Properties

Label 2100.2.x.d.1693.5
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 1693.5
Root \(2.36188 + 0.195512i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.d.1357.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(-2.12678 - 1.57379i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +(-2.12678 - 1.57379i) q^{7} -1.00000i q^{9} -2.40009 q^{11} +(-0.697349 + 0.697349i) q^{13} +(-3.50236 - 3.50236i) q^{17} +0.306326 q^{19} +(-2.61670 + 0.391024i) q^{21} +(2.63050 + 2.63050i) q^{23} +(-0.707107 - 0.707107i) q^{27} +4.12741i q^{29} +9.14119i q^{31} +(-1.69712 + 1.69712i) q^{33} +(-6.11361 + 6.11361i) q^{37} +0.986201i q^{39} -3.06840i q^{41} +(-3.56679 - 3.56679i) q^{43} +(0.325841 + 0.325841i) q^{47} +(2.04638 + 6.69420i) q^{49} -4.95308 q^{51} +(9.00299 + 9.00299i) q^{53} +(0.216605 - 0.216605i) q^{57} -13.0580 q^{59} -6.43890i q^{61} +(-1.57379 + 2.12678i) q^{63} +(-9.48058 + 9.48058i) q^{67} +3.72008 q^{69} -9.39392 q^{71} +(-3.69517 + 3.69517i) q^{73} +(5.10446 + 3.77724i) q^{77} -12.2548i q^{79} -1.00000 q^{81} +(2.89901 - 2.89901i) q^{83} +(2.91852 + 2.91852i) q^{87} +1.75957 q^{89} +(2.58059 - 0.385628i) q^{91} +(6.46380 + 6.46380i) q^{93} +(7.96812 + 7.96812i) q^{97} +2.40009i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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
\(7\) −2.12678 1.57379i −0.803847 0.594836i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −2.40009 −0.723655 −0.361827 0.932245i \(-0.617847\pi\)
−0.361827 + 0.932245i \(0.617847\pi\)
\(12\) 0 0
\(13\) −0.697349 + 0.697349i −0.193410 + 0.193410i −0.797168 0.603758i \(-0.793669\pi\)
0.603758 + 0.797168i \(0.293669\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.50236 3.50236i −0.849447 0.849447i 0.140617 0.990064i \(-0.455091\pi\)
−0.990064 + 0.140617i \(0.955091\pi\)
\(18\) 0 0
\(19\) 0.306326 0.0702760 0.0351380 0.999382i \(-0.488813\pi\)
0.0351380 + 0.999382i \(0.488813\pi\)
\(20\) 0 0
\(21\) −2.61670 + 0.391024i −0.571010 + 0.0853283i
\(22\) 0 0
\(23\) 2.63050 + 2.63050i 0.548496 + 0.548496i 0.926006 0.377509i \(-0.123219\pi\)
−0.377509 + 0.926006i \(0.623219\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 4.12741i 0.766441i 0.923657 + 0.383221i \(0.125185\pi\)
−0.923657 + 0.383221i \(0.874815\pi\)
\(30\) 0 0
\(31\) 9.14119i 1.64181i 0.571068 + 0.820903i \(0.306529\pi\)
−0.571068 + 0.820903i \(0.693471\pi\)
\(32\) 0 0
\(33\) −1.69712 + 1.69712i −0.295431 + 0.295431i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.11361 + 6.11361i −1.00507 + 1.00507i −0.00508461 + 0.999987i \(0.501618\pi\)
−0.999987 + 0.00508461i \(0.998382\pi\)
\(38\) 0 0
\(39\) 0.986201i 0.157919i
\(40\) 0 0
\(41\) 3.06840i 0.479204i −0.970871 0.239602i \(-0.922983\pi\)
0.970871 0.239602i \(-0.0770169\pi\)
\(42\) 0 0
\(43\) −3.56679 3.56679i −0.543930 0.543930i 0.380748 0.924679i \(-0.375666\pi\)
−0.924679 + 0.380748i \(0.875666\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.325841 + 0.325841i 0.0475287 + 0.0475287i 0.730472 0.682943i \(-0.239300\pi\)
−0.682943 + 0.730472i \(0.739300\pi\)
\(48\) 0 0
\(49\) 2.04638 + 6.69420i 0.292340 + 0.956314i
\(50\) 0 0
\(51\) −4.95308 −0.693570
\(52\) 0 0
\(53\) 9.00299 + 9.00299i 1.23666 + 1.23666i 0.961359 + 0.275296i \(0.0887759\pi\)
0.275296 + 0.961359i \(0.411224\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.216605 0.216605i 0.0286900 0.0286900i
\(58\) 0 0
\(59\) −13.0580 −1.70001 −0.850003 0.526778i \(-0.823400\pi\)
−0.850003 + 0.526778i \(0.823400\pi\)
\(60\) 0 0
\(61\) 6.43890i 0.824417i −0.911090 0.412208i \(-0.864758\pi\)
0.911090 0.412208i \(-0.135242\pi\)
\(62\) 0 0
\(63\) −1.57379 + 2.12678i −0.198279 + 0.267949i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.48058 + 9.48058i −1.15824 + 1.15824i −0.173383 + 0.984854i \(0.555470\pi\)
−0.984854 + 0.173383i \(0.944530\pi\)
\(68\) 0 0
\(69\) 3.72008 0.447845
\(70\) 0 0
\(71\) −9.39392 −1.11485 −0.557427 0.830226i \(-0.688211\pi\)
−0.557427 + 0.830226i \(0.688211\pi\)
\(72\) 0 0
\(73\) −3.69517 + 3.69517i −0.432487 + 0.432487i −0.889474 0.456986i \(-0.848929\pi\)
0.456986 + 0.889474i \(0.348929\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.10446 + 3.77724i 0.581708 + 0.430456i
\(78\) 0 0
\(79\) 12.2548i 1.37878i −0.724393 0.689388i \(-0.757880\pi\)
0.724393 0.689388i \(-0.242120\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 2.89901 2.89901i 0.318207 0.318207i −0.529871 0.848078i \(-0.677760\pi\)
0.848078 + 0.529871i \(0.177760\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.91852 + 2.91852i 0.312898 + 0.312898i
\(88\) 0 0
\(89\) 1.75957 0.186514 0.0932570 0.995642i \(-0.470272\pi\)
0.0932570 + 0.995642i \(0.470272\pi\)
\(90\) 0 0
\(91\) 2.58059 0.385628i 0.270519 0.0404248i
