Properties

Label 2100.2.x.d.1693.3
Level $2100$
Weight $2$
Character 2100.1693
Analytic conductor $16.769$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1357,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1357");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.x (of order \(4\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1693.3
Root \(-0.522506 + 1.01508i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.d.1357.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(0.235858 - 2.63522i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(0.235858 - 2.63522i) q^{7} -1.00000i q^{9} +4.99261 q^{11} +(2.63111 - 2.63111i) q^{13} +(-4.14626 - 4.14626i) q^{17} -4.66126 q^{19} +(1.69660 + 2.03016i) q^{21} +(-4.41755 - 4.41755i) q^{23} +(0.707107 + 0.707107i) q^{27} +2.34895i q^{29} +2.57124i q^{31} +(-3.53031 + 3.53031i) q^{33} +(-7.06989 + 7.06989i) q^{37} +3.72095i q^{39} -7.87389i q^{41} +(2.59202 + 2.59202i) q^{43} +(-0.813264 - 0.813264i) q^{47} +(-6.88874 - 1.24308i) q^{49} +5.86369 q^{51} +(0.0317294 + 0.0317294i) q^{53} +(3.29601 - 3.29601i) q^{57} +0.858136 q^{59} +1.59654i q^{61} +(-2.63522 - 0.235858i) q^{63} +(10.5074 - 10.5074i) q^{67} +6.24736 q^{69} -12.5404 q^{71} +(-7.92331 + 7.92331i) q^{73} +(1.17755 - 13.1566i) q^{77} -8.69789i q^{79} -1.00000 q^{81} +(-5.50896 + 5.50896i) q^{83} +(-1.66096 - 1.66096i) q^{87} +1.62052 q^{89} +(-6.31297 - 7.55411i) q^{91} +(-1.81814 - 1.81814i) q^{93} +(-3.51923 - 3.51923i) q^{97} -4.99261i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{11} + 8 q^{21} - 8 q^{23} - 16 q^{37} - 48 q^{43} - 16 q^{51} + 40 q^{53} + 8 q^{57} + 48 q^{67} - 32 q^{71} + 24 q^{77} - 16 q^{81} + 32 q^{91} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.235858 2.63522i 0.0891461 0.996019i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 4.99261 1.50533 0.752665 0.658404i \(-0.228768\pi\)
0.752665 + 0.658404i \(0.228768\pi\)
\(12\) 0 0
\(13\) 2.63111 2.63111i 0.729738 0.729738i −0.240830 0.970567i \(-0.577420\pi\)
0.970567 + 0.240830i \(0.0774195\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.14626 4.14626i −1.00561 1.00561i −0.999984 0.00563041i \(-0.998208\pi\)
−0.00563041 0.999984i \(-0.501792\pi\)
\(18\) 0 0
\(19\) −4.66126 −1.06937 −0.534684 0.845052i \(-0.679569\pi\)
−0.534684 + 0.845052i \(0.679569\pi\)
\(20\) 0 0
\(21\) 1.69660 + 2.03016i 0.370229 + 0.443017i
\(22\) 0 0
\(23\) −4.41755 4.41755i −0.921123 0.921123i 0.0759861 0.997109i \(-0.475790\pi\)
−0.997109 + 0.0759861i \(0.975790\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 2.34895i 0.436188i 0.975928 + 0.218094i \(0.0699840\pi\)
−0.975928 + 0.218094i \(0.930016\pi\)
\(30\) 0 0
\(31\) 2.57124i 0.461808i 0.972976 + 0.230904i \(0.0741684\pi\)
−0.972976 + 0.230904i \(0.925832\pi\)
\(32\) 0 0
\(33\) −3.53031 + 3.53031i −0.614548 + 0.614548i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.06989 + 7.06989i −1.16228 + 1.16228i −0.178309 + 0.983975i \(0.557063\pi\)
−0.983975 + 0.178309i \(0.942937\pi\)
\(38\) 0 0
\(39\) 3.72095i 0.595828i
\(40\) 0 0
\(41\) 7.87389i 1.22969i −0.788646 0.614847i \(-0.789218\pi\)
0.788646 0.614847i \(-0.210782\pi\)
\(42\) 0 0
\(43\) 2.59202 + 2.59202i 0.395280 + 0.395280i 0.876564 0.481285i \(-0.159830\pi\)
−0.481285 + 0.876564i \(0.659830\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.813264 0.813264i −0.118627 0.118627i 0.645301 0.763928i \(-0.276732\pi\)
−0.763928 + 0.645301i \(0.776732\pi\)
\(48\) 0 0
\(49\) −6.88874 1.24308i −0.984106 0.177582i
\(50\) 0 0
\(51\) 5.86369 0.821081
\(52\) 0 0
\(53\) 0.0317294 + 0.0317294i 0.00435837 + 0.00435837i 0.709283 0.704924i \(-0.249019\pi\)
−0.704924 + 0.709283i \(0.749019\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.29601 3.29601i 0.436567 0.436567i
\(58\) 0 0
\(59\) 0.858136 0.111720 0.0558599 0.998439i \(-0.482210\pi\)
0.0558599 + 0.998439i \(0.482210\pi\)
\(60\) 0 0
\(61\) 1.59654i 0.204416i 0.994763 + 0.102208i \(0.0325907\pi\)
−0.994763 + 0.102208i \(0.967409\pi\)
\(62\) 0 0
\(63\) −2.63522 0.235858i −0.332006 0.0297154i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.5074 10.5074i 1.28368 1.28368i 0.345119 0.938559i \(-0.387838\pi\)
0.938559 0.345119i \(-0.112162\pi\)
\(68\) 0 0
\(69\) 6.24736 0.752094
\(70\) 0 0
\(71\) −12.5404 −1.48827 −0.744134 0.668030i \(-0.767138\pi\)
−0.744134 + 0.668030i \(0.767138\pi\)
\(72\) 0 0
\(73\) −7.92331 + 7.92331i −0.927353 + 0.927353i −0.997534 0.0701812i \(-0.977642\pi\)
0.0701812 + 0.997534i \(0.477642\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.17755 13.1566i 0.134194 1.49934i
\(78\) 0 0
\(79\) 8.69789i 0.978589i −0.872119 0.489295i \(-0.837254\pi\)
0.872119 0.489295i \(-0.162746\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −5.50896 + 5.50896i −0.604686 + 0.604686i −0.941553 0.336866i \(-0.890633\pi\)
