(145\) 0 0
\(146\) 7.16515 0.592992
\(147\) 6.00000 0.494872
\(148\) 4.83485 0.397422
\(149\) 16.7477 1.37203 0.686014 0.727589i \(-0.259359\pi\)
0.686014 + 0.727589i \(0.259359\pi\)
\(150\) 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
\(153\) 3.00000 0.242536
\(154\) 8.95644 0.721730
\(155\) 0 0
\(156\) −5.53901 −0.443476
\(157\) −10.7477 −0.857762 −0.428881 0.903361i \(-0.641092\pi\)
−0.428881 + 0.903361i \(0.641092\pi\)
\(158\) −13.5826 −1.08057
\(159\) 0.417424 0.0331039
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) −1.79129 −0.140737
\(163\) −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 \(0.5\pi\)
\(168\) 1.41742 0.109357
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) 3.58258 0.273966
\(172\) 11.5826 0.883163
\(173\) 3.16515 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(174\) 1.79129 0.135797
\(175\) 0 0
\(176\) −24.7822 −1.86803
\(177\) 7.58258 0.569941
\(178\) −2.53901 −0.190307
\(179\) 4.74773 0.354862 0.177431 0.984133i \(-0.443221\pi\)
0.177431 + 0.984133i \(0.443221\pi\)
\(180\) 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
\(183\) −12.7477 −0.942339
\(184\) 5.66970 0.417976
\(185\) 0 0
\(186\) 7.16515 0.525374
\(187\) 15.0000 1.09691
\(188\) −12.7913 −0.932901
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0.912878 0.0658813
\(193\) 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 \(-0.302371\pi\)
0.581743 + 0.813373i \(0.302371\pi\)
\(198\) −8.95644 −0.636506
\(199\) 13.4174 0.951136 0.475568 0.879679i \(-0.342243\pi\)
0.475568 + 0.879679i \(0.342243\pi\)
\(200\) 0 0
\(201\) −4.16515 −0.293787
\(202\) 1.04356 0.0734247
\(203\) −1.00000 −0.0701862
\(204\) −3.62614 −0.253880
\(205\) 0 0
\(206\) 27.1652 1.89269
\(207\) 4.00000 0.278019
\(208\) −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
\(213\) 9.58258 0.656587
\(214\) 9.25227 0.632472
\(215\) 0 0
\(216\) −1.41742 −0.0964435
\(217\) −4.00000 −0.271538
\(218\) −25.3739 −1.71853
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 13.7477 0.924772
\(222\) 7.16515 0.480893
\(223\) −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 \(-0.419031\pi\)
0.251637 + 0.967822i \(0.419031\pi\)
\(228\) −4.33030 −0.286781
\(229\) −17.1652 −1.13431 −0.567153 0.823613i \(-0.691955\pi\)
−0.567153 + 0.823613i \(0.691955\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 1.41742 0.0930585
\(233\) 13.1652 0.862478 0.431239 0.902238i \(-0.358077\pi\)
0.431239 + 0.902238i \(0.358077\pi\)
\(234\) −8.20871 −0.536620
\(235\) 0 0
\(236\) −9.16515 −0.596601
\(237\) −7.58258 −0.492541
\(238\) 5.37386 0.348336
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 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
\(243\) −1.00000 −0.0641500
\(244\) 15.4083 0.986417
\(245\) 0 0
\(246\) −16.4174 −1.04674
\(247\) 16.4174 1.04462
\(248\) 5.66970 0.360026
\(249\) −11.5826 −0.734016
\(250\) 0 0
\(251\) 16.1652 1.02034 0.510168 0.860075i \(-0.329583\pi\)
0.510168 + 0.860075i \(0.329583\pi\)
\(252\) −1.20871 −0.0761417
\(253\) 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 \(0.630154\pi\)
\(258\) 17.1652 1.06866
