Properties

Label 1305.2.a.m.1.2
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 1.2
Root \(2.79129\) 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)
 
\(f(q)\) \(=\) \(q+2.79129 q^{2} +5.79129 q^{4} -1.00000 q^{5} +1.00000 q^{7} +10.5826 q^{8} +O(q^{10})\) \(q+2.79129 q^{2} +5.79129 q^{4} -1.00000 q^{5} +1.00000 q^{7} +10.5826 q^{8} -2.79129 q^{10} -5.00000 q^{11} +4.58258 q^{13} +2.79129 q^{14} +17.9564 q^{16} +3.00000 q^{17} -5.58258 q^{19} -5.79129 q^{20} -13.9564 q^{22} +4.00000 q^{23} +1.00000 q^{25} +12.7913 q^{26} +5.79129 q^{28} -1.00000 q^{29} +4.00000 q^{31} +28.9564 q^{32} +8.37386 q^{34} -1.00000 q^{35} -4.00000 q^{37} -15.5826 q^{38} -10.5826 q^{40} -9.16515 q^{41} -0.417424 q^{43} -28.9564 q^{44} +11.1652 q^{46} -1.41742 q^{47} -6.00000 q^{49} +2.79129 q^{50} +26.5390 q^{52} -9.58258 q^{53} +5.00000 q^{55} +10.5826 q^{56} -2.79129 q^{58} -1.58258 q^{59} -14.7477 q^{61} +11.1652 q^{62} +44.9129 q^{64} -4.58258 q^{65} +14.1652 q^{67} +17.3739 q^{68} -2.79129 q^{70} +0.417424 q^{71} +4.00000 q^{73} -11.1652 q^{74} -32.3303 q^{76} -5.00000 q^{77} -1.58258 q^{79} -17.9564 q^{80} -25.5826 q^{82} +2.41742 q^{83} -3.00000 q^{85} -1.16515 q^{86} -52.9129 q^{88} -10.5826 q^{89} +4.58258 q^{91} +23.1652 q^{92} -3.95644 q^{94} +5.58258 q^{95} +2.41742 q^{97} -16.7477 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 7 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.79129 1.97374 0.986869 0.161521i \(-0.0516399\pi\)
0.986869 + 0.161521i \(0.0516399\pi\)
\(3\) 0 0
\(4\) 5.79129 2.89564
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 10.5826 3.74151
\(9\) 0 0
\(10\) −2.79129 −0.882683
\(11\) −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 \(0.280799\pi\)
\(14\) 2.79129 0.746003
\(15\) 0 0
\(16\) 17.9564 4.48911
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −5.58258 −1.28073 −0.640365 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) −5.79129 −1.29497
\(21\) 0 0
\(22\) −13.9564 −2.97552
\(23\) 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
\(26\) 12.7913 2.50858
\(27\) 0 0
\(28\) 5.79129 1.09445
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 28.9564 5.11882
\(33\) 0 0
\(34\) 8.37386 1.43611
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −15.5826 −2.52783
\(39\) 0 0
\(40\) −10.5826 −1.67325
\(41\) −9.16515 −1.43136 −0.715678 0.698430i \(-0.753882\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) 0 0
\(43\) −0.417424 −0.0636566 −0.0318283 0.999493i \(-0.510133\pi\)
−0.0318283 + 0.999493i \(0.510133\pi\)
\(44\) −28.9564 −4.36535
\(45\) 0 0
\(46\) 11.1652 1.64621
\(47\) −1.41742 −0.206753 −0.103376 0.994642i \(-0.532965\pi\)
−0.103376 + 0.994642i \(0.532965\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 2.79129 0.394748
\(51\) 0 0
\(52\) 26.5390 3.68030
\(53\) −9.58258 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 10.5826 1.41416
\(57\) 0 0
\(58\) −2.79129 −0.366514
\(59\) −1.58258 −0.206034 −0.103017 0.994680i \(-0.532850\pi\)
−0.103017 + 0.994680i \(0.532850\pi\)
\(60\) 0 0
\(61\) −14.7477 −1.88825 −0.944126 0.329583i \(-0.893092\pi\)
−0.944126 + 0.329583i \(0.893092\pi\)
\(62\) 11.1652 1.41798
\(63\) 0 0
\(64\) 44.9129 5.61411
\(65\) −4.58258 −0.568399
\(66\) 0 0
\(67\) 14.1652 1.73055 0.865274 0.501299i \(-0.167144\pi\)
0.865274 + 0.501299i \(0.167144\pi\)
\(68\) 17.3739 2.10689
\(69\) 0 0
\(70\) −2.79129 −0.333623
\(71\) 0.417424 0.0495392 0.0247696 0.999693i \(-0.492115\pi\)
0.0247696 + 0.999693i \(0.492115\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −11.1652 −1.29792
\(75\) 0 0
\(76\) −32.3303 −3.70854
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) −1.58258 −0.178054 −0.0890268 0.996029i \(-0.528376\pi\)
−0.0890268 + 0.996029i \(0.528376\pi\)
\(80\) −17.9564 −2.00759
\(81\) 0 0
\(82\) −25.5826 −2.82512
\(83\) 2.41742 0.265347 0.132673 0.991160i \(-0.457644\pi\)
0.132673 + 0.991160i \(0.457644\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −1.16515 −0.125642
\(87\) 0 0
\(88\) −52.9129 −5.64053
\(89\) −10.5826 −1.12175 −0.560875 0.827900i \(-0.689535\pi\)
−0.560875 + 0.827900i \(0.689535\pi\)
\(90\) 0 0
\(91\) 4.58258 0.480384
\(92\) 23.1652 2.41513