0.336866 + 0.941553i \(0.390633\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.66096 1.66096i −0.178073 0.178073i
\(88\) 0 0
\(89\) 1.62052 0.171775 0.0858875 0.996305i \(-0.472627\pi\)
0.0858875 + 0.996305i \(0.472627\pi\)
\(90\) 0 0
\(91\) −6.31297 7.55411i −0.661779 0.791886i
\(92\) 0 0
\(93\) −1.81814 1.81814i −0.188532 0.188532i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.51923 3.51923i −0.357324 0.357324i 0.505502 0.862826i \(-0.331307\pi\)
−0.862826 + 0.505502i \(0.831307\pi\)
\(98\) 0 0
\(99\) 4.99261i 0.501777i
\(100\) 0 0
\(101\) 15.6729i 1.55951i −0.626083 0.779757i \(-0.715343\pi\)
0.626083 0.779757i \(-0.284657\pi\)
\(102\) 0 0
\(103\) 1.83284 1.83284i 0.180595 0.180595i −0.611020 0.791615i \(-0.709241\pi\)
0.791615 + 0.611020i \(0.209241\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.869787 0.869787i 0.0840855 0.0840855i −0.663813 0.747899i \(-0.731063\pi\)
0.747899 + 0.663813i \(0.231063\pi\)
\(108\) 0 0
\(109\) 11.7274i 1.12328i −0.827382 0.561640i \(-0.810171\pi\)
0.827382 0.561640i \(-0.189829\pi\)
\(110\) 0 0
\(111\) 9.99834i 0.949000i
\(112\) 0 0
\(113\) −10.3172 10.3172i −0.970562 0.970562i 0.0290168 0.999579i \(-0.490762\pi\)
−0.999579 + 0.0290168i \(0.990762\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.63111 2.63111i −0.243246 0.243246i
\(118\) 0 0
\(119\) −11.9042 + 9.94835i −1.09126 + 0.911964i
\(120\) 0 0
\(121\) 13.9262 1.26602
\(122\) 0 0
\(123\) 5.56768 + 5.56768i 0.502021 + 0.502021i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.4843 11.4843i 1.01907 1.01907i 0.0192525 0.999815i \(-0.493871\pi\)
0.999815 0.0192525i \(-0.00612864\pi\)
\(128\) 0 0
\(129\) −3.66567 −0.322745
\(130\) 0 0
\(131\) 11.2928i 0.986658i −0.869843 0.493329i \(-0.835780\pi\)
0.869843 0.493329i \(-0.164220\pi\)
\(132\) 0 0
\(133\) −1.09940 + 12.2834i −0.0953299 + 1.06511i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.75911 + 9.75911i −0.833777 + 0.833777i −0.988031 0.154254i \(-0.950703\pi\)
0.154254 + 0.988031i \(0.450703\pi\)
\(138\) 0 0
\(139\) 7.88433 0.668740 0.334370 0.942442i \(-0.391476\pi\)
0.334370 + 0.942442i \(0.391476\pi\)
\(140\) 0 0
\(141\) 1.15013 0.0968584
\(142\) 0 0
\(143\) 13.1361 13.1361i 1.09850 1.09850i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.75006 3.99209i 0.474257 0.329262i
\(148\) 0 0
\(149\) 11.4991i 0.942041i −0.882122 0.471020i \(-0.843886\pi\)
0.882122 0.471020i \(-0.156114\pi\)
\(150\) 0 0
\(151\) 19.7274 1.60539 0.802696 0.596389i \(-0.203398\pi\)
0.802696 + 0.596389i \(0.203398\pi\)
\(152\) 0 0
\(153\) −4.14626 + 4.14626i −0.335205 + 0.335205i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.1764 + 12.1764i 0.971782 + 0.971782i 0.999613 0.0278306i \(-0.00885990\pi\)
−0.0278306 + 0.999613i \(0.508860\pi\)
\(158\) 0 0
\(159\) −0.0448722 −0.00355859
\(160\) 0 0
\(161\) −12.6831 + 10.5993i −0.999570 + 0.835341i
\(162\) 0 0
\(163\) 1.75692 + 1.75692i 0.137613 + 0.137613i 0.772558 0.634945i \(-0.218977\pi\)
−0.634945 + 0.772558i \(0.718977\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.14217 2.14217i −0.165766 0.165766i 0.619349 0.785116i \(-0.287396\pi\)
−0.785116 + 0.619349i \(0.787396\pi\)
\(168\) 0 0
\(169\) 0.845444i 0.0650341i
\(170\) 0 0
\(171\) 4.66126i 0.356456i
\(172\) 0 0
\(173\) −6.06440 + 6.06440i −0.461068 + 0.461068i −0.899005 0.437938i \(-0.855709\pi\)
0.437938 + 0.899005i \(0.355709\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.606794 + 0.606794i −0.0456094 + 0.0456094i
\(178\) 0 0
\(179\) 12.1280i 0.906487i 0.891387 + 0.453244i \(0.149733\pi\)
−0.891387 + 0.453244i \(0.850267\pi\)
\(180\) 0 0
\(181\) 17.9366i 1.33322i 0.745407 + 0.666610i \(0.232255\pi\)
−0.745407 + 0.666610i \(0.767745\pi\)
\(182\) 0 0
\(183\) −1.12892 1.12892i −0.0834525 0.0834525i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −20.7007 20.7007i −1.51378 1.51378i
\(188\) 0 0
\(189\) 2.03016 1.69660i 0.147672 0.123410i
\(190\) 0 0
\(191\) 19.8764 1.43821 0.719103 0.694904i \(-0.244553\pi\)
0.719103 + 0.694904i \(0.244553\pi\)
\(192\) 0 0
\(193\) −4.35723 4.35723i −0.313640 0.313640i 0.532678 0.846318i \(-0.321186\pi\)
−0.846318 + 0.532678i \(0.821186\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.75911 1.75911i 0.125331 0.125331i −0.641659 0.766990i \(-0.721753\pi\)
0.766990 + 0.641659i \(0.221753\pi\)
\(198\) 0 0
\(199\) 25.9162 1.83715 0.918574 0.395248i \(-0.129341\pi\)
0.918574 + 0.395248i \(0.129341\pi\)
\(200\) 0 0
\(201\) 14.8596i 1.04812i
\(202\) 0 0
\(203\) 6.18998 + 0.554019i 0.434452 + 0.0388845i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.41755 + 4.41755i −0.307041 + 0.307041i
\(208\) 0 0
\(209\) −23.2719 −1.60975
\(210\) 0 0
\(211\) −0.412405 −0.0283911 −0.0141956 0.999899i \(-0.504519\pi\)