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 26.8693 1.65999
\(263\) −30.3303 −1.87025 −0.935123 0.354322i \(-0.884712\pi\)
−0.935123 + 0.354322i \(0.884712\pi\)
\(264\) −7.08712 −0.436182
\(265\) 0 0
\(266\) 6.41742 0.393478
\(267\) −1.41742 −0.0867450
\(268\) 5.03447 0.307529
\(269\) 22.5826 1.37688 0.688442 0.725291i \(-0.258295\pi\)
0.688442 + 0.725291i \(0.258295\pi\)
\(270\) 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
\(273\) 4.58258 0.277350
\(274\) 29.2523 1.76719
\(275\) 0 0
\(276\) −4.83485 −0.291024
\(277\) 24.9129 1.49687 0.748435 0.663208i \(-0.230806\pi\)
0.748435 + 0.663208i \(0.230806\pi\)
\(278\) 16.8693 1.01175
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −0.417424 −0.0249014 −0.0124507 0.999922i \(-0.503963\pi\)
−0.0124507 + 0.999922i \(0.503963\pi\)
\(282\) −18.9564 −1.12884
\(283\) 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
\(288\) 6.04356 0.356120
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 11.5826 0.678983
\(292\) −4.83485 −0.282938
\(293\) −30.1652 −1.76227 −0.881133 0.472868i \(-0.843219\pi\)
−0.881133 + 0.472868i \(0.843219\pi\)
\(294\) −10.7477 −0.626820
\(295\) 0 0
\(296\) 5.66970 0.329544
\(297\) −5.00000 −0.290129
\(298\) −30.0000 −1.73785
\(299\) 18.3303 1.06007
\(300\) 0 0
\(301\) −9.58258 −0.552330
\(302\) −12.8348 −0.738563
\(303\) 0.582576 0.0334681
\(304\) −17.7568 −1.01842
\(305\) 0 0
\(306\) −5.37386 −0.307203
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −6.04356 −0.344364
\(309\) 15.1652 0.862715
\(310\) 0 0
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) −6.49545 −0.367733
\(313\) −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 \(-0.252259\pi\)
0.702070 + 0.712108i \(0.252259\pi\)
\(318\) −0.747727 −0.0419305
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) 5.16515 0.288291
\(322\) 7.16515 0.399298
\(323\) 10.7477 0.598020
\(324\) 1.20871 0.0671507
\(325\) 0 0
\(326\) 13.5826 0.752269
\(327\) −14.1652 −0.783335
\(328\) −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
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) −4.95644 −0.270396
\(337\) 21.1652 1.15294 0.576470 0.817119i \(-0.304430\pi\)
0.576470 + 0.817119i \(0.304430\pi\)
\(338\) −14.3303 −0.779466
\(339\) −14.1652 −0.769345
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) −6.41742 −0.347015
\(343\) 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 \(0.561602\pi\)
\(348\) −1.20871 −0.0647938
\(349\) −4.33030 −0.231796 −0.115898 0.993261i \(-0.536975\pi\)
−0.115898 + 0.993261i \(0.536975\pi\)
\(350\) 0 0
\(351\) −4.58258 −0.244600
\(352\) 30.2178 1.61061
\(353\) 14.8348 0.789579 0.394790 0.918772i \(-0.370817\pi\)
0.394790 + 0.918772i \(0.370817\pi\)
\(354\) −13.5826 −0.721906
\(355\) 0 0
\(356\) 1.71326 0.0908025
\(357\) 3.00000 0.158777
\(358\) −8.50455 −0.449479
\(359\) −27.1652 −1.43372 −0.716861 0.697216i \(-0.754422\pi\)
−0.716861 + 0.697216i \(0.754422\pi\)
\(360\) 0 0
\(361\) −6.16515 −0.324482
\(362\) −28.9564 −1.52192
\(363\) −14.0000 −0.734809
\(364\) −5.53901 −0.290323
\(365\) 0 0
\(366\) 22.8348 1.19360
\(367\) 37.4955 1.95725 0.978623 0.205661i \(-0.0659343\pi\)