\(92\) 0 0
\(93\) 6.46380 + 6.46380i 0.670265 + 0.670265i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.96812 + 7.96812i 0.809040 + 0.809040i 0.984489 0.175449i \(-0.0561376\pi\)
−0.175449 + 0.984489i \(0.556138\pi\)
\(98\) 0 0
\(99\) 2.40009i 0.241218i
\(100\) 0 0
\(101\) 13.0817i 1.30168i −0.759215 0.650840i \(-0.774417\pi\)
0.759215 0.650840i \(-0.225583\pi\)
\(102\) 0 0
\(103\) 2.52210 2.52210i 0.248510 0.248510i −0.571849 0.820359i \(-0.693774\pi\)
0.820359 + 0.571849i \(0.193774\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.4245 + 10.4245i −1.00778 + 1.00778i −0.00780589 + 0.999970i \(0.502485\pi\)
−0.999970 + 0.00780589i \(0.997515\pi\)
\(108\) 0 0
\(109\) 9.90616i 0.948838i 0.880299 + 0.474419i \(0.157342\pi\)
−0.880299 + 0.474419i \(0.842658\pi\)
\(110\) 0 0
\(111\) 8.64595i 0.820638i
\(112\) 0 0
\(113\) −3.12442 3.12442i −0.293921 0.293921i 0.544706 0.838627i \(-0.316641\pi\)
−0.838627 + 0.544706i \(0.816641\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.697349 + 0.697349i 0.0644700 + 0.0644700i
\(118\) 0 0
\(119\) 1.93677 + 12.9607i 0.177544 + 1.18811i
\(120\) 0 0
\(121\) −5.23956 −0.476324
\(122\) 0 0
\(123\) −2.16969 2.16969i −0.195634 0.195634i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.21196 + 2.21196i −0.196280 + 0.196280i −0.798403 0.602123i \(-0.794322\pi\)
0.602123 + 0.798403i \(0.294322\pi\)
\(128\) 0 0
\(129\) −5.04420 −0.444117
\(130\) 0 0
\(131\) 9.61691i 0.840233i −0.907470 0.420117i \(-0.861989\pi\)
0.907470 0.420117i \(-0.138011\pi\)
\(132\) 0 0
\(133\) −0.651488 0.482092i −0.0564911 0.0418027i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.90318 2.90318i 0.248035 0.248035i −0.572129 0.820164i \(-0.693882\pi\)
0.820164 + 0.572129i \(0.193882\pi\)
\(138\) 0 0
\(139\) −7.39691 −0.627398 −0.313699 0.949523i \(-0.601568\pi\)
−0.313699 + 0.949523i \(0.601568\pi\)
\(140\) 0 0
\(141\) 0.460808 0.0388070
\(142\) 0 0
\(143\) 1.67370 1.67370i 0.139962 0.139962i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.18052 + 3.28651i 0.509761 + 0.271066i
\(148\) 0 0
\(149\) 12.5882i 1.03127i −0.856809 0.515634i \(-0.827557\pi\)
0.856809 0.515634i \(-0.172443\pi\)
\(150\) 0 0
\(151\) −1.90616 −0.155121 −0.0775607 0.996988i \(-0.524713\pi\)
−0.0775607 + 0.996988i \(0.524713\pi\)
\(152\) 0 0
\(153\) −3.50236 + 3.50236i −0.283149 + 0.283149i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0128 11.0128i −0.878915 0.878915i 0.114507 0.993422i \(-0.463471\pi\)
−0.993422 + 0.114507i \(0.963471\pi\)
\(158\) 0 0
\(159\) 12.7321 1.00973
\(160\) 0 0
\(161\) −1.45464 9.73433i −0.114642 0.767172i
\(162\) 0 0
\(163\) 9.69420 + 9.69420i 0.759308 + 0.759308i 0.976196 0.216888i \(-0.0695907\pi\)
−0.216888 + 0.976196i \(0.569591\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.6702 15.6702i −1.21260 1.21260i −0.970170 0.242425i \(-0.922057\pi\)
−0.242425 0.970170i \(-0.577943\pi\)
\(168\) 0 0
\(169\) 12.0274i 0.925185i
\(170\) 0 0
\(171\) 0.306326i 0.0234253i
\(172\) 0 0
\(173\) 11.3858 11.3858i 0.865645 0.865645i −0.126342 0.991987i \(-0.540324\pi\)
0.991987 + 0.126342i \(0.0403237\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9.23339 + 9.23339i −0.694024 + 0.694024i
\(178\) 0 0
\(179\) 10.7395i 0.802705i −0.915924 0.401353i \(-0.868540\pi\)
0.915924 0.401353i \(-0.131460\pi\)
\(180\) 0 0
\(181\) 2.05717i 0.152908i −0.997073 0.0764542i \(-0.975640\pi\)
0.997073 0.0764542i \(-0.0243599\pi\)
\(182\) 0 0
\(183\) −4.55299 4.55299i −0.336567 0.336567i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.40598 + 8.40598i 0.614706 + 0.614706i
\(188\) 0 0
\(189\) 0.391024 + 2.61670i 0.0284428 + 0.190337i
\(190\) 0 0
\(191\) 1.54471 0.111771 0.0558857 0.998437i \(-0.482202\pi\)
0.0558857 + 0.998437i \(0.482202\pi\)
\(192\) 0 0
\(193\) 14.9414 + 14.9414i 1.07551 + 1.07551i 0.996906 + 0.0785987i \(0.0250446\pi\)
0.0785987 + 0.996906i \(0.474955\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.9032 + 10.9032i −0.776819 + 0.776819i −0.979289 0.202469i \(-0.935103\pi\)
0.202469 + 0.979289i \(0.435103\pi\)
\(198\) 0 0
\(199\) 14.0635 0.996934 0.498467 0.866909i \(-0.333897\pi\)
0.498467 + 0.866909i \(0.333897\pi\)
\(200\) 0 0
\(201\) 13.4076i 0.945697i
\(202\) 0 0
\(203\) 6.49567 8.77809i 0.455907 0.616101i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.63050 2.63050i 0.182832 0.182832i
\(208\) 0 0
\(209\) −0.735210 −0.0508555
\(210\) 0 0
\(211\) −20.1334 −1.38604 −0.693020 0.720919i \(-0.743720\pi\)
−0.693020 + 0.720919i \(0.743720\pi\)
\(212\) 0 0
\(213\) −6.64251 + 6.64251i −0.455137 + 0.455137i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 14.3863 19.4413i 0.976606 1.31976i
\(218\) 0 0
\(219\) 5.22576i 0.353124i
