Properties

Label 3150.2.bf.f.1151.4
Level $3150$
Weight $2$
Character 3150.1151
Analytic conductor $25.153$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.4
Character \(\chi\) \(=\) 3150.1151
Dual form 3150.2.bf.f.1601.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.07665 + 1.63937i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.07665 + 1.63937i) q^{7} +1.00000i q^{8} +(5.48223 + 3.16517i) q^{11} -1.05290i q^{13} +(-2.61811 - 0.381412i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.31842 - 4.01562i) q^{17} +(-5.35129 + 3.08957i) q^{19} -6.33033 q^{22} +(6.10091 - 3.52236i) q^{23} +(0.526449 + 0.911836i) q^{26} +(2.45806 - 0.978745i) q^{28} +2.98101i q^{29} +(5.46948 + 3.15781i) q^{31} +(0.866025 + 0.500000i) q^{32} +4.63684i q^{34} +(1.73991 + 3.01361i) q^{37} +(3.08957 - 5.35129i) q^{38} +6.97007 q^{41} -2.58650 q^{43} +(5.48223 - 3.16517i) q^{44} +(-3.52236 + 6.10091i) q^{46} +(-4.07711 - 7.06176i) q^{47} +(1.62493 + 6.80879i) q^{49} +(-0.911836 - 0.526449i) q^{52} +(7.68480 + 4.43682i) q^{53} +(-1.63937 + 2.07665i) q^{56} +(-1.49051 - 2.58163i) q^{58} +(0.452296 - 0.783400i) q^{59} +(-8.81047 + 5.08673i) q^{61} -6.31561 q^{62} -1.00000 q^{64} +(4.91612 - 8.51497i) q^{67} +(-2.31842 - 4.01562i) q^{68} -14.4282i q^{71} +(-7.24574 - 4.18333i) q^{73} +(-3.01361 - 1.73991i) q^{74} +6.17914i q^{76} +(6.19578 + 15.5603i) q^{77} +(-2.73283 - 4.73340i) q^{79} +(-6.03626 + 3.48504i) q^{82} -14.6297 q^{83} +(2.23997 - 1.29325i) q^{86} +(-3.16517 + 5.48223i) q^{88} +(1.91982 + 3.32522i) q^{89} +(1.72609 - 2.18650i) q^{91} -7.04472i q^{92} +(7.06176 + 4.07711i) q^{94} -5.87891i q^{97} +(-4.81163 - 5.08412i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{4} - 16 q^{16} - 48 q^{19} + 24 q^{31} - 16 q^{46} + 56 q^{49} + 48 q^{61} - 32 q^{64} - 8 q^{79} - 56 q^{91} + 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 0.866025i 0.250000 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.07665 + 1.63937i 0.784899 + 0.619624i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.48223 + 3.16517i 1.65295 + 0.954334i 0.975849 + 0.218448i \(0.0700994\pi\)
0.677106 + 0.735886i \(0.263234\pi\)
\(12\) 0 0
\(13\) 1.05290i 0.292021i −0.989283 0.146011i \(-0.953357\pi\)
0.989283 0.146011i \(-0.0466434\pi\)
\(14\) −2.61811 0.381412i −0.699721 0.101937i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) 2.31842 4.01562i 0.562299 0.973931i −0.434996 0.900432i \(-0.643250\pi\)
0.997295 0.0734985i \(-0.0234164\pi\)
\(18\) 0 0
\(19\) −5.35129 + 3.08957i −1.22767 + 0.708795i −0.966542 0.256509i \(-0.917428\pi\)
−0.261128 + 0.965304i \(0.584094\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.33033 −1.34963
\(23\) 6.10091 3.52236i 1.27213 0.734463i 0.296739 0.954959i \(-0.404101\pi\)
0.975388 + 0.220496i \(0.0707676\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.526449 + 0.911836i 0.103245 + 0.178826i
\(27\) 0 0
\(28\) 2.45806 0.978745i 0.464530 0.184965i
\(29\) 2.98101i 0.553560i 0.960933 + 0.276780i \(0.0892674\pi\)
−0.960933 + 0.276780i \(0.910733\pi\)
\(30\) 0 0
\(31\) 5.46948 + 3.15781i 0.982348 + 0.567159i 0.902978 0.429686i \(-0.141376\pi\)
0.0793697 + 0.996845i \(0.474709\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 4.63684i 0.795211i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.73991 + 3.01361i 0.286039 + 0.495434i 0.972861 0.231392i \(-0.0743279\pi\)
−0.686821 + 0.726826i \(0.740995\pi\)
\(38\) 3.08957 5.35129i 0.501194 0.868094i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.97007 1.08854 0.544271 0.838909i \(-0.316806\pi\)
0.544271 + 0.838909i \(0.316806\pi\)
\(42\) 0 0
\(43\) −2.58650 −0.394437 −0.197218 0.980360i \(-0.563191\pi\)
−0.197218 + 0.980360i \(0.563191\pi\)
\(44\) 5.48223 3.16517i 0.826477 0.477167i
\(45\) 0 0
\(46\) −3.52236 + 6.10091i −0.519344 + 0.899529i
\(47\) −4.07711 7.06176i −0.594707 1.03006i −0.993588 0.113060i \(-0.963935\pi\)
0.398881 0.917003i \(-0.369399\pi\)
\(48\) 0 0
\(49\) 1.62493 + 6.80879i 0.232133 + 0.972684i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.911836 0.526449i −0.126449 0.0730053i
\(53\) 7.68480 + 4.43682i 1.05559 + 0.609445i 0.924209 0.381887i \(-0.124726\pi\)
0.131380 + 0.991332i \(0.458059\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.63937 + 2.07665i −0.219070 + 0.277504i
\(57\) 0 0
\(58\) −1.49051 2.58163i −0.195713 0.338985i
\(59\) 0.452296 0.783400i 0.0588839 0.101990i −0.835081 0.550127i \(-0.814579\pi\)
0.893965 + 0.448137i \(0.147912\pi\)
\(60\) 0 0
\(61\) −8.81047 + 5.08673i −1.12807 + 0.651289i −0.943448 0.331522i \(-0.892438\pi\)
−0.184618 + 0.982810i \(0.559105\pi\)
\(62\) −6.31561 −0.802084
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.91612 8.51497i 0.600600 1.04027i −0.392131 0.919909i \(-0.628262\pi\)
0.992730 0.120360i \(-0.0384047\pi\)
\(68\) −2.31842 4.01562i −0.281150 0.486965i
\(69\) 0 0
\(70\) 0 0
\(71\) 14.4282i 1.71232i −0.516713 0.856159i \(-0.672844\pi\)
0.516713 0.856159i \(-0.327156\pi\)
\(72\) 0 0
\(73\) −7.24574 4.18333i −0.848050 0.489622i 0.0119423 0.999929i \(-0.496199\pi\)
−0.859992 + 0.510307i \(0.829532\pi\)
\(74\) −3.01361 1.73991i −0.350325 0.202260i
\(75\) 0 0
\(76\) 6.17914i 0.708795i
\(77\) 6.19578 + 15.5603i 0.706075 + 1.77327i
\(78\) 0 0
\(79\) −2.73283 4.73340i −0.307467 0.532549i 0.670340 0.742054i \(-0.266148\pi\)
−0.977808 + 0.209505i \(0.932815\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.03626 + 3.48504i −0.666593 + 0.384858i
\(83\) −14.6297 −1.60581 −0.802907 0.596105i \(-0.796714\pi\)