0.978623 + 0.205661i \(0.0659343\pi\)
\(368\) −19.8258 −1.03349
\(369\) −9.16515 −0.477119
\(370\) 0 0
\(371\) 0.417424 0.0216716
\(372\) −4.83485 −0.250675
\(373\) −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
\(378\) −1.79129 −0.0921339
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 7.16515 0.366601
\(383\) −3.58258 −0.183061 −0.0915305 0.995802i \(-0.529176\pi\)
−0.0915305 + 0.995802i \(0.529176\pi\)
\(384\) 10.4519 0.533371
\(385\) 0 0
\(386\) −36.4174 −1.85360
\(387\) 9.58258 0.487110
\(388\) −14.0000 −0.710742
\(389\) −6.58258 −0.333750 −0.166875 0.985978i \(-0.553368\pi\)
−0.166875 + 0.985978i \(0.553368\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) −8.50455 −0.429544
\(393\) 15.0000 0.756650
\(394\) −29.2523 −1.47371
\(395\) 0 0
\(396\) 6.04356 0.303700
\(397\) −10.8348 −0.543785 −0.271893 0.962328i \(-0.587650\pi\)
−0.271893 + 0.962328i \(0.587650\pi\)
\(398\) −24.0345 −1.20474
\(399\) 3.58258 0.179353
\(400\) 0 0
\(401\) 12.4174 0.620097 0.310048 0.950721i \(-0.399655\pi\)
0.310048 + 0.950721i \(0.399655\pi\)
\(402\) 7.46099 0.372120
\(403\) 18.3303 0.913097
\(404\) −0.704166 −0.0350336
\(405\) 0 0
\(406\) 1.79129 0.0889001
\(407\) 20.0000 0.991363
\(408\) −4.25227 −0.210519
\(409\) −2.74773 −0.135866 −0.0679332 0.997690i \(-0.521640\pi\)
−0.0679332 + 0.997690i \(0.521640\pi\)
\(410\) 0 0
\(411\) 16.3303 0.805514
\(412\) −18.3303 −0.903069
\(413\) 7.58258 0.373114
\(414\) −7.16515 −0.352148
\(415\) 0 0
\(416\) 27.6951 1.35786
\(417\) 9.41742 0.461173
\(418\) −32.0871 −1.56943
\(419\) −37.1652 −1.81564 −0.907818 0.419364i \(-0.862253\pi\)
−0.907818 + 0.419364i \(0.862253\pi\)
\(420\) 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
\(423\) −10.5826 −0.514542
\(424\) −0.591667 −0.0287339
\(425\) 0 0
\(426\) −17.1652 −0.831654
\(427\) −12.7477 −0.616906
\(428\) −6.24318 −0.301776
\(429\) −22.9129 −1.10624
\(430\) 0 0
\(431\) −39.1652 −1.88652 −0.943259 0.332057i \(-0.892258\pi\)
−0.943259 + 0.332057i \(0.892258\pi\)
\(432\) 4.95644 0.238467
\(433\) −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
\(438\) −7.16515 −0.342364
\(439\) −16.9129 −0.807208 −0.403604 0.914934i \(-0.632243\pi\)
−0.403604 + 0.914934i \(0.632243\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −24.6261 −1.17135
\(443\) −0.582576 −0.0276790 −0.0138395 0.999904i \(-0.504405\pi\)
−0.0138395 + 0.999904i \(0.504405\pi\)
\(444\) −4.83485 −0.229452
\(445\) 0 0
\(446\) 12.5390 0.593740
\(447\) −16.7477 −0.792140
\(448\) 0.912878 0.0431295
\(449\) −30.0780 −1.41947 −0.709735 0.704469i \(-0.751185\pi\)
−0.709735 + 0.704469i \(0.751185\pi\)
\(450\) 0 0
\(451\) −45.8258 −2.15785
\(452\) 17.1216 0.805332
\(453\) −7.16515 −0.336648
\(454\) −13.5826 −0.637462
\(455\) 0 0
\(456\) −5.07803 −0.237801
\(457\) 3.74773 0.175311 0.0876556 0.996151i \(-0.472062\pi\)
0.0876556 + 0.996151i \(0.472062\pi\)
\(458\) 30.7477 1.43675
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −9.16515 −0.426864 −0.213432 0.976958i \(-0.568464\pi\)
−0.213432 + 0.976958i \(0.568464\pi\)
\(462\) −8.95644 −0.416691
