Properties

Label 475.2.b.b.324.6
Level $475$
Weight $2$
Character 475.324
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.6
Root \(-0.273891i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.b.324.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.37720i q^{2} -1.27389i q^{3} -3.65109 q^{4} +3.02830 q^{6} +0.726109i q^{7} -3.92498i q^{8} +1.37720 q^{9} +O(q^{10})\) \(q+2.37720i q^{2} -1.27389i q^{3} -3.65109 q^{4} +3.02830 q^{6} +0.726109i q^{7} -3.92498i q^{8} +1.37720 q^{9} -0.273891 q^{11} +4.65109i q^{12} +5.95328i q^{13} -1.72611 q^{14} +2.02830 q^{16} +5.27389i q^{17} +3.27389i q^{18} -1.00000 q^{19} +0.924984 q^{21} -0.651093i q^{22} +3.67939i q^{23} -5.00000 q^{24} -14.1522 q^{26} -5.57608i q^{27} -2.65109i q^{28} +2.27389 q^{29} +3.19887 q^{31} -3.02830i q^{32} +0.348907i q^{33} -12.5371 q^{34} -5.02830 q^{36} +8.12386i q^{37} -2.37720i q^{38} +7.58383 q^{39} -9.43380 q^{41} +2.19887i q^{42} -9.81100i q^{43} +1.00000 q^{44} -8.74666 q^{46} +12.1599i q^{47} -2.58383i q^{48} +6.47277 q^{49} +6.71836 q^{51} -21.7360i q^{52} -5.69781i q^{53} +13.2555 q^{54} +2.84997 q^{56} +1.27389i q^{57} +5.40550i q^{58} +4.20662 q^{59} -0.103312 q^{61} +7.60437i q^{62} +1.00000i q^{63} +11.2555 q^{64} -0.829422 q^{66} -11.7827i q^{67} -19.2555i q^{68} +4.68714 q^{69} +5.75441 q^{71} -5.40550i q^{72} -6.67939i q^{73} -19.3121 q^{74} +3.65109 q^{76} -0.198875i q^{77} +18.0283i q^{78} -3.87826 q^{79} -2.97170 q^{81} -22.4260i q^{82} +0.488265i q^{83} -3.37720 q^{84} +23.3227 q^{86} -2.89669i q^{87} +1.07502i q^{88} +16.4338 q^{89} -4.32273 q^{91} -13.4338i q^{92} -4.07502i q^{93} -28.9066 q^{94} -3.85772 q^{96} -4.44447i q^{97} +15.3871i q^{98} -0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37720i 1.68094i 0.541861 + 0.840468i \(0.317720\pi\)
−0.541861 + 0.840468i \(0.682280\pi\)
\(3\) − 1.27389i − 0.735481i −0.929928 0.367741i \(-0.880131\pi\)
0.929928 0.367741i \(-0.119869\pi\)
\(4\) −3.65109 −1.82555
\(5\) 0 0
\(6\) 3.02830 1.23630
\(7\) 0.726109i 0.274444i 0.990540 + 0.137222i \(0.0438173\pi\)
−0.990540 + 0.137222i \(0.956183\pi\)
\(8\) − 3.92498i − 1.38769i
\(9\) 1.37720 0.459068
\(10\) 0 0
\(11\) −0.273891 −0.0825811 −0.0412906 0.999147i \(-0.513147\pi\)
−0.0412906 + 0.999147i \(0.513147\pi\)
\(12\) 4.65109i 1.34266i
\(13\) 5.95328i 1.65114i 0.564298 + 0.825571i \(0.309147\pi\)
−0.564298 + 0.825571i \(0.690853\pi\)
\(14\) −1.72611 −0.461322
\(15\) 0 0
\(16\) 2.02830 0.507074
\(17\) 5.27389i 1.27911i 0.768747 + 0.639553i \(0.220881\pi\)
−0.768747 + 0.639553i \(0.779119\pi\)
\(18\) 3.27389i 0.771663i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.924984 0.201848
\(22\) − 0.651093i − 0.138814i
\(23\) 3.67939i 0.767206i 0.923498 + 0.383603i \(0.125317\pi\)
−0.923498 + 0.383603i \(0.874683\pi\)
\(24\) −5.00000 −1.02062
\(25\) 0 0
\(26\) −14.1522 −2.77547
\(27\) − 5.57608i − 1.07312i
\(28\) − 2.65109i − 0.501010i
\(29\) 2.27389 0.422251 0.211125 0.977459i \(-0.432287\pi\)
0.211125 + 0.977459i \(0.432287\pi\)
\(30\) 0 0
\(31\) 3.19887 0.574535 0.287267 0.957850i \(-0.407253\pi\)
0.287267 + 0.957850i \(0.407253\pi\)
\(32\) − 3.02830i − 0.535332i
\(33\) 0.348907i 0.0607368i
\(34\) −12.5371 −2.15010
\(35\) 0 0
\(36\) −5.02830 −0.838049
\(37\) 8.12386i 1.33555i 0.744361 + 0.667777i \(0.232754\pi\)
−0.744361 + 0.667777i \(0.767246\pi\)
\(38\) − 2.37720i − 0.385633i
\(39\) 7.58383 1.21438
\(40\) 0 0
\(41\) −9.43380 −1.47331 −0.736656 0.676268i \(-0.763596\pi\)
−0.736656 + 0.676268i \(0.763596\pi\)
\(42\) 2.19887i 0.339294i
\(43\) − 9.81100i − 1.49616i −0.663607 0.748082i \(-0.730975\pi\)
0.663607 0.748082i \(-0.269025\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.74666 −1.28962
\(47\) 12.1599i 1.77370i 0.462053 + 0.886852i \(0.347113\pi\)
−0.462053 + 0.886852i \(0.652887\pi\)
\(48\) − 2.58383i − 0.372943i
\(49\) 6.47277 0.924681
\(50\) 0 0
\(51\) 6.71836 0.940758
\(52\) − 21.7360i − 3.01424i
\(53\) − 5.69781i − 0.782655i −0.920252 0.391327i \(-0.872016\pi\)
0.920252 0.391327i \(-0.127984\pi\)
\(54\) 13.2555 1.80384
\(55\) 0 0
\(56\) 2.84997 0.380843
\(57\) 1.27389i 0.168731i
\(58\) 5.40550i 0.709777i
\(59\) 4.20662 0.547656 0.273828 0.961779i \(-0.411710\pi\)
0.273828 + 0.961779i \(0.411710\pi\)
\(60\) 0 0
\(61\) −0.103312 −0.0132278 −0.00661389 0.999978i \(-0.502105\pi\)
−0.00661389 + 0.999978i \(0.502105\pi\)
\(62\) 7.60437i 0.965756i
\(63\) 1.00000i 0.125988i
\(64\) 11.2555 1.40693
\(65\) 0 0
\(66\) −0.829422 −0.102095
\(67\) − 11.7827i − 1.43949i −0.694241 0.719743i \(-0.744260\pi\)
0.694241 0.719743i \(-0.255740\pi\)
\(68\) − 19.2555i − 2.33507i
\(69\) 4.68714 0.564265
\(70\) 0 0
\(71\) 5.75441 0.682922 0.341461 0.939896i \(-0.389078\pi\)
0.341461 + 0.939896i \(0.389078\pi\)
\(72\) − 5.40550i − 0.637044i
\(73\) − 6.67939i − 0.781763i −0.920441 0.390882i \(-0.872170\pi\)
0.920441 0.390882i \(-0.127830\pi\)
\(74\) −19.3121 −2.24498
\(75\) 0 0
\(76\) 3.65109 0.418809