−0.802907 + 0.596105i \(0.796714\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.23997 1.29325i 0.241542 0.139455i
\(87\) 0 0
\(88\) −3.16517 + 5.48223i −0.337408 + 0.584408i
\(89\) 1.91982 + 3.32522i 0.203500 + 0.352473i 0.949654 0.313301i \(-0.101435\pi\)
−0.746154 + 0.665774i \(0.768102\pi\)
\(90\) 0 0
\(91\) 1.72609 2.18650i 0.180943 0.229207i
\(92\) 7.04472i 0.734463i
\(93\) 0 0
\(94\) 7.06176 + 4.07711i 0.728365 + 0.420522i
\(95\) 0 0
\(96\) 0 0
\(97\) 5.87891i 0.596913i −0.954423 0.298457i \(-0.903528\pi\)
0.954423 0.298457i \(-0.0964718\pi\)
\(98\) −4.81163 5.08412i −0.486048 0.513573i
\(99\) 0 0
\(100\) 0 0
\(101\) −3.35408 + 5.80944i −0.333744 + 0.578061i −0.983243 0.182301i \(-0.941645\pi\)
0.649499 + 0.760362i \(0.274979\pi\)
\(102\) 0 0
\(103\) 4.30789 2.48716i 0.424469 0.245067i −0.272519 0.962151i \(-0.587857\pi\)
0.696988 + 0.717083i \(0.254523\pi\)
\(104\) 1.05290 0.103245
\(105\) 0 0
\(106\) −8.87365 −0.861885
\(107\) −6.35602 + 3.66965i −0.614460 + 0.354759i −0.774709 0.632318i \(-0.782104\pi\)
0.160249 + 0.987077i \(0.448770\pi\)
\(108\) 0 0
\(109\) 1.00000 1.73205i 0.0957826 0.165900i −0.814152 0.580651i \(-0.802798\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.381412 2.61811i 0.0360400 0.247389i
\(113\) 0.184551i 0.0173611i 0.999962 + 0.00868054i \(0.00276314\pi\)
−0.999962 + 0.00868054i \(0.997237\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.58163 + 1.49051i 0.239699 + 0.138390i
\(117\) 0 0
\(118\) 0.904592i 0.0832745i
\(119\) 11.3976 4.53828i 1.04482 0.416024i
\(120\) 0 0
\(121\) 14.5366 + 25.1781i 1.32151 + 2.28891i
\(122\) 5.08673 8.81047i 0.460531 0.797663i
\(123\) 0 0
\(124\) 5.46948 3.15781i 0.491174 0.283579i
\(125\) 0 0
\(126\) 0 0
\(127\) −6.70276 −0.594774 −0.297387 0.954757i \(-0.596115\pi\)
−0.297387 + 0.954757i \(0.596115\pi\)
\(128\) 0.866025 0.500000i 0.0765466 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.24116 + 3.88180i 0.195811 + 0.339155i 0.947166 0.320743i \(-0.103933\pi\)
−0.751355 + 0.659898i \(0.770599\pi\)
\(132\) 0 0
\(133\) −16.1777 2.35680i −1.40278 0.204360i
\(134\) 9.83224i 0.849376i
\(135\) 0 0
\(136\) 4.01562 + 2.31842i 0.344337 + 0.198803i
\(137\) 15.2098 + 8.78137i 1.29946 + 0.750243i 0.980311 0.197461i \(-0.0632697\pi\)
0.319149 + 0.947705i \(0.396603\pi\)
\(138\) 0 0
\(139\) 6.02268i 0.510837i 0.966831 + 0.255418i \(0.0822132\pi\)
−0.966831 + 0.255418i \(0.917787\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.21412 + 12.4952i 0.605396 + 1.04858i
\(143\) 3.33260 5.77223i 0.278686 0.482698i
\(144\) 0 0
\(145\) 0 0
\(146\) 8.36666 0.692430
\(147\) 0 0
\(148\) 3.47982 0.286039
\(149\) 6.06548 3.50191i 0.496903 0.286887i −0.230530 0.973065i \(-0.574046\pi\)
0.727434 + 0.686178i \(0.240713\pi\)
\(150\) 0 0
\(151\) −6.43540 + 11.1464i −0.523706 + 0.907085i 0.475913 + 0.879492i \(0.342118\pi\)
−0.999619 + 0.0275929i \(0.991216\pi\)
\(152\) −3.08957 5.35129i −0.250597 0.434047i
\(153\) 0 0
\(154\) −13.1459 10.3778i −1.05932 0.836263i
\(155\) 0 0
\(156\) 0 0
\(157\) −9.56427 5.52193i −0.763312 0.440698i 0.0671718 0.997741i \(-0.478602\pi\)
−0.830484 + 0.557043i \(0.811936\pi\)
\(158\) 4.73340 + 2.73283i 0.376569 + 0.217412i
\(159\) 0 0
\(160\) 0 0
\(161\) 18.4439 + 2.68694i 1.45358 + 0.211760i
\(162\) 0 0
\(163\) 3.62287 + 6.27500i 0.283765 + 0.491496i 0.972309 0.233699i \(-0.0750830\pi\)
−0.688544 + 0.725195i \(0.741750\pi\)
\(164\) 3.48504 6.03626i 0.272136 0.471353i
\(165\) 0 0
\(166\) 12.6697 7.31483i 0.983356 0.567741i
\(167\) 13.8606 1.07257 0.536283 0.844038i \(-0.319828\pi\)
0.536283 + 0.844038i \(0.319828\pi\)
\(168\) 0 0
\(169\) 11.8914 0.914724
\(170\) 0 0
\(171\) 0 0
\(172\) −1.29325 + 2.23997i −0.0986092 + 0.170796i
\(173\) −0.479402 0.830348i −0.0364482 0.0631302i 0.847226 0.531233i \(-0.178271\pi\)
−0.883674 + 0.468103i \(0.844938\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.33033i 0.477167i
\(177\) 0 0
\(178\) −3.32522 1.91982i −0.249236 0.143896i
\(179\) 20.2531 + 11.6931i 1.51379 + 0.873984i 0.999870 + 0.0161436i \(0.00513888\pi\)
0.513916 + 0.857841i \(0.328194\pi\)
\(180\) 0 0
\(181\) 19.2865i 1.43355i 0.697303 + 0.716776i \(0.254383\pi\)
−0.697303 + 0.716776i \(0.745617\pi\)
\(182\) −0.401588 + 2.75661i −0.0297677 + 0.204333i
\(183\) 0 0
\(184\) 3.52236 + 6.10091i 0.259672 + 0.449765i
\(185\) 0 0
\(186\) 0 0
\(187\) 25.4202 14.6764i 1.85891 1.07324i
\(188\) −8.15421 −0.594707
\(189\) 0 0
\(190\) 0 0
\(191\) 2.44949 1.41421i 0.177239 0.102329i −0.408756 0.912644i \(-0.634037\pi\)
0.585995 + 0.810315i \(0.300704\pi\)
\(192\) 0 0
\(193\) 4.10862 7.11634i 0.295745 0.512246i −0.679413 0.733756i \(-0.737765\pi\)
0.975158 + 0.221511i \(0.0710987\pi\)
\(194\) 2.93946 + 5.09129i 0.211041 + 0.365533i
\(195\) 0 0
\(196\) 6.70905 + 1.99716i 0.479218 + 0.142654i
\(197\) 3.51321i 0.250306i −0.992137 0.125153i \(-0.960058\pi\)
0.992137 0.125153i \(-0.0399422\pi\)
\(198\) 0 0
\(199\) 3.00000 + 1.73205i 0.212664 + 0.122782i 0.602549 0.798082i \(-0.294152\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.70816i 0.471985i
\(203\) −4.88698 + 6.19052i −0.342999 + 0.434489i
\(204\) 0 0
\(205\) 0 0
\(206\) −2.48716 + 4.30789i −0.173289 + 0.300145i
\(207\) 0 0
\(208\) −0.911836 + 0.526449i −0.0632245 + 0.0365027i
\(209\) −39.1160 −2.70571
\(210\) 0 0
\(211\) −9.96592 −0.686082 −0.343041 0.939320i \(-0.611457\pi\)
−0.343041 + 0.939320i \(0.611457\pi\)
\(212\) 7.68480 4.43682i 0.527795 0.304722i
\(213\) 0 0
\(214\) 3.66965 6.35602i 0.250852 0.434489i
\(215\) 0 0
\(216\) 0 0