\(463\) 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 \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 5.53901 0.256041
\(469\) −4.16515 −0.192329
\(470\) 0 0
\(471\) 10.7477 0.495229
\(472\) −10.7477 −0.494704
\(473\) 47.9129 2.20304
\(474\) 13.5826 0.623868
\(475\) 0 0
\(476\) −3.62614 −0.166204
\(477\) −0.417424 −0.0191125
\(478\) 46.5735 2.13022
\(479\) −19.1652 −0.875678 −0.437839 0.899053i \(-0.644256\pi\)
−0.437839 + 0.899053i \(0.644256\pi\)
\(480\) 0 0
\(481\) 18.3303 0.835790
\(482\) −52.5390 −2.39309
\(483\) 4.00000 0.182006
\(484\) 16.9220 0.769180
\(485\) 0 0
\(486\) 1.79129 0.0812545
\(487\) −2.33030 −0.105596 −0.0527980 0.998605i \(-0.516814\pi\)
−0.0527980 + 0.998605i \(0.516814\pi\)
\(488\) 18.0689 0.817942
\(489\) 7.58258 0.342896
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 11.0780 0.499436
\(493\) 3.00000 0.135113
\(494\) −29.4083 −1.32314
\(495\) 0 0
\(496\) −19.8258 −0.890203
\(497\) 9.58258 0.429837
\(498\) 20.7477 0.929728
\(499\) −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 \(-0.329343\pi\)
0.510817 + 0.859689i \(0.329343\pi\)
\(504\) −1.41742 −0.0631371
\(505\) 0 0
\(506\) −35.8258 −1.59265
\(507\) −8.00000 −0.355292
\(508\) −2.41742 −0.107256
\(509\) 26.7477 1.18557 0.592786 0.805360i \(-0.298028\pi\)
0.592786 + 0.805360i \(0.298028\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −15.9038 −0.702855
\(513\) −3.58258 −0.158175
\(514\) 22.8348 1.00720
\(515\) 0 0
\(516\) −11.5826 −0.509894
\(517\) −52.9129 −2.32711
\(518\) 7.16515 0.314819
\(519\) −3.16515 −0.138935
\(520\) 0 0
\(521\) 43.0780 1.88728 0.943641 0.330970i \(-0.107376\pi\)
0.943641 + 0.330970i \(0.107376\pi\)
\(522\) −1.79129 −0.0784025
\(523\) 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
\(528\) 24.7822 1.07851
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −7.58258 −0.329056
\(532\) −4.33030 −0.187742
\(533\) −42.0000 −1.81922
\(534\) 2.53901 0.109874
\(535\) 0 0
\(536\) 5.90379 0.255005
\(537\) −4.74773 −0.204880
\(538\) −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
\(543\) −16.1652 −0.693713
\(544\) 18.1307 0.777347
\(545\) 0 0
\(546\) −8.20871 −0.351300
\(547\) −4.16515 −0.178089 −0.0890445 0.996028i \(-0.528381\pi\)
−0.0890445 + 0.996028i \(0.528381\pi\)
\(548\) −19.7386 −0.843193
\(549\) 12.7477 0.544060
\(550\) 0 0
\(551\) 3.58258 0.152623
\(552\) −5.66970 −0.241318
\(553\) −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 \(0.339937\pi\)
\(558\) −7.16515 −0.303325
\(559\) 43.9129 1.85732
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0.747727 0.0315410
\(563\) 14.5826 0.614582 0.307291 0.951616i \(-0.400577\pi\)
0.307291 + 0.951616i \(0.400577\pi\)
\(564\) 12.7913 0.538610
\(565\) 0 0
\(566\) −1.49545 −0.0628586
\(567\) −1.00000 −0.0419961
\(568\) −13.5826 −0.569912
\(569\) 28.5826 1.19824 0.599122 0.800658i \(-0.295516\pi\)
0.599122 + 0.800658i \(0.295516\pi\)
\(570\) 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
\(573\) 4.00000 0.167102
\(574\) −16.4174 −0.685250
\(575\) 0 0
\(576\) −0.912878 −0.0380366
\(577\) 19.1652 0.797856 0.398928 0.916982i \(-0.369382\pi\)