\(77\) − 0.198875i − 0.0226639i
\(78\) 18.0283i 2.04130i
\(79\) −3.87826 −0.436339 −0.218169 0.975911i \(-0.570009\pi\)
−0.218169 + 0.975911i \(0.570009\pi\)
\(80\) 0 0
\(81\) −2.97170 −0.330189
\(82\) − 22.4260i − 2.47654i
\(83\) 0.488265i 0.0535941i 0.999641 + 0.0267970i \(0.00853078\pi\)
−0.999641 + 0.0267970i \(0.991469\pi\)
\(84\) −3.37720 −0.368483
\(85\) 0 0
\(86\) 23.3227 2.51495
\(87\) − 2.89669i − 0.310558i
\(88\) 1.07502i 0.114597i
\(89\) 16.4338 1.74198 0.870989 0.491302i \(-0.163479\pi\)
0.870989 + 0.491302i \(0.163479\pi\)
\(90\) 0 0
\(91\) −4.32273 −0.453146
\(92\) − 13.4338i − 1.40057i
\(93\) − 4.07502i − 0.422559i
\(94\) −28.9066 −2.98148
\(95\) 0 0
\(96\) −3.85772 −0.393727
\(97\) − 4.44447i − 0.451267i −0.974212 0.225634i \(-0.927555\pi\)
0.974212 0.225634i \(-0.0724453\pi\)
\(98\) 15.3871i 1.55433i
\(99\) −0.377203 −0.0379103
\(100\) 0 0
\(101\) 4.38495 0.436319 0.218160 0.975913i \(-0.429995\pi\)
0.218160 + 0.975913i \(0.429995\pi\)
\(102\) 15.9709i 1.58136i
\(103\) − 3.33048i − 0.328162i −0.986447 0.164081i \(-0.947534\pi\)
0.986447 0.164081i \(-0.0524659\pi\)
\(104\) 23.3665 2.29128
\(105\) 0 0
\(106\) 13.5449 1.31559
\(107\) 16.4904i 1.59419i 0.603857 + 0.797093i \(0.293630\pi\)
−0.603857 + 0.797093i \(0.706370\pi\)
\(108\) 20.3588i 1.95902i
\(109\) −7.79045 −0.746190 −0.373095 0.927793i \(-0.621703\pi\)
−0.373095 + 0.927793i \(0.621703\pi\)
\(110\) 0 0
\(111\) 10.3489 0.982275
\(112\) 1.47277i 0.139163i
\(113\) 0.142282i 0.0133848i 0.999978 + 0.00669238i \(0.00213027\pi\)
−0.999978 + 0.00669238i \(0.997870\pi\)
\(114\) −3.02830 −0.283626
\(115\) 0 0
\(116\) −8.30219 −0.770839
\(117\) 8.19887i 0.757986i
\(118\) 10.0000i 0.920575i
\(119\) −3.82942 −0.351043
\(120\) 0 0
\(121\) −10.9250 −0.993180
\(122\) − 0.245594i − 0.0222351i
\(123\) 12.0176i 1.08359i
\(124\) −11.6794 −1.04884
\(125\) 0 0
\(126\) −2.37720 −0.211778
\(127\) − 15.1316i − 1.34271i −0.741135 0.671357i \(-0.765712\pi\)
0.741135 0.671357i \(-0.234288\pi\)
\(128\) 20.6999i 1.82963i
\(129\) −12.4981 −1.10040
\(130\) 0 0
\(131\) 5.58383 0.487861 0.243931 0.969793i \(-0.421563\pi\)
0.243931 + 0.969793i \(0.421563\pi\)
\(132\) − 1.27389i − 0.110878i
\(133\) − 0.726109i − 0.0629617i
\(134\) 28.0099 2.41968
\(135\) 0 0
\(136\) 20.6999 1.77500
\(137\) 12.8294i 1.09609i 0.836448 + 0.548046i \(0.184628\pi\)
−0.836448 + 0.548046i \(0.815372\pi\)
\(138\) 11.1423i 0.948494i
\(139\) 15.2477 1.29329 0.646647 0.762789i \(-0.276171\pi\)
0.646647 + 0.762789i \(0.276171\pi\)
\(140\) 0 0
\(141\) 15.4904 1.30453
\(142\) 13.6794i 1.14795i
\(143\) − 1.63055i − 0.136353i
\(144\) 2.79338 0.232781
\(145\) 0 0
\(146\) 15.8783 1.31409
\(147\) − 8.24559i − 0.680085i
\(148\) − 29.6610i − 2.43812i
\(149\) −13.8315 −1.13312 −0.566562 0.824019i \(-0.691727\pi\)
−0.566562 + 0.824019i \(0.691727\pi\)
\(150\) 0 0
\(151\) −11.7077 −0.952758 −0.476379 0.879240i \(-0.658051\pi\)
−0.476379 + 0.879240i \(0.658051\pi\)
\(152\) 3.92498i 0.318358i
\(153\) 7.26322i 0.587196i
\(154\) 0.472765 0.0380965
\(155\) 0 0
\(156\) −27.6893 −2.21692
\(157\) − 4.79045i − 0.382320i −0.981559 0.191160i \(-0.938775\pi\)
0.981559 0.191160i \(-0.0612249\pi\)
\(158\) − 9.21942i − 0.733458i
\(159\) −7.25839 −0.575628
\(160\) 0 0
\(161\) −2.67164 −0.210555
\(162\) − 7.06434i − 0.555027i
\(163\) − 12.8011i − 1.00266i −0.865256 0.501331i \(-0.832844\pi\)
0.865256 0.501331i \(-0.167156\pi\)
\(164\) 34.4437 2.68960
\(165\) 0 0
\(166\) −1.16071 −0.0900882
\(167\) − 20.9426i − 1.62059i −0.586024 0.810294i \(-0.699308\pi\)
0.586024 0.810294i \(-0.300692\pi\)
\(168\) − 3.63055i − 0.280103i
\(169\) −22.4415 −1.72627
\(170\) 0 0
\(171\) −1.37720 −0.105317
\(172\) 35.8209i 2.73132i
\(173\) − 15.7282i − 1.19580i −0.801572 0.597898i \(-0.796003\pi\)
0.801572 0.597898i \(-0.203997\pi\)
\(174\) 6.88601 0.522027
\(175\) 0 0
\(176\) −0.555531 −0.0418747
\(177\) − 5.35878i − 0.402791i
\(178\) 39.0665i 2.92816i
\(179\) −3.41325 −0.255118 −0.127559 0.991831i \(-0.540714\pi\)
−0.127559 + 0.991831i \(0.540714\pi\)
\(180\) 0 0
\(181\) 23.5109 1.74755 0.873777 0.486327i \(-0.161664\pi\)
0.873777 + 0.486327i \(0.161664\pi\)
\(182\) − 10.2760i − 0.761709i
\(183\) 0.131609i 0.00972878i
\(184\) 14.4415 1.06464
\(185\) 0 0
\(186\) 9.68714 0.710296
\(187\) − 1.44447i − 0.105630i
\(188\) − 44.3969i − 3.23798i
\(189\) 4.04884 0.294510
\(190\) 0 0
\(191\) 12.4650 0.901937 0.450968 0.892540i \(-0.351079\pi\)
0.450968 + 0.892540i \(0.351079\pi\)
\(192\) − 14.3382i − 1.03477i
\(193\) − 19.2993i − 1.38919i −0.719400 0.694596i \(-0.755583\pi\)
0.719400 0.694596i \(-0.244417\pi\)
\(194\) 10.5654 0.758552
\(195\) 0 0
\(196\) −23.6327 −1.68805
\(197\) 6.63055i 0.472407i 0.971704 + 0.236203i \(0.0759032\pi\)
−0.971704 + 0.236203i \(0.924097\pi\)
\(198\) − 0.896688i − 0.0637248i
\(199\) 23.0849 1.63644 0.818222 0.574902i \(-0.194960\pi\)
0.818222 + 0.574902i \(0.194960\pi\)
\(200\) 0 0