−0.0141956 + 0.999899i \(0.504519\pi\)
\(212\) 0 0
\(213\) 8.86739 8.86739i 0.607583 0.607583i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.77578 + 0.606449i 0.459970 + 0.0411684i
\(218\) 0 0
\(219\) 11.2053i 0.757181i
\(220\) 0 0
\(221\) −21.8185 −1.46767
\(222\) 0 0
\(223\) 12.0779 12.0779i 0.808797 0.808797i −0.175655 0.984452i \(-0.556204\pi\)
0.984452 + 0.175655i \(0.0562042\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.84390 + 7.84390i 0.520618 + 0.520618i 0.917758 0.397140i \(-0.129997\pi\)
−0.397140 + 0.917758i \(0.629997\pi\)
\(228\) 0 0
\(229\) −13.7475 −0.908460 −0.454230 0.890884i \(-0.650086\pi\)
−0.454230 + 0.890884i \(0.650086\pi\)
\(230\) 0 0
\(231\) 8.47048 + 10.1358i 0.557317 + 0.666886i
\(232\) 0 0
\(233\) −13.7104 13.7104i −0.898200 0.898200i 0.0970772 0.995277i \(-0.469051\pi\)
−0.995277 + 0.0970772i \(0.969051\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.15034 + 6.15034i 0.399507 + 0.399507i
\(238\) 0 0
\(239\) 2.69236i 0.174154i −0.996202 0.0870770i \(-0.972247\pi\)
0.996202 0.0870770i \(-0.0277526\pi\)
\(240\) 0 0
\(241\) 27.2501i 1.75533i −0.479273 0.877666i \(-0.659100\pi\)
0.479273 0.877666i \(-0.340900\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.2643 + 12.2643i −0.780358 + 0.780358i
\(248\) 0 0
\(249\) 7.79084i 0.493724i
\(250\) 0 0
\(251\) 6.91873i 0.436706i −0.975870 0.218353i \(-0.929932\pi\)
0.975870 0.218353i \(-0.0700685\pi\)
\(252\) 0 0
\(253\) −22.0551 22.0551i −1.38659 1.38659i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.405721 0.405721i −0.0253082 0.0253082i 0.694339 0.719648i \(-0.255697\pi\)
−0.719648 + 0.694339i \(0.755697\pi\)
\(258\) 0 0
\(259\) 16.9632 + 20.2982i 1.05404 + 1.26127i
\(260\) 0 0
\(261\) 2.34895 0.145396
\(262\) 0 0
\(263\) −5.57949 5.57949i −0.344046 0.344046i 0.513840 0.857886i \(-0.328223\pi\)
−0.857886 + 0.513840i \(0.828223\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.14588 + 1.14588i −0.0701268 + 0.0701268i
\(268\) 0 0
\(269\) 8.43131 0.514066 0.257033 0.966403i \(-0.417255\pi\)
0.257033 + 0.966403i \(0.417255\pi\)
\(270\) 0 0
\(271\) 15.2157i 0.924287i 0.886805 + 0.462143i \(0.152919\pi\)
−0.886805 + 0.462143i \(0.847081\pi\)
\(272\) 0 0
\(273\) 9.80550 + 0.877617i 0.593456 + 0.0531158i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.76521 + 1.76521i −0.106061 + 0.106061i −0.758146 0.652085i \(-0.773895\pi\)
0.652085 + 0.758146i \(0.273895\pi\)
\(278\) 0 0
\(279\) 2.57124 0.153936
\(280\) 0 0
\(281\) −16.7230 −0.997608 −0.498804 0.866715i \(-0.666227\pi\)
−0.498804 + 0.866715i \(0.666227\pi\)
\(282\) 0 0
\(283\) −7.00974 + 7.00974i −0.416686 + 0.416686i −0.884060 0.467374i \(-0.845200\pi\)
0.467374 + 0.884060i \(0.345200\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.7494 1.85712i −1.22480 0.109622i
\(288\) 0 0
\(289\) 17.3829i 1.02252i
\(290\) 0 0
\(291\) 4.97694 0.291754
\(292\) 0 0
\(293\) 0.405721 0.405721i 0.0237025 0.0237025i −0.695156 0.718859i \(-0.744665\pi\)
0.718859 + 0.695156i \(0.244665\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.53031 + 3.53031i 0.204849 + 0.204849i
\(298\) 0 0
\(299\) −23.2461 −1.34436
\(300\) 0 0
\(301\) 7.44189 6.21919i 0.428944 0.358468i
\(302\) 0 0
\(303\) 11.0824 + 11.0824i 0.636669 + 0.636669i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.01258 3.01258i −0.171937 0.171937i 0.615893 0.787830i \(-0.288795\pi\)
−0.787830 + 0.615893i \(0.788795\pi\)
\(308\) 0 0
\(309\) 2.59202i 0.147455i
\(310\) 0 0
\(311\) 1.49770i 0.0849266i −0.999098 0.0424633i \(-0.986479\pi\)
0.999098 0.0424633i \(-0.0135206\pi\)
\(312\) 0 0
\(313\) −16.0870 + 16.0870i −0.909290 + 0.909290i −0.996215 0.0869246i \(-0.972296\pi\)
0.0869246 + 0.996215i \(0.472296\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.95350 + 1.95350i −0.109719 + 0.109719i −0.759835 0.650116i \(-0.774720\pi\)
0.650116 + 0.759835i \(0.274720\pi\)
\(318\) 0 0
\(319\) 11.7274i 0.656607i
\(320\) 0 0
\(321\) 1.23006i 0.0686555i
\(322\) 0 0
\(323\) 19.3268 + 19.3268i 1.07537 + 1.07537i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.29251 + 8.29251i 0.458577 + 0.458577i
\(328\) 0 0
\(329\) −2.33494 + 1.95131i −0.128730 + 0.107579i
\(330\) 0 0
\(331\) −25.4271 −1.39760 −0.698801 0.715317i \(-0.746282\pi\)
−0.698801 + 0.715317i \(0.746282\pi\)
\(332\) 0 0
\(333\) 7.06989 + 7.06989i 0.387428 + 0.387428i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.43562 + 8.43562i −0.459517 + 0.459517i −0.898497 0.438980i \(-0.855340\pi\)
0.438980 + 0.898497i \(0.355340\pi\)
\(338\) 0 0
\(339\) 14.5907 0.792461
\(340\) 0 0
\(341\) 12.8372i 0.695174i
\(342\) 0 0
\(343\) −4.90055 + 17.8601i −0.264605 + 0.964357i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.66801 7.66801i 0.411640 0.411640i −0.470669 0.882310i \(-0.655987\pi\)