\(217\) 6.18138 + 15.5242i 0.419619 + 1.05385i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −4.22804 2.44106i −0.284408 0.164203i
\(222\) 0 0
\(223\) 11.1087i 0.743894i 0.928254 + 0.371947i \(0.121310\pi\)
−0.928254 + 0.371947i \(0.878690\pi\)
\(224\) 0.978745 + 2.45806i 0.0653952 + 0.164236i
\(225\) 0 0
\(226\) −0.0922754 0.159826i −0.00613807 0.0106314i
\(227\) −11.7619 + 20.3722i −0.780665 + 1.35215i 0.150889 + 0.988551i \(0.451786\pi\)
−0.931555 + 0.363602i \(0.881547\pi\)
\(228\) 0 0
\(229\) 8.21579 4.74339i 0.542915 0.313452i −0.203345 0.979107i \(-0.565181\pi\)
0.746259 + 0.665655i \(0.231848\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.98101 −0.195713
\(233\) 8.98361 5.18669i 0.588536 0.339791i −0.175982 0.984393i \(-0.556310\pi\)
0.764518 + 0.644602i \(0.222977\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.452296 0.783400i −0.0294420 0.0509950i
\(237\) 0 0
\(238\) −7.60149 + 9.62908i −0.492732 + 0.624161i
\(239\) 12.3746i 0.800444i −0.916418 0.400222i \(-0.868933\pi\)
0.916418 0.400222i \(-0.131067\pi\)
\(240\) 0 0
\(241\) −10.8208 6.24737i −0.697027 0.402429i 0.109212 0.994018i \(-0.465167\pi\)
−0.806239 + 0.591590i \(0.798501\pi\)
\(242\) −25.1781 14.5366i −1.61851 0.934445i
\(243\) 0 0
\(244\) 10.1735i 0.651289i
\(245\) 0 0
\(246\) 0 0
\(247\) 3.25300 + 5.63436i 0.206983 + 0.358506i
\(248\) −3.15781 + 5.46948i −0.200521 + 0.347312i
\(249\) 0 0
\(250\) 0 0
\(251\) −13.5304 −0.854034 −0.427017 0.904244i \(-0.640436\pi\)
−0.427017 + 0.904244i \(0.640436\pi\)
\(252\) 0 0
\(253\) 44.5954 2.80369
\(254\) 5.80476 3.35138i 0.364223 0.210284i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) 2.20899 + 3.82609i 0.137793 + 0.238665i 0.926661 0.375898i \(-0.122666\pi\)
−0.788868 + 0.614563i \(0.789332\pi\)
\(258\) 0 0
\(259\) −1.32724 + 9.11056i −0.0824709 + 0.566103i
\(260\) 0 0
\(261\) 0 0
\(262\) −3.88180 2.24116i −0.239819 0.138459i
\(263\) 5.30558 + 3.06318i 0.327156 + 0.188884i 0.654578 0.755995i \(-0.272846\pi\)
−0.327422 + 0.944878i \(0.606180\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 15.1887 6.04780i 0.931278 0.370814i
\(267\) 0 0
\(268\) −4.91612 8.51497i −0.300300 0.520135i
\(269\) 2.09370 3.62639i 0.127655 0.221105i −0.795113 0.606462i \(-0.792588\pi\)
0.922768 + 0.385357i \(0.125922\pi\)
\(270\) 0 0
\(271\) 0.253692 0.146469i 0.0154107 0.00889737i −0.492275 0.870440i \(-0.663835\pi\)
0.507686 + 0.861542i \(0.330501\pi\)
\(272\) −4.63684 −0.281150
\(273\) 0 0
\(274\) −17.5627 −1.06100
\(275\) 0 0
\(276\) 0 0
\(277\) −7.31099 + 12.6630i −0.439275 + 0.760847i −0.997634 0.0687531i \(-0.978098\pi\)
0.558359 + 0.829600i \(0.311431\pi\)
\(278\) −3.01134 5.21579i −0.180608 0.312822i
\(279\) 0 0
\(280\) 0 0
\(281\) 18.1691i 1.08388i −0.840419 0.541938i \(-0.817691\pi\)
0.840419 0.541938i \(-0.182309\pi\)
\(282\) 0 0
\(283\) 18.3712 + 10.6066i 1.09206 + 0.630499i 0.934123 0.356951i \(-0.116184\pi\)
0.157933 + 0.987450i \(0.449517\pi\)
\(284\) −12.4952 7.21412i −0.741455 0.428079i
\(285\) 0 0
\(286\) 6.66519i 0.394121i
\(287\) 14.4744 + 11.4265i 0.854396 + 0.674486i
\(288\) 0 0
\(289\) −2.25013 3.89734i −0.132361 0.229256i
\(290\) 0 0
\(291\) 0 0
\(292\) −7.24574 + 4.18333i −0.424025 + 0.244811i
\(293\) 7.64481 0.446615 0.223307 0.974748i \(-0.428315\pi\)
0.223307 + 0.974748i \(0.428315\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.01361 + 1.73991i −0.175163 + 0.101130i
\(297\) 0 0
\(298\) −3.50191 + 6.06548i −0.202860 + 0.351364i
\(299\) −3.70868 6.42363i −0.214479 0.371488i
\(300\) 0 0
\(301\) −5.37124 4.24022i −0.309593 0.244402i
\(302\) 12.8708i 0.740632i
\(303\) 0 0
\(304\) 5.35129 + 3.08957i 0.306917 + 0.177199i
\(305\) 0 0
\(306\) 0 0
\(307\) 13.1561i 0.750859i −0.926851 0.375429i \(-0.877495\pi\)
0.926851 0.375429i \(-0.122505\pi\)
\(308\) 16.5735 + 2.41446i 0.944365 + 0.137577i
\(309\) 0 0
\(310\) 0 0
\(311\) 3.13043 5.42207i 0.177511 0.307457i −0.763517 0.645788i \(-0.776529\pi\)
0.941027 + 0.338331i \(0.109862\pi\)
\(312\) 0 0
\(313\) 15.1392 8.74065i 0.855721 0.494051i −0.00685609 0.999976i \(-0.502182\pi\)
0.862577 + 0.505926i \(0.168849\pi\)
\(314\) 11.0439 0.623241
\(315\) 0 0
\(316\) −5.46566 −0.307467
\(317\) 1.32832 0.766906i 0.0746058 0.0430737i −0.462233 0.886758i \(-0.652952\pi\)
0.536839 + 0.843685i \(0.319618\pi\)
\(318\) 0 0
\(319\) −9.43540 + 16.3426i −0.528281 + 0.915010i
\(320\) 0 0
\(321\) 0 0
\(322\) −17.3163 + 6.89498i −0.965002 + 0.384242i
\(323\) 28.6516i 1.59422i
\(324\) 0 0
\(325\) 0 0
\(326\) −6.27500 3.62287i −0.347540 0.200652i
\(327\) 0 0
\(328\) 6.97007i 0.384858i
\(329\) 3.11011 21.3487i 0.171466 1.17699i
\(330\) 0 0
\(331\) 0.756607 + 1.31048i 0.0415869 + 0.0720306i 0.886070 0.463552i \(-0.153425\pi\)
−0.844483 + 0.535583i \(0.820092\pi\)
\(332\) −7.31483 + 12.6697i −0.401453 + 0.695337i
\(333\) 0 0
\(334\) −12.0036 + 6.93030i −0.656810 + 0.379209i
\(335\) 0 0
\(336\) 0 0
\(337\) −3.14064 −0.171082 −0.0855408 0.996335i \(-0.527262\pi\)
−0.0855408 + 0.996335i \(0.527262\pi\)
\(338\) −10.2983 + 5.94570i −0.560152 + 0.323404i
\(339\) 0 0
\(340\) 0 0
\(341\) 19.9900 + 34.6236i 1.08252 + 1.87498i
\(342\) 0 0
\(343\) −7.78771 + 16.8033i −0.420497 + 0.907294i
\(344\) 2.58650i 0.139455i
\(345\) 0 0
\(346\) 0.830348 + 0.479402i 0.0446398 + 0.0257728i
\(347\) −23.7336 13.7026i −1.27408 0.735593i −0.298330 0.954463i \(-0.596430\pi\)
−0.975754 + 0.218870i \(0.929763\pi\)
\(348\) 0 0
\(349\) 19.5594i 1.04699i 0.852028 + 0.523496i \(0.175373\pi\)
−0.852028 + 0.523496i \(0.824627\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.16517 + 5.48223i 0.168704 + 0.292204i