\(201\) −15.0099 −1.05871
\(202\) 10.4239i 0.733425i
\(203\) 1.65109i 0.115884i
\(204\) −24.5294 −1.71740
\(205\) 0 0
\(206\) 7.91723 0.551620
\(207\) 5.06727i 0.352199i
\(208\) 12.0750i 0.837252i
\(209\) 0.273891 0.0189454
\(210\) 0 0
\(211\) −7.54778 −0.519611 −0.259805 0.965661i \(-0.583658\pi\)
−0.259805 + 0.965661i \(0.583658\pi\)
\(212\) 20.8032i 1.42877i
\(213\) − 7.33048i − 0.502276i
\(214\) −39.2010 −2.67973
\(215\) 0 0
\(216\) −21.8860 −1.48915
\(217\) 2.32273i 0.157677i
\(218\) − 18.5195i − 1.25430i
\(219\) −8.50881 −0.574972
\(220\) 0 0
\(221\) −31.3969 −2.11199
\(222\) 24.6015i 1.65114i
\(223\) 1.09344i 0.0732221i 0.999330 + 0.0366111i \(0.0116563\pi\)
−0.999330 + 0.0366111i \(0.988344\pi\)
\(224\) 2.19887 0.146918
\(225\) 0 0
\(226\) −0.338233 −0.0224989
\(227\) − 20.1316i − 1.33618i −0.744080 0.668091i \(-0.767112\pi\)
0.744080 0.668091i \(-0.232888\pi\)
\(228\) − 4.65109i − 0.308026i
\(229\) 5.51656 0.364545 0.182272 0.983248i \(-0.441655\pi\)
0.182272 + 0.983248i \(0.441655\pi\)
\(230\) 0 0
\(231\) −0.253344 −0.0166688
\(232\) − 8.92498i − 0.585954i
\(233\) 18.1805i 1.19104i 0.803340 + 0.595520i \(0.203054\pi\)
−0.803340 + 0.595520i \(0.796946\pi\)
\(234\) −19.4904 −1.27413
\(235\) 0 0
\(236\) −15.3588 −0.999771
\(237\) 4.94048i 0.320919i
\(238\) − 9.10331i − 0.590080i
\(239\) −21.9164 −1.41766 −0.708828 0.705381i \(-0.750776\pi\)
−0.708828 + 0.705381i \(0.750776\pi\)
\(240\) 0 0
\(241\) −28.1882 −1.81576 −0.907881 0.419228i \(-0.862301\pi\)
−0.907881 + 0.419228i \(0.862301\pi\)
\(242\) − 25.9709i − 1.66947i
\(243\) − 12.9426i − 0.830269i
\(244\) 0.377203 0.0241479
\(245\) 0 0
\(246\) −28.5683 −1.82145
\(247\) − 5.95328i − 0.378798i
\(248\) − 12.5555i − 0.797277i
\(249\) 0.621996 0.0394174
\(250\) 0 0
\(251\) 9.00987 0.568698 0.284349 0.958721i \(-0.408223\pi\)
0.284349 + 0.958721i \(0.408223\pi\)
\(252\) − 3.65109i − 0.229997i
\(253\) − 1.00775i − 0.0633567i
\(254\) 35.9709 2.25702
\(255\) 0 0
\(256\) −26.6970 −1.66856
\(257\) 6.86064i 0.427955i 0.976839 + 0.213978i \(0.0686419\pi\)
−0.976839 + 0.213978i \(0.931358\pi\)
\(258\) − 29.7106i − 1.84970i
\(259\) −5.89881 −0.366534
\(260\) 0 0
\(261\) 3.13161 0.193842
\(262\) 13.2739i 0.820064i
\(263\) 9.25547i 0.570717i 0.958421 + 0.285358i \(0.0921126\pi\)
−0.958421 + 0.285358i \(0.907887\pi\)
\(264\) 1.36945 0.0842840
\(265\) 0 0
\(266\) 1.72611 0.105835
\(267\) − 20.9349i − 1.28119i
\(268\) 43.0197i 2.62785i
\(269\) −0.498939 −0.0304208 −0.0152104 0.999884i \(-0.504842\pi\)
−0.0152104 + 0.999884i \(0.504842\pi\)
\(270\) 0 0
\(271\) 3.71061 0.225403 0.112702 0.993629i \(-0.464050\pi\)
0.112702 + 0.993629i \(0.464050\pi\)
\(272\) 10.6970i 0.648602i
\(273\) 5.50669i 0.333280i
\(274\) −30.4981 −1.84246
\(275\) 0 0
\(276\) −17.1132 −1.03009
\(277\) 4.58675i 0.275591i 0.990461 + 0.137796i \(0.0440017\pi\)
−0.990461 + 0.137796i \(0.955998\pi\)
\(278\) 36.2469i 2.17395i
\(279\) 4.40550 0.263750
\(280\) 0 0
\(281\) 27.2653 1.62651 0.813257 0.581905i \(-0.197692\pi\)
0.813257 + 0.581905i \(0.197692\pi\)
\(282\) 36.8238i 2.19283i
\(283\) 10.2661i 0.610259i 0.952311 + 0.305129i \(0.0986997\pi\)
−0.952311 + 0.305129i \(0.901300\pi\)
\(284\) −21.0099 −1.24671
\(285\) 0 0
\(286\) 3.87614 0.229201
\(287\) − 6.84997i − 0.404341i
\(288\) − 4.17058i − 0.245754i
\(289\) −10.8139 −0.636113
\(290\) 0 0
\(291\) −5.66177 −0.331899
\(292\) 24.3871i 1.42715i
\(293\) 1.87051i 0.109277i 0.998506 + 0.0546383i \(0.0174006\pi\)
−0.998506 + 0.0546383i \(0.982599\pi\)
\(294\) 19.6015 1.14318
\(295\) 0 0
\(296\) 31.8860 1.85334
\(297\) 1.52723i 0.0886192i
\(298\) − 32.8804i − 1.90471i
\(299\) −21.9044 −1.26677
\(300\) 0 0
\(301\) 7.12386 0.410612
\(302\) − 27.8315i − 1.60153i
\(303\) − 5.58595i − 0.320904i
\(304\) −2.02830 −0.116331
\(305\) 0 0
\(306\) −17.2661 −0.987040
\(307\) 0.227171i 0.0129653i 0.999979 + 0.00648266i \(0.00206351\pi\)
−0.999979 + 0.00648266i \(0.997936\pi\)
\(308\) 0.726109i 0.0413739i
\(309\) −4.24267 −0.241357
\(310\) 0 0
\(311\) 20.9554 1.18827 0.594136 0.804365i \(-0.297494\pi\)
0.594136 + 0.804365i \(0.297494\pi\)
\(312\) − 29.7664i − 1.68519i
\(313\) 11.2349i 0.635035i 0.948252 + 0.317518i \(0.102849\pi\)
−0.948252 + 0.317518i \(0.897151\pi\)
\(314\) 11.3879 0.642655
\(315\) 0 0
\(316\) 14.1599 0.796557
\(317\) 18.6228i 1.04596i 0.852345 + 0.522980i \(0.175180\pi\)
−0.852345 + 0.522980i \(0.824820\pi\)
\(318\) − 17.2547i − 0.967594i
\(319\) −0.622797 −0.0348699
\(320\) 0 0
\(321\) 21.0069 1.17249
\(322\) − 6.35103i − 0.353929i
\(323\) − 5.27389i − 0.293447i
\(324\) 10.8500 0.602776
\(325\) 0 0
\(326\) 30.4309 1.68541
\(327\) 9.92418i 0.548809i
\(328\) 37.0275i 2.04450i
\(329\) −8.82942 −0.486782
\(330\) 0 0
\(331\) −14.1054 −0.775305 −0.387652 0.921806i \(-0.626714\pi\)
−0.387652 + 0.921806i \(0.626714\pi\)
\(332\) − 1.78270i − 0.0978385i
\(333\) 11.1882i 0.613110i
\(334\) 49.7848 2.72410
\(335\) 0 0