0.882310 + 0.470669i \(0.155987\pi\)
\(348\) 0 0
\(349\) −13.2122 −0.707234 −0.353617 0.935390i \(-0.615048\pi\)
−0.353617 + 0.935390i \(0.615048\pi\)
\(350\) 0 0
\(351\) 3.72095 0.198609
\(352\) 0 0
\(353\) 18.4154 18.4154i 0.980153 0.980153i −0.0196540 0.999807i \(-0.506256\pi\)
0.999807 + 0.0196540i \(0.00625647\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.38300 15.4521i 0.0731962 0.817812i
\(358\) 0 0
\(359\) 17.5636i 0.926971i 0.886105 + 0.463485i \(0.153401\pi\)
−0.886105 + 0.463485i \(0.846599\pi\)
\(360\) 0 0
\(361\) 2.72738 0.143546
\(362\) 0 0
\(363\) −9.84731 + 9.84731i −0.516850 + 0.516850i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 18.5241 + 18.5241i 0.966948 + 0.966948i 0.999471 0.0325233i \(-0.0103543\pi\)
−0.0325233 + 0.999471i \(0.510354\pi\)
\(368\) 0 0
\(369\) −7.87389 −0.409898
\(370\) 0 0
\(371\) 0.0910975 0.0761302i 0.00472955 0.00395249i
\(372\) 0 0
\(373\) 14.7274 + 14.7274i 0.762555 + 0.762555i 0.976784 0.214229i \(-0.0687238\pi\)
−0.214229 + 0.976784i \(0.568724\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.18033 + 6.18033i 0.318303 + 0.318303i
\(378\) 0 0
\(379\) 22.2943i 1.14518i −0.819841 0.572592i \(-0.805938\pi\)
0.819841 0.572592i \(-0.194062\pi\)
\(380\) 0 0
\(381\) 16.2413i 0.832065i
\(382\) 0 0
\(383\) 18.8978 18.8978i 0.965633 0.965633i −0.0337961 0.999429i \(-0.510760\pi\)
0.999429 + 0.0337961i \(0.0107597\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.59202 2.59202i 0.131760 0.131760i
\(388\) 0 0
\(389\) 2.77164i 0.140528i −0.997528 0.0702639i \(-0.977616\pi\)
0.997528 0.0702639i \(-0.0223841\pi\)
\(390\) 0 0
\(391\) 36.6326i 1.85259i
\(392\) 0 0
\(393\) 7.98523 + 7.98523i 0.402801 + 0.402801i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.6199 + 20.6199i 1.03488 + 1.03488i 0.999369 + 0.0355137i \(0.0113067\pi\)
0.0355137 + 0.999369i \(0.488693\pi\)
\(398\) 0 0
\(399\) −7.90831 9.46310i −0.395911 0.473747i
\(400\) 0 0
\(401\) −28.3385 −1.41516 −0.707580 0.706633i \(-0.750213\pi\)
−0.707580 + 0.706633i \(0.750213\pi\)
\(402\) 0 0
\(403\) 6.76521 + 6.76521i 0.336999 + 0.336999i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −35.2972 + 35.2972i −1.74962 + 1.74962i
\(408\) 0 0
\(409\) 38.1691 1.88734 0.943671 0.330885i \(-0.107347\pi\)
0.943671 + 0.330885i \(0.107347\pi\)
\(410\) 0 0
\(411\) 13.8015i 0.680776i
\(412\) 0 0
\(413\) 0.202399 2.26138i 0.00995939 0.111275i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.57506 + 5.57506i −0.273012 + 0.273012i
\(418\) 0 0
\(419\) 33.3495 1.62923 0.814616 0.580001i \(-0.196948\pi\)
0.814616 + 0.580001i \(0.196948\pi\)
\(420\) 0 0
\(421\) 7.41055 0.361168 0.180584 0.983560i \(-0.442201\pi\)
0.180584 + 0.983560i \(0.442201\pi\)
\(422\) 0 0
\(423\) −0.813264 + 0.813264i −0.0395423 + 0.0395423i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.20723 + 0.376557i 0.203602 + 0.0182229i
\(428\) 0 0
\(429\) 18.5773i 0.896918i
\(430\) 0 0
\(431\) −15.4005 −0.741818 −0.370909 0.928669i \(-0.620954\pi\)
−0.370909 + 0.928669i \(0.620954\pi\)
\(432\) 0 0
\(433\) 6.12490 6.12490i 0.294344 0.294344i −0.544450 0.838794i \(-0.683261\pi\)
0.838794 + 0.544450i \(0.183261\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.5914 + 20.5914i 0.985019 + 0.985019i
\(438\) 0 0
\(439\) −4.18867 −0.199914 −0.0999572 0.994992i \(-0.531871\pi\)
−0.0999572 + 0.994992i \(0.531871\pi\)
\(440\) 0 0
\(441\) −1.24308 + 6.88874i −0.0591941 + 0.328035i
\(442\) 0 0
\(443\) 2.45442 + 2.45442i 0.116613 + 0.116613i 0.763005 0.646392i \(-0.223723\pi\)
−0.646392 + 0.763005i \(0.723723\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.13107 + 8.13107i 0.384587 + 0.384587i
\(448\) 0 0
\(449\) 31.6407i 1.49321i 0.665265 + 0.746607i \(0.268319\pi\)
−0.665265 + 0.746607i \(0.731681\pi\)
\(450\) 0 0
\(451\) 39.3113i 1.85110i
\(452\) 0 0
\(453\) −13.9494 + 13.9494i −0.655398 + 0.655398i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.4887 11.4887i 0.537420 0.537420i −0.385350 0.922770i \(-0.625919\pi\)
0.922770 + 0.385350i \(0.125919\pi\)
\(458\) 0 0
\(459\) 5.86369i 0.273694i
\(460\) 0 0
\(461\) 37.4080i 1.74227i −0.491048 0.871133i \(-0.663386\pi\)
0.491048 0.871133i \(-0.336614\pi\)
\(462\) 0 0
\(463\) 21.4695 + 21.4695i 0.997774 + 0.997774i 0.999998 0.00222376i \(-0.000707846\pi\)
−0.00222376 + 0.999998i \(0.500708\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.1279 21.1279i −0.977683 0.977683i 0.0220738 0.999756i \(-0.492973\pi\)
−0.999756 + 0.0220738i \(0.992973\pi\)
\(468\) 0 0
\(469\) −25.2109 30.1674i −1.16413 1.39300i
\(470\) 0 0
\(471\) −17.2200 −0.793457
\(472\) 0 0
\(473\) 12.9410 + 12.9410i 0.595026 + 0.595026i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.0317294 0.0317294i 0.00145279 0.00145279i