\(93\) 0 0
\(94\) −3.95644 −0.408076
\(95\) 5.58258 0.572760
\(96\) 0 0
\(97\) 2.41742 0.245452 0.122726 0.992441i \(-0.460836\pi\)
0.122726 + 0.992441i \(0.460836\pi\)
\(98\) −16.7477 −1.69178
\(99\) 0 0
\(100\) 5.79129 0.579129
\(101\) −8.58258 −0.853998 −0.426999 0.904252i \(-0.640429\pi\)
−0.426999 + 0.904252i \(0.640429\pi\)
\(102\) 0 0
\(103\) −3.16515 −0.311872 −0.155936 0.987767i \(-0.549839\pi\)
−0.155936 + 0.987767i \(0.549839\pi\)
\(104\) 48.4955 4.75537
\(105\) 0 0
\(106\) −26.7477 −2.59797
\(107\) 13.1652 1.27272 0.636362 0.771391i \(-0.280439\pi\)
0.636362 + 0.771391i \(0.280439\pi\)
\(108\) 0 0
\(109\) −4.16515 −0.398949 −0.199475 0.979903i \(-0.563924\pi\)
−0.199475 + 0.979903i \(0.563924\pi\)
\(110\) 13.9564 1.33069
\(111\) 0 0
\(112\) 17.9564 1.69672
\(113\) −4.16515 −0.391824 −0.195912 0.980621i \(-0.562767\pi\)
−0.195912 + 0.980621i \(0.562767\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −5.79129 −0.537708
\(117\) 0 0
\(118\) −4.41742 −0.406657
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −41.1652 −3.72692
\(123\) 0 0
\(124\) 23.1652 2.08029
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 67.4519 5.96196
\(129\) 0 0
\(130\) −12.7913 −1.12187
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −5.58258 −0.484071
\(134\) 39.5390 3.41565
\(135\) 0 0
\(136\) 31.7477 2.72235
\(137\) 20.3303 1.73693 0.868467 0.495746i \(-0.165105\pi\)
0.868467 + 0.495746i \(0.165105\pi\)
\(138\) 0 0
\(139\) −18.5826 −1.57615 −0.788077 0.615577i \(-0.788923\pi\)
−0.788077 + 0.615577i \(0.788923\pi\)
\(140\) −5.79129 −0.489453
\(141\) 0 0
\(142\) 1.16515 0.0977773
\(143\) −22.9129 −1.91607
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 11.1652 0.924035
\(147\) 0 0
\(148\) −23.1652 −1.90416
\(149\) 10.7477 0.880488 0.440244 0.897878i \(-0.354892\pi\)
0.440244 + 0.897878i \(0.354892\pi\)
\(150\) 0 0
\(151\) −11.1652 −0.908607 −0.454304 0.890847i \(-0.650112\pi\)
−0.454304 + 0.890847i \(0.650112\pi\)
\(152\) −59.0780 −4.79186
\(153\) 0 0
\(154\) −13.9564 −1.12464
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −16.7477 −1.33661 −0.668307 0.743886i \(-0.732981\pi\)
−0.668307 + 0.743886i \(0.732981\pi\)
\(158\) −4.41742 −0.351431
\(159\) 0 0
\(160\) −28.9564 −2.28921
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −1.58258 −0.123957 −0.0619784 0.998077i \(-0.519741\pi\)
−0.0619784 + 0.998077i \(0.519741\pi\)
\(164\) −53.0780 −4.14470
\(165\) 0 0
\(166\) 6.74773 0.523725
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 8.00000 0.615385
\(170\) −8.37386 −0.642246
\(171\) 0 0
\(172\) −2.41742 −0.184327
\(173\) −15.1652 −1.15299 −0.576493 0.817102i \(-0.695579\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −89.7822 −6.76759
\(177\) 0 0
\(178\) −29.5390 −2.21404
\(179\) 22.7477 1.70024 0.850122 0.526585i \(-0.176528\pi\)
0.850122 + 0.526585i \(0.176528\pi\)
\(180\) 0 0
\(181\) −2.16515 −0.160934 −0.0804672 0.996757i \(-0.525641\pi\)
−0.0804672 + 0.996757i \(0.525641\pi\)
\(182\) 12.7913 0.948153
\(183\) 0 0
\(184\) 42.3303 3.12063
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) −8.20871 −0.598682
\(189\) 0 0
\(190\) 15.5826 1.13048
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 16.3303 1.17548 0.587740 0.809050i \(-0.300018\pi\)
0.587740 + 0.809050i \(0.300018\pi\)
\(194\) 6.74773 0.484459
\(195\) 0 0
\(196\) −34.7477 −2.48198
\(197\) −20.3303 −1.44847 −0.724237 0.689551i \(-0.757808\pi\)
−0.724237 + 0.689551i \(0.757808\pi\)
\(198\) 0 0
\(199\) 22.5826 1.60084 0.800418 0.599442i \(-0.204611\pi\)
0.800418 + 0.599442i \(0.204611\pi\)
\(200\) 10.5826 0.748301
\(201\) 0 0
\(202\) −23.9564 −1.68557
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 9.16515 0.640122
\(206\) −8.83485 −0.615553
\(207\) 0 0
\(208\) 82.2867 5.70556
\(209\) 27.9129 1.93077
\(210\) 0 0
\(211\) −16.3303 −1.12422 −0.562112 0.827061i \(-0.690011\pi\)
−0.562112 + 0.827061i \(0.690011\pi\)
\(212\) −55.4955 −3.81144
\(213\) 0 0
\(214\) 36.7477 2.51202
\(215\) 0.417424 0.0284681
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −11.6261 −0.787421
\(219\) 0 0
\(220\) 28.9564 1.95224
\(221\) 13.7477 0.924772
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 28.9564 1.93473