\(336\) 1.87614 0.102352
\(337\) 22.9709i 1.25130i 0.780102 + 0.625652i \(0.215167\pi\)
−0.780102 + 0.625652i \(0.784833\pi\)
\(338\) − 53.3481i − 2.90175i
\(339\) 0.181252 0.00984424
\(340\) 0 0
\(341\) −0.876142 −0.0474457
\(342\) − 3.27389i − 0.177032i
\(343\) 9.78270i 0.528216i
\(344\) −38.5080 −2.07621
\(345\) 0 0
\(346\) 37.3892 2.01006
\(347\) − 3.93273i − 0.211120i −0.994413 0.105560i \(-0.966336\pi\)
0.994413 0.105560i \(-0.0336635\pi\)
\(348\) 10.5761i 0.566937i
\(349\) 34.4252 1.84274 0.921371 0.388685i \(-0.127071\pi\)
0.921371 + 0.388685i \(0.127071\pi\)
\(350\) 0 0
\(351\) 33.1960 1.77187
\(352\) 0.829422i 0.0442083i
\(353\) 4.25547i 0.226496i 0.993567 + 0.113248i \(0.0361254\pi\)
−0.993567 + 0.113248i \(0.963875\pi\)
\(354\) 12.7389 0.677065
\(355\) 0 0
\(356\) −60.0013 −3.18006
\(357\) 4.87826i 0.258185i
\(358\) − 8.11399i − 0.428837i
\(359\) 20.2944 1.07110 0.535550 0.844504i \(-0.320104\pi\)
0.535550 + 0.844504i \(0.320104\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 55.8903i 2.93753i
\(363\) 13.9172i 0.730465i
\(364\) 15.7827 0.827238
\(365\) 0 0
\(366\) −0.312860 −0.0163535
\(367\) 3.85289i 0.201119i 0.994931 + 0.100560i \(0.0320633\pi\)
−0.994931 + 0.100560i \(0.967937\pi\)
\(368\) 7.46289i 0.389030i
\(369\) −12.9922 −0.676350
\(370\) 0 0
\(371\) 4.13724 0.214795
\(372\) 14.8783i 0.771402i
\(373\) − 14.6356i − 0.757802i −0.925437 0.378901i \(-0.876302\pi\)
0.925437 0.378901i \(-0.123698\pi\)
\(374\) 3.43380 0.177557
\(375\) 0 0
\(376\) 47.7274 2.46135
\(377\) 13.5371i 0.697197i
\(378\) 9.62492i 0.495052i
\(379\) 22.0099 1.13057 0.565286 0.824895i \(-0.308766\pi\)
0.565286 + 0.824895i \(0.308766\pi\)
\(380\) 0 0
\(381\) −19.2760 −0.987540
\(382\) 29.6319i 1.51610i
\(383\) 3.08569i 0.157671i 0.996888 + 0.0788357i \(0.0251202\pi\)
−0.996888 + 0.0788357i \(0.974880\pi\)
\(384\) 26.3695 1.34566
\(385\) 0 0
\(386\) 45.8783 2.33514
\(387\) − 13.5117i − 0.686840i
\(388\) 16.2272i 0.823810i
\(389\) −8.77203 −0.444760 −0.222380 0.974960i \(-0.571382\pi\)
−0.222380 + 0.974960i \(0.571382\pi\)
\(390\) 0 0
\(391\) −19.4047 −0.981338
\(392\) − 25.4055i − 1.28317i
\(393\) − 7.11319i − 0.358813i
\(394\) −15.7622 −0.794086
\(395\) 0 0
\(396\) 1.37720 0.0692070
\(397\) − 1.59450i − 0.0800257i −0.999199 0.0400129i \(-0.987260\pi\)
0.999199 0.0400129i \(-0.0127399\pi\)
\(398\) 54.8775i 2.75076i
\(399\) −0.924984 −0.0463071
\(400\) 0 0
\(401\) −17.5526 −0.876535 −0.438268 0.898844i \(-0.644408\pi\)
−0.438268 + 0.898844i \(0.644408\pi\)
\(402\) − 35.6815i − 1.77963i
\(403\) 19.0438i 0.948639i
\(404\) −16.0099 −0.796521
\(405\) 0 0
\(406\) −3.92498 −0.194794
\(407\) − 2.22505i − 0.110292i
\(408\) − 26.3695i − 1.30548i
\(409\) −36.6815 −1.81378 −0.906892 0.421363i \(-0.861552\pi\)
−0.906892 + 0.421363i \(0.861552\pi\)
\(410\) 0 0
\(411\) 16.3433 0.806155
\(412\) 12.1599i 0.599076i
\(413\) 3.05447i 0.150301i
\(414\) −12.0459 −0.592025
\(415\) 0 0
\(416\) 18.0283 0.883910
\(417\) − 19.4239i − 0.951194i
\(418\) 0.651093i 0.0318460i
\(419\) 18.8187 0.919356 0.459678 0.888086i \(-0.347965\pi\)
0.459678 + 0.888086i \(0.347965\pi\)
\(420\) 0 0
\(421\) −33.7819 −1.64643 −0.823215 0.567730i \(-0.807822\pi\)
−0.823215 + 0.567730i \(0.807822\pi\)
\(422\) − 17.9426i − 0.873432i
\(423\) 16.7467i 0.814250i
\(424\) −22.3638 −1.08608
\(425\) 0 0
\(426\) 17.4260 0.844295
\(427\) − 0.0750160i − 0.00363028i
\(428\) − 60.2079i − 2.91026i
\(429\) −2.07714 −0.100285
\(430\) 0 0
\(431\) −12.7651 −0.614872 −0.307436 0.951569i \(-0.599471\pi\)
−0.307436 + 0.951569i \(0.599471\pi\)
\(432\) − 11.3099i − 0.544150i
\(433\) 16.0771i 0.772618i 0.922369 + 0.386309i \(0.126250\pi\)
−0.922369 + 0.386309i \(0.873750\pi\)
\(434\) −5.52161 −0.265046
\(435\) 0 0
\(436\) 28.4437 1.36220
\(437\) − 3.67939i − 0.176009i
\(438\) − 20.2272i − 0.966492i
\(439\) −1.36945 −0.0653604 −0.0326802 0.999466i \(-0.510404\pi\)
−0.0326802 + 0.999466i \(0.510404\pi\)
\(440\) 0 0
\(441\) 8.91431 0.424491
\(442\) − 74.6369i − 3.55012i
\(443\) − 4.62280i − 0.219636i −0.993952 0.109818i \(-0.964973\pi\)
0.993952 0.109818i \(-0.0350268\pi\)
\(444\) −37.7848 −1.79319
\(445\) 0 0
\(446\) −2.59933 −0.123082
\(447\) 17.6199i 0.833391i
\(448\) 8.17270i 0.386124i
\(449\) −23.2555 −1.09749 −0.548747 0.835989i \(-0.684895\pi\)
−0.548747 + 0.835989i \(0.684895\pi\)
\(450\) 0 0
\(451\) 2.58383 0.121668
\(452\) − 0.519485i − 0.0244345i
\(453\) 14.9143i 0.700735i
\(454\) 47.8569 2.24604
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) − 35.8443i − 1.67673i −0.545111 0.838364i \(-0.683513\pi\)
0.545111 0.838364i \(-0.316487\pi\)
\(458\) 13.1140i 0.612776i
\(459\) 29.4076 1.37263
\(460\) 0 0
\(461\) −14.8812 −0.693086 −0.346543 0.938034i \(-0.612645\pi\)
−0.346543 + 0.938034i \(0.612645\pi\)
\(462\) − 0.602251i − 0.0280193i
\(463\) 29.9554i 1.39215i 0.717971 + 0.696073i \(0.245071\pi\)
−0.717971 + 0.696073i \(0.754929\pi\)
\(464\) 4.61212 0.214112
