Properties

Label 2028.2.q.c.361.1
Level $2028$
Weight $2$
Character 2028.361
Analytic conductor $16.194$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2028,2,Mod(361,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.q (of order \(6\), degree \(2\), 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: \(16.1936615299\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2028.361
Dual form 2028.2.q.c.1837.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +3.46410i q^{5} +(-0.500000 + 0.866025i) q^{9} +(3.00000 - 1.73205i) q^{11} +(-3.00000 + 1.73205i) q^{15} +(-3.00000 + 5.19615i) q^{17} +(6.00000 + 3.46410i) q^{19} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 + 5.19615i) q^{29} -6.92820i q^{31} +(3.00000 + 1.73205i) q^{33} +(-3.00000 + 1.73205i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(-3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(-3.50000 - 6.06218i) q^{49} -6.00000 q^{51} +6.00000 q^{53} +(6.00000 + 10.3923i) q^{55} +6.92820i q^{57} +(3.00000 + 1.73205i) q^{59} +(-5.00000 + 8.66025i) q^{61} +(12.0000 - 6.92820i) q^{67} +(-9.00000 - 5.19615i) q^{71} -6.92820i q^{73} +(-3.50000 - 6.06218i) q^{75} -8.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} -3.46410i q^{83} +(-18.0000 - 10.3923i) q^{85} +(-3.00000 + 5.19615i) q^{87} +(-15.0000 + 8.66025i) q^{89} +(6.00000 - 3.46410i) q^{93} +(-12.0000 + 20.7846i) q^{95} +(-6.00000 - 3.46410i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{9} + 6 q^{11} - 6 q^{15} - 6 q^{17} + 12 q^{19} - 14 q^{25} - 2 q^{27} + 6 q^{29} + 6 q^{33} - 6 q^{41} - 8 q^{43} - 6 q^{45} - 7 q^{49} - 12 q^{51} + 12 q^{53} + 12 q^{55} + 6 q^{59}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.00000 1.73205i 0.904534 0.522233i 0.0258656 0.999665i \(-0.491766\pi\)
0.878668 + 0.477432i \(0.158432\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −3.00000 + 1.73205i −0.774597 + 0.447214i
\(16\) 0 0
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 6.00000 + 3.46410i 1.37649 + 0.794719i 0.991736 0.128298i \(-0.0409513\pi\)
0.384759 + 0.923017i \(0.374285\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) 6.92820i 1.24434i −0.782881 0.622171i \(-0.786251\pi\)
0.782881 0.622171i \(-0.213749\pi\)
\(32\) 0 0
\(33\) 3.00000 + 1.73205i 0.522233 + 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 1.73205i −0.468521 + 0.270501i −0.715621 0.698489i \(-0.753856\pi\)
0.247099 + 0.968990i \(0.420523\pi\)
\(42\) 0 0
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) −3.00000 1.73205i −0.447214 0.258199i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 6.00000 + 10.3923i 0.809040 + 1.40130i
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) 3.00000 + 1.73205i 0.390567 + 0.225494i 0.682406 0.730974i \(-0.260934\pi\)
−0.291839 + 0.956467i \(0.594267\pi\)
\(60\) 0 0
\(61\) −5.00000 + 8.66025i −0.640184 + 1.10883i 0.345207 + 0.938527i \(0.387809\pi\)
−0.985391 + 0.170305i \(0.945525\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 6.92820i 1.46603 0.846415i 0.466755 0.884387i \(-0.345423\pi\)
0.999279 + 0.0379722i \(0.0120898\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.00000 5.19615i −1.06810 0.616670i −0.140441 0.990089i \(-0.544852\pi\)
−0.927663 + 0.373419i \(0.878185\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) −3.50000 6.06218i −0.404145 0.700000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) −18.0000 10.3923i −1.95237 1.12720i
\(86\) 0 0
\(87\) −3.00000 + 5.19615i −0.321634 + 0.557086i
\(88\) 0 0
\(89\) −15.0000 + 8.66025i −1.59000 + 0.917985i −0.596691 + 0.802471i \(0.703518\pi\)
−0.993306 + 0.115514i \(0.963148\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 3.46410i 0.622171 0.359211i
\(94\) 0 0
\(95\) −12.0000 + 20.7846i −1.23117 + 2.13246i
\(96\) 0 0
\(97\) −6.00000 3.46410i −0.609208 0.351726i 0.163448 0.986552i \(-0.447739\pi\)
−0.772655 + 0.634826i \(0.781072\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 + 10.3923i 0.580042 + 1.00466i 0.995474 + 0.0950377i \(0.0302972\pi\)