\(220\) 0 0
\(221\) 4.88473 0.328583
\(222\) 0 0
\(223\) −7.62816 + 7.62816i −0.510819 + 0.510819i −0.914777 0.403958i \(-0.867634\pi\)
0.403958 + 0.914777i \(0.367634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.71882 2.71882i −0.180454 0.180454i 0.611099 0.791554i \(-0.290728\pi\)
−0.791554 + 0.611099i \(0.790728\pi\)
\(228\) 0 0
\(229\) 9.87998 0.652887 0.326444 0.945217i \(-0.394150\pi\)
0.326444 + 0.945217i \(0.394150\pi\)
\(230\) 0 0
\(231\) 6.28031 0.938492i 0.413214 0.0617483i
\(232\) 0 0
\(233\) 2.10897 + 2.10897i 0.138163 + 0.138163i 0.772806 0.634643i \(-0.218853\pi\)
−0.634643 + 0.772806i \(0.718853\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.66547 8.66547i −0.562883 0.562883i
\(238\) 0 0
\(239\) 3.32171i 0.214863i 0.994212 + 0.107432i \(0.0342627\pi\)
−0.994212 + 0.107432i \(0.965737\pi\)
\(240\) 0 0
\(241\) 29.3877i 1.89303i 0.322666 + 0.946513i \(0.395421\pi\)
−0.322666 + 0.946513i \(0.604579\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.213616 + 0.213616i −0.0135921 + 0.0135921i
\(248\) 0 0
\(249\) 4.09981i 0.259815i
\(250\) 0 0
\(251\) 3.74084i 0.236120i −0.993006 0.118060i \(-0.962333\pi\)
0.993006 0.118060i \(-0.0376675\pi\)
\(252\) 0 0
\(253\) −6.31343 6.31343i −0.396922 0.396922i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.40308 5.40308i −0.337035 0.337035i 0.518215 0.855250i \(-0.326597\pi\)
−0.855250 + 0.518215i \(0.826597\pi\)
\(258\) 0 0
\(259\) 22.6238 3.38077i 1.40578 0.210071i
\(260\) 0 0
\(261\) 4.12741 0.255480
\(262\) 0 0
\(263\) −18.7970 18.7970i −1.15907 1.15907i −0.984675 0.174398i \(-0.944202\pi\)
−0.174398 0.984675i \(-0.555798\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.24420 1.24420i 0.0761440 0.0761440i
\(268\) 0 0
\(269\) −22.4710 −1.37008 −0.685040 0.728506i \(-0.740215\pi\)
−0.685040 + 0.728506i \(0.740215\pi\)
\(270\) 0 0
\(271\) 2.69150i 0.163497i 0.996653 + 0.0817484i \(0.0260504\pi\)
−0.996653 + 0.0817484i \(0.973950\pi\)
\(272\) 0 0
\(273\) 1.55207 2.09743i 0.0939357 0.126942i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.3746 11.3746i 0.683434 0.683434i −0.277339 0.960772i \(-0.589452\pi\)
0.960772 + 0.277339i \(0.0894524\pi\)
\(278\) 0 0
\(279\) 9.14119 0.547269
\(280\) 0 0
\(281\) 18.4728 1.10199 0.550996 0.834508i \(-0.314248\pi\)
0.550996 + 0.834508i \(0.314248\pi\)
\(282\) 0 0
\(283\) −18.5376 + 18.5376i −1.10195 + 1.10195i −0.107773 + 0.994176i \(0.534372\pi\)
−0.994176 + 0.107773i \(0.965628\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.82901 + 6.52581i −0.285048 + 0.385207i
\(288\) 0 0
\(289\) 7.53302i 0.443119i
\(290\) 0 0
\(291\) 11.2686 0.660578
\(292\) 0 0
\(293\) 5.40308 5.40308i 0.315652 0.315652i −0.531443 0.847094i \(-0.678350\pi\)
0.847094 + 0.531443i \(0.178350\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.69712 + 1.69712i 0.0984769 + 0.0984769i
\(298\) 0 0
\(299\) −3.66875 −0.212169
\(300\) 0 0
\(301\) 1.97240 + 13.1991i 0.113687 + 0.760786i
\(302\) 0 0
\(303\) −9.25018 9.25018i −0.531409 0.531409i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.9851 + 18.9851i 1.08354 + 1.08354i 0.996177 + 0.0873616i \(0.0278436\pi\)
0.0873616 + 0.996177i \(0.472156\pi\)
\(308\) 0 0
\(309\) 3.56679i 0.202908i
\(310\) 0 0
\(311\) 7.69244i 0.436198i 0.975927 + 0.218099i \(0.0699856\pi\)
−0.975927 + 0.218099i \(0.930014\pi\)
\(312\) 0 0
\(313\) −3.65913 + 3.65913i −0.206826 + 0.206826i −0.802917 0.596091i \(-0.796720\pi\)
0.596091 + 0.802917i \(0.296720\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.8032 21.8032i 1.22459 1.22459i 0.258605 0.965983i \(-0.416737\pi\)
0.965983 0.258605i \(-0.0832628\pi\)
\(318\) 0 0
\(319\) 9.90616i 0.554639i
\(320\) 0 0
\(321\) 14.7425i 0.822845i
\(322\) 0 0
\(323\) −1.07286 1.07286i −0.0596957 0.0596957i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.00472 + 7.00472i 0.387362 + 0.387362i
\(328\) 0 0
\(329\) −0.180187 1.20580i −0.00993402 0.0664776i
\(330\) 0 0
\(331\) −5.17222 −0.284291 −0.142145 0.989846i \(-0.545400\pi\)
−0.142145 + 0.989846i \(0.545400\pi\)
\(332\) 0 0
\(333\) 6.11361 + 6.11361i 0.335024 + 0.335024i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.41776 8.41776i 0.458545 0.458545i −0.439633 0.898178i \(-0.644891\pi\)
0.898178 + 0.439633i \(0.144891\pi\)
\(338\) 0 0
\(339\) −4.41860 −0.239985
\(340\) 0 0
\(341\) 21.9397i 1.18810i
\(342\) 0 0
\(343\) 6.18306 17.4577i 0.333854 0.942625i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.0753948 0.0753948i 0.00404740 0.00404740i −0.705080 0.709128i \(-0.749089\pi\)
0.709128 + 0.705080i \(0.249089\pi\)
\(348\) 0 0
\(349\) −34.9899 −1.87296 −0.936482 0.350714i \(-0.885939\pi\)
−0.936482 + 0.350714i \(0.885939\pi\)
\(350\) 0 0
\(351\) 0.986201 0.0526395
\(352\) 0 0