\(478\) 0 0
\(479\) 36.6835 1.67611 0.838055 0.545586i \(-0.183693\pi\)
0.838055 + 0.545586i \(0.183693\pi\)
\(480\) 0 0
\(481\) 37.2033i 1.69632i
\(482\) 0 0
\(483\) 1.47349 16.4631i 0.0670462 0.749099i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.19439 + 2.19439i −0.0994373 + 0.0994373i −0.755075 0.655638i \(-0.772400\pi\)
0.655638 + 0.755075i \(0.272400\pi\)
\(488\) 0 0
\(489\) −2.48466 −0.112360
\(490\) 0 0
\(491\) −13.6566 −0.616313 −0.308157 0.951336i \(-0.599712\pi\)
−0.308157 + 0.951336i \(0.599712\pi\)
\(492\) 0 0
\(493\) 9.73933 9.73933i 0.438637 0.438637i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.95775 + 33.0466i −0.132673 + 1.48234i
\(498\) 0 0
\(499\) 3.27077i 0.146420i 0.997317 + 0.0732099i \(0.0233243\pi\)
−0.997317 + 0.0732099i \(0.976676\pi\)
\(500\) 0 0
\(501\) 3.02949 0.135348
\(502\) 0 0
\(503\) 9.91904 9.91904i 0.442268 0.442268i −0.450506 0.892774i \(-0.648756\pi\)
0.892774 + 0.450506i \(0.148756\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.597819 + 0.597819i 0.0265501 + 0.0265501i
\(508\) 0 0
\(509\) −20.6081 −0.913437 −0.456718 0.889611i \(-0.650975\pi\)
−0.456718 + 0.889611i \(0.650975\pi\)
\(510\) 0 0
\(511\) 19.0109 + 22.7484i 0.840991 + 1.00633i
\(512\) 0 0
\(513\) −3.29601 3.29601i −0.145522 0.145522i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.06031 4.06031i −0.178572 0.178572i
\(518\) 0 0
\(519\) 8.57635i 0.376460i
\(520\) 0 0
\(521\) 14.3938i 0.630603i 0.948991 + 0.315302i \(0.102106\pi\)
−0.948991 + 0.315302i \(0.897894\pi\)
\(522\) 0 0
\(523\) 14.7222 14.7222i 0.643757 0.643757i −0.307720 0.951477i \(-0.599566\pi\)
0.951477 + 0.307720i \(0.0995661\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.6610 10.6610i 0.464401 0.464401i
\(528\) 0 0
\(529\) 16.0295i 0.696934i
\(530\) 0 0
\(531\) 0.858136i 0.0372399i
\(532\) 0 0
\(533\) −20.7170 20.7170i −0.897354 0.897354i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.57577 8.57577i −0.370072 0.370072i
\(538\) 0 0
\(539\) −34.3928 6.20620i −1.48140 0.267320i
\(540\) 0 0
\(541\) −12.5138 −0.538010 −0.269005 0.963139i \(-0.586695\pi\)
−0.269005 + 0.963139i \(0.586695\pi\)
\(542\) 0 0
\(543\) −12.6831 12.6831i −0.544285 0.544285i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.66191 + 4.66191i −0.199329 + 0.199329i −0.799712 0.600383i \(-0.795015\pi\)
0.600383 + 0.799712i \(0.295015\pi\)
\(548\) 0 0
\(549\) 1.59654 0.0681387
\(550\) 0 0
\(551\) 10.9491i 0.466445i
\(552\) 0 0
\(553\) −22.9208 2.05147i −0.974693 0.0872374i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.2626 18.2626i 0.773812 0.773812i −0.204959 0.978771i \(-0.565706\pi\)
0.978771 + 0.204959i \(0.0657061\pi\)
\(558\) 0 0
\(559\) 13.6398 0.576901
\(560\) 0 0
\(561\) 29.2751 1.23600
\(562\) 0 0
\(563\) 0.858136 0.858136i 0.0361661 0.0361661i −0.688792 0.724959i \(-0.741859\pi\)
0.724959 + 0.688792i \(0.241859\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.235858 + 2.63522i −0.00990512 + 0.110669i
\(568\) 0 0
\(569\) 24.5451i 1.02899i −0.857495 0.514493i \(-0.827980\pi\)
0.857495 0.514493i \(-0.172020\pi\)
\(570\) 0 0
\(571\) −0.528564 −0.0221197 −0.0110599 0.999939i \(-0.503521\pi\)
−0.0110599 + 0.999939i \(0.503521\pi\)
\(572\) 0 0
\(573\) −14.0547 + 14.0547i −0.587145 + 0.587145i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.90536 + 8.90536i 0.370735 + 0.370735i 0.867745 0.497010i \(-0.165569\pi\)
−0.497010 + 0.867745i \(0.665569\pi\)
\(578\) 0 0
\(579\) 6.16205 0.256086
\(580\) 0 0
\(581\) 13.2180 + 15.8166i 0.548373 + 0.656184i
\(582\) 0 0
\(583\) 0.158413 + 0.158413i 0.00656078 + 0.00656078i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.0394 + 17.0394i 0.703292 + 0.703292i 0.965116 0.261824i \(-0.0843239\pi\)
−0.261824 + 0.965116i \(0.584324\pi\)
\(588\) 0 0
\(589\) 11.9852i 0.493843i
\(590\) 0 0
\(591\) 2.48776i 0.102333i
\(592\) 0 0
\(593\) −9.25748 + 9.25748i −0.380159 + 0.380159i −0.871159 0.491000i \(-0.836631\pi\)
0.491000 + 0.871159i \(0.336631\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.3255 + 18.3255i −0.750013 + 0.750013i
\(598\) 0 0
\(599\) 27.6462i 1.12959i 0.825230 + 0.564796i \(0.191045\pi\)
−0.825230 + 0.564796i \(0.808955\pi\)
\(600\) 0 0
\(601\) 20.7142i 0.844949i 0.906375 + 0.422475i \(0.138838\pi\)
−0.906375 + 0.422475i \(0.861162\pi\)
\(602\) 0 0
\(603\) −10.5074 10.5074i −0.427893 0.427893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.59685 4.59685i −0.186580 0.186580i 0.607636 0.794216i \(-0.292118\pi\)
−0.794216 + 0.607636i \(0.792118\pi\)
\(608\) 0 0
\(609\) −4.76873 + 3.98523i −0.193239 + 0.161490i
\(610\) 0 0
\(611\) −4.27957 −0.173133
\(612\) 0 0
\(613\) 29.5964 + 29.5964i 1.19539 + 1.19539i 0.975533 + 0.219854i \(0.0705582\pi\)