\(465\) 0 0
\(466\) −43.2186 −2.00206
\(467\) − 6.73598i − 0.311704i −0.987780 0.155852i \(-0.950188\pi\)
0.987780 0.155852i \(-0.0498123\pi\)
\(468\) − 29.9349i − 1.38374i
\(469\) 8.55553 0.395058
\(470\) 0 0
\(471\) −6.10251 −0.281189
\(472\) − 16.5109i − 0.759977i
\(473\) 2.68714i 0.123555i
\(474\) −11.7445 −0.539444
\(475\) 0 0
\(476\) 13.9816 0.640845
\(477\) − 7.84704i − 0.359291i
\(478\) − 52.0998i − 2.38299i
\(479\) −16.6978 −0.762943 −0.381471 0.924381i \(-0.624582\pi\)
−0.381471 + 0.924381i \(0.624582\pi\)
\(480\) 0 0
\(481\) −48.3636 −2.20519
\(482\) − 67.0091i − 3.05218i
\(483\) 3.40338i 0.154859i
\(484\) 39.8881 1.81310
\(485\) 0 0
\(486\) 30.7672 1.39563
\(487\) 3.64042i 0.164963i 0.996593 + 0.0824816i \(0.0262846\pi\)
−0.996593 + 0.0824816i \(0.973715\pi\)
\(488\) 0.405499i 0.0183561i
\(489\) −16.3072 −0.737439
\(490\) 0 0
\(491\) −33.3249 −1.50393 −0.751965 0.659203i \(-0.770894\pi\)
−0.751965 + 0.659203i \(0.770894\pi\)
\(492\) − 43.8775i − 1.97815i
\(493\) 11.9922i 0.540104i
\(494\) 14.1522 0.636736
\(495\) 0 0
\(496\) 6.48827 0.291332
\(497\) 4.17833i 0.187424i
\(498\) 1.47861i 0.0662582i
\(499\) 37.9914 1.70073 0.850365 0.526193i \(-0.176381\pi\)
0.850365 + 0.526193i \(0.176381\pi\)
\(500\) 0 0
\(501\) −26.6786 −1.19191
\(502\) 21.4183i 0.955945i
\(503\) 42.1826i 1.88083i 0.340032 + 0.940414i \(0.389562\pi\)
−0.340032 + 0.940414i \(0.610438\pi\)
\(504\) 3.92498 0.174833
\(505\) 0 0
\(506\) 2.39563 0.106499
\(507\) 28.5881i 1.26964i
\(508\) 55.2469i 2.45119i
\(509\) 21.9971 0.975003 0.487502 0.873122i \(-0.337908\pi\)
0.487502 + 0.873122i \(0.337908\pi\)
\(510\) 0 0
\(511\) 4.84997 0.214550
\(512\) − 22.0643i − 0.975115i
\(513\) 5.57608i 0.246190i
\(514\) −16.3091 −0.719365
\(515\) 0 0
\(516\) 45.6319 2.00883
\(517\) − 3.33048i − 0.146474i
\(518\) − 14.0227i − 0.616121i
\(519\) −20.0360 −0.879485
\(520\) 0 0
\(521\) 20.0977 0.880496 0.440248 0.897876i \(-0.354891\pi\)
0.440248 + 0.897876i \(0.354891\pi\)
\(522\) 7.44447i 0.325836i
\(523\) 4.64817i 0.203250i 0.994823 + 0.101625i \(0.0324042\pi\)
−0.994823 + 0.101625i \(0.967596\pi\)
\(524\) −20.3871 −0.890614
\(525\) 0 0
\(526\) −22.0021 −0.959338
\(527\) 16.8705i 0.734891i
\(528\) 0.707686i 0.0307981i
\(529\) 9.46209 0.411395
\(530\) 0 0
\(531\) 5.79338 0.251411
\(532\) 2.65109i 0.114939i
\(533\) − 56.1620i − 2.43265i
\(534\) 49.7664 2.15360
\(535\) 0 0
\(536\) −46.2469 −1.99756
\(537\) 4.34811i 0.187635i
\(538\) − 1.18608i − 0.0511355i
\(539\) −1.77283 −0.0763612
\(540\) 0 0
\(541\) 20.0673 0.862759 0.431380 0.902171i \(-0.358027\pi\)
0.431380 + 0.902171i \(0.358027\pi\)
\(542\) 8.82087i 0.378889i
\(543\) − 29.9504i − 1.28529i
\(544\) 15.9709 0.684747
\(545\) 0 0
\(546\) −13.0905 −0.560222
\(547\) − 37.2010i − 1.59060i −0.606216 0.795300i \(-0.707313\pi\)
0.606216 0.795300i \(-0.292687\pi\)
\(548\) − 46.8414i − 2.00097i
\(549\) −0.142282 −0.00607245
\(550\) 0 0
\(551\) −2.27389 −0.0968710
\(552\) − 18.3969i − 0.783026i
\(553\) − 2.81604i − 0.119750i
\(554\) −10.9036 −0.463251
\(555\) 0 0
\(556\) −55.6708 −2.36097
\(557\) 44.8393i 1.89990i 0.312399 + 0.949951i \(0.398867\pi\)
−0.312399 + 0.949951i \(0.601133\pi\)
\(558\) 10.4728i 0.443347i
\(559\) 58.4076 2.47038
\(560\) 0 0
\(561\) −1.84010 −0.0776889
\(562\) 64.8152i 2.73407i
\(563\) 21.9172i 0.923701i 0.886958 + 0.461851i \(0.152814\pi\)
−0.886958 + 0.461851i \(0.847186\pi\)
\(564\) −56.5569 −2.38147
\(565\) 0 0
\(566\) −24.4047 −1.02581
\(567\) − 2.15778i − 0.0906183i
\(568\) − 22.5860i − 0.947685i
\(569\) −9.90656 −0.415305 −0.207652 0.978203i \(-0.566582\pi\)
−0.207652 + 0.978203i \(0.566582\pi\)
\(570\) 0 0
\(571\) 17.6404 0.738229 0.369114 0.929384i \(-0.379661\pi\)
0.369114 + 0.929384i \(0.379661\pi\)
\(572\) 5.95328i 0.248919i
\(573\) − 15.8791i − 0.663357i
\(574\) 16.2838 0.679671
\(575\) 0 0
\(576\) 15.5011 0.645878
\(577\) − 12.7048i − 0.528906i −0.964399 0.264453i \(-0.914809\pi\)
0.964399 0.264453i \(-0.0851914\pi\)
\(578\) − 25.7069i − 1.06927i
\(579\) −24.5851 −1.02172
\(580\) 0 0
\(581\) −0.354534 −0.0147085
\(582\) − 13.4592i − 0.557900i
\(583\) 1.56058i 0.0646325i
\(584\) −26.2165 −1.08485
\(585\) 0 0
\(586\) −4.44659 −0.183687
\(587\) 15.0438i 0.620924i 0.950586 + 0.310462i \(0.100484\pi\)
−0.950586 + 0.310462i \(0.899516\pi\)
\(588\) 30.1054i 1.24153i
\(589\) −3.19887 −0.131807
\(590\) 0 0
\(591\) 8.44659 0.347446
\(592\) 16.4776i 0.677225i
\(593\) 16.4231i 0.674417i 0.941430 + 0.337208i \(0.109483\pi\)
−0.941430 + 0.337208i \(0.890517\pi\)
\(594\) −3.63055 −0.148963
\(595\) 0 0
\(596\) 50.5003 2.06857
\(597\) − 29.4076i − 1.20357i
\(598\) − 52.0713i − 2.12935i
\(599\) 19.1260 0.781466 0.390733 0.920504i \(-0.372222\pi\)
0.390733 + 0.920504i \(0.372222\pi\)
\(600\) 0 0
\(601\) 31.4124 1.28134 0.640670 0.767816i \(-0.278657\pi\)
0.640670 + 0.767816i \(0.278657\pi\)
\(602\) 16.9349i 0.690213i
\(603\) − 16.2272i − 0.660821i
\(604\) 42.7459 1.73930