\(353\) 7.53299 7.53299i 0.400941 0.400941i −0.477624 0.878564i \(-0.658502\pi\)
0.878564 + 0.477624i \(0.158502\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.5341 + 7.79510i 0.557524 + 0.412561i
\(358\) 0 0
\(359\) 22.1572i 1.16941i −0.811245 0.584707i \(-0.801210\pi\)
0.811245 0.584707i \(-0.198790\pi\)
\(360\) 0 0
\(361\) −18.9062 −0.995061
\(362\) 0 0
\(363\) −3.70493 + 3.70493i −0.194458 + 0.194458i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −21.8093 21.8093i −1.13844 1.13844i −0.988730 0.149707i \(-0.952167\pi\)
−0.149707 0.988730i \(-0.547833\pi\)
\(368\) 0 0
\(369\) −3.06840 −0.159735
\(370\) 0 0
\(371\) −4.97857 33.3162i −0.258474 1.72969i
\(372\) 0 0
\(373\) −6.90616 6.90616i −0.357588 0.357588i 0.505335 0.862923i \(-0.331369\pi\)
−0.862923 + 0.505335i \(0.831369\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.87825 2.87825i −0.148237 0.148237i
\(378\) 0 0
\(379\) 33.2546i 1.70818i −0.520129 0.854088i \(-0.674116\pi\)
0.520129 0.854088i \(-0.325884\pi\)
\(380\) 0 0
\(381\) 3.12819i 0.160262i
\(382\) 0 0
\(383\) −5.14086 + 5.14086i −0.262686 + 0.262686i −0.826144 0.563459i \(-0.809470\pi\)
0.563459 + 0.826144i \(0.309470\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.56679 + 3.56679i −0.181310 + 0.181310i
\(388\) 0 0
\(389\) 29.2670i 1.48389i 0.670458 + 0.741947i \(0.266097\pi\)
−0.670458 + 0.741947i \(0.733903\pi\)
\(390\) 0 0
\(391\) 18.4259i 0.931836i
\(392\) 0 0
\(393\) −6.80018 6.80018i −0.343024 0.343024i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.3632 + 22.3632i 1.12238 + 1.12238i 0.991383 + 0.130992i \(0.0418163\pi\)
0.130992 + 0.991383i \(0.458184\pi\)
\(398\) 0 0
\(399\) −0.801562 + 0.119781i −0.0401283 + 0.00599653i
\(400\) 0 0
\(401\) 25.8675 1.29176 0.645881 0.763438i \(-0.276490\pi\)
0.645881 + 0.763438i \(0.276490\pi\)
\(402\) 0 0
\(403\) −6.37460 6.37460i −0.317542 0.317542i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.6732 14.6732i 0.727325 0.727325i
\(408\) 0 0
\(409\) −36.4392 −1.80180 −0.900902 0.434023i \(-0.857094\pi\)
−0.900902 + 0.434023i \(0.857094\pi\)
\(410\) 0 0
\(411\) 4.10571i 0.202520i
\(412\) 0 0
\(413\) 27.7715 + 20.5505i 1.36654 + 1.01122i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.23040 + 5.23040i −0.256134 + 0.256134i
\(418\) 0 0
\(419\) −22.9097 −1.11921 −0.559607 0.828758i \(-0.689048\pi\)
−0.559607 + 0.828758i \(0.689048\pi\)
\(420\) 0 0
\(421\) 29.3098 1.42847 0.714237 0.699904i \(-0.246774\pi\)
0.714237 + 0.699904i \(0.246774\pi\)
\(422\) 0 0
\(423\) 0.325841 0.325841i 0.0158429 0.0158429i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10.1335 + 13.6941i −0.490393 + 0.662705i
\(428\) 0 0
\(429\) 2.36697i 0.114279i
\(430\) 0 0
\(431\) 40.5947 1.95538 0.977688 0.210063i \(-0.0673669\pi\)
0.977688 + 0.210063i \(0.0673669\pi\)
\(432\) 0 0
\(433\) −1.09377 + 1.09377i −0.0525632 + 0.0525632i −0.732900 0.680337i \(-0.761834\pi\)
0.680337 + 0.732900i \(0.261834\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.805789 + 0.805789i 0.0385461 + 0.0385461i
\(438\) 0 0
\(439\) 18.2283 0.869991 0.434995 0.900433i \(-0.356750\pi\)
0.434995 + 0.900433i \(0.356750\pi\)
\(440\) 0 0
\(441\) 6.69420 2.04638i 0.318771 0.0974467i
\(442\) 0 0
\(443\) −22.3914 22.3914i −1.06385 1.06385i −0.997818 0.0660294i \(-0.978967\pi\)
−0.0660294 0.997818i \(-0.521033\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −8.90122 8.90122i −0.421013 0.421013i
\(448\) 0 0
\(449\) 26.1223i 1.23279i −0.787437 0.616395i \(-0.788593\pi\)
0.787437 0.616395i \(-0.211407\pi\)
\(450\) 0 0
\(451\) 7.36444i 0.346778i
\(452\) 0 0
\(453\) −1.34786 + 1.34786i −0.0633281 + 0.0633281i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.3546 11.3546i 0.531147 0.531147i −0.389766 0.920914i \(-0.627444\pi\)
0.920914 + 0.389766i \(0.127444\pi\)
\(458\) 0 0
\(459\) 4.95308i 0.231190i
\(460\) 0 0
\(461\) 10.9957i 0.512120i 0.966661 + 0.256060i \(0.0824244\pi\)
−0.966661 + 0.256060i \(0.917576\pi\)
\(462\) 0 0
\(463\) −7.01215 7.01215i −0.325882 0.325882i 0.525136 0.851018i \(-0.324014\pi\)
−0.851018 + 0.525136i \(0.824014\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.80297 + 3.80297i 0.175980 + 0.175980i 0.789601 0.613621i \(-0.210288\pi\)
−0.613621 + 0.789601i \(0.710288\pi\)
\(468\) 0 0
\(469\) 35.0835 5.24268i 1.62001 0.242084i
\(470\) 0 0
\(471\) −15.5744 −0.717631
\(472\) 0 0
\(473\) 8.56062 + 8.56062i 0.393618 + 0.393618i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 9.00299 9.00299i 0.412219 0.412219i
\(478\) 0 0
\(479\) −33.5693 −1.53382 −0.766910 0.641755i \(-0.778207\pi\)
−0.766910 + 0.641755i \(0.778207\pi\)
\(480\) 0 0
\(481\) 8.52665i 0.388782i
\(482\) 0 0