\(225\) 0 0
\(226\) −11.6261 −0.773359
\(227\) −1.58258 −0.105039 −0.0525196 0.998620i \(-0.516725\pi\)
−0.0525196 + 0.998620i \(0.516725\pi\)
\(228\) 0 0
\(229\) 1.16515 0.0769954 0.0384977 0.999259i \(-0.487743\pi\)
0.0384977 + 0.999259i \(0.487743\pi\)
\(230\) −11.1652 −0.736208
\(231\) 0 0
\(232\) −10.5826 −0.694780
\(233\) −5.16515 −0.338380 −0.169190 0.985583i \(-0.554115\pi\)
−0.169190 + 0.985583i \(0.554115\pi\)
\(234\) 0 0
\(235\) 1.41742 0.0924626
\(236\) −9.16515 −0.596601
\(237\) 0 0
\(238\) 8.37386 0.542797
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −7.33030 −0.472186 −0.236093 0.971730i \(-0.575867\pi\)
−0.236093 + 0.971730i \(0.575867\pi\)
\(242\) 39.0780 2.51203
\(243\) 0 0
\(244\) −85.4083 −5.46771
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −25.5826 −1.62778
\(248\) 42.3303 2.68798
\(249\) 0 0
\(250\) −2.79129 −0.176537
\(251\) 2.16515 0.136663 0.0683316 0.997663i \(-0.478232\pi\)
0.0683316 + 0.997663i \(0.478232\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 5.58258 0.350282
\(255\) 0 0
\(256\) 98.4519 6.15324
\(257\) 14.7477 0.919938 0.459969 0.887935i \(-0.347861\pi\)
0.459969 + 0.887935i \(0.347861\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −26.5390 −1.64588
\(261\) 0 0
\(262\) 41.8693 2.58670
\(263\) 6.33030 0.390343 0.195172 0.980769i \(-0.437474\pi\)
0.195172 + 0.980769i \(0.437474\pi\)
\(264\) 0 0
\(265\) 9.58258 0.588653
\(266\) −15.5826 −0.955429
\(267\) 0 0
\(268\) 82.0345 5.01105
\(269\) −13.4174 −0.818075 −0.409037 0.912518i \(-0.634135\pi\)
−0.409037 + 0.912518i \(0.634135\pi\)
\(270\) 0 0
\(271\) −17.1652 −1.04271 −0.521354 0.853340i \(-0.674573\pi\)
−0.521354 + 0.853340i \(0.674573\pi\)
\(272\) 53.8693 3.26631
\(273\) 0 0
\(274\) 56.7477 3.42826
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) 20.9129 1.25653 0.628267 0.777998i \(-0.283765\pi\)
0.628267 + 0.777998i \(0.283765\pi\)
\(278\) −51.8693 −3.11091
\(279\) 0 0
\(280\) −10.5826 −0.632430
\(281\) 9.58258 0.571649 0.285824 0.958282i \(-0.407733\pi\)
0.285824 + 0.958282i \(0.407733\pi\)
\(282\) 0 0
\(283\) −19.1652 −1.13925 −0.569625 0.821905i \(-0.692912\pi\)
−0.569625 + 0.821905i \(0.692912\pi\)
\(284\) 2.41742 0.143448
\(285\) 0 0
\(286\) −63.9564 −3.78182
\(287\) −9.16515 −0.541002
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 2.79129 0.163910
\(291\) 0 0
\(292\) 23.1652 1.35564
\(293\) −11.8348 −0.691399 −0.345700 0.938345i \(-0.612358\pi\)
−0.345700 + 0.938345i \(0.612358\pi\)
\(294\) 0 0
\(295\) 1.58258 0.0921411
\(296\) −42.3303 −2.46040
\(297\) 0 0
\(298\) 30.0000 1.73785
\(299\) 18.3303 1.06007
\(300\) 0 0
\(301\) −0.417424 −0.0240599
\(302\) −31.1652 −1.79335
\(303\) 0 0
\(304\) −100.243 −5.74934
\(305\) 14.7477 0.844452
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −28.9564 −1.64995
\(309\) 0 0
\(310\) −11.1652 −0.634138
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 1.41742 0.0801176 0.0400588 0.999197i \(-0.487245\pi\)
0.0400588 + 0.999197i \(0.487245\pi\)
\(314\) −46.7477 −2.63813
\(315\) 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
\(320\) −44.9129 −2.51071
\(321\) 0 0
\(322\) 11.1652 0.622210
\(323\) −16.7477 −0.931868
\(324\) 0 0
\(325\) 4.58258 0.254196
\(326\) −4.41742 −0.244659
\(327\) 0 0
\(328\) −96.9909 −5.35543
\(329\) −1.41742 −0.0781451
\(330\) 0 0
\(331\) 28.3303 1.55717 0.778587 0.627537i \(-0.215937\pi\)
0.778587 + 0.627537i \(0.215937\pi\)
\(332\) 14.0000 0.768350
\(333\) 0 0
\(334\) 0 0
\(335\) −14.1652 −0.773925
\(336\) 0 0
\(337\) −2.83485 −0.154424 −0.0772120 0.997015i \(-0.524602\pi\)
−0.0772120 + 0.997015i \(0.524602\pi\)
\(338\) 22.3303 1.21461
\(339\) 0 0
\(340\) −17.3739 −0.942230
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −4.41742 −0.238172
\(345\) 0 0
\(346\) −42.3303 −2.27569
\(347\) 11.1652 0.599377 0.299688 0.954037i \(-0.403117\pi\)
0.299688 + 0.954037i \(0.403117\pi\)
\(348\) 0 0
\(349\) 32.3303 1.73060 0.865301 0.501253i \(-0.167127\pi\)
0.865301 + 0.501253i \(0.167127\pi\)
\(350\) 2.79129 0.149201
\(351\) 0 0
\(352\) −144.782 −7.71692
\(353\) 33.1652 1.76520 0.882601 0.470122i \(-0.155790\pi\)
0.882601 + 0.470122i \(0.155790\pi\)