−0.415432 + 0.909624i \(0.636370\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i −0.943349 0.331801i \(-0.892344\pi\)
0.943349 0.331801i \(-0.107656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 15.5885i 0.846649 1.46644i −0.0375328 0.999295i \(-0.511950\pi\)
0.884182 0.467143i \(-0.154717\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 0.866025i 0.0454545 0.0787296i
\(122\) 0 0
\(123\) −3.00000 1.73205i −0.270501 0.156174i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) −8.00000 13.8564i −0.709885 1.22956i −0.964899 0.262620i \(-0.915413\pi\)
0.255014 0.966937i \(-0.417920\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.46410i 0.298142i
\(136\) 0 0
\(137\) 9.00000 + 5.19615i 0.768922 + 0.443937i 0.832490 0.554040i \(-0.186915\pi\)
−0.0635680 + 0.997978i \(0.520248\pi\)
\(138\) 0 0
\(139\) −4.00000 + 6.92820i −0.339276 + 0.587643i −0.984297 0.176522i \(-0.943515\pi\)
0.645021 + 0.764165i \(0.276849\pi\)
\(140\) 0 0
\(141\) −3.00000 + 1.73205i −0.252646 + 0.145865i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −18.0000 + 10.3923i −1.49482 + 0.863034i
\(146\) 0 0
\(147\) 3.50000 6.06218i 0.288675 0.500000i
\(148\) 0 0
\(149\) 15.0000 + 8.66025i 1.22885 + 0.709476i 0.966789 0.255576i \(-0.0822652\pi\)
0.262059 + 0.965052i \(0.415599\pi\)
\(150\) 0 0
\(151\) 6.92820i 0.563809i 0.959442 + 0.281905i \(0.0909662\pi\)
−0.959442 + 0.281905i \(0.909034\pi\)
\(152\) 0 0
\(153\) −3.00000 5.19615i −0.242536 0.420084i
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 3.00000 + 5.19615i 0.237915 + 0.412082i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.0000 + 10.3923i 1.40987 + 0.813988i 0.995375 0.0960641i \(-0.0306254\pi\)
0.414494 + 0.910052i \(0.363959\pi\)
\(164\) 0 0
\(165\) −6.00000 + 10.3923i −0.467099 + 0.809040i
\(166\) 0 0
\(167\) 15.0000 8.66025i 1.16073 0.670151i 0.209255 0.977861i \(-0.432896\pi\)
0.951480 + 0.307711i \(0.0995628\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −6.00000 + 3.46410i −0.458831 + 0.264906i
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.46410i 0.260378i
\(178\) 0 0
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 12.0000 6.92820i 0.863779 0.498703i −0.00149702 0.999999i \(-0.500477\pi\)
0.865276 + 0.501296i \(0.167143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 8.66025i 1.06871 0.617018i 0.140878 0.990027i \(-0.455007\pi\)
0.927828 + 0.373009i \(0.121674\pi\)
\(198\) 0 0
\(199\) 4.00000 6.92820i 0.283552 0.491127i −0.688705 0.725042i \(-0.741820\pi\)
0.972257 + 0.233915i \(0.0751537\pi\)
\(200\) 0 0
\(201\) 12.0000 + 6.92820i 0.846415 + 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 10.3923i −0.419058 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −2.00000 3.46410i −0.137686 0.238479i 0.788935 0.614477i \(-0.210633\pi\)
−0.926620 + 0.375999i \(0.877300\pi\)
\(212\) 0 0
\(213\) 10.3923i 0.712069i
\(214\) 0 0
\(215\) −24.0000 13.8564i −1.63679 0.944999i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.00000 3.46410i 0.405442 0.234082i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −6.00000 + 3.46410i −0.401790 + 0.231973i −0.687256 0.726415i \(-0.741185\pi\)
0.285466 + 0.958389i \(0.407852\pi\)
\(224\) 0 0
\(225\) 3.50000 6.06218i 0.233333 0.404145i
\(226\) 0 0
\(227\) 9.00000 + 5.19615i 0.597351 + 0.344881i 0.767999 0.640451i \(-0.221253\pi\)
−0.170648 + 0.985332i \(0.554586\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) −4.00000 6.92820i −0.259828 0.450035i
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) 18.0000 + 10.3923i 1.15948 + 0.669427i 0.951180 0.308637i \(-0.0998729\pi\)
0.208302 + 0.978065i \(0.433206\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 21.0000 12.1244i 1.34164 0.774597i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.00000 1.73205i 0.190117 0.109764i
\(250\) 0 0
\(251\) −6.00000 + 10.3923i −0.378717 + 0.655956i −0.990876 0.134778i \(-0.956968\pi\)