0.398928 + 0.916982i \(0.369382\pi\)
\(578\) 14.3303 0.596062
\(579\) −20.3303 −0.844899
\(580\) 0 0
\(581\) −11.5826 −0.480526
\(582\) −20.7477 −0.860021
\(583\) −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 \(0.423170\pi\)
\(588\) 7.25227 0.299079
\(589\) 14.3303 0.590470
\(590\) 0 0
\(591\) −16.3303 −0.671739
\(592\) −19.8258 −0.814834
\(593\) −8.41742 −0.345662 −0.172831 0.984951i \(-0.555291\pi\)
−0.172831 + 0.984951i \(0.555291\pi\)
\(594\) 8.95644 0.367487
\(595\) 0 0
\(596\) 20.2432 0.829193
\(597\) −13.4174 −0.549139
\(598\) −32.8348 −1.34272
\(599\) 0.165151 0.00674790 0.00337395 0.999994i \(-0.498926\pi\)
0.00337395 + 0.999994i \(0.498926\pi\)
\(600\) 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
\(603\) 4.16515 0.169618
\(604\) 8.66061 0.352395
\(605\) 0 0
\(606\) −1.04356 −0.0423918
\(607\) −3.58258 −0.145412 −0.0727061 0.997353i \(-0.523164\pi\)
−0.0727061 + 0.997353i \(0.523164\pi\)
\(608\) 21.6515 0.878085
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) −48.4955 −1.96192
\(612\) 3.62614 0.146578
\(613\) 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 \(-0.600969\pi\)
−0.311912 + 0.950111i \(0.600969\pi\)
\(618\) −27.1652 −1.09274
\(619\) 13.1652 0.529152 0.264576 0.964365i \(-0.414768\pi\)
0.264576 + 0.964365i \(0.414768\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 5.37386 0.215472
\(623\) −1.41742 −0.0567879
\(624\) 22.7133 0.909258
\(625\) 0 0
\(626\) 18.9564 0.757652
\(627\) −17.9129 −0.715371
\(628\) −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
\(633\) −20.3303 −0.808057
\(634\) −44.7822 −1.77853
\(635\) 0 0
\(636\) 0.504546 0.0200065
\(637\) −27.4955 −1.08941
\(638\) −8.95644 −0.354589
\(639\) −9.58258 −0.379081
\(640\) 0 0
\(641\) −6.58258 −0.259996 −0.129998 0.991514i \(-0.541497\pi\)
−0.129998 + 0.991514i \(0.541497\pi\)
\(642\) −9.25227 −0.365158
\(643\) −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 \(-0.661715\pi\)
−0.486467 + 0.873699i \(0.661715\pi\)
\(648\) 1.41742 0.0556817
\(649\) −37.9129 −1.48821
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −9.16515 −0.358935
\(653\) −26.1652 −1.02392 −0.511961 0.859009i \(-0.671081\pi\)
−0.511961 + 0.859009i \(0.671081\pi\)
\(654\) 25.3739 0.992197
\(655\) 0 0
\(656\) 45.4265 1.77361
\(657\) −4.00000 −0.156055
\(658\) −18.9564 −0.738999
\(659\) 18.1652 0.707614 0.353807 0.935318i \(-0.384887\pi\)
0.353807 + 0.935318i \(0.384887\pi\)
\(660\) 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
\(663\) −13.7477 −0.533917
\(664\) 16.4174 0.637120
\(665\) 0 0
\(666\) −7.16515 −0.277644
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 7.00000 0.270636
\(670\) 0 0
\(671\) 63.7386 2.46060
\(672\) 6.04356 0.233135
\(673\) −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 \(-0.830410\pi\)
−0.861397 + 0.507932i \(0.830410\pi\)
\(678\) 25.3739 0.974477
\(679\) 11.5826 0.444498
\(680\) 0 0
\(681\) −7.58258 −0.290565
\(682\) −35.8258 −1.37184
\(683\) −7.16515 −0.274167 −0.137083 0.990560i \(-0.543773\pi\)
−0.137083 + 0.990560i \(0.543773\pi\)
\(684\) 4.33030 0.165573
\(685\) 0 0
\(686\) −23.2867 −0.889092