\(354\) 0 0
\(355\) −0.417424 −0.0221546
\(356\) −61.2867 −3.24819
\(357\) 0 0
\(358\) 63.4955 3.35584
\(359\) 8.83485 0.466285 0.233143 0.972443i \(-0.425099\pi\)
0.233143 + 0.972443i \(0.425099\pi\)
\(360\) 0 0
\(361\) 12.1652 0.640271
\(362\) −6.04356 −0.317643
\(363\) 0 0
\(364\) 26.5390 1.39102
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 17.4955 0.913255 0.456628 0.889658i \(-0.349057\pi\)
0.456628 + 0.889658i \(0.349057\pi\)
\(368\) 71.8258 3.74418
\(369\) 0 0
\(370\) 11.1652 0.580449
\(371\) −9.58258 −0.497503
\(372\) 0 0
\(373\) −8.33030 −0.431327 −0.215663 0.976468i \(-0.569191\pi\)
−0.215663 + 0.976468i \(0.569191\pi\)
\(374\) −41.8693 −2.16501
\(375\) 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 \(0.732749\pi\)
\(380\) 32.3303 1.65851
\(381\) 0 0
\(382\) 11.1652 0.571259
\(383\) 5.58258 0.285256 0.142628 0.989776i \(-0.454445\pi\)
0.142628 + 0.989776i \(0.454445\pi\)
\(384\) 0 0
\(385\) 5.00000 0.254824
\(386\) 45.5826 2.32009
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −2.58258 −0.130942 −0.0654709 0.997854i \(-0.520855\pi\)
−0.0654709 + 0.997854i \(0.520855\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) −63.4955 −3.20700
\(393\) 0 0
\(394\) −56.7477 −2.85891
\(395\) 1.58258 0.0796280
\(396\) 0 0
\(397\) 29.1652 1.46376 0.731878 0.681435i \(-0.238644\pi\)
0.731878 + 0.681435i \(0.238644\pi\)
\(398\) 63.0345 3.15963
\(399\) 0 0
\(400\) 17.9564 0.897822
\(401\) −21.5826 −1.07778 −0.538891 0.842375i \(-0.681157\pi\)
−0.538891 + 0.842375i \(0.681157\pi\)
\(402\) 0 0
\(403\) 18.3303 0.913097
\(404\) −49.7042 −2.47287
\(405\) 0 0
\(406\) −2.79129 −0.138529
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 24.7477 1.22370 0.611848 0.790975i \(-0.290426\pi\)
0.611848 + 0.790975i \(0.290426\pi\)
\(410\) 25.5826 1.26343
\(411\) 0 0
\(412\) −18.3303 −0.903069
\(413\) −1.58258 −0.0778735
\(414\) 0 0
\(415\) −2.41742 −0.118667
\(416\) 132.695 6.50591
\(417\) 0 0
\(418\) 77.9129 3.81084
\(419\) 18.8348 0.920143 0.460071 0.887882i \(-0.347824\pi\)
0.460071 + 0.887882i \(0.347824\pi\)
\(420\) 0 0
\(421\) 13.5826 0.661974 0.330987 0.943635i \(-0.392618\pi\)
0.330987 + 0.943635i \(0.392618\pi\)
\(422\) −45.5826 −2.21893
\(423\) 0 0
\(424\) −101.408 −4.92482
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) −14.7477 −0.713693
\(428\) 76.2432 3.68535
\(429\) 0 0
\(430\) 1.16515 0.0561886
\(431\) 20.8348 1.00358 0.501790 0.864990i \(-0.332675\pi\)
0.501790 + 0.864990i \(0.332675\pi\)
\(432\) 0 0
\(433\) −39.0780 −1.87797 −0.938985 0.343958i \(-0.888232\pi\)
−0.938985 + 0.343958i \(0.888232\pi\)
\(434\) 11.1652 0.535944
\(435\) 0 0
\(436\) −24.1216 −1.15521
\(437\) −22.3303 −1.06820
\(438\) 0 0
\(439\) 28.9129 1.37994 0.689968 0.723840i \(-0.257624\pi\)
0.689968 + 0.723840i \(0.257624\pi\)
\(440\) 52.9129 2.52252
\(441\) 0 0
\(442\) 38.3739 1.82526
\(443\) 8.58258 0.407770 0.203885 0.978995i \(-0.434643\pi\)
0.203885 + 0.978995i \(0.434643\pi\)
\(444\) 0 0
\(445\) 10.5826 0.501662
\(446\) 19.5390 0.925199
\(447\) 0 0
\(448\) 44.9129 2.12193
\(449\) −34.0780 −1.60824 −0.804121 0.594466i \(-0.797364\pi\)
−0.804121 + 0.594466i \(0.797364\pi\)
\(450\) 0 0
\(451\) 45.8258 2.15785
\(452\) −24.1216 −1.13458
\(453\) 0 0
\(454\) −4.41742 −0.207320
\(455\) −4.58258 −0.214834
\(456\) 0 0
\(457\) 23.7477 1.11087 0.555436 0.831559i \(-0.312551\pi\)
0.555436 + 0.831559i \(0.312551\pi\)
\(458\) 3.25227 0.151969
\(459\) 0 0
\(460\) −23.1652 −1.08008
\(461\) −9.16515 −0.426864 −0.213432 0.976958i \(-0.568464\pi\)
−0.213432 + 0.976958i \(0.568464\pi\)
\(462\) 0 0
\(463\) 18.1652 0.844206 0.422103 0.906548i \(-0.361292\pi\)
0.422103 + 0.906548i \(0.361292\pi\)
\(464\) −17.9564 −0.833607
\(465\) 0 0
\(466\) −14.4174 −0.667874
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 14.1652 0.654086
\(470\) 3.95644 0.182497
\(471\) 0 0
\(472\) −16.7477 −0.770877
\(473\) 2.08712 0.0959659
\(474\) 0 0
\(475\) −5.58258 −0.256146
\(476\) 17.3739 0.796330
\(477\) 0 0
\(478\) 72.5735 3.31943
\(479\) 0.834849 0.0381452 0.0190726 0.999818i \(-0.493929\pi\)
0.0190726 + 0.999818i \(0.493929\pi\)
\(480\) 0 0
\(481\) −18.3303 −0.835790
\(482\) −20.4610 −0.931972