0.612159 + 0.790735i \(0.290301\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 20.7846i 1.30158i
\(256\) 0 0
\(257\) 3.00000 + 5.19615i 0.187135 + 0.324127i 0.944294 0.329104i \(-0.106747\pi\)
−0.757159 + 0.653231i \(0.773413\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) −15.0000 8.66025i −0.917985 0.529999i
\(268\) 0 0
\(269\) 15.0000 25.9808i 0.914566 1.58408i 0.107031 0.994256i \(-0.465866\pi\)
0.807535 0.589819i \(-0.200801\pi\)
\(270\) 0 0
\(271\) −24.0000 + 13.8564i −1.45790 + 0.841717i −0.998908 0.0467255i \(-0.985121\pi\)
−0.458988 + 0.888442i \(0.651788\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −21.0000 + 12.1244i −1.26635 + 0.731126i
\(276\) 0 0
\(277\) 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i \(-0.736206\pi\)
0.976231 + 0.216731i \(0.0695395\pi\)
\(278\) 0 0
\(279\) 6.00000 + 3.46410i 0.359211 + 0.207390i
\(280\) 0 0
\(281\) 17.3205i 1.03325i 0.856210 + 0.516627i \(0.172813\pi\)
−0.856210 + 0.516627i \(0.827187\pi\)
\(282\) 0 0
\(283\) 10.0000 + 17.3205i 0.594438 + 1.02960i 0.993626 + 0.112728i \(0.0359589\pi\)
−0.399188 + 0.916869i \(0.630708\pi\)
\(284\) 0 0
\(285\) −24.0000 −1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 6.92820i 0.406138i
\(292\) 0 0
\(293\) −3.00000 1.73205i −0.175262 0.101187i 0.409803 0.912174i \(-0.365598\pi\)
−0.585065 + 0.810987i \(0.698931\pi\)
\(294\) 0 0
\(295\) −6.00000 + 10.3923i −0.349334 + 0.605063i
\(296\) 0 0
\(297\) −3.00000 + 1.73205i −0.174078 + 0.100504i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.00000 5.19615i 0.172345 0.298511i
\(304\) 0 0
\(305\) −30.0000 17.3205i −1.71780 0.991769i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −2.00000 3.46410i −0.113776 0.197066i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.2487i 1.36194i −0.732310 0.680972i \(-0.761558\pi\)
0.732310 0.680972i \(-0.238442\pi\)
\(318\) 0 0
\(319\) 18.0000 + 10.3923i 1.00781 + 0.581857i
\(320\) 0 0
\(321\) −6.00000 + 10.3923i −0.334887 + 0.580042i
\(322\) 0 0
\(323\) −36.0000 + 20.7846i −2.00309 + 1.15649i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.00000 3.46410i 0.331801 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 6.92820i −0.659580 0.380808i 0.132537 0.991178i \(-0.457688\pi\)
−0.792117 + 0.610370i \(0.791021\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.0000 + 41.5692i 1.31126 + 2.27117i
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −12.0000 20.7846i −0.649836 1.12555i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.00000 5.19615i 0.479022 0.276563i −0.240987 0.970528i \(-0.577471\pi\)
0.720009 + 0.693965i \(0.244138\pi\)
\(354\) 0 0
\(355\) 18.0000 31.1769i 0.955341 1.65470i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.2487i 1.27980i −0.768459 0.639899i \(-0.778976\pi\)
0.768459 0.639899i \(-0.221024\pi\)
\(360\) 0 0
\(361\) 14.5000 + 25.1147i 0.763158 + 1.32183i
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) 14.0000 + 24.2487i 0.730794 + 1.26577i 0.956544 + 0.291587i \(0.0941834\pi\)
−0.225750 + 0.974185i \(0.572483\pi\)
\(368\) 0 0
\(369\) 3.46410i 0.180334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.00000 1.73205i 0.0517780 0.0896822i −0.838975 0.544170i \(-0.816844\pi\)
0.890753 + 0.454488i \(0.150178\pi\)
\(374\) 0 0
\(375\) 6.00000 3.46410i 0.309839 0.178885i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 18.0000 10.3923i 0.924598 0.533817i 0.0394989 0.999220i \(-0.487424\pi\)
0.885099 + 0.465403i \(0.154091\pi\)
\(380\) 0 0
\(381\) 8.00000 13.8564i 0.409852 0.709885i
\(382\) 0 0
\(383\) 21.0000 + 12.1244i 1.07305 + 0.619526i 0.929013 0.370047i \(-0.120658\pi\)
0.144037 + 0.989572i \(0.453992\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 6.92820i −0.203331 0.352180i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −6.00000 10.3923i −0.302660 0.524222i
\(394\) 0 0
\(395\) 27.7128i 1.39438i
\(396\) 0 0
\(397\) −12.0000 6.92820i −0.602263 0.347717i 0.167668 0.985843i \(-0.446376\pi\)