\(687\) 17.1652 0.654891
\(688\) −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
\(693\) −5.00000 −0.189934
\(694\) 12.8348 0.487204
\(695\) 0 0
\(696\) −1.41742 −0.0537273
\(697\) −27.4955 −1.04146
\(698\) 7.75682 0.293600
\(699\) −13.1652 −0.497952
\(700\) 0 0
\(701\) −23.0780 −0.871645 −0.435823 0.900033i \(-0.643542\pi\)
−0.435823 + 0.900033i \(0.643542\pi\)
\(702\) 8.20871 0.309818
\(703\) 14.3303 0.540478
\(704\) −4.56439 −0.172027
\(705\) 0 0
\(706\) −26.5735 −1.00011
\(707\) 0.582576 0.0219100
\(708\) 9.16515 0.344447
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 7.58258 0.284369
\(712\) 2.00909 0.0752939
\(713\) 16.0000 0.599205
\(714\) −5.37386 −0.201112
\(715\) 0 0
\(716\) 5.73864 0.214463
\(717\) 26.0000 0.970988
\(718\) 48.6606 1.81600
\(719\) 7.91288 0.295101 0.147550 0.989055i \(-0.452861\pi\)
0.147550 + 0.989055i \(0.452861\pi\)
\(720\) 0 0
\(721\) 15.1652 0.564780
\(722\) 11.0436 0.410999
\(723\) −29.3303 −1.09081
\(724\) 19.5390 0.726162
\(725\) 0 0
\(726\) 25.0780 0.930733
\(727\) −9.66970 −0.358629 −0.179315 0.983792i \(-0.557388\pi\)
−0.179315 + 0.983792i \(0.557388\pi\)
\(728\) −6.49545 −0.240738
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.7477 1.06327
\(732\) −15.4083 −0.569508
\(733\) 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
\(738\) 16.4174 0.604334
\(739\) 19.2523 0.708206 0.354103 0.935206i \(-0.384786\pi\)
0.354103 + 0.935206i \(0.384786\pi\)
\(740\) 0 0
\(741\) −16.4174 −0.603109
\(742\) −0.747727 −0.0274499
\(743\) −29.7477 −1.09134 −0.545669 0.838001i \(-0.683724\pi\)
−0.545669 + 0.838001i \(0.683724\pi\)
\(744\) −5.66970 −0.207861
\(745\) 0 0
\(746\) 50.7477 1.85801
\(747\) 11.5826 0.423784
\(748\) 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
\(753\) −16.1652 −0.589091
\(754\) −8.20871 −0.298944
\(755\) 0 0
\(756\) 1.20871 0.0439604
\(757\) 2.33030 0.0846963 0.0423481 0.999103i \(-0.486516\pi\)
0.0423481 + 0.999103i \(0.486516\pi\)
\(758\) 46.5735 1.69163
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) 36.4174 1.32013 0.660065 0.751208i \(-0.270529\pi\)
0.660065 + 0.751208i \(0.270529\pi\)
\(762\) −3.58258 −0.129783
\(763\) −14.1652 −0.512813
\(764\) −4.83485 −0.174919
\(765\) 0 0
\(766\) 6.41742 0.231871
\(767\) −34.7477 −1.25467
\(768\) −20.5481 −0.741466
\(769\) 25.5826 0.922531 0.461266 0.887262i \(-0.347396\pi\)
0.461266 + 0.887262i \(0.347396\pi\)
\(770\) 0 0
\(771\) 12.7477 0.459098
\(772\) 24.5735 0.884419
\(773\) 5.16515 0.185778 0.0928888 0.995676i \(-0.470390\pi\)
0.0928888 + 0.995676i \(0.470390\pi\)
\(774\) −17.1652 −0.616989
\(775\) 0 0
\(776\) −16.4174 −0.589351
\(777\) 4.00000 0.143499
\(778\) 11.7913 0.422738
\(779\) −32.8348 −1.17643
\(780\) 0 0
\(781\) −47.9129 −1.71446
\(782\) −21.4955 −0.768676
\(783\) −1.00000 −0.0357371
\(784\) 29.7386 1.06209
\(785\) 0 0
\(786\) −26.8693 −0.958397
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 19.7386 0.703160
\(789\) 30.3303 1.07979
\(790\) 0 0
\(791\) −14.1652 −0.503655
\(792\) 7.08712 0.251830
\(793\) 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 \(0.694065\pi\)