\(483\) 0 0
\(484\) 81.0780 3.68536
\(485\) −2.41742 −0.109770
\(486\) 0 0
\(487\) −34.3303 −1.55565 −0.777827 0.628478i \(-0.783678\pi\)
−0.777827 + 0.628478i \(0.783678\pi\)
\(488\) −156.069 −7.06491
\(489\) 0 0
\(490\) 16.7477 0.756585
\(491\) −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
\(494\) −71.4083 −3.21281
\(495\) 0 0
\(496\) 71.8258 3.22507
\(497\) 0.417424 0.0187240
\(498\) 0 0
\(499\) −22.5826 −1.01093 −0.505467 0.862846i \(-0.668680\pi\)
−0.505467 + 0.862846i \(0.668680\pi\)
\(500\) −5.79129 −0.258994
\(501\) 0 0
\(502\) 6.04356 0.269737
\(503\) −22.9129 −1.02163 −0.510817 0.859689i \(-0.670657\pi\)
−0.510817 + 0.859689i \(0.670657\pi\)
\(504\) 0 0
\(505\) 8.58258 0.381920
\(506\) −55.8258 −2.48176
\(507\) 0 0
\(508\) 11.5826 0.513894
\(509\) 0.747727 0.0331424 0.0165712 0.999863i \(-0.494725\pi\)
0.0165712 + 0.999863i \(0.494725\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 139.904 6.18293
\(513\) 0 0
\(514\) 41.1652 1.81572
\(515\) 3.16515 0.139473
\(516\) 0 0
\(517\) 7.08712 0.311691
\(518\) −11.1652 −0.490569
\(519\) 0 0
\(520\) −48.4955 −2.12667
\(521\) 21.0780 0.923445 0.461723 0.887024i \(-0.347232\pi\)
0.461723 + 0.887024i \(0.347232\pi\)
\(522\) 0 0
\(523\) 3.33030 0.145624 0.0728120 0.997346i \(-0.476803\pi\)
0.0728120 + 0.997346i \(0.476803\pi\)
\(524\) 86.8693 3.79490
\(525\) 0 0
\(526\) 17.6697 0.770435
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 26.7477 1.16185
\(531\) 0 0
\(532\) −32.3303 −1.40170
\(533\) −42.0000 −1.81922
\(534\) 0 0
\(535\) −13.1652 −0.569179
\(536\) 149.904 6.47486
\(537\) 0 0
\(538\) −37.4519 −1.61467
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 23.4955 1.01015 0.505074 0.863076i \(-0.331465\pi\)
0.505074 + 0.863076i \(0.331465\pi\)
\(542\) −47.9129 −2.05803
\(543\) 0 0
\(544\) 86.8693 3.72449
\(545\) 4.16515 0.178415
\(546\) 0 0
\(547\) −14.1652 −0.605658 −0.302829 0.953045i \(-0.597931\pi\)
−0.302829 + 0.953045i \(0.597931\pi\)
\(548\) 117.739 5.02955
\(549\) 0 0
\(550\) −13.9564 −0.595105
\(551\) 5.58258 0.237826
\(552\) 0 0
\(553\) −1.58258 −0.0672980
\(554\) 58.3739 2.48007
\(555\) 0 0
\(556\) −107.617 −4.56398
\(557\) −4.74773 −0.201168 −0.100584 0.994929i \(-0.532071\pi\)
−0.100584 + 0.994929i \(0.532071\pi\)
\(558\) 0 0
\(559\) −1.91288 −0.0809061
\(560\) −17.9564 −0.758798
\(561\) 0 0
\(562\) 26.7477 1.12828
\(563\) 5.41742 0.228317 0.114159 0.993463i \(-0.463583\pi\)
0.114159 + 0.993463i \(0.463583\pi\)
\(564\) 0 0
\(565\) 4.16515 0.175229
\(566\) −53.4955 −2.24858
\(567\) 0 0
\(568\) 4.41742 0.185351
\(569\) −19.4174 −0.814021 −0.407010 0.913424i \(-0.633429\pi\)
−0.407010 + 0.913424i \(0.633429\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −132.695 −5.54826
\(573\) 0 0
\(574\) −25.5826 −1.06780
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −0.834849 −0.0347552 −0.0173776 0.999849i \(-0.505532\pi\)
−0.0173776 + 0.999849i \(0.505532\pi\)
\(578\) −22.3303 −0.928818
\(579\) 0 0
\(580\) 5.79129 0.240470
\(581\) 2.41742 0.100292
\(582\) 0 0
\(583\) 47.9129 1.98435
\(584\) 42.3303 1.75164
\(585\) 0 0
\(586\) −33.0345 −1.36464
\(587\) 2.41742 0.0997778 0.0498889 0.998755i \(-0.484113\pi\)
0.0498889 + 0.998755i \(0.484113\pi\)
\(588\) 0 0
\(589\) −22.3303 −0.920104
\(590\) 4.41742 0.181862
\(591\) 0 0
\(592\) −71.8258 −2.95202
\(593\) −17.5826 −0.722030 −0.361015 0.932560i \(-0.617570\pi\)
−0.361015 + 0.932560i \(0.617570\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 62.2432 2.54958
\(597\) 0 0
\(598\) 51.1652 2.09230
\(599\) 18.1652 0.742208 0.371104 0.928591i \(-0.378979\pi\)
0.371104 + 0.928591i \(0.378979\pi\)
\(600\) 0 0
\(601\) 4.33030 0.176637 0.0883184 0.996092i \(-0.471851\pi\)
0.0883184 + 0.996092i \(0.471851\pi\)
\(602\) −1.16515 −0.0474880
\(603\) 0 0
\(604\) −64.6606 −2.63100
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) −5.58258 −0.226590 −0.113295 0.993561i \(-0.536140\pi\)
−0.113295 + 0.993561i \(0.536140\pi\)
\(608\) −161.652 −6.55583
\(609\) 0 0
\(610\) 41.1652 1.66673
\(611\) −6.49545 −0.262778
\(612\) 0 0
\(613\) −16.2523 −0.656423 −0.328212 0.944604i \(-0.606446\pi\)