−0.769931 + 0.638127i \(0.779710\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 5.19615i 0.449439 0.259483i −0.258154 0.966104i \(-0.583114\pi\)
0.707593 + 0.706620i \(0.249781\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.00000 1.73205i 0.149071 0.0860663i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.0000 10.3923i −0.890043 0.513866i −0.0160862 0.999871i \(-0.505121\pi\)
−0.873956 + 0.486004i \(0.838454\pi\)
\(410\) 0 0
\(411\) 10.3923i 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −6.00000 10.3923i −0.293119 0.507697i 0.681426 0.731887i \(-0.261360\pi\)
−0.974546 + 0.224189i \(0.928027\pi\)
\(420\) 0 0
\(421\) 20.7846i 1.01298i −0.862246 0.506490i \(-0.830943\pi\)
0.862246 0.506490i \(-0.169057\pi\)
\(422\) 0 0
\(423\) −3.00000 1.73205i −0.145865 0.0842152i
\(424\) 0 0
\(425\) 21.0000 36.3731i 1.01865 1.76435i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.0000 15.5885i 1.30054 0.750870i 0.320047 0.947402i \(-0.396301\pi\)
0.980497 + 0.196532i \(0.0629680\pi\)
\(432\) 0 0
\(433\) 1.00000 1.73205i 0.0480569 0.0832370i −0.840996 0.541041i \(-0.818030\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) −18.0000 10.3923i −0.863034 0.498273i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −10.0000 17.3205i −0.477274 0.826663i 0.522387 0.852709i \(-0.325042\pi\)
−0.999661 + 0.0260459i \(0.991708\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −30.0000 51.9615i −1.42214 2.46321i
\(446\) 0 0
\(447\) 17.3205i 0.819232i
\(448\) 0 0
\(449\) 3.00000 + 1.73205i 0.141579 + 0.0817405i 0.569116 0.822257i \(-0.307285\pi\)
−0.427537 + 0.903998i \(0.640619\pi\)
\(450\) 0 0
\(451\) −6.00000 + 10.3923i −0.282529 + 0.489355i
\(452\) 0 0
\(453\) −6.00000 + 3.46410i −0.281905 + 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 3.46410i 0.280668 0.162044i −0.353058 0.935602i \(-0.614858\pi\)
0.633726 + 0.773558i \(0.281525\pi\)
\(458\) 0 0
\(459\) 3.00000 5.19615i 0.140028 0.242536i
\(460\) 0 0
\(461\) 9.00000 + 5.19615i 0.419172 + 0.242009i 0.694723 0.719277i \(-0.255527\pi\)
−0.275551 + 0.961286i \(0.588860\pi\)
\(462\) 0 0
\(463\) 13.8564i 0.643962i 0.946746 + 0.321981i \(0.104349\pi\)
−0.946746 + 0.321981i \(0.895651\pi\)
\(464\) 0 0
\(465\) 12.0000 + 20.7846i 0.556487 + 0.963863i
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.00000 + 1.73205i 0.0460776 + 0.0798087i
\(472\) 0 0
\(473\) 27.7128i 1.27424i
\(474\) 0 0
\(475\) −42.0000 24.2487i −1.92709 1.11261i
\(476\) 0 0
\(477\) −3.00000 + 5.19615i −0.137361 + 0.237915i
\(478\) 0 0
\(479\) −3.00000 + 1.73205i −0.137073 + 0.0791394i −0.566969 0.823739i \(-0.691884\pi\)
0.429895 + 0.902879i \(0.358551\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 20.7846i 0.544892 0.943781i
\(486\) 0 0
\(487\) −6.00000 3.46410i −0.271886 0.156973i 0.357858 0.933776i \(-0.383507\pi\)
−0.629744 + 0.776802i \(0.716840\pi\)
\(488\) 0 0
\(489\) 20.7846i 0.939913i
\(490\) 0 0
\(491\) 18.0000 + 31.1769i 0.812329 + 1.40699i 0.911230 + 0.411897i \(0.135134\pi\)
−0.0989017 + 0.995097i \(0.531533\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 15.0000 + 8.66025i 0.670151 + 0.386912i
\(502\) 0 0
\(503\) −12.0000 + 20.7846i −0.535054 + 0.926740i 0.464107 + 0.885779i \(0.346375\pi\)
−0.999161 + 0.0409609i \(0.986958\pi\)
\(504\) 0 0
\(505\) 18.0000 10.3923i 0.800989 0.462451i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.0000 + 12.1244i −0.930809 + 0.537403i −0.887067 0.461640i \(-0.847261\pi\)
−0.0437414 + 0.999043i \(0.513928\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.00000 3.46410i −0.264906 0.152944i
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) 6.00000 + 10.3923i 0.263880 + 0.457053i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −4.00000 6.92820i −0.174908 0.302949i 0.765222 0.643767i \(-0.222629\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 36.0000 + 20.7846i 1.56818 + 0.905392i