0.219854 + 0.975533i \(0.429442\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.2730 + 23.2730i −0.936934 + 0.936934i −0.998126 0.0611921i \(-0.980510\pi\)
0.0611921 + 0.998126i \(0.480510\pi\)
\(618\) 0 0
\(619\) −19.8683 −0.798575 −0.399288 0.916826i \(-0.630743\pi\)
−0.399288 + 0.916826i \(0.630743\pi\)
\(620\) 0 0
\(621\) 6.24736i 0.250698i
\(622\) 0 0
\(623\) 0.382214 4.27043i 0.0153131 0.171091i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 16.4557 16.4557i 0.657178 0.657178i
\(628\) 0 0
\(629\) 58.6272 2.33762
\(630\) 0 0
\(631\) 8.05496 0.320663 0.160332 0.987063i \(-0.448744\pi\)
0.160332 + 0.987063i \(0.448744\pi\)
\(632\) 0 0
\(633\) 0.291614 0.291614i 0.0115906 0.0115906i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −21.3957 + 14.8543i −0.847728 + 0.588551i
\(638\) 0 0
\(639\) 12.5404i 0.496090i
\(640\) 0 0
\(641\) 10.5201 0.415518 0.207759 0.978180i \(-0.433383\pi\)
0.207759 + 0.978180i \(0.433383\pi\)
\(642\) 0 0
\(643\) 24.2466 24.2466i 0.956192 0.956192i −0.0428876 0.999080i \(-0.513656\pi\)
0.999080 + 0.0428876i \(0.0136558\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.1304 15.1304i −0.594836 0.594836i 0.344098 0.938934i \(-0.388185\pi\)
−0.938934 + 0.344098i \(0.888185\pi\)
\(648\) 0 0
\(649\) 4.28434 0.168175
\(650\) 0 0
\(651\) −5.22002 + 4.36237i −0.204589 + 0.170975i
\(652\) 0 0
\(653\) −27.7591 27.7591i −1.08630 1.08630i −0.995906 0.0903916i \(-0.971188\pi\)
−0.0903916 0.995906i \(-0.528812\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.92331 + 7.92331i 0.309118 + 0.309118i
\(658\) 0 0
\(659\) 5.15936i 0.200980i 0.994938 + 0.100490i \(0.0320411\pi\)
−0.994938 + 0.100490i \(0.967959\pi\)
\(660\) 0 0
\(661\) 8.62844i 0.335608i 0.985820 + 0.167804i \(0.0536675\pi\)
−0.985820 + 0.167804i \(0.946332\pi\)
\(662\) 0 0
\(663\) 15.4280 15.4280i 0.599174 0.599174i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3766 10.3766i 0.401783 0.401783i
\(668\) 0 0
\(669\) 17.0808i 0.660380i
\(670\) 0 0
\(671\) 7.97091i 0.307713i
\(672\) 0 0
\(673\) 1.88585 + 1.88585i 0.0726941 + 0.0726941i 0.742519 0.669825i \(-0.233631\pi\)
−0.669825 + 0.742519i \(0.733631\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.56673 5.56673i −0.213947 0.213947i 0.591995 0.805942i \(-0.298341\pi\)
−0.805942 + 0.591995i \(0.798341\pi\)
\(678\) 0 0
\(679\) −10.1040 + 8.44390i −0.387755 + 0.324047i
\(680\) 0 0
\(681\) −11.0929 −0.425083
\(682\) 0 0
\(683\) −25.4526 25.4526i −0.973916 0.973916i 0.0257527 0.999668i \(-0.491802\pi\)
−0.999668 + 0.0257527i \(0.991802\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 9.72095 9.72095i 0.370877 0.370877i
\(688\) 0 0
\(689\) 0.166967 0.00636093
\(690\) 0 0
\(691\) 15.5594i 0.591909i 0.955202 + 0.295955i \(0.0956377\pi\)
−0.955202 + 0.295955i \(0.904362\pi\)
\(692\) 0 0
\(693\) −13.1566 1.17755i −0.499779 0.0447314i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −32.6471 + 32.6471i −1.23660 + 1.23660i
\(698\) 0 0
\(699\) 19.3895 0.733377
\(700\) 0 0
\(701\) 8.61966 0.325560 0.162780 0.986662i \(-0.447954\pi\)
0.162780 + 0.986662i \(0.447954\pi\)
\(702\) 0 0
\(703\) 32.9546 32.9546i 1.24291 1.24291i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −41.3015 3.69659i −1.55330 0.139025i
\(708\) 0 0
\(709\) 8.58353i 0.322361i −0.986925 0.161181i \(-0.948470\pi\)
0.986925 0.161181i \(-0.0515302\pi\)
\(710\) 0 0
\(711\) −8.69789 −0.326196
\(712\) 0 0
\(713\) 11.3586 11.3586i 0.425382 0.425382i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.90378 + 1.90378i 0.0710981 + 0.0710981i
\(718\) 0 0
\(719\) −17.8287 −0.664898 −0.332449 0.943121i \(-0.607875\pi\)
−0.332449 + 0.943121i \(0.607875\pi\)
\(720\) 0 0
\(721\) −4.39763 5.26221i −0.163776 0.195975i
\(722\) 0 0
\(723\) 19.2687 + 19.2687i 0.716611 + 0.716611i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.5253 + 18.5253i 0.687066 + 0.687066i 0.961582 0.274516i \(-0.0885177\pi\)
−0.274516 + 0.961582i \(0.588518\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 21.4944i 0.794998i
\(732\) 0 0
\(733\) 10.8206 10.8206i 0.399668 0.399668i −0.478448 0.878116i \(-0.658801\pi\)
0.878116 + 0.478448i \(0.158801\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.4592 52.4592i 1.93236 1.93236i
\(738\) 0 0
\(739\) 26.7993i 0.985827i −0.870078 0.492914i \(-0.835932\pi\)
0.870078 0.492914i \(-0.164068\pi\)
\(740\) 0 0
\(741\) 17.3443i 0.637159i
\(742\) 0 0
\(743\) 13.5456 + 13.5456i 0.496939 + 0.496939i 0.910484 0.413545i \(-0.135709\pi\)
−0.413545 + 0.910484i \(0.635709\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.50896 + 5.50896i 0.201562 + 0.201562i
\(748\) 0 0
\(749\) −2.08693 2.49723i −0.0762548 0.0912466i
\(750\) 0 0
\(751\) −27.3386 −0.997600 −0.498800 0.866717i \(-0.666226\pi\)
−0.498800 + 0.866717i \(0.666226\pi\)
\(752\) 0 0