\(353\) −7.40515 + 12.8261i −0.394136 + 0.682664i −0.992991 0.118194i \(-0.962290\pi\)
0.598854 + 0.800858i \(0.295623\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.83964 0.203500
\(357\) 0 0
\(358\) −23.3862 −1.23600
\(359\) −14.3492 + 8.28450i −0.757321 + 0.437239i −0.828333 0.560236i \(-0.810710\pi\)
0.0710122 + 0.997475i \(0.477377\pi\)
\(360\) 0 0
\(361\) 9.59086 16.6119i 0.504782 0.874308i
\(362\) −9.64324 16.7026i −0.506837 0.877868i
\(363\) 0 0
\(364\) −1.03052 2.58809i −0.0540138 0.135653i
\(365\) 0 0
\(366\) 0 0
\(367\) 12.1357 + 7.00655i 0.633479 + 0.365739i 0.782098 0.623155i \(-0.214150\pi\)
−0.148619 + 0.988894i \(0.547483\pi\)
\(368\) −6.10091 3.52236i −0.318032 0.183616i
\(369\) 0 0
\(370\) 0 0
\(371\) 8.68504 + 21.8120i 0.450905 + 1.13242i
\(372\) 0 0
\(373\) −7.31099 12.6630i −0.378549 0.655666i 0.612303 0.790624i \(-0.290243\pi\)
−0.990851 + 0.134958i \(0.956910\pi\)
\(374\) −14.6764 + 25.4202i −0.758897 + 1.31445i
\(375\) 0 0
\(376\) 7.06176 4.07711i 0.364182 0.210261i
\(377\) 3.13870 0.161651
\(378\) 0 0
\(379\) 4.68145 0.240470 0.120235 0.992745i \(-0.461635\pi\)
0.120235 + 0.992745i \(0.461635\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.41421 + 2.44949i −0.0723575 + 0.125327i
\(383\) 5.83621 + 10.1086i 0.298216 + 0.516526i 0.975728 0.218986i \(-0.0702749\pi\)
−0.677511 + 0.735512i \(0.736942\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.21725i 0.418247i
\(387\) 0 0
\(388\) −5.09129 2.93946i −0.258471 0.149228i
\(389\) 11.3452 + 6.55018i 0.575226 + 0.332107i 0.759234 0.650818i \(-0.225574\pi\)
−0.184008 + 0.982925i \(0.558907\pi\)
\(390\) 0 0
\(391\) 32.6652i 1.65195i
\(392\) −6.80879 + 1.62493i −0.343896 + 0.0820715i
\(393\) 0 0
\(394\) 1.75661 + 3.04253i 0.0884966 + 0.153281i
\(395\) 0 0
\(396\) 0 0
\(397\) 26.5700 15.3402i 1.33351 0.769903i 0.347675 0.937615i \(-0.386971\pi\)
0.985836 + 0.167712i \(0.0536378\pi\)
\(398\) −3.46410 −0.173640
\(399\) 0 0
\(400\) 0 0
\(401\) −6.88085 + 3.97266i −0.343613 + 0.198385i −0.661869 0.749620i \(-0.730236\pi\)
0.318255 + 0.948005i \(0.396903\pi\)
\(402\) 0 0
\(403\) 3.32485 5.75880i 0.165622 0.286866i
\(404\) 3.35408 + 5.80944i 0.166872 + 0.289030i
\(405\) 0 0
\(406\) 1.13699 7.80464i 0.0564281 0.387338i
\(407\) 22.0284i 1.09191i
\(408\) 0 0
\(409\) −34.4957 19.9161i −1.70570 0.984789i −0.939729 0.341919i \(-0.888923\pi\)
−0.765975 0.642870i \(-0.777744\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.97432i 0.245067i
\(413\) 2.22354 0.885365i 0.109413 0.0435660i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.526449 0.911836i 0.0258113 0.0447064i
\(417\) 0 0
\(418\) 33.8754 19.5580i 1.65690 0.956613i
\(419\) 11.0357 0.539131 0.269566 0.962982i \(-0.413120\pi\)
0.269566 + 0.962982i \(0.413120\pi\)
\(420\) 0 0
\(421\) −7.78421 −0.379379 −0.189690 0.981844i \(-0.560748\pi\)
−0.189690 + 0.981844i \(0.560748\pi\)
\(422\) 8.63074 4.98296i 0.420138 0.242567i
\(423\) 0 0
\(424\) −4.43682 + 7.68480i −0.215471 + 0.373207i
\(425\) 0 0
\(426\) 0 0
\(427\) −26.6353 3.88028i −1.28897 0.187780i
\(428\) 7.33930i 0.354759i
\(429\) 0 0
\(430\) 0 0
\(431\) 8.86273 + 5.11690i 0.426903 + 0.246472i 0.698026 0.716072i \(-0.254062\pi\)
−0.271124 + 0.962545i \(0.587395\pi\)
\(432\) 0 0
\(433\) 40.3902i 1.94103i −0.241047 0.970513i \(-0.577491\pi\)
0.241047 0.970513i \(-0.422509\pi\)
\(434\) −13.1153 10.3536i −0.629555 0.496990i
\(435\) 0 0
\(436\) −1.00000 1.73205i −0.0478913 0.0829502i
\(437\) −21.7651 + 37.6983i −1.04117 + 1.80336i
\(438\) 0 0
\(439\) 11.2331 6.48543i 0.536126 0.309533i −0.207381 0.978260i \(-0.566494\pi\)
0.743508 + 0.668728i \(0.233161\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.88212 0.232219
\(443\) −7.49547 + 4.32751i −0.356120 + 0.205606i −0.667378 0.744720i \(-0.732583\pi\)
0.311257 + 0.950326i \(0.399250\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5.55435 9.62042i −0.263006 0.455540i
\(447\) 0 0
\(448\) −2.07665 1.63937i −0.0981124 0.0774529i
\(449\) 25.4407i 1.20062i 0.799768 + 0.600310i \(0.204956\pi\)
−0.799768 + 0.600310i \(0.795044\pi\)
\(450\) 0 0
\(451\) 38.2115 + 22.0614i 1.79931 + 1.03883i
\(452\) 0.159826 + 0.0922754i 0.00751757 + 0.00434027i
\(453\) 0 0
\(454\) 23.5238i 1.10403i
\(455\) 0 0
\(456\) 0 0
\(457\) 15.2697 + 26.4480i 0.714289 + 1.23718i 0.963233 + 0.268666i \(0.0865829\pi\)
−0.248945 + 0.968518i \(0.580084\pi\)
\(458\) −4.74339 + 8.21579i −0.221644 + 0.383899i
\(459\) 0 0
\(460\) 0 0
\(461\) −15.6680 −0.729734 −0.364867 0.931060i \(-0.618886\pi\)
−0.364867 + 0.931060i \(0.618886\pi\)
\(462\) 0 0
\(463\) −32.0462 −1.48931 −0.744656 0.667448i \(-0.767387\pi\)
−0.744656 + 0.667448i \(0.767387\pi\)
\(464\) 2.58163 1.49051i 0.119849 0.0691950i
\(465\) 0 0
\(466\) −5.18669 + 8.98361i −0.240269 + 0.416158i
\(467\) −0.349579 0.605489i −0.0161766 0.0280187i 0.857824 0.513944i \(-0.171816\pi\)
−0.874000 + 0.485925i \(0.838483\pi\)
\(468\) 0 0
\(469\) 24.1682 9.62326i 1.11599 0.444361i
\(470\) 0 0
\(471\) 0 0
\(472\) 0.783400 + 0.452296i 0.0360589 + 0.0208186i
\(473\) −14.1798 8.18669i −0.651986 0.376424i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.76855 12.1398i 0.0810611 0.556426i
\(477\) 0 0
\(478\) 6.18728 + 10.7167i 0.283000 + 0.490170i
\(479\) 4.27280 7.40071i 0.195229 0.338147i −0.751746 0.659452i \(-0.770788\pi\)
0.946976 + 0.321305i \(0.104122\pi\)
\(480\) 0 0
\(481\) 3.17302 1.83195i 0.144677 0.0835295i
\(482\) 12.4947 0.569120
\(483\) 0 0
\(484\) 29.0731 1.32151
\(485\) 0 0
\(486\) 0 0
\(487\) −11.9089 + 20.6268i −0.539643 + 0.934689i 0.459280 + 0.888292i \(0.348108\pi\)