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) −3.00000 + 1.73205i −0.130189 + 0.0751646i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −36.0000 + 20.7846i −1.55642 + 0.898597i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) −21.0000 12.1244i −0.904534 0.522233i
\(540\) 0 0
\(541\) 20.7846i 0.893600i −0.894634 0.446800i \(-0.852564\pi\)
0.894634 0.446800i \(-0.147436\pi\)
\(542\) 0 0
\(543\) −11.0000 19.0526i −0.472055 0.817624i
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) −5.00000 8.66025i −0.213395 0.369611i
\(550\) 0 0
\(551\) 41.5692i 1.77091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21.0000 + 12.1244i −0.889799 + 0.513725i −0.873877 0.486148i \(-0.838402\pi\)
−0.0159220 + 0.999873i \(0.505068\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 + 10.3923i −0.759961 + 0.438763i
\(562\) 0 0
\(563\) 18.0000 31.1769i 0.758610 1.31395i −0.184950 0.982748i \(-0.559212\pi\)
0.943560 0.331202i \(-0.107454\pi\)
\(564\) 0 0
\(565\) 54.0000 + 31.1769i 2.27180 + 1.31162i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.00000 15.5885i −0.377300 0.653502i 0.613369 0.789797i \(-0.289814\pi\)
−0.990668 + 0.136295i \(0.956481\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.8564i 0.576850i −0.957503 0.288425i \(-0.906868\pi\)
0.957503 0.288425i \(-0.0931316\pi\)
\(578\) 0 0
\(579\) 12.0000 + 6.92820i 0.498703 + 0.287926i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 10.3923i 0.745484 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.0000 + 15.5885i −1.11441 + 0.643404i −0.939968 0.341263i \(-0.889145\pi\)
−0.174441 + 0.984668i \(0.555812\pi\)
\(588\) 0 0
\(589\) 24.0000 41.5692i 0.988903 1.71283i
\(590\) 0 0
\(591\) 15.0000 + 8.66025i 0.617018 + 0.356235i
\(592\) 0 0
\(593\) 24.2487i 0.995775i 0.867242 + 0.497888i \(0.165891\pi\)
−0.867242 + 0.497888i \(0.834109\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −5.00000 8.66025i −0.203954 0.353259i 0.745845 0.666120i \(-0.232046\pi\)
−0.949799 + 0.312861i \(0.898713\pi\)
\(602\) 0 0
\(603\) 13.8564i 0.564276i
\(604\) 0 0
\(605\) 3.00000 + 1.73205i 0.121967 + 0.0704179i
\(606\) 0 0
\(607\) 14.0000 24.2487i 0.568242 0.984225i −0.428497 0.903543i \(-0.640957\pi\)
0.996740 0.0806818i \(-0.0257098\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12.0000 6.92820i 0.484675 0.279827i −0.237687 0.971342i \(-0.576389\pi\)
0.722363 + 0.691514i \(0.243056\pi\)
\(614\) 0 0
\(615\) 6.00000 10.3923i 0.241943 0.419058i
\(616\) 0 0
\(617\) 27.0000 + 15.5885i 1.08698 + 0.627568i 0.932771 0.360471i \(-0.117384\pi\)
0.154209 + 0.988038i \(0.450717\pi\)
\(618\) 0 0
\(619\) 27.7128i 1.11387i −0.830555 0.556936i \(-0.811977\pi\)
0.830555 0.556936i \(-0.188023\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 12.0000 + 20.7846i 0.479234 + 0.830057i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −18.0000 10.3923i −0.716569 0.413711i 0.0969198 0.995292i \(-0.469101\pi\)
−0.813488 + 0.581581i \(0.802434\pi\)
\(632\) 0 0
\(633\) 2.00000 3.46410i 0.0794929 0.137686i
\(634\) 0 0
\(635\) 48.0000 27.7128i 1.90482 1.09975i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 9.00000 5.19615i 0.356034 0.205557i
\(640\) 0 0
\(641\) −15.0000 + 25.9808i −0.592464 + 1.02618i 0.401435 + 0.915888i \(0.368512\pi\)
−0.993899 + 0.110291i \(0.964822\pi\)
\(642\) 0 0
\(643\) 36.0000 + 20.7846i 1.41970 + 0.819665i 0.996272 0.0862642i \(-0.0274929\pi\)
0.423429 + 0.905929i \(0.360826\pi\)
\(644\) 0 0
\(645\) 27.7128i 1.09119i
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 + 5.19615i 0.117399 + 0.203341i 0.918736 0.394872i \(-0.129211\pi\)
−0.801337 + 0.598213i \(0.795878\pi\)
\(654\) 0 0
\(655\) 41.5692i 1.62424i
\(656\) 0 0
\(657\) 6.00000 + 3.46410i 0.234082 + 0.135147i
\(658\) 0 0
\(659\) −6.00000 + 10.3923i −0.233727 + 0.404827i −0.958902 0.283738i \(-0.908425\pi\)
0.725175 + 0.688565i \(0.241759\pi\)
\(660\) 0 0
\(661\) −12.0000 + 6.92820i −0.466746 + 0.269476i −0.714877 0.699251i \(-0.753517\pi\)
0.248131 + 0.968727i \(0.420184\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −6.00000 3.46410i −0.231973 0.133930i