\(483\) −7.91179 5.85462i −0.359999 0.266395i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.90000 8.90000i 0.403297 0.403297i −0.476096 0.879393i \(-0.657949\pi\)
0.879393 + 0.476096i \(0.157949\pi\)
\(488\) 0 0
\(489\) 13.7097 0.619973
\(490\) 0 0
\(491\) −21.4491 −0.967985 −0.483993 0.875072i \(-0.660814\pi\)
−0.483993 + 0.875072i \(0.660814\pi\)
\(492\) 0 0
\(493\) 14.4557 14.4557i 0.651051 0.651051i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.9788 + 14.7840i 0.896172 + 0.663155i
\(498\) 0 0
\(499\) 27.0826i 1.21238i 0.795318 + 0.606192i \(0.207304\pi\)
−0.795318 + 0.606192i \(0.792696\pi\)
\(500\) 0 0
\(501\) −22.1610 −0.990080
\(502\) 0 0
\(503\) 6.35303 6.35303i 0.283268 0.283268i −0.551143 0.834411i \(-0.685808\pi\)
0.834411 + 0.551143i \(0.185808\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.50466 + 8.50466i 0.377705 + 0.377705i
\(508\) 0 0
\(509\) 28.8456 1.27856 0.639279 0.768975i \(-0.279233\pi\)
0.639279 + 0.768975i \(0.279233\pi\)
\(510\) 0 0
\(511\) 13.6742 2.04340i 0.604912 0.0903945i
\(512\) 0 0
\(513\) −0.216605 0.216605i −0.00956335 0.00956335i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.782047 0.782047i −0.0343944 0.0343944i
\(518\) 0 0
\(519\) 16.1019i 0.706796i
\(520\) 0 0
\(521\) 2.35064i 0.102983i 0.998673 + 0.0514917i \(0.0163976\pi\)
−0.998673 + 0.0514917i \(0.983602\pi\)
\(522\) 0 0
\(523\) 5.70011 5.70011i 0.249248 0.249248i −0.571414 0.820662i \(-0.693605\pi\)
0.820662 + 0.571414i \(0.193605\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.0157 32.0157i 1.39463 1.39463i
\(528\) 0 0
\(529\) 9.16099i 0.398304i
\(530\) 0 0
\(531\) 13.0580i 0.566668i
\(532\) 0 0
\(533\) 2.13975 + 2.13975i 0.0926828 + 0.0926828i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −7.59395 7.59395i −0.327703 0.327703i
\(538\) 0 0
\(539\) −4.91150 16.0667i −0.211553 0.692042i
\(540\) 0 0
\(541\) 26.3730 1.13386 0.566931 0.823765i \(-0.308131\pi\)
0.566931 + 0.823765i \(0.308131\pi\)
\(542\) 0 0
\(543\) −1.45464 1.45464i −0.0624246 0.0624246i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.45318 2.45318i 0.104890 0.104890i −0.652714 0.757604i \(-0.726370\pi\)
0.757604 + 0.652714i \(0.226370\pi\)
\(548\) 0 0
\(549\) −6.43890 −0.274806
\(550\) 0 0
\(551\) 1.26433i 0.0538624i
\(552\) 0 0
\(553\) −19.2865 + 26.0633i −0.820145 + 1.10832i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.2516 20.2516i 0.858090 0.858090i −0.133023 0.991113i \(-0.542468\pi\)
0.991113 + 0.133023i \(0.0424685\pi\)
\(558\) 0 0
\(559\) 4.97460 0.210403
\(560\) 0 0
\(561\) 11.8879 0.501905
\(562\) 0 0
\(563\) −13.0580 + 13.0580i −0.550329 + 0.550329i −0.926536 0.376207i \(-0.877228\pi\)
0.376207 + 0.926536i \(0.377228\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.12678 + 1.57379i 0.0893163 + 0.0660929i
\(568\) 0 0
\(569\) 41.7104i 1.74859i 0.485395 + 0.874295i \(0.338676\pi\)
−0.485395 + 0.874295i \(0.661324\pi\)
\(570\) 0 0
\(571\) 23.5728 0.986490 0.493245 0.869891i \(-0.335811\pi\)
0.493245 + 0.869891i \(0.335811\pi\)
\(572\) 0 0
\(573\) 1.09228 1.09228i 0.0456305 0.0456305i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −20.3191 20.3191i −0.845897 0.845897i 0.143721 0.989618i \(-0.454093\pi\)
−0.989618 + 0.143721i \(0.954093\pi\)
\(578\) 0 0
\(579\) 21.1303 0.878146
\(580\) 0 0
\(581\) −10.7280 + 1.60312i −0.445071 + 0.0665088i
\(582\) 0 0
\(583\) −21.6080 21.6080i −0.894912 0.894912i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.5042 21.5042i −0.887575 0.887575i 0.106715 0.994290i \(-0.465967\pi\)
−0.994290 + 0.106715i \(0.965967\pi\)
\(588\) 0 0
\(589\) 2.80018i 0.115380i
\(590\) 0 0
\(591\) 15.4194i 0.634270i
\(592\) 0 0
\(593\) 24.2636 24.2636i 0.996386 0.996386i −0.00360761 0.999993i \(-0.501148\pi\)
0.999993 + 0.00360761i \(0.00114834\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9.94438 9.94438i 0.406996 0.406996i
\(598\) 0 0
\(599\) 20.5458i 0.839479i −0.907645 0.419740i \(-0.862121\pi\)
0.907645 0.419740i \(-0.137879\pi\)
\(600\) 0 0
\(601\) 10.2578i 0.418424i 0.977870 + 0.209212i \(0.0670899\pi\)
−0.977870 + 0.209212i \(0.932910\pi\)
\(602\) 0 0
\(603\) 9.48058 + 9.48058i 0.386079 + 0.386079i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.82671 + 3.82671i 0.155321 + 0.155321i 0.780490 0.625169i \(-0.214970\pi\)
−0.625169 + 0.780490i \(0.714970\pi\)
\(608\) 0 0
\(609\) −1.61392 10.8002i −0.0653991 0.437645i
\(610\) 0 0
\(611\) −0.454449 −0.0183851
\(612\) 0 0
\(613\) −17.7615 17.7615i −0.717382 0.717382i 0.250686 0.968068i \(-0.419344\pi\)
−0.968068 + 0.250686i \(0.919344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.4852 + 26.4852i −1.06626 + 1.06626i −0.0686116 + 0.997643i \(0.521857\pi\)
−0.997643 + 0.0686116i \(0.978143\pi\)
\(618\) 0 0