−0.998923 + 0.0463978i \(0.985226\pi\)
\(488\) −5.08673 8.81047i −0.230265 0.398831i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.9104i 0.582638i −0.956626 0.291319i \(-0.905906\pi\)
0.956626 0.291319i \(-0.0940941\pi\)
\(492\) 0 0
\(493\) 11.9706 + 6.91124i 0.539129 + 0.311267i
\(494\) −5.63436 3.25300i −0.253502 0.146359i
\(495\) 0 0
\(496\) 6.31561i 0.283579i
\(497\) 23.6532 29.9624i 1.06099 1.34400i
\(498\) 0 0
\(499\) −8.16823 14.1478i −0.365660 0.633342i 0.623222 0.782045i \(-0.285823\pi\)
−0.988882 + 0.148703i \(0.952490\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 11.7177 6.76522i 0.522987 0.301947i
\(503\) −28.3413 −1.26368 −0.631838 0.775100i \(-0.717699\pi\)
−0.631838 + 0.775100i \(0.717699\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −38.6208 + 22.2977i −1.71690 + 0.991254i
\(507\) 0 0
\(508\) −3.35138 + 5.80476i −0.148693 + 0.257545i
\(509\) 1.18911 + 2.05959i 0.0527062 + 0.0912898i 0.891175 0.453660i \(-0.149882\pi\)
−0.838469 + 0.544950i \(0.816549\pi\)
\(510\) 0 0
\(511\) −8.18883 20.5658i −0.362253 0.909776i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −3.82609 2.20899i −0.168762 0.0974346i
\(515\) 0 0
\(516\) 0 0
\(517\) 51.6189i 2.27020i
\(518\) −3.40585 8.55360i −0.149645 0.375824i
\(519\) 0 0
\(520\) 0 0
\(521\) 4.14185 7.17389i 0.181458 0.314294i −0.760919 0.648846i \(-0.775252\pi\)
0.942377 + 0.334552i \(0.108585\pi\)
\(522\) 0 0
\(523\) −36.2532 + 20.9308i −1.58524 + 0.915239i −0.591166 + 0.806550i \(0.701332\pi\)
−0.994076 + 0.108689i \(0.965335\pi\)
\(524\) 4.48232 0.195811
\(525\) 0 0
\(526\) −6.12635 −0.267122
\(527\) 25.3611 14.6422i 1.10475 0.637826i
\(528\) 0 0
\(529\) 13.3140 23.0606i 0.578871 1.00263i
\(530\) 0 0
\(531\) 0 0
\(532\) −10.1299 + 12.8319i −0.439186 + 0.556333i
\(533\) 7.33877i 0.317877i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.51497 + 4.91612i 0.367791 + 0.212344i
\(537\) 0 0
\(538\) 4.18740i 0.180531i
\(539\) −12.6427 + 42.4705i −0.544559 + 1.82933i
\(540\) 0 0
\(541\) −22.9652 39.7769i −0.987352 1.71014i −0.630978 0.775801i \(-0.717346\pi\)
−0.356374 0.934343i \(-0.615987\pi\)
\(542\) −0.146469 + 0.253692i −0.00629139 + 0.0108970i
\(543\) 0 0
\(544\) 4.01562 2.31842i 0.172168 0.0994014i
\(545\) 0 0
\(546\) 0 0
\(547\) −4.88688 −0.208948 −0.104474 0.994528i \(-0.533316\pi\)
−0.104474 + 0.994528i \(0.533316\pi\)
\(548\) 15.2098 8.78137i 0.649730 0.375122i
\(549\) 0 0
\(550\) 0 0
\(551\) −9.21004 15.9523i −0.392361 0.679589i
\(552\) 0 0
\(553\) 2.08467 14.3097i 0.0886490 0.608511i
\(554\) 14.6220i 0.621229i
\(555\) 0 0
\(556\) 5.21579 + 3.01134i 0.221199 + 0.127709i
\(557\) 25.4332 + 14.6839i 1.07764 + 0.622175i 0.930258 0.366905i \(-0.119583\pi\)
0.147380 + 0.989080i \(0.452916\pi\)
\(558\) 0 0
\(559\) 2.72332i 0.115184i
\(560\) 0 0
\(561\) 0 0
\(562\) 9.08453 + 15.7349i 0.383208 + 0.663735i
\(563\) 1.36961 2.37223i 0.0577220 0.0999775i −0.835720 0.549155i \(-0.814950\pi\)
0.893442 + 0.449178i \(0.148283\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −21.2133 −0.891660
\(567\) 0 0
\(568\) 14.4282 0.605396
\(569\) −0.798865 + 0.461225i −0.0334902 + 0.0193356i −0.516652 0.856196i \(-0.672822\pi\)
0.483161 + 0.875531i \(0.339488\pi\)
\(570\) 0 0
\(571\) 12.2761 21.2629i 0.513740 0.889824i −0.486133 0.873885i \(-0.661593\pi\)
0.999873 0.0159389i \(-0.00507372\pi\)
\(572\) −3.33260 5.77223i −0.139343 0.241349i
\(573\) 0 0
\(574\) −18.2484 2.65847i −0.761675 0.110962i
\(575\) 0 0
\(576\) 0 0
\(577\) −23.4982 13.5667i −0.978241 0.564788i −0.0765024 0.997069i \(-0.524375\pi\)
−0.901739 + 0.432282i \(0.857709\pi\)
\(578\) 3.89734 + 2.25013i 0.162108 + 0.0935932i
\(579\) 0 0
\(580\) 0 0
\(581\) −30.3806 23.9834i −1.26040 0.995000i
\(582\) 0 0
\(583\) 28.0866 + 48.6474i 1.16323 + 2.01477i
\(584\) 4.18333 7.24574i 0.173108 0.299831i
\(585\) 0 0
\(586\) −6.62060 + 3.82240i −0.273494 + 0.157902i
\(587\) −8.71353 −0.359646 −0.179823 0.983699i \(-0.557552\pi\)
−0.179823 + 0.983699i \(0.557552\pi\)
\(588\) 0 0
\(589\) −39.0250 −1.60800
\(590\) 0 0
\(591\) 0 0
\(592\) 1.73991 3.01361i 0.0715098 0.123859i
\(593\) 9.13432 + 15.8211i 0.375102 + 0.649695i 0.990342 0.138644i \(-0.0442744\pi\)
−0.615241 + 0.788339i \(0.710941\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.00381i 0.286887i
\(597\) 0 0
\(598\) 6.42363 + 3.70868i 0.262682 + 0.151659i
\(599\) −9.94576 5.74219i −0.406373 0.234619i 0.282857 0.959162i \(-0.408718\pi\)
−0.689230 + 0.724543i \(0.742051\pi\)
\(600\) 0 0
\(601\) 34.4022i 1.40329i 0.712525 + 0.701647i \(0.247552\pi\)
−0.712525 + 0.701647i \(0.752448\pi\)
\(602\) 6.77174 + 0.986520i 0.275996 + 0.0402076i
\(603\) 0 0
\(604\) 6.43540 + 11.1464i 0.261853 + 0.453543i
\(605\) 0 0
\(606\) 0 0
\(607\) 39.4152 22.7564i 1.59981 0.923653i 0.608293 0.793713i \(-0.291855\pi\)
0.991522 0.129941i \(-0.0414787\pi\)
\(608\) −6.17914 −0.250597
\(609\) 0 0
\(610\) 0 0
\(611\) −7.43531 + 4.29278i −0.300800 + 0.173667i
\(612\) 0 0
\(613\) −5.85205 + 10.1361i −0.236362 + 0.409391i −0.959668 0.281137i \(-0.909289\pi\)
0.723306 + 0.690528i \(0.242622\pi\)
\(614\) 6.57805 + 11.3935i 0.265469 + 0.459805i
\(615\) 0 0
\(616\) −15.5603 + 6.19578i −0.626944 + 0.249635i
\(617\) 24.0582i 0.968547i 0.874917 + 0.484273i \(0.160916\pi\)
−0.874917 + 0.484273i \(0.839084\pi\)
\(618\) 0 0
\(619\) −3.32794 1.92139i −0.133761 0.0772271i 0.431626 0.902053i \(-0.357940\pi\)
−0.565387 + 0.824825i \(0.691273\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.26087i 0.251038i
\(623\) −1.46448 + 10.0526i −0.0586733 + 0.402749i