−0.328212 + 0.944604i \(0.606446\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −52.9129 −2.13192
\(617\) 39.4955 1.59003 0.795014 0.606592i \(-0.207464\pi\)
0.795014 + 0.606592i \(0.207464\pi\)
\(618\) 0 0
\(619\) −5.16515 −0.207605 −0.103802 0.994598i \(-0.533101\pi\)
−0.103802 + 0.994598i \(0.533101\pi\)
\(620\) −23.1652 −0.930335
\(621\) 0 0
\(622\) 8.37386 0.335761
\(623\) −10.5826 −0.423982
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.95644 0.158131
\(627\) 0 0
\(628\) −96.9909 −3.87036
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −34.9129 −1.38986 −0.694930 0.719078i \(-0.744565\pi\)
−0.694930 + 0.719078i \(0.744565\pi\)
\(632\) −16.7477 −0.666189
\(633\) 0 0
\(634\) 69.7822 2.77141
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −27.4955 −1.08941
\(638\) 13.9564 0.552541
\(639\) 0 0
\(640\) −67.4519 −2.66627
\(641\) −2.58258 −0.102006 −0.0510028 0.998699i \(-0.516242\pi\)
−0.0510028 + 0.998699i \(0.516242\pi\)
\(642\) 0 0
\(643\) −29.6606 −1.16970 −0.584850 0.811141i \(-0.698847\pi\)
−0.584850 + 0.811141i \(0.698847\pi\)
\(644\) 23.1652 0.912835
\(645\) 0 0
\(646\) −46.7477 −1.83926
\(647\) 2.74773 0.108024 0.0540121 0.998540i \(-0.482799\pi\)
0.0540121 + 0.998540i \(0.482799\pi\)
\(648\) 0 0
\(649\) 7.91288 0.310608
\(650\) 12.7913 0.501716
\(651\) 0 0
\(652\) −9.16515 −0.358935
\(653\) −7.83485 −0.306601 −0.153301 0.988180i \(-0.548990\pi\)
−0.153301 + 0.988180i \(0.548990\pi\)
\(654\) 0 0
\(655\) −15.0000 −0.586098
\(656\) −164.573 −6.42552
\(657\) 0 0
\(658\) −3.95644 −0.154238
\(659\) 0.165151 0.00643338 0.00321669 0.999995i \(-0.498976\pi\)
0.00321669 + 0.999995i \(0.498976\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 79.0780 3.07345
\(663\) 0 0
\(664\) 25.5826 0.992796
\(665\) 5.58258 0.216483
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 0 0
\(669\) 0 0
\(670\) −39.5390 −1.52753
\(671\) 73.7386 2.84665
\(672\) 0 0
\(673\) −25.7477 −0.992502 −0.496251 0.868179i \(-0.665290\pi\)
−0.496251 + 0.868179i \(0.665290\pi\)
\(674\) −7.91288 −0.304793
\(675\) 0 0
\(676\) 46.3303 1.78193
\(677\) 46.8258 1.79966 0.899830 0.436241i \(-0.143690\pi\)
0.899830 + 0.436241i \(0.143690\pi\)
\(678\) 0 0
\(679\) 2.41742 0.0927722
\(680\) −31.7477 −1.21747
\(681\) 0 0
\(682\) −55.8258 −2.13768
\(683\) 11.1652 0.427223 0.213611 0.976919i \(-0.431477\pi\)
0.213611 + 0.976919i \(0.431477\pi\)
\(684\) 0 0
\(685\) −20.3303 −0.776781
\(686\) −36.2867 −1.38543
\(687\) 0 0
\(688\) −7.49545 −0.285762
\(689\) −43.9129 −1.67295
\(690\) 0 0
\(691\) −2.91288 −0.110811 −0.0554056 0.998464i \(-0.517645\pi\)
−0.0554056 + 0.998464i \(0.517645\pi\)
\(692\) −87.8258 −3.33863
\(693\) 0 0
\(694\) 31.1652 1.18301
\(695\) 18.5826 0.704877
\(696\) 0 0
\(697\) −27.4955 −1.04146
\(698\) 90.2432 3.41575
\(699\) 0 0
\(700\) 5.79129 0.218890
\(701\) −41.0780 −1.55150 −0.775748 0.631043i \(-0.782627\pi\)
−0.775748 + 0.631043i \(0.782627\pi\)
\(702\) 0 0
\(703\) 22.3303 0.842203
\(704\) −224.564 −8.46359
\(705\) 0 0
\(706\) 92.5735 3.48405
\(707\) −8.58258 −0.322781
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −1.16515 −0.0437274
\(711\) 0 0
\(712\) −111.991 −4.19704
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 22.9129 0.856893
\(716\) 131.739 4.92330
\(717\) 0 0
\(718\) 24.6606 0.920326
\(719\) 37.9129 1.41391 0.706956 0.707258i \(-0.250068\pi\)
0.706956 + 0.707258i \(0.250068\pi\)
\(720\) 0 0
\(721\) −3.16515 −0.117876
\(722\) 33.9564 1.26373
\(723\) 0 0
\(724\) −12.5390 −0.466009
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 46.3303 1.71830 0.859148 0.511727i \(-0.170994\pi\)
0.859148 + 0.511727i \(0.170994\pi\)
\(728\) 48.4955 1.79736
\(729\) 0 0
\(730\) −11.1652 −0.413241
\(731\) −1.25227 −0.0463170
\(732\) 0 0
\(733\) −47.5826 −1.75750 −0.878751 0.477280i \(-0.841623\pi\)
−0.878751 + 0.477280i \(0.841623\pi\)
\(734\) 48.8348 1.80253
\(735\) 0 0
\(736\) 115.826 4.26939
\(737\) −70.8258 −2.60890
\(738\) 0 0
\(739\) 46.7477 1.71964 0.859821 0.510595i \(-0.170575\pi\)
0.859821 + 0.510595i \(0.170575\pi\)
\(740\) 23.1652 0.851568
\(741\) 0 0
\(742\) −26.7477 −0.981940
\(743\) −2.25227 −0.0826279 −0.0413139 0.999146i \(-0.513154\pi\)
−0.0413139 + 0.999146i \(0.513154\pi\)