\(753\) 4.89228 + 4.89228i 0.178285 + 0.178285i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.47988 9.47988i 0.344552 0.344552i −0.513524 0.858076i \(-0.671660\pi\)
0.858076 + 0.513524i \(0.171660\pi\)
\(758\) 0 0
\(759\) 31.1907 1.13215
\(760\) 0 0
\(761\) 21.2685i 0.770983i 0.922711 + 0.385492i \(0.125968\pi\)
−0.922711 + 0.385492i \(0.874032\pi\)
\(762\) 0 0
\(763\) −30.9042 2.76600i −1.11881 0.100136i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.25785 2.25785i 0.0815262 0.0815262i
\(768\) 0 0
\(769\) −18.1127 −0.653161 −0.326580 0.945169i \(-0.605896\pi\)
−0.326580 + 0.945169i \(0.605896\pi\)
\(770\) 0 0
\(771\) 0.573776 0.0206640
\(772\) 0 0
\(773\) −22.5967 + 22.5967i −0.812747 + 0.812747i −0.985045 0.172298i \(-0.944881\pi\)
0.172298 + 0.985045i \(0.444881\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −26.3478 2.35819i −0.945222 0.0845997i
\(778\) 0 0
\(779\) 36.7023i 1.31500i
\(780\) 0 0
\(781\) −62.6093 −2.24034
\(782\) 0 0
\(783\) −1.66096 + 1.66096i −0.0593577 + 0.0593577i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.4776 + 18.4776i 0.658656 + 0.658656i 0.955062 0.296406i \(-0.0957882\pi\)
−0.296406 + 0.955062i \(0.595788\pi\)
\(788\) 0 0
\(789\) 7.89059 0.280913
\(790\) 0 0
\(791\) −29.6215 + 24.7547i −1.05322 + 0.880176i
\(792\) 0 0
\(793\) 4.20067 + 4.20067i 0.149170 + 0.149170i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.01770 + 6.01770i 0.213158 + 0.213158i 0.805608 0.592450i \(-0.201839\pi\)
−0.592450 + 0.805608i \(0.701839\pi\)
\(798\) 0 0
\(799\) 6.74400i 0.238586i
\(800\) 0 0
\(801\) 1.62052i 0.0572583i
\(802\) 0 0
\(803\) −39.5580 + 39.5580i −1.39597 + 1.39597i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.96184 + 5.96184i −0.209867 + 0.209867i
\(808\) 0 0
\(809\) 55.4002i 1.94777i −0.227051 0.973883i \(-0.572908\pi\)
0.227051 0.973883i \(-0.427092\pi\)
\(810\) 0 0
\(811\) 54.3175i 1.90734i −0.300847 0.953672i \(-0.597269\pi\)
0.300847 0.953672i \(-0.402731\pi\)
\(812\) 0 0
\(813\) −10.7591 10.7591i −0.377338 0.377338i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.0821 12.0821i −0.422699 0.422699i
\(818\) 0 0
\(819\) −7.55411 + 6.31297i −0.263962 + 0.220593i
\(820\) 0 0
\(821\) 33.2327 1.15983 0.579915 0.814677i \(-0.303086\pi\)
0.579915 + 0.814677i \(0.303086\pi\)
\(822\) 0 0
\(823\) −14.3976 14.3976i −0.501870 0.501870i 0.410149 0.912019i \(-0.365477\pi\)
−0.912019 + 0.410149i \(0.865477\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.5824 18.5824i 0.646173 0.646173i −0.305893 0.952066i \(-0.598955\pi\)
0.952066 + 0.305893i \(0.0989549\pi\)
\(828\) 0 0
\(829\) 36.0402 1.25173 0.625865 0.779932i \(-0.284746\pi\)
0.625865 + 0.779932i \(0.284746\pi\)
\(830\) 0 0
\(831\) 2.49638i 0.0865984i
\(832\) 0 0
\(833\) 23.4084 + 33.7166i 0.811052 + 1.16821i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.81814 + 1.81814i −0.0628442 + 0.0628442i
\(838\) 0 0
\(839\) −37.2603 −1.28637 −0.643185 0.765711i \(-0.722387\pi\)
−0.643185 + 0.765711i \(0.722387\pi\)
\(840\) 0 0
\(841\) 23.4825 0.809740
\(842\) 0 0
\(843\) 11.8249 11.8249i 0.407272 0.407272i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.28461 36.6986i 0.112861 1.26098i
\(848\) 0 0
\(849\) 9.91328i 0.340223i
\(850\) 0 0
\(851\) 62.4632 2.14121
\(852\) 0 0
\(853\) 5.65231 5.65231i 0.193531 0.193531i −0.603689 0.797220i \(-0.706303\pi\)
0.797220 + 0.603689i \(0.206303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.5964 + 31.5964i 1.07931 + 1.07931i 0.996571 + 0.0827404i \(0.0263672\pi\)
0.0827404 + 0.996571i \(0.473633\pi\)
\(858\) 0 0
\(859\) −23.7273 −0.809565 −0.404783 0.914413i \(-0.632653\pi\)
−0.404783 + 0.914413i \(0.632653\pi\)
\(860\) 0 0
\(861\) 15.9852 13.3589i 0.544775 0.455269i
\(862\) 0 0
\(863\) 36.0121 + 36.0121i 1.22587 + 1.22587i 0.965514 + 0.260352i \(0.0838387\pi\)
0.260352 + 0.965514i \(0.416161\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −12.2915 12.2915i −0.417443 0.417443i
\(868\) 0 0
\(869\) 43.4252i 1.47310i
\(870\) 0 0
\(871\) 55.2920i 1.87350i
\(872\) 0 0
\(873\) −3.51923 + 3.51923i −0.119108 + 0.119108i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.97231 9.97231i 0.336741 0.336741i −0.518398 0.855139i \(-0.673472\pi\)
0.855139 + 0.518398i \(0.173472\pi\)
\(878\) 0 0
\(879\) 0.573776i 0.0193530i
\(880\) 0 0
\(881\) 35.5199i 1.19669i −0.801237 0.598347i \(-0.795824\pi\)
0.801237 0.598347i \(-0.204176\pi\)
\(882\) 0 0
\(883\) −15.9006 15.9006i −0.535097 0.535097i 0.386988 0.922085i \(-0.373515\pi\)
−0.922085 + 0.386988i \(0.873515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.98186 + 8.98186i 0.301581 + 0.301581i 0.841632 0.540051i \(-0.181595\pi\)
−0.540051 + 0.841632i \(0.681595\pi\)
\(888\) 0 0
\(889\) −27.5550 32.9723i −0.924164 1.10586i
\(890\) 0 0
\(891\) −4.99261 −0.167259