\(624\) 0 0
\(625\) 0 0
\(626\) −8.74065 + 15.1392i −0.349347 + 0.605086i
\(627\) 0 0
\(628\) −9.56427 + 5.52193i −0.381656 + 0.220349i
\(629\) 16.1353 0.643358
\(630\) 0 0
\(631\) −25.4387 −1.01270 −0.506349 0.862328i \(-0.669005\pi\)
−0.506349 + 0.862328i \(0.669005\pi\)
\(632\) 4.73340 2.73283i 0.188284 0.108706i
\(633\) 0 0
\(634\) −0.766906 + 1.32832i −0.0304577 + 0.0527543i
\(635\) 0 0
\(636\) 0 0
\(637\) 7.16896 1.71089i 0.284044 0.0677879i
\(638\) 18.8708i 0.747102i
\(639\) 0 0
\(640\) 0 0
\(641\) −34.2380 19.7673i −1.35232 0.780762i −0.363745 0.931499i \(-0.618502\pi\)
−0.988574 + 0.150737i \(0.951835\pi\)
\(642\) 0 0
\(643\) 12.2816i 0.484341i −0.970234 0.242170i \(-0.922141\pi\)
0.970234 0.242170i \(-0.0778593\pi\)
\(644\) 11.5489 14.6294i 0.455090 0.576479i
\(645\) 0 0
\(646\) −14.3258 24.8131i −0.563642 0.976257i
\(647\) −6.69470 + 11.5956i −0.263196 + 0.455869i −0.967089 0.254437i \(-0.918110\pi\)
0.703893 + 0.710306i \(0.251443\pi\)
\(648\) 0 0
\(649\) 4.95918 2.86319i 0.194665 0.112390i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.24574 0.283765
\(653\) 42.9355 24.7888i 1.68020 0.970062i 0.718670 0.695352i \(-0.244751\pi\)
0.961527 0.274710i \(-0.0885819\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.48504 6.03626i −0.136068 0.235676i
\(657\) 0 0
\(658\) 7.98090 + 20.0435i 0.311128 + 0.781379i
\(659\) 40.7622i 1.58787i −0.608002 0.793936i \(-0.708029\pi\)
0.608002 0.793936i \(-0.291971\pi\)
\(660\) 0 0
\(661\) −30.9652 17.8778i −1.20441 0.695365i −0.242875 0.970057i \(-0.578091\pi\)
−0.961532 + 0.274692i \(0.911424\pi\)
\(662\) −1.31048 0.756607i −0.0509333 0.0294064i
\(663\) 0 0
\(664\) 14.6297i 0.567741i
\(665\) 0 0
\(666\) 0 0
\(667\) 10.5002 + 18.1869i 0.406569 + 0.704199i
\(668\) 6.93030 12.0036i 0.268141 0.464435i
\(669\) 0 0
\(670\) 0 0
\(671\) −64.4014 −2.48619
\(672\) 0 0
\(673\) −35.1987 −1.35681 −0.678406 0.734687i \(-0.737329\pi\)
−0.678406 + 0.734687i \(0.737329\pi\)
\(674\) 2.71987 1.57032i 0.104766 0.0604865i
\(675\) 0 0
\(676\) 5.94570 10.2983i 0.228681 0.396087i
\(677\) −19.2211 33.2919i −0.738727 1.27951i −0.953069 0.302754i \(-0.902094\pi\)
0.214341 0.976759i \(-0.431239\pi\)
\(678\) 0 0
\(679\) 9.63771 12.2084i 0.369862 0.468517i
\(680\) 0 0
\(681\) 0 0
\(682\) −34.6236 19.9900i −1.32581 0.765455i
\(683\) −33.2574 19.2012i −1.27256 0.734712i −0.297089 0.954850i \(-0.596016\pi\)
−0.975469 + 0.220138i \(0.929349\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.65731 18.4460i −0.0632764 0.704270i
\(687\) 0 0
\(688\) 1.29325 + 2.23997i 0.0493046 + 0.0853981i
\(689\) 4.67152 8.09131i 0.177971 0.308254i
\(690\) 0 0
\(691\) 17.6209 10.1735i 0.670332 0.387016i −0.125870 0.992047i \(-0.540172\pi\)
0.796203 + 0.605030i \(0.206839\pi\)
\(692\) −0.958803 −0.0364482
\(693\) 0 0
\(694\) 27.4052 1.04029
\(695\) 0 0
\(696\) 0 0
\(697\) 16.1595 27.9892i 0.612086 1.06016i
\(698\) −9.77972 16.9390i −0.370168 0.641149i
\(699\) 0 0
\(700\) 0 0
\(701\) 6.98574i 0.263848i −0.991260 0.131924i \(-0.957885\pi\)
0.991260 0.131924i \(-0.0421154\pi\)
\(702\) 0 0
\(703\) −18.6215 10.7511i −0.702323 0.405487i
\(704\) −5.48223 3.16517i −0.206619 0.119292i
\(705\) 0 0
\(706\) 14.8103i 0.557393i
\(707\) −16.4891 + 6.56558i −0.620135 + 0.246924i
\(708\) 0 0
\(709\) 6.01348 + 10.4157i 0.225841 + 0.391168i 0.956571 0.291498i \(-0.0941537\pi\)
−0.730730 + 0.682666i \(0.760820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.32522 + 1.91982i −0.124618 + 0.0719482i
\(713\) 44.4917 1.66623
\(714\) 0 0
\(715\) 0 0
\(716\) 20.2531 11.6931i 0.756893 0.436992i
\(717\) 0 0
\(718\) 8.28450 14.3492i 0.309175 0.535507i
\(719\) −0.835694 1.44746i −0.0311661 0.0539813i 0.850022 0.526748i \(-0.176589\pi\)
−0.881188 + 0.472766i \(0.843255\pi\)
\(720\) 0 0
\(721\) 13.0233 + 1.89727i 0.485015 + 0.0706579i
\(722\) 19.1817i 0.713869i
\(723\) 0 0
\(724\) 16.7026 + 9.64324i 0.620746 + 0.358388i
\(725\) 0 0
\(726\) 0 0
\(727\) 42.5043i 1.57640i 0.615422 + 0.788198i \(0.288986\pi\)
−0.615422 + 0.788198i \(0.711014\pi\)
\(728\) 2.18650 + 1.72609i 0.0810370 + 0.0639731i
\(729\) 0 0
\(730\) 0 0
\(731\) −5.99658 + 10.3864i −0.221792 + 0.384154i
\(732\) 0 0
\(733\) 27.6053 15.9379i 1.01963 0.588681i 0.105630 0.994405i \(-0.466314\pi\)
0.913996 + 0.405724i \(0.132981\pi\)
\(734\) −14.0131 −0.517233
\(735\) 0 0
\(736\) 7.04472 0.259672
\(737\) 53.9026 31.1207i 1.98553 1.14634i
\(738\) 0 0
\(739\) 8.20932 14.2190i 0.301985 0.523053i −0.674601 0.738183i \(-0.735684\pi\)
0.976585 + 0.215130i \(0.0690176\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −18.4274 14.5472i −0.676493 0.534044i
\(743\) 4.52385i 0.165964i −0.996551 0.0829821i \(-0.973556\pi\)
0.996551 0.0829821i \(-0.0264444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.6630 + 7.31099i 0.463626 + 0.267674i
\(747\) 0 0
\(748\) 29.3527i 1.07324i
\(749\) −19.2151 2.79930i −0.702106 0.102284i
\(750\) 0 0
\(751\) 8.69875 + 15.0667i 0.317422 + 0.549791i 0.979949 0.199247i \(-0.0638495\pi\)
−0.662527 + 0.749038i \(0.730516\pi\)
\(752\) −4.07711 + 7.06176i −0.148677 + 0.257516i
\(753\) 0 0
\(754\) −2.71820 + 1.56935i −0.0989908 + 0.0571524i
\(755\) 0 0
\(756\) 0 0
\(757\) −5.25454 −0.190980 −0.0954898 0.995430i \(-0.530442\pi\)
−0.0954898 + 0.995430i \(0.530442\pi\)
\(758\) −4.05425 + 2.34072i −0.147257 + 0.0850189i
\(759\) 0 0
\(760\) 0 0
\(761\) 7.31402 + 12.6683i 0.265133 + 0.459224i 0.967598 0.252494i \(-0.0812508\pi\)
−0.702465 + 0.711718i \(0.747917\pi\)
\(762\) 0 0
\(763\) 4.91612 1.95749i 0.177975 0.0708659i