\(670\) 0 0
\(671\) 34.6410i 1.33730i
\(672\) 0 0
\(673\) −7.00000 12.1244i −0.269830 0.467360i 0.698988 0.715134i \(-0.253634\pi\)
−0.968818 + 0.247774i \(0.920301\pi\)
\(674\) 0 0
\(675\) 7.00000 0.269430
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.3923i 0.398234i
\(682\) 0 0
\(683\) −3.00000 1.73205i −0.114792 0.0662751i 0.441505 0.897259i \(-0.354445\pi\)
−0.556297 + 0.830984i \(0.687778\pi\)
\(684\) 0 0
\(685\) −18.0000 + 31.1769i −0.687745 + 1.19121i
\(686\) 0 0
\(687\) −6.00000 + 3.46410i −0.228914 + 0.132164i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 24.0000 13.8564i 0.913003 0.527123i 0.0316069 0.999500i \(-0.489938\pi\)
0.881396 + 0.472378i \(0.156604\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 13.8564i −0.910372 0.525603i
\(696\) 0 0
\(697\) 20.7846i 0.787273i
\(698\) 0 0
\(699\) 9.00000 + 15.5885i 0.340411 + 0.589610i
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −6.00000 10.3923i −0.225973 0.391397i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000 + 3.46410i 0.225335 + 0.130097i 0.608418 0.793617i \(-0.291804\pi\)
−0.383083 + 0.923714i \(0.625138\pi\)
\(710\) 0 0
\(711\) 4.00000 6.92820i 0.150012 0.259828i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.00000 + 5.19615i −0.336111 + 0.194054i
\(718\) 0 0
\(719\) −12.0000 + 20.7846i −0.447524 + 0.775135i −0.998224 0.0595683i \(-0.981028\pi\)
0.550700 + 0.834703i \(0.314361\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20.7846i 0.772988i
\(724\) 0 0
\(725\) −21.0000 36.3731i −0.779920 1.35086i
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 41.5692i −0.887672 1.53749i
\(732\) 0 0
\(733\) 6.92820i 0.255899i 0.991781 + 0.127950i \(0.0408395\pi\)
−0.991781 + 0.127950i \(0.959160\pi\)
\(734\) 0 0
\(735\) 21.0000 + 12.1244i 0.774597 + 0.447214i
\(736\) 0 0
\(737\) 24.0000 41.5692i 0.884051 1.53122i
\(738\) 0 0
\(739\) 24.0000 13.8564i 0.882854 0.509716i 0.0112558 0.999937i \(-0.496417\pi\)
0.871598 + 0.490221i \(0.163084\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 5.19615i 0.330178 0.190628i −0.325742 0.945459i \(-0.605614\pi\)
0.655920 + 0.754830i \(0.272281\pi\)
\(744\) 0 0
\(745\) −30.0000 + 51.9615i −1.09911 + 1.90372i
\(746\) 0 0
\(747\) 3.00000 + 1.73205i 0.109764 + 0.0633724i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.0000 + 24.2487i 0.510867 + 0.884848i 0.999921 + 0.0125942i \(0.00400897\pi\)
−0.489053 + 0.872254i \(0.662658\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 19.0000 + 32.9090i 0.690567 + 1.19610i 0.971652 + 0.236414i \(0.0759722\pi\)
−0.281086 + 0.959683i \(0.590695\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 15.5885i −0.978749 0.565081i −0.0768569 0.997042i \(-0.524488\pi\)
−0.901892 + 0.431961i \(0.857822\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 18.0000 10.3923i 0.650791 0.375735i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 24.0000 13.8564i 0.865462 0.499675i −0.000375472 1.00000i \(-0.500120\pi\)
0.865838 + 0.500325i \(0.166786\pi\)
\(770\) 0 0
\(771\) −3.00000 + 5.19615i −0.108042 + 0.187135i
\(772\) 0 0
\(773\) −45.0000 25.9808i −1.61854 0.934463i −0.987299 0.158874i \(-0.949213\pi\)
−0.631239 0.775589i \(-0.717453\pi\)
\(774\) 0 0
\(775\) 48.4974i 1.74208i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) −3.00000 5.19615i −0.107211 0.185695i
\(784\) 0 0
\(785\) 6.92820i 0.247278i
\(786\) 0 0
\(787\) 24.0000 + 13.8564i 0.855508 + 0.493928i 0.862505 0.506048i \(-0.168894\pi\)
−0.00699773 + 0.999976i \(0.502227\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −18.0000 + 10.3923i −0.638394 + 0.368577i
\(796\) 0 0
\(797\) 15.0000 25.9808i 0.531327 0.920286i −0.468004 0.883726i \(-0.655027\pi\)
0.999331 0.0365596i \(-0.0116399\pi\)
\(798\) 0 0
\(799\) −18.0000 10.3923i −0.636794 0.367653i
\(800\) 0 0
\(801\) 17.3205i 0.611990i
\(802\) 0 0
\(803\) −12.0000 20.7846i −0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 9.00000 + 15.5885i 0.316423 + 0.548061i 0.979739 0.200279i \(-0.0641847\pi\)