\(744\) 0 0
\(745\) −10.7477 −0.393766
\(746\) −23.2523 −0.851326
\(747\) 0 0
\(748\) −86.8693 −3.17626
\(749\) 13.1652 0.481044
\(750\) 0 0
\(751\) 37.4955 1.36823 0.684114 0.729375i \(-0.260189\pi\)
0.684114 + 0.729375i \(0.260189\pi\)
\(752\) −25.4519 −0.928135
\(753\) 0 0
\(754\) −12.7913 −0.465831
\(755\) 11.1652 0.406341
\(756\) 0 0
\(757\) 34.3303 1.24776 0.623878 0.781522i \(-0.285556\pi\)
0.623878 + 0.781522i \(0.285556\pi\)
\(758\) −72.5735 −2.63599
\(759\) 0 0
\(760\) 59.0780 2.14299
\(761\) −45.5826 −1.65237 −0.826184 0.563401i \(-0.809493\pi\)
−0.826184 + 0.563401i \(0.809493\pi\)
\(762\) 0 0
\(763\) −4.16515 −0.150789
\(764\) 23.1652 0.838086
\(765\) 0 0
\(766\) 15.5826 0.563021
\(767\) −7.25227 −0.261864
\(768\) 0 0
\(769\) 16.4174 0.592027 0.296014 0.955184i \(-0.404343\pi\)
0.296014 + 0.955184i \(0.404343\pi\)
\(770\) 13.9564 0.502955
\(771\) 0 0
\(772\) 94.5735 3.40377
\(773\) −13.1652 −0.473518 −0.236759 0.971568i \(-0.576085\pi\)
−0.236759 + 0.971568i \(0.576085\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 25.5826 0.918361
\(777\) 0 0
\(778\) −7.20871 −0.258445
\(779\) 51.1652 1.83318
\(780\) 0 0
\(781\) −2.08712 −0.0746831
\(782\) 33.4955 1.19779
\(783\) 0 0
\(784\) −107.739 −3.84781
\(785\) 16.7477 0.597752
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −117.739 −4.19427
\(789\) 0 0
\(790\) 4.41742 0.157165
\(791\) −4.16515 −0.148096
\(792\) 0 0
\(793\) −67.5826 −2.39993
\(794\) 81.4083 2.88907
\(795\) 0 0
\(796\) 130.782 4.63545
\(797\) 4.33030 0.153387 0.0766936 0.997055i \(-0.475564\pi\)
0.0766936 + 0.997055i \(0.475564\pi\)
\(798\) 0 0
\(799\) −4.25227 −0.150435
\(800\) 28.9564 1.02376
\(801\) 0 0
\(802\) −60.2432 −2.12726
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 51.1652 1.80222
\(807\) 0 0
\(808\) −90.8258 −3.19524
\(809\) 20.0780 0.705906 0.352953 0.935641i \(-0.385178\pi\)
0.352953 + 0.935641i \(0.385178\pi\)
\(810\) 0 0
\(811\) 23.4174 0.822297 0.411148 0.911568i \(-0.365128\pi\)
0.411148 + 0.911568i \(0.365128\pi\)
\(812\) −5.79129 −0.203234
\(813\) 0 0
\(814\) 55.8258 1.95669
\(815\) 1.58258 0.0554352
\(816\) 0 0
\(817\) 2.33030 0.0815270
\(818\) 69.0780 2.41526
\(819\) 0 0
\(820\) 53.0780 1.85357
\(821\) −23.4955 −0.819997 −0.409999 0.912086i \(-0.634471\pi\)
−0.409999 + 0.912086i \(0.634471\pi\)
\(822\) 0 0
\(823\) −13.0780 −0.455871 −0.227936 0.973676i \(-0.573198\pi\)
−0.227936 + 0.973676i \(0.573198\pi\)
\(824\) −33.4955 −1.16687
\(825\) 0 0
\(826\) −4.41742 −0.153702
\(827\) −47.8258 −1.66306 −0.831532 0.555476i \(-0.812536\pi\)
−0.831532 + 0.555476i \(0.812536\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) −6.74773 −0.234217
\(831\) 0 0
\(832\) 205.817 7.13541
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 161.652 5.59083
\(837\) 0 0
\(838\) 52.5735 1.81612
\(839\) −6.49545 −0.224248 −0.112124 0.993694i \(-0.535765\pi\)
−0.112124 + 0.993694i \(0.535765\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.9129 1.30656
\(843\) 0 0
\(844\) −94.5735 −3.25535
\(845\) −8.00000 −0.275208
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) −172.069 −5.90887
\(849\) 0 0
\(850\) 8.37386 0.287221
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −6.74773 −0.231038 −0.115519 0.993305i \(-0.536853\pi\)
−0.115519 + 0.993305i \(0.536853\pi\)
\(854\) −41.1652 −1.40864
\(855\) 0 0
\(856\) 139.321 4.76190
\(857\) −26.6606 −0.910709 −0.455354 0.890310i \(-0.650487\pi\)
−0.455354 + 0.890310i \(0.650487\pi\)
\(858\) 0 0
\(859\) −38.4174 −1.31079 −0.655393 0.755288i \(-0.727497\pi\)
−0.655393 + 0.755288i \(0.727497\pi\)
\(860\) 2.41742 0.0824335
\(861\) 0 0
\(862\) 58.1561 1.98080
\(863\) 15.5826 0.530437 0.265219 0.964188i \(-0.414556\pi\)
0.265219 + 0.964188i \(0.414556\pi\)
\(864\) 0 0
\(865\) 15.1652 0.515631
\(866\) −109.078 −3.70662
\(867\) 0 0
\(868\) 23.1652 0.786276
\(869\) 7.91288 0.268426
\(870\) 0 0
\(871\) 64.9129 2.19949
\(872\) −44.0780 −1.49267
\(873\) 0 0
\(874\) −62.3303 −2.10835
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 56.3303 1.90214 0.951070 0.308977i \(-0.0999865\pi\)
0.951070 + 0.308977i \(0.0999865\pi\)
\(878\) 80.7042 2.72363
\(879\) 0 0
\(880\) 89.7822 3.02656