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) −10.1086 5.83621i −0.365239 0.210871i
\(767\) −0.824840 0.476222i −0.0297832 0.0171954i
\(768\) 0 0
\(769\) 12.4548i 0.449131i −0.974459 0.224566i \(-0.927904\pi\)
0.974459 0.224566i \(-0.0720963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.10862 7.11634i −0.147873 0.256123i
\(773\) 4.15288 7.19300i 0.149369 0.258714i −0.781626 0.623748i \(-0.785609\pi\)
0.930994 + 0.365034i \(0.118943\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.87891 0.211041
\(777\) 0 0
\(778\) −13.1004 −0.469670
\(779\) −37.2989 + 21.5345i −1.33637 + 0.771554i
\(780\) 0 0
\(781\) 45.6678 79.0990i 1.63412 2.83038i
\(782\) 16.3326 + 28.2889i 0.584053 + 1.01161i
\(783\) 0 0
\(784\) 5.08412 4.81163i 0.181576 0.171844i
\(785\) 0 0
\(786\) 0 0
\(787\) −43.8271 25.3036i −1.56227 0.901975i −0.997027 0.0770479i \(-0.975451\pi\)
−0.565239 0.824927i \(-0.691216\pi\)
\(788\) −3.04253 1.75661i −0.108386 0.0625765i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.302547 + 0.383247i −0.0107573 + 0.0136267i
\(792\) 0 0
\(793\) 5.35580 + 9.27652i 0.190190 + 0.329419i
\(794\) −15.3402 + 26.5700i −0.544404 + 0.942935i
\(795\) 0 0
\(796\) 3.00000 1.73205i 0.106332 0.0613909i
\(797\) −36.6354 −1.29769 −0.648846 0.760920i \(-0.724748\pi\)
−0.648846 + 0.760920i \(0.724748\pi\)
\(798\) 0 0
\(799\) −37.8098 −1.33761
\(800\) 0 0
\(801\) 0 0
\(802\) 3.97266 6.88085i 0.140280 0.242971i
\(803\) −26.4819 45.8680i −0.934525 1.61865i
\(804\) 0 0
\(805\) 0 0
\(806\) 6.64969i 0.234225i
\(807\) 0 0
\(808\) −5.80944 3.35408i −0.204375 0.117996i
\(809\) 37.7055 + 21.7693i 1.32566 + 0.765368i 0.984624 0.174685i \(-0.0558906\pi\)
0.341031 + 0.940052i \(0.389224\pi\)
\(810\) 0 0
\(811\) 13.8915i 0.487795i −0.969801 0.243897i \(-0.921574\pi\)
0.969801 0.243897i \(-0.0784260\pi\)
\(812\) 2.91765 + 7.32751i 0.102390 + 0.257145i
\(813\) 0 0
\(814\) −11.0142 19.0772i −0.386048 0.668654i
\(815\) 0 0
\(816\) 0 0
\(817\) 13.8411 7.99115i 0.484238 0.279575i
\(818\) 39.8323 1.39270
\(819\) 0 0
\(820\) 0 0
\(821\) 8.26634 4.77257i 0.288497 0.166564i −0.348767 0.937210i \(-0.613399\pi\)
0.637264 + 0.770646i \(0.280066\pi\)
\(822\) 0 0
\(823\) −5.39798 + 9.34957i −0.188162 + 0.325905i −0.944637 0.328116i \(-0.893586\pi\)
0.756476 + 0.654022i \(0.226920\pi\)
\(824\) 2.48716 + 4.30789i 0.0866444 + 0.150072i
\(825\) 0 0
\(826\) −1.48296 + 1.87852i −0.0515988 + 0.0653621i
\(827\) 10.5599i 0.367204i −0.983001 0.183602i \(-0.941224\pi\)
0.983001 0.183602i \(-0.0587757\pi\)
\(828\) 0 0
\(829\) 30.7288 + 17.7413i 1.06726 + 0.616181i 0.927431 0.373995i \(-0.122012\pi\)
0.139826 + 0.990176i \(0.455346\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.05290i 0.0365027i
\(833\) 31.1088 + 9.26051i 1.07786 + 0.320858i
\(834\) 0 0
\(835\) 0 0
\(836\) −19.5580 + 33.8754i −0.676427 + 1.17161i
\(837\) 0 0
\(838\) −9.55723 + 5.51787i −0.330149 + 0.190612i
\(839\) −32.5932 −1.12524 −0.562621 0.826715i \(-0.690207\pi\)
−0.562621 + 0.826715i \(0.690207\pi\)
\(840\) 0 0
\(841\) 20.1136 0.693571
\(842\) 6.74132 3.89211i 0.232321 0.134131i
\(843\) 0 0
\(844\) −4.98296 + 8.63074i −0.171521 + 0.297082i
\(845\) 0 0
\(846\) 0 0
\(847\) −11.0888 + 76.1168i −0.381017 + 2.61540i
\(848\) 8.87365i 0.304722i
\(849\) 0 0
\(850\) 0 0
\(851\) 21.2300 + 12.2572i 0.727756 + 0.420170i
\(852\) 0 0
\(853\) 20.2310i 0.692695i 0.938106 + 0.346347i \(0.112578\pi\)
−0.938106 + 0.346347i \(0.887422\pi\)
\(854\) 25.0070 9.95722i 0.855721 0.340729i
\(855\) 0 0
\(856\) −3.66965 6.35602i −0.125426 0.217244i
\(857\) 22.4855 38.9461i 0.768091 1.33037i −0.170506 0.985357i \(-0.554540\pi\)
0.938597 0.345016i \(-0.112126\pi\)
\(858\) 0 0
\(859\) 37.8490 21.8521i 1.29139 0.745585i 0.312490 0.949921i \(-0.398837\pi\)
0.978901 + 0.204336i \(0.0655036\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −10.2338 −0.348565
\(863\) −30.8636 + 17.8191i −1.05061 + 0.606569i −0.922819 0.385233i \(-0.874121\pi\)
−0.127788 + 0.991801i \(0.540788\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 20.1951 + 34.9789i 0.686257 + 1.18863i
\(867\) 0 0
\(868\) 16.5350 + 2.40885i 0.561235 + 0.0817617i
\(869\) 34.5994i 1.17371i
\(870\) 0 0
\(871\) −8.96539 5.17617i −0.303781 0.175388i
\(872\) 1.73205 + 1.00000i 0.0586546 + 0.0338643i
\(873\) 0 0
\(874\) 43.5303i 1.47243i
\(875\) 0 0
\(876\) 0 0
\(877\) 18.2453 + 31.6018i 0.616100 + 1.06712i 0.990190 + 0.139724i \(0.0446216\pi\)
−0.374091 + 0.927392i \(0.622045\pi\)
\(878\) −6.48543 + 11.2331i −0.218873 + 0.379099i
\(879\) 0 0
\(880\) 0 0
\(881\) −0.487935 −0.0164390 −0.00821948 0.999966i \(-0.502616\pi\)
−0.00821948 + 0.999966i \(0.502616\pi\)
\(882\) 0 0
\(883\) −6.26720 −0.210908 −0.105454 0.994424i \(-0.533630\pi\)
−0.105454 + 0.994424i \(0.533630\pi\)
\(884\) −4.22804 + 2.44106i −0.142204 + 0.0821017i
\(885\) 0 0
\(886\) 4.32751 7.49547i 0.145386 0.251815i
\(887\) −28.4598 49.2939i −0.955588 1.65513i −0.733017 0.680210i \(-0.761888\pi\)
−0.222571 0.974917i \(-0.571445\pi\)
\(888\) 0 0
\(889\) −13.9193 10.9883i −0.466838 0.368536i
\(890\) 0 0
\(891\) 0 0
\(892\) 9.62042 + 5.55435i 0.322115 + 0.185973i
\(893\) 43.6356 + 25.1930i 1.46021 + 0.843052i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.61811 + 0.381412i 0.0874651 + 0.0127421i
\(897\) 0 0
\(898\) −12.7203 22.0323i −0.424483 0.735226i
\(899\) −9.41346 + 16.3046i −0.313957 + 0.543789i
\(900\) 0 0
\(901\) 35.6332 20.5728i 1.18711 0.685380i
\(902\) −44.1229 −1.46913
\(903\) 0 0
\(904\) −0.184551 −0.00613807
\(905\) 0 0
\(906\) 0 0
\(907\) 8.33031 14.4285i 0.276603 0.479091i −0.693935 0.720038i \(-0.744124\pi\)