\(798\) −6.41742 −0.227174
\(799\) −31.7477 −1.12315
\(800\) 0 0
\(801\) 1.41742 0.0500822
\(802\) −22.2432 −0.785434
\(803\) −20.0000 −0.705785
\(804\) −5.03447 −0.177552
\(805\) 0 0
\(806\) −32.8348 −1.15656
\(807\) −22.5826 −0.794944
\(808\) −0.825757 −0.0290500
\(809\) 44.0780 1.54970 0.774851 0.632145i \(-0.217825\pi\)
0.774851 + 0.632145i \(0.217825\pi\)
\(810\) 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
\(813\) −1.16515 −0.0408636
\(814\) −35.8258 −1.25569
\(815\) 0 0
\(816\) 14.8693 0.520530
\(817\) 34.3303 1.20107
\(818\) 4.92197 0.172093
\(819\) −4.58258 −0.160128
\(820\) 0 0
\(821\) −31.4955 −1.09920 −0.549599 0.835428i \(-0.685220\pi\)
−0.549599 + 0.835428i \(0.685220\pi\)
\(822\) −29.2523 −1.02029
\(823\) −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 \(-0.224225\pi\)
0.761985 + 0.647594i \(0.224225\pi\)
\(828\) 4.83485 0.168023
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −24.9129 −0.864218
\(832\) −4.18333 −0.145031
\(833\) −18.0000 −0.623663
\(834\) −16.8693 −0.584137
\(835\) 0 0
\(836\) 21.6515 0.748833
\(837\) −4.00000 −0.138260
\(838\) 66.5735 2.29974
\(839\) −48.4955 −1.67425 −0.837125 0.547012i \(-0.815765\pi\)
−0.837125 + 0.547012i \(0.815765\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −7.91288 −0.272696
\(843\) 0.417424 0.0143769
\(844\) 24.5735 0.845854
\(845\) 0 0
\(846\) 18.9564 0.651736
\(847\) −14.0000 −0.481046
\(848\) 2.06894 0.0710476
\(849\) −0.834849 −0.0286519
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 11.5826 0.396813
\(853\) −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 \(-0.206446\pi\)
0.796948 + 0.604048i \(0.206446\pi\)
\(858\) 41.0436 1.40120
\(859\) −47.5826 −1.62350 −0.811748 0.584007i \(-0.801484\pi\)
−0.811748 + 0.584007i \(0.801484\pi\)
\(860\) 0 0
\(861\) −9.16515 −0.312348
\(862\) 70.1561 2.38952
\(863\) 6.41742 0.218452 0.109226 0.994017i \(-0.465163\pi\)
0.109226 + 0.994017i \(0.465163\pi\)
\(864\) −6.04356 −0.205606
\(865\) 0 0
\(866\) 44.9220 1.52651
\(867\) 8.00000 0.271694
\(868\) −4.83485 −0.164105
\(869\) 37.9129 1.28611
\(870\) 0 0
\(871\) 19.0871 0.646742
\(872\) 20.0780 0.679928
\(873\) −11.5826 −0.392011
\(874\) −25.6697 −0.868290
\(875\) 0 0
\(876\) 4.83485 0.163354
\(877\) −19.6697 −0.664198 −0.332099 0.943244i \(-0.607757\pi\)
−0.332099 + 0.943244i \(0.607757\pi\)
\(878\) 30.2958 1.02243
\(879\) 30.1652 1.01745
\(880\) 0 0
\(881\) −32.0780 −1.08074 −0.540368 0.841429i \(-0.681715\pi\)
−0.540368 + 0.841429i \(0.681715\pi\)
\(882\) 10.7477 0.361895
\(883\) −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 \(-0.228115\pi\)
0.754013 + 0.656859i \(0.228115\pi\)
\(888\) −5.66970 −0.190263
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) −8.46099 −0.283295
\(893\) −37.9129 −1.26871
\(894\) 30.0000 1.00335
\(895\) 0 0
\(896\) 10.4519 0.349173
\(897\) −18.3303 −0.612031
\(898\) 53.8784 1.79795
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −1.25227 −0.0417193
\(902\) 82.0871 2.73320
\(903\) 9.58258 0.318888
\(904\) 20.0780 0.667785
\(905\) 0 0
\(906\) 12.8348 0.426409
\(907\) −23.5826 −0.783047 −0.391523 0.920168i \(-0.628052\pi\)