\(619\) −1.71847 −0.0690712 −0.0345356 0.999403i \(-0.510995\pi\)
−0.0345356 + 0.999403i \(0.510995\pi\)
\(620\) 0 0
\(621\) 3.72008i 0.149282i
\(622\) 0 0
\(623\) −3.74221 2.76919i −0.149929 0.110945i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.519872 + 0.519872i −0.0207617 + 0.0207617i
\(628\) 0 0
\(629\) 42.8241 1.70751
\(630\) 0 0
\(631\) 22.5960 0.899531 0.449765 0.893147i \(-0.351508\pi\)
0.449765 + 0.893147i \(0.351508\pi\)
\(632\) 0 0
\(633\) −14.2365 + 14.2365i −0.565848 + 0.565848i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.09524 3.24116i −0.241502 0.128419i
\(638\) 0 0
\(639\) 9.39392i 0.371618i
\(640\) 0 0
\(641\) −16.9828 −0.670780 −0.335390 0.942079i \(-0.608868\pi\)
−0.335390 + 0.942079i \(0.608868\pi\)
\(642\) 0 0
\(643\) 16.3153 16.3153i 0.643413 0.643413i −0.307980 0.951393i \(-0.599653\pi\)
0.951393 + 0.307980i \(0.0996530\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.1017 20.1017i −0.790281 0.790281i 0.191259 0.981540i \(-0.438743\pi\)
−0.981540 + 0.191259i \(0.938743\pi\)
\(648\) 0 0
\(649\) 31.3404 1.23022
\(650\) 0 0
\(651\) −3.57442 23.9197i −0.140093 0.937488i
\(652\) 0 0
\(653\) −15.0968 15.0968i −0.590784 0.590784i 0.347059 0.937843i \(-0.387180\pi\)
−0.937843 + 0.347059i \(0.887180\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.69517 + 3.69517i 0.144162 + 0.144162i
\(658\) 0 0
\(659\) 9.68446i 0.377253i 0.982049 + 0.188627i \(0.0604036\pi\)
−0.982049 + 0.188627i \(0.939596\pi\)
\(660\) 0 0
\(661\) 31.4292i 1.22245i 0.791455 + 0.611227i \(0.209324\pi\)
−0.791455 + 0.611227i \(0.790676\pi\)
\(662\) 0 0
\(663\) 3.45403 3.45403i 0.134143 0.134143i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.8571 + 10.8571i −0.420390 + 0.420390i
\(668\) 0 0
\(669\) 10.7878i 0.417082i
\(670\) 0 0
\(671\) 15.4540i 0.596593i
\(672\) 0 0
\(673\) 13.2472 + 13.2472i 0.510642 + 0.510642i 0.914723 0.404081i \(-0.132409\pi\)
−0.404081 + 0.914723i \(0.632409\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.0979 + 17.0979i 0.657127 + 0.657127i 0.954699 0.297573i \(-0.0961770\pi\)
−0.297573 + 0.954699i \(0.596177\pi\)
\(678\) 0 0
\(679\) −4.40630 29.4866i −0.169098 1.13159i
\(680\) 0 0
\(681\) −3.84499 −0.147340
\(682\) 0 0
\(683\) −2.78505 2.78505i −0.106567 0.106567i 0.651813 0.758380i \(-0.274009\pi\)
−0.758380 + 0.651813i \(0.774009\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.98620 6.98620i 0.266540 0.266540i
\(688\) 0 0
\(689\) −12.5565 −0.478363
\(690\) 0 0
\(691\) 13.5727i 0.516331i 0.966101 + 0.258166i \(0.0831180\pi\)
−0.966101 + 0.258166i \(0.916882\pi\)
\(692\) 0 0
\(693\) 3.77724 5.10446i 0.143485 0.193903i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10.7466 + 10.7466i −0.407058 + 0.407058i
\(698\) 0 0
\(699\) 2.98253 0.112810
\(700\) 0 0
\(701\) −20.5513 −0.776213 −0.388107 0.921614i \(-0.626871\pi\)
−0.388107 + 0.921614i \(0.626871\pi\)
\(702\) 0 0
\(703\) −1.87276 + 1.87276i −0.0706324 + 0.0706324i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.5879 + 27.8220i −0.774287 + 1.04635i
\(708\) 0 0
\(709\) 0.976812i 0.0366850i 0.999832 + 0.0183425i \(0.00583892\pi\)
−0.999832 + 0.0183425i \(0.994161\pi\)
\(710\) 0 0
\(711\) −12.2548 −0.459592
\(712\) 0 0
\(713\) −24.0459 + 24.0459i −0.900524 + 0.900524i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.34880 + 2.34880i 0.0877176 + 0.0877176i
\(718\) 0 0
\(719\) 30.0286 1.11988 0.559938 0.828534i \(-0.310825\pi\)
0.559938 + 0.828534i \(0.310825\pi\)
\(720\) 0 0
\(721\) −9.33321 + 1.39470i −0.347587 + 0.0519413i
\(722\) 0 0
\(723\) 20.7802 + 20.7802i 0.772825 + 0.772825i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.4518 + 18.4518i 0.684338 + 0.684338i 0.960975 0.276636i \(-0.0892197\pi\)
−0.276636 + 0.960975i \(0.589220\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 24.9843i 0.924080i
\(732\) 0 0
\(733\) −3.66693 + 3.66693i −0.135441 + 0.135441i −0.771577 0.636136i \(-0.780532\pi\)
0.636136 + 0.771577i \(0.280532\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.7543 22.7543i 0.838164 0.838164i
\(738\) 0 0
\(739\) 28.2515i 1.03925i 0.854395 + 0.519624i \(0.173928\pi\)
−0.854395 + 0.519624i \(0.826072\pi\)
\(740\) 0 0
\(741\) 0.302099i 0.0110979i
\(742\) 0 0
\(743\) 38.3914 + 38.3914i 1.40844 + 1.40844i 0.768032 + 0.640412i \(0.221236\pi\)
0.640412 + 0.768032i \(0.278764\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.89901 2.89901i −0.106069 0.106069i
\(748\) 0 0
\(749\) 38.5766 5.76466i 1.40956 0.210636i
\(750\) 0 0
\(751\) −27.8938 −1.01786 −0.508930 0.860808i \(-0.669959\pi\)
−0.508930 + 0.860808i \(0.669959\pi\)
\(752\) 0 0
\(753\) −2.64517 2.64517i −0.0963955 0.0963955i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17.7786 + 17.7786i −0.646173 + 0.646173i −0.952066 0.305893i \(-0.901045\pi\)