\(892\) 0 0
\(893\) 3.79084 + 3.79084i 0.126856 + 0.126856i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 16.4375 16.4375i 0.548831 0.548831i
\(898\) 0 0
\(899\) −6.03970 −0.201435
\(900\) 0 0
\(901\) 0.263116i 0.00876568i
\(902\) 0 0
\(903\) −0.864580 + 9.65985i −0.0287714 + 0.321460i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.91534 9.91534i 0.329233 0.329233i −0.523062 0.852295i \(-0.675210\pi\)
0.852295 + 0.523062i \(0.175210\pi\)
\(908\) 0 0
\(909\) −15.6729 −0.519838
\(910\) 0 0
\(911\) 26.7476 0.886189 0.443094 0.896475i \(-0.353881\pi\)
0.443094 + 0.896475i \(0.353881\pi\)
\(912\) 0 0
\(913\) −27.5041 + 27.5041i −0.910252 + 0.910252i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −29.7590 2.66351i −0.982730 0.0879567i
\(918\) 0 0
\(919\) 4.81405i 0.158801i −0.996843 0.0794004i \(-0.974699\pi\)
0.996843 0.0794004i \(-0.0253006\pi\)
\(920\) 0 0
\(921\) 4.26043 0.140386
\(922\) 0 0
\(923\) −32.9951 + 32.9951i −1.08605 + 1.08605i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.83284 1.83284i −0.0601983 0.0601983i
\(928\) 0 0
\(929\) 19.1224 0.627385 0.313693 0.949525i \(-0.398434\pi\)
0.313693 + 0.949525i \(0.398434\pi\)
\(930\) 0 0
\(931\) 32.1102 + 5.79431i 1.05237 + 0.189901i
\(932\) 0 0
\(933\) 1.05903 + 1.05903i 0.0346711 + 0.0346711i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18.2759 18.2759i −0.597046 0.597046i 0.342479 0.939525i \(-0.388733\pi\)
−0.939525 + 0.342479i \(0.888733\pi\)
\(938\) 0 0
\(939\) 22.7504i 0.742432i
\(940\) 0 0
\(941\) 3.37097i 0.109890i 0.998489 + 0.0549452i \(0.0174984\pi\)
−0.998489 + 0.0549452i \(0.982502\pi\)
\(942\) 0 0
\(943\) −34.7833 + 34.7833i −1.13270 + 1.13270i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.6046 35.6046i 1.15699 1.15699i 0.171873 0.985119i \(-0.445018\pi\)
0.985119 0.171873i \(-0.0549820\pi\)
\(948\) 0 0
\(949\) 41.6941i 1.35345i
\(950\) 0 0
\(951\) 2.76266i 0.0895855i
\(952\) 0 0
\(953\) 24.2600 + 24.2600i 0.785860 + 0.785860i 0.980813 0.194953i \(-0.0624554\pi\)
−0.194953 + 0.980813i \(0.562455\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.29251 8.29251i −0.268059 0.268059i
\(958\) 0 0
\(959\) 23.4156 + 28.0191i 0.756129 + 0.904785i
\(960\) 0 0
\(961\) 24.3887 0.786733
\(962\) 0 0
\(963\) −0.869787 0.869787i −0.0280285 0.0280285i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0.00833799 0.00833799i 0.000268132 0.000268132i −0.706973 0.707241i \(-0.749940\pi\)
0.707241 + 0.706973i \(0.249940\pi\)
\(968\) 0 0
\(969\) −27.3322 −0.878037
\(970\) 0 0
\(971\) 23.9127i 0.767396i 0.923459 + 0.383698i \(0.125350\pi\)
−0.923459 + 0.383698i \(0.874650\pi\)
\(972\) 0 0
\(973\) 1.85959 20.7769i 0.0596156 0.666078i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.23497 + 4.23497i −0.135489 + 0.135489i −0.771599 0.636110i \(-0.780543\pi\)
0.636110 + 0.771599i \(0.280543\pi\)
\(978\) 0 0
\(979\) 8.09064 0.258578
\(980\) 0 0
\(981\) −11.7274 −0.374426
\(982\) 0 0
\(983\) 8.01578 8.01578i 0.255664 0.255664i −0.567624 0.823288i \(-0.692137\pi\)
0.823288 + 0.567624i \(0.192137\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.271268 3.03084i 0.00863455 0.0964727i
\(988\) 0 0
\(989\) 22.9008i 0.728202i
\(990\) 0 0
\(991\) −55.1397 −1.75157 −0.875786 0.482700i \(-0.839656\pi\)
−0.875786 + 0.482700i \(0.839656\pi\)
\(992\) 0 0
\(993\) 17.9797 17.9797i 0.570568 0.570568i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −37.6893 37.6893i −1.19363 1.19363i −0.976040 0.217592i \(-0.930180\pi\)
−0.217592 0.976040i \(-0.569820\pi\)
\(998\) 0 0
\(999\) −9.99834 −0.316333
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 2100.2.x.d.1693.3 16
5.2 odd 4 inner 2100.2.x.d.1357.8 16
5.3 odd 4 420.2.x.a.97.3 yes 16
5.4 even 2 420.2.x.a.13.6 yes 16
7.6 odd 2 inner 2100.2.x.d.1693.8 16
15.8 even 4 1260.2.ba.b.937.3 16
15.14 odd 2 1260.2.ba.b.433.6 16
20.3 even 4 1680.2.cz.c.97.7 16
20.19 odd 2 1680.2.cz.c.433.2 16
35.13 even 4 420.2.x.a.97.6 yes 16
35.27 even 4 inner 2100.2.x.d.1357.3 16
35.34 odd 2 420.2.x.a.13.3 16
105.83 odd 4 1260.2.ba.b.937.6 16
105.104 even 2 1260.2.ba.b.433.3 16
140.83 odd 4 1680.2.cz.c.97.2 16
140.139 even 2 1680.2.cz.c.433.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.x.a.13.3 16 35.34 odd 2
420.2.x.a.13.6 yes 16 5.4 even 2
420.2.x.a.97.3 yes 16 5.3 odd 4
420.2.x.a.97.6 yes 16 35.13 even 4
1260.2.ba.b.433.3 16 105.104 even 2
1260.2.ba.b.433.6 16 15.14 odd 2
1260.2.ba.b.937.3 16 15.8 even 4
1260.2.ba.b.937.6 16 105.83 odd 4
1680.2.cz.c.97.2 16 140.83 odd 4
1680.2.cz.c.97.7 16 20.3 even 4
1680.2.cz.c.433.2 16 20.19 odd 2
1680.2.cz.c.433.7 16 140.139 even 2
2100.2.x.d.1357.3 16 35.27 even 4 inner
2100.2.x.d.1357.8 16 5.2 odd 4 inner
2100.2.x.d.1693.3 16 1.1 even 1 trivial
2100.2.x.d.1693.8 16 7.6 odd 2 inner