\(605\) 0 0
\(606\) 13.2789 0.539420
\(607\) − 41.5315i − 1.68571i −0.538140 0.842855i \(-0.680873\pi\)
0.538140 0.842855i \(-0.319127\pi\)
\(608\) 3.02830i 0.122814i
\(609\) 2.10331 0.0852305
\(610\) 0 0
\(611\) −72.3913 −2.92864
\(612\) − 26.5187i − 1.07195i
\(613\) − 21.7274i − 0.877563i −0.898594 0.438781i \(-0.855410\pi\)
0.898594 0.438781i \(-0.144590\pi\)
\(614\) −0.540031 −0.0217939
\(615\) 0 0
\(616\) −0.780579 −0.0314504
\(617\) 33.6065i 1.35295i 0.736467 + 0.676473i \(0.236493\pi\)
−0.736467 + 0.676473i \(0.763507\pi\)
\(618\) − 10.0857i − 0.405706i
\(619\) 27.6036 1.10948 0.554741 0.832023i \(-0.312817\pi\)
0.554741 + 0.832023i \(0.312817\pi\)
\(620\) 0 0
\(621\) 20.5166 0.823301
\(622\) 49.8152i 1.99741i
\(623\) 11.9327i 0.478075i
\(624\) 15.3822 0.615783
\(625\) 0 0
\(626\) −26.7077 −1.06745
\(627\) − 0.348907i − 0.0139340i
\(628\) 17.4904i 0.697942i
\(629\) −42.8443 −1.70832
\(630\) 0 0
\(631\) 1.94048 0.0772495 0.0386247 0.999254i \(-0.487702\pi\)
0.0386247 + 0.999254i \(0.487702\pi\)
\(632\) 15.2221i 0.605504i
\(633\) 9.61505i 0.382164i
\(634\) −44.2702 −1.75819
\(635\) 0 0
\(636\) 26.5011 1.05084
\(637\) 38.5342i 1.52678i
\(638\) − 1.48052i − 0.0586142i
\(639\) 7.92498 0.313508
\(640\) 0 0
\(641\) 1.01975 0.0402775 0.0201388 0.999797i \(-0.493589\pi\)
0.0201388 + 0.999797i \(0.493589\pi\)
\(642\) 49.9378i 1.97089i
\(643\) − 36.9866i − 1.45861i −0.684189 0.729305i \(-0.739844\pi\)
0.684189 0.729305i \(-0.260156\pi\)
\(644\) 9.75441 0.384377
\(645\) 0 0
\(646\) 12.5371 0.493266
\(647\) 24.1182i 0.948186i 0.880475 + 0.474093i \(0.157224\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(648\) 11.6639i 0.458201i
\(649\) −1.15215 −0.0452260
\(650\) 0 0
\(651\) 2.95891 0.115969
\(652\) 46.7381i 1.83041i
\(653\) − 37.2603i − 1.45811i −0.684456 0.729054i \(-0.739960\pi\)
0.684456 0.729054i \(-0.260040\pi\)
\(654\) −23.5918 −0.922512
\(655\) 0 0
\(656\) −19.1345 −0.747078
\(657\) − 9.19887i − 0.358882i
\(658\) − 20.9893i − 0.818249i
\(659\) −21.4386 −0.835130 −0.417565 0.908647i \(-0.637116\pi\)
−0.417565 + 0.908647i \(0.637116\pi\)
\(660\) 0 0
\(661\) 0.783503 0.0304747 0.0152374 0.999884i \(-0.495150\pi\)
0.0152374 + 0.999884i \(0.495150\pi\)
\(662\) − 33.5315i − 1.30324i
\(663\) 39.9963i 1.55333i
\(664\) 1.91643 0.0743720
\(665\) 0 0
\(666\) −26.5966 −1.03060
\(667\) 8.36653i 0.323953i
\(668\) 76.4634i 2.95846i
\(669\) 1.39292 0.0538535
\(670\) 0 0
\(671\) 0.0282963 0.00109237
\(672\) − 2.80113i − 0.108056i
\(673\) 50.1903i 1.93469i 0.253454 + 0.967347i \(0.418433\pi\)
−0.253454 + 0.967347i \(0.581567\pi\)
\(674\) −54.6065 −2.10336
\(675\) 0 0
\(676\) 81.9362 3.15139
\(677\) − 29.8804i − 1.14840i −0.818716 0.574198i \(-0.805314\pi\)
0.818716 0.574198i \(-0.194686\pi\)
\(678\) 0.430872i 0.0165475i
\(679\) 3.22717 0.123847
\(680\) 0 0
\(681\) −25.6455 −0.982736
\(682\) − 2.08277i − 0.0797532i
\(683\) − 12.3326i − 0.471894i −0.971766 0.235947i \(-0.924181\pi\)
0.971766 0.235947i \(-0.0758192\pi\)
\(684\) 5.02830 0.192262
\(685\) 0 0
\(686\) −23.2555 −0.887898
\(687\) − 7.02750i − 0.268116i
\(688\) − 19.8996i − 0.758666i
\(689\) 33.9207 1.29227
\(690\) 0 0
\(691\) 3.62200 0.137787 0.0688936 0.997624i \(-0.478053\pi\)
0.0688936 + 0.997624i \(0.478053\pi\)
\(692\) 57.4252i 2.18298i
\(693\) − 0.273891i − 0.0104042i
\(694\) 9.34891 0.354880
\(695\) 0 0
\(696\) −11.3695 −0.430958
\(697\) − 49.7528i − 1.88452i
\(698\) 81.8358i 3.09753i
\(699\) 23.1599 0.875988
\(700\) 0 0
\(701\) 34.1209 1.28873 0.644365 0.764718i \(-0.277122\pi\)
0.644365 + 0.764718i \(0.277122\pi\)
\(702\) 78.9135i 2.97840i
\(703\) − 8.12386i − 0.306397i
\(704\) −3.08277 −0.116186
\(705\) 0 0
\(706\) −10.1161 −0.380725
\(707\) 3.18396i 0.119745i
\(708\) 19.5654i 0.735313i
\(709\) 17.1209 0.642990 0.321495 0.946911i \(-0.395815\pi\)
0.321495 + 0.946911i \(0.395815\pi\)
\(710\) 0 0
\(711\) −5.34116 −0.200309
\(712\) − 64.5024i − 2.41733i
\(713\) 11.7699i 0.440786i
\(714\) −11.5966 −0.433993
\(715\) 0 0
\(716\) 12.4621 0.465730
\(717\) 27.9191i 1.04266i
\(718\) 48.2440i 1.80045i
\(719\) 7.02750 0.262081 0.131041 0.991377i \(-0.458168\pi\)
0.131041 + 0.991377i \(0.458168\pi\)
\(720\) 0 0
\(721\) 2.41830 0.0900620
\(722\) 2.37720i 0.0884703i
\(723\) 35.9087i 1.33546i
\(724\) −85.8406 −3.19024
\(725\) 0 0
\(726\) −33.0841 −1.22787
\(727\) − 11.8938i − 0.441115i −0.975374 0.220558i \(-0.929212\pi\)
0.975374 0.220558i \(-0.0707877\pi\)
\(728\) 16.9667i 0.628826i
\(729\) −25.4026 −0.940836
\(730\) 0 0
\(731\) 51.7421 1.91375
\(732\) − 0.480515i − 0.0177604i
\(733\) − 20.7154i − 0.765142i −0.923926 0.382571i \(-0.875039\pi\)
0.923926 0.382571i \(-0.124961\pi\)
\(734\) −9.15910 −0.338069
\(735\) 0 0
\(736\) 11.1423 0.410710
\(737\) 3.22717i 0.118874i
\(738\) − 30.8852i − 1.13690i
\(739\) −33.8620 −1.24563 −0.622816 0.782368i \(-0.714012\pi\)
−0.622816 + 0.782368i \(0.714012\pi\)
\(740\) 0 0