0.305893 + 0.952066i \(0.401045\pi\)
\(758\) 0 0
\(759\) −8.92854 −0.324085
\(760\) 0 0
\(761\) 44.4107i 1.60989i −0.593353 0.804943i \(-0.702196\pi\)
0.593353 0.804943i \(-0.297804\pi\)
\(762\) 0 0
\(763\) 15.5902 21.0682i 0.564403 0.762721i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.10598 9.10598i 0.328798 0.328798i
\(768\) 0 0
\(769\) −12.1042 −0.436489 −0.218244 0.975894i \(-0.570033\pi\)
−0.218244 + 0.975894i \(0.570033\pi\)
\(770\) 0 0
\(771\) −7.64112 −0.275188
\(772\) 0 0
\(773\) −28.8991 + 28.8991i −1.03943 + 1.03943i −0.0402368 + 0.999190i \(0.512811\pi\)
−0.999190 + 0.0402368i \(0.987189\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 13.6069 18.3880i 0.488145 0.659667i
\(778\) 0 0
\(779\) 0.939931i 0.0336765i
\(780\) 0 0
\(781\) 22.5463 0.806769
\(782\) 0 0
\(783\) 2.91852 2.91852i 0.104299 0.104299i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.7200 14.7200i −0.524711 0.524711i 0.394280 0.918990i \(-0.370994\pi\)
−0.918990 + 0.394280i \(0.870994\pi\)
\(788\) 0 0
\(789\) −26.5830 −0.946379
\(790\) 0 0
\(791\) 1.72778 + 11.5621i 0.0614327 + 0.411102i
\(792\) 0 0
\(793\) 4.49016 + 4.49016i 0.159450 + 0.159450i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.4784 + 12.4784i 0.442007 + 0.442007i 0.892686 0.450679i \(-0.148818\pi\)
−0.450679 + 0.892686i \(0.648818\pi\)
\(798\) 0 0
\(799\) 2.28242i 0.0807462i
\(800\) 0 0
\(801\) 1.75957i 0.0621713i
\(802\) 0 0
\(803\) 8.86875 8.86875i 0.312971 0.312971i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −15.8894 + 15.8894i −0.559333 + 0.559333i
\(808\) 0 0
\(809\) 21.3149i 0.749392i −0.927148 0.374696i \(-0.877747\pi\)
0.927148 0.374696i \(-0.122253\pi\)
\(810\) 0 0
\(811\) 40.4684i 1.42104i −0.703678 0.710519i \(-0.748460\pi\)
0.703678 0.710519i \(-0.251540\pi\)
\(812\) 0 0
\(813\) 1.90318 + 1.90318i 0.0667473 + 0.0667473i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.09260 1.09260i −0.0382252 0.0382252i
\(818\) 0 0
\(819\) −0.385628 2.58059i −0.0134749 0.0901731i
\(820\) 0 0
\(821\) 24.0722 0.840126 0.420063 0.907495i \(-0.362008\pi\)
0.420063 + 0.907495i \(0.362008\pi\)
\(822\) 0 0
\(823\) −19.3332 19.3332i −0.673913 0.673913i 0.284703 0.958616i \(-0.408105\pi\)
−0.958616 + 0.284703i \(0.908105\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.1309 + 29.1309i −1.01298 + 1.01298i −0.0130645 + 0.999915i \(0.504159\pi\)
−0.999915 + 0.0130645i \(0.995841\pi\)
\(828\) 0 0
\(829\) −16.6708 −0.579001 −0.289500 0.957178i \(-0.593489\pi\)
−0.289500 + 0.957178i \(0.593489\pi\)
\(830\) 0 0
\(831\) 16.0861i 0.558021i
\(832\) 0 0
\(833\) 16.2783 30.6126i 0.564011 1.06067i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.46380 6.46380i 0.223422 0.223422i
\(838\) 0 0
\(839\) −34.5881 −1.19411 −0.597057 0.802198i \(-0.703664\pi\)
−0.597057 + 0.802198i \(0.703664\pi\)
\(840\) 0 0
\(841\) 11.9645 0.412568
\(842\) 0 0
\(843\) 13.0622 13.0622i 0.449887 0.449887i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11.1434 + 8.24596i 0.382891 + 0.283335i
\(848\) 0 0
\(849\) 26.2162i 0.899737i
\(850\) 0 0
\(851\) −32.1637 −1.10256
\(852\) 0 0
\(853\) −19.0158 + 19.0158i −0.651088 + 0.651088i −0.953255 0.302167i \(-0.902290\pi\)
0.302167 + 0.953255i \(0.402290\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.52545 3.52545i −0.120427 0.120427i 0.644325 0.764752i \(-0.277139\pi\)
−0.764752 + 0.644325i \(0.777139\pi\)
\(858\) 0 0
\(859\) −22.2575 −0.759415 −0.379707 0.925107i \(-0.623975\pi\)
−0.379707 + 0.925107i \(0.623975\pi\)
\(860\) 0 0
\(861\) 1.19982 + 8.02907i 0.0408897 + 0.273630i
\(862\) 0 0
\(863\) −16.2156 16.2156i −0.551985 0.551985i 0.375028 0.927013i \(-0.377633\pi\)
−0.927013 + 0.375028i \(0.877633\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5.32665 + 5.32665i 0.180903 + 0.180903i
\(868\) 0 0
\(869\) 29.4127i 0.997757i
\(870\) 0 0
\(871\) 13.2226i 0.448029i
\(872\) 0 0
\(873\) 7.96812 7.96812i 0.269680 0.269680i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −21.7768 + 21.7768i −0.735350 + 0.735350i −0.971674 0.236324i \(-0.924057\pi\)
0.236324 + 0.971674i \(0.424057\pi\)
\(878\) 0 0
\(879\) 7.64112i 0.257728i
\(880\) 0 0
\(881\) 9.67968i 0.326117i −0.986616 0.163058i \(-0.947864\pi\)
0.986616 0.163058i \(-0.0521359\pi\)
\(882\) 0 0
\(883\) 12.7140 + 12.7140i 0.427859 + 0.427859i 0.887899 0.460039i \(-0.152165\pi\)
−0.460039 + 0.887899i \(0.652165\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.6352 + 17.6352i 0.592132 + 0.592132i 0.938207 0.346075i \(-0.112486\pi\)
−0.346075 + 0.938207i \(0.612486\pi\)
\(888\) 0 0
\(889\) 8.18552 1.22320i 0.274534 0.0410247i
\(890\) 0 0
\(891\) 2.40009 0.0804061
\(892\) 0 0