−0.663316 + 0.748340i \(0.730851\pi\)
\(810\) 0 0
\(811\) 6.92820i 0.243282i 0.992574 + 0.121641i \(0.0388157\pi\)
−0.992574 + 0.121641i \(0.961184\pi\)
\(812\) 0 0
\(813\) −24.0000 13.8564i −0.841717 0.485965i
\(814\) 0 0
\(815\) −36.0000 + 62.3538i −1.26102 + 2.18416i
\(816\) 0 0
\(817\) −48.0000 + 27.7128i −1.67931 + 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 + 1.73205i −0.104701 + 0.0604490i −0.551436 0.834217i \(-0.685920\pi\)
0.446735 + 0.894666i \(0.352587\pi\)
\(822\) 0 0
\(823\) 2.00000 3.46410i 0.0697156 0.120751i −0.829060 0.559159i \(-0.811124\pi\)
0.898776 + 0.438408i \(0.144457\pi\)
\(824\) 0 0
\(825\) −21.0000 12.1244i −0.731126 0.422116i
\(826\) 0 0
\(827\) 17.3205i 0.602293i 0.953578 + 0.301147i \(0.0973693\pi\)
−0.953578 + 0.301147i \(0.902631\pi\)
\(828\) 0 0
\(829\) −17.0000 29.4449i −0.590434 1.02266i −0.994174 0.107788i \(-0.965623\pi\)
0.403739 0.914874i \(-0.367710\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 0 0
\(835\) 30.0000 + 51.9615i 1.03819 + 1.79820i
\(836\) 0 0
\(837\) 6.92820i 0.239474i
\(838\) 0 0
\(839\) −9.00000 5.19615i −0.310715 0.179391i 0.336532 0.941672i \(-0.390746\pi\)
−0.647246 + 0.762281i \(0.724079\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) −15.0000 + 8.66025i −0.516627 + 0.298275i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.0000 + 17.3205i −0.343199 + 0.594438i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 13.8564i 0.474434i 0.971457 + 0.237217i \(0.0762353\pi\)
−0.971457 + 0.237217i \(0.923765\pi\)
\(854\) 0 0
\(855\) −12.0000 20.7846i −0.410391 0.710819i
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.1051i 1.29711i −0.761166 0.648557i \(-0.775373\pi\)
0.761166 0.648557i \(-0.224627\pi\)
\(864\) 0 0
\(865\) −18.0000 10.3923i −0.612018 0.353349i
\(866\) 0 0
\(867\) 9.50000 16.4545i 0.322637 0.558824i
\(868\) 0 0
\(869\) −24.0000 + 13.8564i −0.814144 + 0.470046i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 6.00000 3.46410i 0.203069 0.117242i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.0000 20.7846i −1.21563 0.701846i −0.251653 0.967818i \(-0.580974\pi\)
−0.963981 + 0.265971i \(0.914307\pi\)
\(878\) 0 0
\(879\) 3.46410i 0.116841i
\(880\) 0 0
\(881\) 3.00000 + 5.19615i 0.101073 + 0.175063i 0.912127 0.409908i \(-0.134439\pi\)
−0.811054 + 0.584971i \(0.801106\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 24.0000 + 41.5692i 0.805841 + 1.39576i 0.915722 + 0.401813i \(0.131620\pi\)
−0.109881 + 0.993945i \(0.535047\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 1.73205i −0.100504 0.0580259i
\(892\) 0 0
\(893\) −12.0000 + 20.7846i −0.401565 + 0.695530i
\(894\) 0 0
\(895\) −36.0000 + 20.7846i −1.20335 + 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 36.0000 20.7846i 1.20067 0.693206i
\(900\) 0 0
\(901\) −18.0000 + 31.1769i −0.599667 + 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 76.2102i 2.53331i
\(906\) 0 0
\(907\) −14.0000 24.2487i −0.464862 0.805165i 0.534333 0.845274i \(-0.320563\pi\)
−0.999195 + 0.0401089i \(0.987230\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −6.00000 10.3923i −0.198571 0.343935i
\(914\) 0 0
\(915\) 34.6410i 1.14520i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 27.7128i 0.527791 0.914161i −0.471684 0.881768i \(-0.656354\pi\)
0.999475 0.0323936i \(-0.0103130\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.00000 3.46410i 0.0656886 0.113776i
\(928\) 0 0
\(929\) 27.0000 + 15.5885i 0.885841 + 0.511441i 0.872580 0.488471i \(-0.162445\pi\)
0.0132613 + 0.999912i \(0.495779\pi\)
\(930\) 0 0
\(931\) 48.4974i 1.58944i
\(932\) 0 0
\(933\) 12.0000 + 20.7846i 0.392862 + 0.680458i
\(934\) 0 0
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 5.00000 + 8.66025i 0.163169 + 0.282617i
\(940\) 0 0
\(941\) 10.3923i 0.338779i 0.985549 + 0.169390i \(0.0541797\pi\)
−0.985549 + 0.169390i \(0.945820\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −39.0000 + 22.5167i −1.26733 + 0.731693i −0.974482 0.224466i \(-0.927936\pi\)