\(741\) −7.58383 −0.278599
\(742\) 9.83505i 0.361056i
\(743\) − 42.7381i − 1.56791i −0.620818 0.783955i \(-0.713200\pi\)
0.620818 0.783955i \(-0.286800\pi\)
\(744\) −15.9944 −0.586382
\(745\) 0 0
\(746\) 34.7918 1.27382
\(747\) 0.672440i 0.0246033i
\(748\) 5.27389i 0.192833i
\(749\) −11.9738 −0.437514
\(750\) 0 0
\(751\) 11.9581 0.436358 0.218179 0.975909i \(-0.429988\pi\)
0.218179 + 0.975909i \(0.429988\pi\)
\(752\) 24.6639i 0.899400i
\(753\) − 11.4776i − 0.418267i
\(754\) −32.1805 −1.17194
\(755\) 0 0
\(756\) −14.7827 −0.537642
\(757\) 29.1103i 1.05803i 0.848612 + 0.529015i \(0.177439\pi\)
−0.848612 + 0.529015i \(0.822561\pi\)
\(758\) 52.3219i 1.90042i
\(759\) −1.28376 −0.0465977
\(760\) 0 0
\(761\) −22.3014 −0.808425 −0.404212 0.914665i \(-0.632454\pi\)
−0.404212 + 0.914665i \(0.632454\pi\)
\(762\) − 45.8230i − 1.65999i
\(763\) − 5.65672i − 0.204787i
\(764\) −45.5109 −1.64653
\(765\) 0 0
\(766\) −7.33531 −0.265036
\(767\) 25.0432i 0.904258i
\(768\) 34.0091i 1.22720i
\(769\) −3.95891 −0.142762 −0.0713809 0.997449i \(-0.522741\pi\)
−0.0713809 + 0.997449i \(0.522741\pi\)
\(770\) 0 0
\(771\) 8.73971 0.314753
\(772\) 70.4634i 2.53603i
\(773\) − 27.8139i − 1.00040i −0.865911 0.500199i \(-0.833260\pi\)
0.865911 0.500199i \(-0.166740\pi\)
\(774\) 32.1201 1.15453
\(775\) 0 0
\(776\) −17.4445 −0.626220
\(777\) 7.51444i 0.269579i
\(778\) − 20.8529i − 0.747612i
\(779\) 9.43380 0.338001
\(780\) 0 0
\(781\) −1.57608 −0.0563965
\(782\) − 46.1289i − 1.64957i
\(783\) − 12.6794i − 0.453124i
\(784\) 13.1287 0.468882
\(785\) 0 0
\(786\) 16.9095 0.603141
\(787\) − 1.82460i − 0.0650398i −0.999471 0.0325199i \(-0.989647\pi\)
0.999471 0.0325199i \(-0.0103532\pi\)
\(788\) − 24.2087i − 0.862401i
\(789\) 11.7905 0.419751
\(790\) 0 0
\(791\) −0.103312 −0.00367336
\(792\) 1.48052i 0.0526078i
\(793\) − 0.615047i − 0.0218410i
\(794\) 3.79045 0.134518
\(795\) 0 0
\(796\) −84.2851 −2.98741
\(797\) − 21.0360i − 0.745135i −0.928005 0.372567i \(-0.878478\pi\)
0.928005 0.372567i \(-0.121522\pi\)
\(798\) − 2.19887i − 0.0778393i
\(799\) −64.1300 −2.26876
\(800\) 0 0
\(801\) 22.6327 0.799686
\(802\) − 41.7261i − 1.47340i
\(803\) 1.82942i 0.0645589i
\(804\) 54.8024 1.93273
\(805\) 0 0
\(806\) −45.2710 −1.59460
\(807\) 0.635593i 0.0223739i
\(808\) − 17.2109i − 0.605476i
\(809\) 0.0819654 0.00288175 0.00144088 0.999999i \(-0.499541\pi\)
0.00144088 + 0.999999i \(0.499541\pi\)
\(810\) 0 0
\(811\) −6.72531 −0.236158 −0.118079 0.993004i \(-0.537674\pi\)
−0.118079 + 0.993004i \(0.537674\pi\)
\(812\) − 6.02830i − 0.211552i
\(813\) − 4.72691i − 0.165780i
\(814\) 5.28939 0.185393
\(815\) 0 0
\(816\) 13.6268 0.477034
\(817\) 9.81100i 0.343243i
\(818\) − 87.1994i − 3.04886i
\(819\) −5.95328 −0.208024
\(820\) 0 0
\(821\) 30.9426 1.07990 0.539952 0.841696i \(-0.318442\pi\)
0.539952 + 0.841696i \(0.318442\pi\)
\(822\) 38.8513i 1.35509i
\(823\) − 26.5908i − 0.926896i −0.886124 0.463448i \(-0.846612\pi\)
0.886124 0.463448i \(-0.153388\pi\)
\(824\) −13.0721 −0.455388
\(825\) 0 0
\(826\) −7.26109 −0.252646
\(827\) − 5.57900i − 0.194001i −0.995284 0.0970004i \(-0.969075\pi\)
0.995284 0.0970004i \(-0.0309248\pi\)
\(828\) − 18.5011i − 0.642956i
\(829\) −18.9765 −0.659082 −0.329541 0.944141i \(-0.606894\pi\)
−0.329541 + 0.944141i \(0.606894\pi\)
\(830\) 0 0
\(831\) 5.84302 0.202692
\(832\) 67.0069i 2.32305i
\(833\) 34.1367i 1.18276i
\(834\) 46.1746 1.59890
\(835\) 0 0
\(836\) −1.00000 −0.0345857
\(837\) − 17.8372i − 0.616543i
\(838\) 44.7360i 1.54538i
\(839\) −12.9143 −0.445852 −0.222926 0.974835i \(-0.571561\pi\)
−0.222926 + 0.974835i \(0.571561\pi\)
\(840\) 0 0
\(841\) −23.8294 −0.821704
\(842\) − 80.3064i − 2.76754i
\(843\) − 34.7331i − 1.19627i
\(844\) 27.5577 0.948574
\(845\) 0 0
\(846\) −39.8102 −1.36870
\(847\) − 7.93273i − 0.272572i
\(848\) − 11.5569i − 0.396864i
\(849\) 13.0779 0.448834
\(850\) 0 0
\(851\) −29.8908 −1.02464
\(852\) 26.7643i 0.916929i
\(853\) − 6.46077i − 0.221213i −0.993864 0.110606i \(-0.964721\pi\)
0.993864 0.110606i \(-0.0352793\pi\)
\(854\) 0.178328 0.00610227
\(855\) 0 0
\(856\) 64.7245 2.21224
\(857\) − 12.4055i − 0.423764i −0.977295 0.211882i \(-0.932041\pi\)
0.977295 0.211882i \(-0.0679592\pi\)
\(858\) − 4.93778i − 0.168573i
\(859\) −40.3425 −1.37647 −0.688234 0.725489i \(-0.741614\pi\)
−0.688234 + 0.725489i \(0.741614\pi\)
\(860\) 0 0
\(861\) −8.72611 −0.297385
\(862\) − 30.3452i − 1.03356i
\(863\) − 1.83235i − 0.0623738i −0.999514 0.0311869i \(-0.990071\pi\)
0.999514 0.0311869i \(-0.00992870\pi\)
\(864\) −16.8860 −0.574474
\(865\) 0 0
\(866\) −38.2186 −1.29872
\(867\) 13.7758i 0.467849i
\(868\) − 8.48052i − 0.287847i
\(869\) 1.06222 0.0360333
\(870\) 0 0
\(871\) 70.1457 2.37680
\(872\) 30.5774i 1.03548i
\(873\) − 6.12094i − 0.207162i
\(874\) 8.74666 0.295860
\(875\) 0 0
\(876\) 31.0665 1.04964
\(877\) − 8.14419i − 0.275010i −0.990501 0.137505i \(-0.956092\pi\)
0.990501 0.137505i \(-0.0439083\pi\)