0.970538 + 0.240946i \(0.0774578\pi\)
\(908\) 11.7619 + 20.3722i 0.390333 + 0.676076i
\(909\) 0 0
\(910\) 0 0
\(911\) 8.02653i 0.265931i 0.991121 + 0.132965i \(0.0424499\pi\)
−0.991121 + 0.132965i \(0.957550\pi\)
\(912\) 0 0
\(913\) −80.2031 46.3053i −2.65434 1.53248i
\(914\) −26.4480 15.2697i −0.874821 0.505078i
\(915\) 0 0
\(916\) 9.48678i 0.313452i
\(917\) −1.70961 + 11.7352i −0.0564563 + 0.387532i
\(918\) 0 0
\(919\) 6.85733 + 11.8772i 0.226202 + 0.391794i 0.956680 0.291143i \(-0.0940356\pi\)
−0.730477 + 0.682937i \(0.760702\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 13.5689 7.83402i 0.446869 0.258000i
\(923\) −15.1915 −0.500033
\(924\) 0 0
\(925\) 0 0
\(926\) 27.7528 16.0231i 0.912014 0.526551i
\(927\) 0 0
\(928\) −1.49051 + 2.58163i −0.0489283 + 0.0847463i
\(929\) −22.9480 39.7470i −0.752898 1.30406i −0.946413 0.322960i \(-0.895322\pi\)
0.193515 0.981097i \(-0.438011\pi\)
\(930\) 0 0
\(931\) −29.7317 31.4154i −0.974417 1.02960i
\(932\) 10.3734i 0.339791i
\(933\) 0 0
\(934\) 0.605489 + 0.349579i 0.0198122 + 0.0114386i
\(935\) 0 0
\(936\) 0 0
\(937\) 6.17552i 0.201746i 0.994899 + 0.100873i \(0.0321635\pi\)
−0.994899 + 0.100873i \(0.967836\pi\)
\(938\) −16.1187 + 20.4181i −0.526293 + 0.666675i
\(939\) 0 0
\(940\) 0 0
\(941\) −16.7048 + 28.9335i −0.544560 + 0.943206i 0.454074 + 0.890964i \(0.349970\pi\)
−0.998634 + 0.0522424i \(0.983363\pi\)
\(942\) 0 0
\(943\) 42.5237 24.5511i 1.38476 0.799494i
\(944\) −0.904592 −0.0294420
\(945\) 0 0
\(946\) 16.3734 0.532345
\(947\) 5.67563 3.27683i 0.184433 0.106483i −0.404941 0.914343i \(-0.632708\pi\)
0.589374 + 0.807860i \(0.299375\pi\)
\(948\) 0 0
\(949\) −4.40462 + 7.62903i −0.142980 + 0.247649i
\(950\) 0 0
\(951\) 0 0
\(952\) 4.53828 + 11.3976i 0.147087 + 0.369399i
\(953\) 46.5253i 1.50710i −0.657389 0.753551i \(-0.728339\pi\)
0.657389 0.753551i \(-0.271661\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10.7167 6.18728i −0.346602 0.200111i
\(957\) 0 0
\(958\) 8.54561i 0.276096i
\(959\) 17.1894 + 43.1703i 0.555076 + 1.39404i
\(960\) 0 0
\(961\) 4.44349 + 7.69635i 0.143338 + 0.248269i
\(962\) −1.83195 + 3.17302i −0.0590643 + 0.102302i
\(963\) 0 0
\(964\) −10.8208 + 6.24737i −0.348514 + 0.201214i
\(965\) 0 0
\(966\) 0 0
\(967\) −5.93169 −0.190750 −0.0953752 0.995441i \(-0.530405\pi\)
−0.0953752 + 0.995441i \(0.530405\pi\)
\(968\) −25.1781 + 14.5366i −0.809253 + 0.467223i
\(969\) 0 0
\(970\) 0 0
\(971\) −0.170069 0.294568i −0.00545777 0.00945314i 0.863284 0.504719i \(-0.168404\pi\)
−0.868741 + 0.495266i \(0.835071\pi\)
\(972\) 0 0
\(973\) −9.87339 + 12.5070i −0.316526 + 0.400955i
\(974\) 23.8178i 0.763171i
\(975\) 0 0
\(976\) 8.81047 + 5.08673i 0.282016 + 0.162822i
\(977\) −17.9168 10.3443i −0.573210 0.330943i 0.185220 0.982697i \(-0.440700\pi\)
−0.758430 + 0.651754i \(0.774033\pi\)
\(978\) 0 0
\(979\) 24.3062i 0.776829i
\(980\) 0 0
\(981\) 0 0
\(982\) 6.45520 + 11.1807i 0.205994 + 0.356791i
\(983\) −10.6129 + 18.3821i −0.338499 + 0.586297i −0.984151 0.177335i \(-0.943252\pi\)
0.645652 + 0.763632i \(0.276586\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −13.8225 −0.440197
\(987\) 0 0
\(988\) 6.50600 0.206983
\(989\) −15.7800 + 9.11057i −0.501774 + 0.289699i
\(990\) 0 0
\(991\) −29.1034 + 50.4085i −0.924499 + 1.60128i −0.132134 + 0.991232i \(0.542183\pi\)
−0.792365 + 0.610048i \(0.791150\pi\)
\(992\) 3.15781 + 5.46948i 0.100260 + 0.173656i
\(993\) 0 0
\(994\) −5.50310 + 37.7748i −0.174548 + 1.19814i
\(995\) 0 0
\(996\) 0 0
\(997\) −33.1542 19.1416i −1.05000 0.606220i −0.127354 0.991857i \(-0.540648\pi\)
−0.922651 + 0.385637i \(0.873982\pi\)
\(998\) 14.1478 + 8.16823i 0.447841 + 0.258561i
\(999\) 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 3150.2.bf.f.1151.4 32
3.2 odd 2 inner 3150.2.bf.f.1151.14 32
5.2 odd 4 630.2.bo.a.269.6 yes 16
5.3 odd 4 630.2.bo.b.269.1 yes 16
5.4 even 2 inner 3150.2.bf.f.1151.13 32
7.5 odd 6 inner 3150.2.bf.f.1601.16 32
15.2 even 4 630.2.bo.b.269.3 yes 16
15.8 even 4 630.2.bo.a.269.8 yes 16
15.14 odd 2 inner 3150.2.bf.f.1151.3 32
21.5 even 6 inner 3150.2.bf.f.1601.4 32
35.3 even 12 4410.2.d.a.4409.6 16
35.12 even 12 630.2.bo.a.89.8 yes 16
35.17 even 12 4410.2.d.b.4409.5 16
35.18 odd 12 4410.2.d.a.4409.11 16
35.19 odd 6 inner 3150.2.bf.f.1601.3 32
35.32 odd 12 4410.2.d.b.4409.12 16
35.33 even 12 630.2.bo.b.89.3 yes 16
105.17 odd 12 4410.2.d.a.4409.12 16
105.32 even 12 4410.2.d.a.4409.5 16
105.38 odd 12 4410.2.d.b.4409.11 16
105.47 odd 12 630.2.bo.b.89.1 yes 16
105.53 even 12 4410.2.d.b.4409.6 16
105.68 odd 12 630.2.bo.a.89.6 16
105.89 even 6 inner 3150.2.bf.f.1601.15 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.6 16 105.68 odd 12
630.2.bo.a.89.8 yes 16 35.12 even 12
630.2.bo.a.269.6 yes 16 5.2 odd 4
630.2.bo.a.269.8 yes 16 15.8 even 4
630.2.bo.b.89.1 yes 16 105.47 odd 12
630.2.bo.b.89.3 yes 16 35.33 even 12
630.2.bo.b.269.1 yes 16 5.3 odd 4
630.2.bo.b.269.3 yes 16 15.2 even 4
3150.2.bf.f.1151.3 32 15.14 odd 2 inner
3150.2.bf.f.1151.4 32 1.1 even 1 trivial
3150.2.bf.f.1151.13 32 5.4 even 2 inner
3150.2.bf.f.1151.14 32 3.2 odd 2 inner
3150.2.bf.f.1601.3 32 35.19 odd 6 inner
3150.2.bf.f.1601.4 32 21.5 even 6 inner
3150.2.bf.f.1601.15 32 105.89 even 6 inner
3150.2.bf.f.1601.16 32 7.5 odd 6 inner
4410.2.d.a.4409.5 16 105.32 even 12
4410.2.d.a.4409.6 16 35.3 even 12
4410.2.d.a.4409.11 16 35.18 odd 12
4410.2.d.a.4409.12 16 105.17 odd 12
4410.2.d.b.4409.5 16 35.17 even 12
4410.2.d.b.4409.6 16 105.53 even 12
4410.2.d.b.4409.11 16 105.38 odd 12
4410.2.d.b.4409.12 16 35.32 odd 12