\(893\) 0.0998134 + 0.0998134i 0.00334013 + 0.00334013i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.59420 + 2.59420i −0.0866177 + 0.0866177i
\(898\) 0 0
\(899\) −37.7294 −1.25835
\(900\) 0 0
\(901\) 63.0634i 2.10095i
\(902\) 0 0
\(903\) 10.7279 + 7.93851i 0.357002 + 0.264177i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.91380 + 3.91380i −0.129955 + 0.129955i −0.769093 0.639137i \(-0.779292\pi\)
0.639137 + 0.769093i \(0.279292\pi\)
\(908\) 0 0
\(909\) −13.0817 −0.433894
\(910\) 0 0
\(911\) −25.2908 −0.837922 −0.418961 0.908004i \(-0.637606\pi\)
−0.418961 + 0.908004i \(0.637606\pi\)
\(912\) 0 0
\(913\) −6.95788 + 6.95788i −0.230272 + 0.230272i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.1350 + 20.4530i −0.499801 + 0.675419i
\(918\) 0 0
\(919\) 35.4513i 1.16943i 0.811238 + 0.584716i \(0.198794\pi\)
−0.811238 + 0.584716i \(0.801206\pi\)
\(920\) 0 0
\(921\) 26.8490 0.884705
\(922\) 0 0
\(923\) 6.55085 6.55085i 0.215624 0.215624i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.52210 2.52210i −0.0828367 0.0828367i
\(928\) 0 0
\(929\) −25.9756 −0.852233 −0.426117 0.904668i \(-0.640119\pi\)
−0.426117 + 0.904668i \(0.640119\pi\)
\(930\) 0 0
\(931\) 0.626859 + 2.05061i 0.0205445 + 0.0672059i
\(932\) 0 0
\(933\) 5.43938 + 5.43938i 0.178077 + 0.178077i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.53485 + 4.53485i 0.148147 + 0.148147i 0.777290 0.629143i \(-0.216594\pi\)
−0.629143 + 0.777290i \(0.716594\pi\)
\(938\) 0 0
\(939\) 5.17479i 0.168873i
\(940\) 0 0
\(941\) 9.84843i 0.321050i −0.987032 0.160525i \(-0.948681\pi\)
0.987032 0.160525i \(-0.0513187\pi\)
\(942\) 0 0
\(943\) 8.07142 8.07142i 0.262841 0.262841i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0694 10.0694i 0.327212 0.327212i −0.524313 0.851525i \(-0.675678\pi\)
0.851525 + 0.524313i \(0.175678\pi\)
\(948\) 0 0
\(949\) 5.15365i 0.167295i
\(950\) 0 0
\(951\) 30.8343i 0.999872i
\(952\) 0 0
\(953\) 10.5086 + 10.5086i 0.340407 + 0.340407i 0.856520 0.516113i \(-0.172622\pi\)
−0.516113 + 0.856520i \(0.672622\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7.00472 7.00472i −0.226430 0.226430i
\(958\) 0 0
\(959\) −10.7434 + 1.60543i −0.346922 + 0.0518420i
\(960\) 0 0
\(961\) −52.5613 −1.69553
\(962\) 0 0
\(963\) 10.4245 + 10.4245i 0.335925 + 0.335925i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.6925 33.6925i 1.08348 1.08348i 0.0872970 0.996182i \(-0.472177\pi\)
0.996182 0.0872970i \(-0.0278229\pi\)
\(968\) 0 0
\(969\) −1.51726 −0.0487413
\(970\) 0 0
\(971\) 46.3619i 1.48782i 0.668278 + 0.743911i \(0.267032\pi\)
−0.668278 + 0.743911i \(0.732968\pi\)
\(972\) 0 0
\(973\) 15.7316 + 11.6412i 0.504332 + 0.373199i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −29.2362 + 29.2362i −0.935349 + 0.935349i −0.998033 0.0626847i \(-0.980034\pi\)
0.0626847 + 0.998033i \(0.480034\pi\)
\(978\) 0 0
\(979\) −4.22313 −0.134972
\(980\) 0 0
\(981\) 9.90616 0.316279
\(982\) 0 0
\(983\) 2.72180 2.72180i 0.0868120 0.0868120i −0.662367 0.749179i \(-0.730448\pi\)
0.749179 + 0.662367i \(0.230448\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.980037 0.725215i −0.0311949 0.0230838i
\(988\) 0 0
\(989\) 18.7648i 0.596687i
\(990\) 0 0
\(991\) 1.53413 0.0487332 0.0243666 0.999703i \(-0.492243\pi\)
0.0243666 + 0.999703i \(0.492243\pi\)
\(992\) 0 0
\(993\) −3.65731 + 3.65731i −0.116061 + 0.116061i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.64616 6.64616i −0.210486 0.210486i 0.593988 0.804474i \(-0.297553\pi\)
−0.804474 + 0.593988i \(0.797553\pi\)
\(998\) 0 0
\(999\) 8.64595 0.273546
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.1693.5 16
5.2 odd 4 inner 2100.2.x.d.1357.4 16
5.3 odd 4 420.2.x.a.97.5 yes 16
5.4 even 2 420.2.x.a.13.4 16
7.6 odd 2 inner 2100.2.x.d.1693.4 16
15.8 even 4 1260.2.ba.b.937.7 16
15.14 odd 2 1260.2.ba.b.433.2 16
20.3 even 4 1680.2.cz.c.97.1 16
20.19 odd 2 1680.2.cz.c.433.8 16
35.13 even 4 420.2.x.a.97.4 yes 16
35.27 even 4 inner 2100.2.x.d.1357.5 16
35.34 odd 2 420.2.x.a.13.5 yes 16
105.83 odd 4 1260.2.ba.b.937.2 16
105.104 even 2 1260.2.ba.b.433.7 16
140.83 odd 4 1680.2.cz.c.97.8 16
140.139 even 2 1680.2.cz.c.433.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.x.a.13.4 16 5.4 even 2
420.2.x.a.13.5 yes 16 35.34 odd 2
420.2.x.a.97.4 yes 16 35.13 even 4
420.2.x.a.97.5 yes 16 5.3 odd 4
1260.2.ba.b.433.2 16 15.14 odd 2
1260.2.ba.b.433.7 16 105.104 even 2
1260.2.ba.b.937.2 16 105.83 odd 4
1260.2.ba.b.937.7 16 15.8 even 4
1680.2.cz.c.97.1 16 20.3 even 4
1680.2.cz.c.97.8 16 140.83 odd 4
1680.2.cz.c.433.1 16 140.139 even 2
1680.2.cz.c.433.8 16 20.19 odd 2
2100.2.x.d.1357.4 16 5.2 odd 4 inner
2100.2.x.d.1357.5 16 35.27 even 4 inner
2100.2.x.d.1693.4 16 7.6 odd 2 inner
2100.2.x.d.1693.5 16 1.1 even 1 trivial