\(878\) − 3.25547i − 0.109867i
\(879\) 2.38283 0.0803709
\(880\) 0 0
\(881\) 12.1706 0.410037 0.205019 0.978758i \(-0.434275\pi\)
0.205019 + 0.978758i \(0.434275\pi\)
\(882\) 21.1911i 0.713542i
\(883\) − 46.7614i − 1.57364i −0.617179 0.786822i \(-0.711725\pi\)
0.617179 0.786822i \(-0.288275\pi\)
\(884\) 114.633 3.85553
\(885\) 0 0
\(886\) 10.9893 0.369194
\(887\) − 34.8804i − 1.17117i −0.810611 0.585584i \(-0.800865\pi\)
0.810611 0.585584i \(-0.199135\pi\)
\(888\) − 40.6193i − 1.36309i
\(889\) 10.9872 0.368499
\(890\) 0 0
\(891\) 0.813922 0.0272674
\(892\) − 3.99225i − 0.133670i
\(893\) − 12.1599i − 0.406916i
\(894\) −41.8860 −1.40088
\(895\) 0 0
\(896\) −15.0304 −0.502131
\(897\) 27.9039i 0.931683i
\(898\) − 55.2830i − 1.84482i
\(899\) 7.27389 0.242598
\(900\) 0 0
\(901\) 30.0496 1.00110
\(902\) 6.14228i 0.204516i
\(903\) − 9.07502i − 0.301998i
\(904\) 0.558455 0.0185739
\(905\) 0 0
\(906\) −35.4543 −1.17789
\(907\) − 25.5080i − 0.846980i −0.905901 0.423490i \(-0.860805\pi\)
0.905901 0.423490i \(-0.139195\pi\)
\(908\) 73.5024i 2.43926i
\(909\) 6.03897 0.200300
\(910\) 0 0
\(911\) −21.5032 −0.712432 −0.356216 0.934404i \(-0.615933\pi\)
−0.356216 + 0.934404i \(0.615933\pi\)
\(912\) 2.58383i 0.0855591i
\(913\) − 0.133731i − 0.00442586i
\(914\) 85.2093 2.81847
\(915\) 0 0
\(916\) −20.1415 −0.665493
\(917\) 4.05447i 0.133890i
\(918\) 69.9079i 2.30730i
\(919\) −37.1386 −1.22509 −0.612544 0.790436i \(-0.709854\pi\)
−0.612544 + 0.790436i \(0.709854\pi\)
\(920\) 0 0
\(921\) 0.289391 0.00953575
\(922\) − 35.3756i − 1.16503i
\(923\) 34.2576i 1.12760i
\(924\) 0.924984 0.0304297
\(925\) 0 0
\(926\) −71.2101 −2.34011
\(927\) − 4.58675i − 0.150649i
\(928\) − 6.88601i − 0.226044i
\(929\) 3.36170 0.110294 0.0551469 0.998478i \(-0.482437\pi\)
0.0551469 + 0.998478i \(0.482437\pi\)
\(930\) 0 0
\(931\) −6.47277 −0.212136
\(932\) − 66.3785i − 2.17430i
\(933\) − 26.6949i − 0.873951i
\(934\) 16.0128 0.523955
\(935\) 0 0
\(936\) 32.1805 1.05185
\(937\) 10.0694i 0.328953i 0.986381 + 0.164476i \(0.0525934\pi\)
−0.986381 + 0.164476i \(0.947407\pi\)
\(938\) 20.3382i 0.664067i
\(939\) 14.3121 0.467056
\(940\) 0 0
\(941\) 20.8139 0.678514 0.339257 0.940694i \(-0.389824\pi\)
0.339257 + 0.940694i \(0.389824\pi\)
\(942\) − 14.5069i − 0.472661i
\(943\) − 34.7106i − 1.13033i
\(944\) 8.53228 0.277702
\(945\) 0 0
\(946\) −6.38788 −0.207688
\(947\) − 30.4904i − 0.990804i −0.868664 0.495402i \(-0.835021\pi\)
0.868664 0.495402i \(-0.164979\pi\)
\(948\) − 18.0382i − 0.585852i
\(949\) 39.7643 1.29080
\(950\) 0 0
\(951\) 23.7234 0.769284
\(952\) 15.0304i 0.487139i
\(953\) − 7.58383i − 0.245664i −0.992427 0.122832i \(-0.960802\pi\)
0.992427 0.122832i \(-0.0391977\pi\)
\(954\) 18.6540 0.603946
\(955\) 0 0
\(956\) 80.0189 2.58800
\(957\) 0.793375i 0.0256462i
\(958\) − 39.6941i − 1.28246i
\(959\) −9.31556 −0.300815
\(960\) 0 0
\(961\) −20.7672 −0.669910
\(962\) − 114.970i − 3.70678i
\(963\) 22.7106i 0.731839i
\(964\) 102.918 3.31476
\(965\) 0 0
\(966\) −8.09052 −0.260308
\(967\) 54.6687i 1.75803i 0.476797 + 0.879014i \(0.341798\pi\)
−0.476797 + 0.879014i \(0.658202\pi\)
\(968\) 42.8804i 1.37823i
\(969\) −6.71836 −0.215825
\(970\) 0 0
\(971\) −39.9632 −1.28248 −0.641239 0.767341i \(-0.721579\pi\)
−0.641239 + 0.767341i \(0.721579\pi\)
\(972\) 47.2547i 1.51569i
\(973\) 11.0715i 0.354936i
\(974\) −8.65402 −0.277293
\(975\) 0 0
\(976\) −0.209548 −0.00670747
\(977\) 15.2400i 0.487570i 0.969829 + 0.243785i \(0.0783891\pi\)
−0.969829 + 0.243785i \(0.921611\pi\)
\(978\) − 38.7656i − 1.23959i
\(979\) −4.50106 −0.143855
\(980\) 0 0
\(981\) −10.7290 −0.342552
\(982\) − 79.2199i − 2.52801i
\(983\) 45.3609i 1.44679i 0.690435 + 0.723394i \(0.257419\pi\)
−0.690435 + 0.723394i \(0.742581\pi\)
\(984\) 47.1690 1.50369
\(985\) 0 0
\(986\) −28.5080 −0.907880
\(987\) 11.2477i 0.358019i
\(988\) 21.7360i 0.691514i
\(989\) 36.0985 1.14787
\(990\) 0 0
\(991\) −16.3537 −0.519493 −0.259747 0.965677i \(-0.583639\pi\)
−0.259747 + 0.965677i \(0.583639\pi\)
\(992\) − 9.68714i − 0.307567i
\(993\) 17.9688i 0.570222i
\(994\) −9.93273 −0.315047
\(995\) 0 0
\(996\) −2.27097 −0.0719583
\(997\) − 33.2037i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(998\) 90.3134i 2.85882i
\(999\) 45.2993 1.43321
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 475.2.b.b.324.6 6
5.2 odd 4 475.2.a.e.1.1 3
5.3 odd 4 475.2.a.g.1.3 yes 3
5.4 even 2 inner 475.2.b.b.324.1 6
15.2 even 4 4275.2.a.bm.1.3 3
15.8 even 4 4275.2.a.ba.1.1 3
20.3 even 4 7600.2.a.bh.1.2 3
20.7 even 4 7600.2.a.cc.1.2 3
95.18 even 4 9025.2.a.y.1.1 3
95.37 even 4 9025.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 3 5.2 odd 4
475.2.a.g.1.3 yes 3 5.3 odd 4
475.2.b.b.324.1 6 5.4 even 2 inner
475.2.b.b.324.6 6 1.1 even 1 trivial
4275.2.a.ba.1.1 3 15.8 even 4
4275.2.a.bm.1.3 3 15.2 even 4
7600.2.a.bh.1.2 3 20.3 even 4
7600.2.a.cc.1.2 3 20.7 even 4
9025.2.a.y.1.1 3 95.18 even 4
9025.2.a.bc.1.3 3 95.37 even 4