−0.292848 + 0.956159i \(0.594603\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 21.0000 12.1244i 0.680972 0.393159i
\(952\) 0 0
\(953\) 3.00000 5.19615i 0.0971795 0.168320i −0.813337 0.581793i \(-0.802351\pi\)
0.910516 + 0.413473i \(0.135685\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.7846i 0.671871i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −17.0000 −0.548387
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 24.0000 + 41.5692i 0.772587 + 1.33816i
\(966\) 0 0
\(967\) 34.6410i 1.11398i −0.830519 0.556990i \(-0.811956\pi\)
0.830519 0.556990i \(-0.188044\pi\)
\(968\) 0 0
\(969\) −36.0000 20.7846i −1.15649 0.667698i
\(970\) 0 0
\(971\) −30.0000 + 51.9615i −0.962746 + 1.66752i −0.247193 + 0.968966i \(0.579508\pi\)
−0.715553 + 0.698558i \(0.753825\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21.0000 + 12.1244i −0.671850 + 0.387893i −0.796777 0.604273i \(-0.793463\pi\)
0.124928 + 0.992166i \(0.460130\pi\)
\(978\) 0 0
\(979\) −30.0000 + 51.9615i −0.958804 + 1.66070i
\(980\) 0 0
\(981\) 6.00000 + 3.46410i 0.191565 + 0.110600i
\(982\) 0 0
\(983\) 31.1769i 0.994389i −0.867639 0.497195i \(-0.834364\pi\)
0.867639 0.497195i \(-0.165636\pi\)
\(984\) 0 0
\(985\) 30.0000 + 51.9615i 0.955879 + 1.65563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −10.0000 17.3205i −0.317660 0.550204i 0.662339 0.749204i \(-0.269564\pi\)
−0.979999 + 0.199000i \(0.936231\pi\)
\(992\) 0 0
\(993\) 13.8564i 0.439720i
\(994\) 0 0
\(995\) 24.0000 + 13.8564i 0.760851 + 0.439278i
\(996\) 0 0
\(997\) −19.0000 + 32.9090i −0.601736 + 1.04224i 0.390822 + 0.920466i \(0.372191\pi\)
−0.992558 + 0.121771i \(0.961143\pi\)
\(998\) 0 0
\(999\) 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 2028.2.q.c.361.1 2
13.2 odd 12 2028.2.a.g.1.2 2
13.3 even 3 156.2.b.a.25.2 yes 2
13.4 even 6 inner 2028.2.q.c.1837.1 2
13.5 odd 4 2028.2.i.i.2005.2 4
13.6 odd 12 2028.2.i.i.529.2 4
13.7 odd 12 2028.2.i.i.529.1 4
13.8 odd 4 2028.2.i.i.2005.1 4
13.9 even 3 2028.2.q.b.1837.1 2
13.10 even 6 156.2.b.a.25.1 2
13.11 odd 12 2028.2.a.g.1.1 2
13.12 even 2 2028.2.q.b.361.1 2
39.2 even 12 6084.2.a.v.1.1 2
39.11 even 12 6084.2.a.v.1.2 2
39.23 odd 6 468.2.b.a.181.2 2
39.29 odd 6 468.2.b.a.181.1 2
52.3 odd 6 624.2.c.f.337.2 2
52.11 even 12 8112.2.a.bs.1.1 2
52.15 even 12 8112.2.a.bs.1.2 2
52.23 odd 6 624.2.c.f.337.1 2
65.3 odd 12 3900.2.j.h.649.2 4
65.23 odd 12 3900.2.j.h.649.1 4
65.29 even 6 3900.2.c.c.3301.2 2
65.42 odd 12 3900.2.j.h.649.4 4
65.49 even 6 3900.2.c.c.3301.1 2
65.62 odd 12 3900.2.j.h.649.3 4
91.55 odd 6 7644.2.e.g.4705.1 2
91.62 odd 6 7644.2.e.g.4705.2 2
104.3 odd 6 2496.2.c.e.961.1 2
104.29 even 6 2496.2.c.l.961.1 2
104.75 odd 6 2496.2.c.e.961.2 2
104.101 even 6 2496.2.c.l.961.2 2
156.23 even 6 1872.2.c.c.1585.2 2
156.107 even 6 1872.2.c.c.1585.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.b.a.25.1 2 13.10 even 6
156.2.b.a.25.2 yes 2 13.3 even 3
468.2.b.a.181.1 2 39.29 odd 6
468.2.b.a.181.2 2 39.23 odd 6
624.2.c.f.337.1 2 52.23 odd 6
624.2.c.f.337.2 2 52.3 odd 6
1872.2.c.c.1585.1 2 156.107 even 6
1872.2.c.c.1585.2 2 156.23 even 6
2028.2.a.g.1.1 2 13.11 odd 12
2028.2.a.g.1.2 2 13.2 odd 12
2028.2.i.i.529.1 4 13.7 odd 12
2028.2.i.i.529.2 4 13.6 odd 12
2028.2.i.i.2005.1 4 13.8 odd 4
2028.2.i.i.2005.2 4 13.5 odd 4
2028.2.q.b.361.1 2 13.12 even 2
2028.2.q.b.1837.1 2 13.9 even 3
2028.2.q.c.361.1 2 1.1 even 1 trivial
2028.2.q.c.1837.1 2 13.4 even 6 inner
2496.2.c.e.961.1 2 104.3 odd 6
2496.2.c.e.961.2 2 104.75 odd 6
2496.2.c.l.961.1 2 104.29 even 6
2496.2.c.l.961.2 2 104.101 even 6
3900.2.c.c.3301.1 2 65.49 even 6
3900.2.c.c.3301.2 2 65.29 even 6
3900.2.j.h.649.1 4 65.23 odd 12
3900.2.j.h.649.2 4 65.3 odd 12
3900.2.j.h.649.3 4 65.62 odd 12
3900.2.j.h.649.4 4 65.42 odd 12
6084.2.a.v.1.1 2 39.2 even 12
6084.2.a.v.1.2 2 39.11 even 12
7644.2.e.g.4705.1 2 91.55 odd 6
7644.2.e.g.4705.2 2 91.62 odd 6
8112.2.a.bs.1.1 2 52.11 even 12
8112.2.a.bs.1.2 2 52.15 even 12