\(881\) −32.0780 −1.08074 −0.540368 0.841429i \(-0.681715\pi\)
−0.540368 + 0.841429i \(0.681715\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 79.6170 2.67781
\(885\) 0 0
\(886\) 23.9564 0.804832
\(887\) −0.912878 −0.0306515 −0.0153257 0.999883i \(-0.504879\pi\)
−0.0153257 + 0.999883i \(0.504879\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 29.5390 0.990150
\(891\) 0 0
\(892\) 40.5390 1.35735
\(893\) 7.91288 0.264794
\(894\) 0 0
\(895\) −22.7477 −0.760373
\(896\) 67.4519 2.25341
\(897\) 0 0
\(898\) −95.1216 −3.17425
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −28.7477 −0.957726
\(902\) 127.913 4.25903
\(903\) 0 0
\(904\) −44.0780 −1.46601
\(905\) 2.16515 0.0719721
\(906\) 0 0
\(907\) 14.4174 0.478723 0.239361 0.970931i \(-0.423062\pi\)
0.239361 + 0.970931i \(0.423062\pi\)
\(908\) −9.16515 −0.304156
\(909\) 0 0
\(910\) −12.7913 −0.424027
\(911\) −40.8258 −1.35262 −0.676309 0.736618i \(-0.736422\pi\)
−0.676309 + 0.736618i \(0.736422\pi\)
\(912\) 0 0
\(913\) −12.0871 −0.400025
\(914\) 66.2867 2.19257
\(915\) 0 0
\(916\) 6.74773 0.222951
\(917\) 15.0000 0.495344
\(918\) 0 0
\(919\) −26.9129 −0.887774 −0.443887 0.896083i \(-0.646401\pi\)
−0.443887 + 0.896083i \(0.646401\pi\)
\(920\) −42.3303 −1.39559
\(921\) 0 0
\(922\) −25.5826 −0.842517
\(923\) 1.91288 0.0629632
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 50.7042 1.66624
\(927\) 0 0
\(928\) −28.9564 −0.950542
\(929\) −52.3303 −1.71690 −0.858451 0.512896i \(-0.828573\pi\)
−0.858451 + 0.512896i \(0.828573\pi\)
\(930\) 0 0
\(931\) 33.4955 1.09777
\(932\) −29.9129 −0.979829
\(933\) 0 0
\(934\) 22.3303 0.730670
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −9.41742 −0.307654 −0.153827 0.988098i \(-0.549160\pi\)
−0.153827 + 0.988098i \(0.549160\pi\)
\(938\) 39.5390 1.29099
\(939\) 0 0
\(940\) 8.20871 0.267739
\(941\) 25.9129 0.844736 0.422368 0.906425i \(-0.361199\pi\)
0.422368 + 0.906425i \(0.361199\pi\)
\(942\) 0 0
\(943\) −36.6606 −1.19383
\(944\) −28.4174 −0.924908
\(945\) 0 0
\(946\) 5.82576 0.189412
\(947\) 9.08712 0.295292 0.147646 0.989040i \(-0.452830\pi\)
0.147646 + 0.989040i \(0.452830\pi\)
\(948\) 0 0
\(949\) 18.3303 0.595027
\(950\) −15.5826 −0.505566
\(951\) 0 0
\(952\) 31.7477 1.02895
\(953\) 28.4174 0.920531 0.460265 0.887781i \(-0.347754\pi\)
0.460265 + 0.887781i \(0.347754\pi\)
\(954\) 0 0
\(955\) −4.00000 −0.129437
\(956\) 150.573 4.86989
\(957\) 0 0
\(958\) 2.33030 0.0752887
\(959\) 20.3303 0.656500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −51.1652 −1.64963
\(963\) 0 0
\(964\) −42.4519 −1.36728
\(965\) −16.3303 −0.525691
\(966\) 0 0
\(967\) 36.7477 1.18173 0.590864 0.806771i \(-0.298787\pi\)
0.590864 + 0.806771i \(0.298787\pi\)
\(968\) 148.156 4.76192
\(969\) 0 0
\(970\) −6.74773 −0.216656
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) −18.5826 −0.595730
\(974\) −95.8258 −3.07046
\(975\) 0 0
\(976\) −264.817 −8.47657
\(977\) 57.4955 1.83944 0.919721 0.392572i \(-0.128415\pi\)
0.919721 + 0.392572i \(0.128415\pi\)
\(978\) 0 0
\(979\) 52.9129 1.69110
\(980\) 34.7477 1.10998
\(981\) 0 0
\(982\) −44.6606 −1.42518
\(983\) −36.8348 −1.17485 −0.587425 0.809279i \(-0.699858\pi\)
−0.587425 + 0.809279i \(0.699858\pi\)
\(984\) 0 0
\(985\) 20.3303 0.647777
\(986\) −8.37386 −0.266678
\(987\) 0 0
\(988\) −148.156 −4.71347
\(989\) −1.66970 −0.0530933
\(990\) 0 0
\(991\) −48.0780 −1.52725 −0.763624 0.645661i \(-0.776582\pi\)
−0.763624 + 0.645661i \(0.776582\pi\)
\(992\) 115.826 3.67747
\(993\) 0 0
\(994\) 1.16515 0.0369564
\(995\) −22.5826 −0.715916
\(996\) 0 0
\(997\) −19.1652 −0.606966 −0.303483 0.952837i \(-0.598150\pi\)
−0.303483 + 0.952837i \(0.598150\pi\)
\(998\) −63.0345 −1.99532
\(999\) 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.1.2 2
3.2 odd 2 435.2.a.f.1.1 2
5.4 even 2 6525.2.a.t.1.1 2
12.11 even 2 6960.2.a.bw.1.2 2
15.2 even 4 2175.2.c.f.349.1 4
15.8 even 4 2175.2.c.f.349.4 4
15.14 odd 2 2175.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.1 2 3.2 odd 2
1305.2.a.m.1.2 2 1.1 even 1 trivial
2175.2.a.r.1.2 2 15.14 odd 2
2175.2.c.f.349.1 4 15.2 even 4
2175.2.c.f.349.4 4 15.8 even 4
6525.2.a.t.1.1 2 5.4 even 2
6960.2.a.bw.1.2 2 12.11 even 2