−0.391523 + 0.920168i \(0.628052\pi\)
\(908\) 9.16515 0.304156
\(909\) −0.582576 −0.0193228
\(910\) 0 0
\(911\) −50.8258 −1.68393 −0.841966 0.539530i \(-0.818602\pi\)
−0.841966 + 0.539530i \(0.818602\pi\)
\(912\) 17.7568 0.587987
\(913\) 57.9129 1.91664
\(914\) −6.71326 −0.222055
\(915\) 0 0
\(916\) −20.7477 −0.685524
\(917\) 15.0000 0.495344
\(918\) 5.37386 0.177364
\(919\) 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
\(924\) 6.04356 0.198819
\(925\) 0 0
\(926\) −0.295834 −0.00972170
\(927\) −15.1652 −0.498089
\(928\) 6.04356 0.198390
\(929\) 15.6697 0.514106 0.257053 0.966397i \(-0.417248\pi\)
0.257053 + 0.966397i \(0.417248\pi\)
\(930\) 0 0
\(931\) −21.4955 −0.704485
\(932\) 15.9129 0.521244
\(933\) 3.00000 0.0982156
\(934\) −14.3303 −0.468902
\(935\) 0 0
\(936\) 6.49545 0.212311
\(937\) 18.5826 0.607066 0.303533 0.952821i \(-0.401834\pi\)
0.303533 + 0.952821i \(0.401834\pi\)
\(938\) 7.46099 0.243610
\(939\) 10.5826 0.345349
\(940\) 0 0
\(941\) 19.9129 0.649141 0.324571 0.945861i \(-0.394780\pi\)
0.324571 + 0.945861i \(0.394780\pi\)
\(942\) −19.2523 −0.627273
\(943\) −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 \(0.149151\pi\)
\(948\) −9.16515 −0.297670
\(949\) −18.3303 −0.595027
\(950\) 0 0
\(951\) −25.0000 −0.810681
\(952\) −4.25227 −0.137817
\(953\) 37.5826 1.21742 0.608710 0.793393i \(-0.291687\pi\)
0.608710 + 0.793393i \(0.291687\pi\)
\(954\) 0.747727 0.0242086
\(955\) 0 0
\(956\) −31.4265 −1.01641
\(957\) −5.00000 −0.161627
\(958\) 34.3303 1.10916
\(959\) 16.3303 0.527333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −32.8348 −1.05864
\(963\) −5.16515 −0.166445
\(964\) 35.4519 1.14183
\(965\) 0 0
\(966\) −7.16515 −0.230535
\(967\) −9.25227 −0.297533 −0.148767 0.988872i \(-0.547530\pi\)
−0.148767 + 0.988872i \(0.547530\pi\)
\(968\) 19.8439 0.637808
\(969\) −10.7477 −0.345267
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.20871 −0.0387695
\(973\) 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 \(0.487244\pi\)
\(978\) −13.5826 −0.434323
\(979\) 7.08712 0.226505
\(980\) 0 0
\(981\) 14.1652 0.452258
\(982\) −28.6606 −0.914597
\(983\) −55.1652 −1.75950 −0.879748 0.475441i \(-0.842288\pi\)
−0.879748 + 0.475441i \(0.842288\pi\)
\(984\) 12.9909 0.414135
\(985\) 0 0
\(986\) −5.37386 −0.171139
\(987\) −10.5826 −0.336847
\(988\) 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
\(993\) 8.33030 0.264354
\(994\) −17.1652 −0.544446
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) 0.834849 0.0264399 0.0132200 0.999913i \(-0.495792\pi\)
0.0132200 + 0.999913i \(0.495792\pi\)
\(998\) 24.0345 0.760798
\(999\) −4.00000 −0.126554
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 2175.2.a.r.1.1 2
3.2 odd 2 6525.2.a.t.1.2 2
5.2 odd 4 2175.2.c.f.349.2 4
5.3 odd 4 2175.2.c.f.349.3 4
5.4 even 2 435.2.a.f.1.2 2
15.14 odd 2 1305.2.a.m.1.1 2
20.19 odd 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 5.4 even 2
1305.2.a.m.1.1 2 15.14 odd 2
2175.2.a.r.1.1 2 1.1 even 1 trivial
2175.2.c.f.349.2 4 5.2 odd 4
2175.2.c.f.349.3 4 5.3 odd 4
6525.2.a.t.1.2 2 3.2 odd 2
6960.2.a.bw.1.1 2 20.19 odd 2