Properties

Label 2100.2.x.d.1693.6
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 1693.6
Root \(-0.919224 + 1.30296i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.d.1357.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(-1.51930 + 2.16604i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +(-1.51930 + 2.16604i) q^{7} -1.00000i q^{9} +1.10553 q^{11} +(-3.52265 + 3.52265i) q^{13} +(-3.23842 - 3.23842i) q^{17} +0.916719 q^{19} +(0.457309 + 2.60593i) q^{21} +(-4.43909 - 4.43909i) q^{23} +(-0.707107 - 0.707107i) q^{27} -8.28530i q^{29} -4.59362i q^{31} +(0.781726 - 0.781726i) q^{33} +(2.30352 - 2.30352i) q^{37} +4.98178i q^{39} -3.15091i q^{41} +(-2.70356 - 2.70356i) q^{43} +(-4.71436 - 4.71436i) q^{47} +(-2.38343 - 6.58174i) q^{49} -4.57981 q^{51} +(6.41894 + 6.41894i) q^{53} +(0.648218 - 0.648218i) q^{57} -4.36339 q^{59} -10.8687i q^{61} +(2.16604 + 1.51930i) q^{63} +(6.81101 - 6.81101i) q^{67} -6.27782 q^{69} +4.79796 q^{71} +(-9.72950 + 9.72950i) q^{73} +(-1.67963 + 2.39461i) q^{77} +12.5706i q^{79} -1.00000 q^{81} +(-0.227509 + 0.227509i) q^{83} +(-5.85859 - 5.85859i) q^{87} -8.93123 q^{89} +(-2.27821 - 12.9822i) q^{91} +(-3.24818 - 3.24818i) q^{93} +(-12.4114 - 12.4114i) q^{97} -1.10553i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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
\(7\) −1.51930 + 2.16604i −0.574243 + 0.818685i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.10553 0.333329 0.166665 0.986014i \(-0.446700\pi\)
0.166665 + 0.986014i \(0.446700\pi\)
\(12\) 0 0
\(13\) −3.52265 + 3.52265i −0.977007 + 0.977007i −0.999742 0.0227347i \(-0.992763\pi\)
0.0227347 + 0.999742i \(0.492763\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.23842 3.23842i −0.785432 0.785432i 0.195310 0.980742i \(-0.437429\pi\)
−0.980742 + 0.195310i \(0.937429\pi\)
\(18\) 0 0
\(19\) 0.916719 0.210310 0.105155 0.994456i \(-0.466466\pi\)
0.105155 + 0.994456i \(0.466466\pi\)
\(20\) 0 0
\(21\) 0.457309 + 2.60593i 0.0997931 + 0.568660i
\(22\) 0 0
\(23\) −4.43909 4.43909i −0.925613 0.925613i 0.0718052 0.997419i \(-0.477124\pi\)
−0.997419 + 0.0718052i \(0.977124\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 8.28530i 1.53854i −0.638923 0.769271i \(-0.720620\pi\)
0.638923 0.769271i \(-0.279380\pi\)
\(30\) 0 0
\(31\) 4.59362i 0.825038i −0.910949 0.412519i \(-0.864649\pi\)
0.910949 0.412519i \(-0.135351\pi\)
\(32\) 0 0
\(33\) 0.781726 0.781726i 0.136081 0.136081i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.30352 2.30352i 0.378697 0.378697i −0.491935 0.870632i \(-0.663710\pi\)
0.870632 + 0.491935i \(0.163710\pi\)
\(38\) 0 0
\(39\) 4.98178i 0.797723i
\(40\) 0 0
\(41\) 3.15091i 0.492090i −0.969258 0.246045i \(-0.920869\pi\)
0.969258 0.246045i \(-0.0791311\pi\)
\(42\) 0 0
\(43\) −2.70356 2.70356i −0.412290 0.412290i 0.470246 0.882535i \(-0.344165\pi\)
−0.882535 + 0.470246i \(0.844165\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.71436 4.71436i −0.687661 0.687661i 0.274054 0.961714i \(-0.411635\pi\)
−0.961714 + 0.274054i \(0.911635\pi\)
\(48\) 0 0
\(49\) −2.38343 6.58174i −0.340490 0.940248i
\(50\) 0 0
\(51\) −4.57981 −0.641302
\(52\) 0 0
\(53\) 6.41894 + 6.41894i 0.881709 + 0.881709i 0.993708 0.111999i \(-0.0357254\pi\)
−0.111999 + 0.993708i \(0.535725\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.648218 0.648218i 0.0858586 0.0858586i
\(58\) 0 0
\(59\) −4.36339 −0.568065 −0.284032 0.958815i \(-0.591672\pi\)
−0.284032 + 0.958815i \(0.591672\pi\)
\(60\) 0 0
\(61\) 10.8687i 1.39160i −0.718237 0.695798i \(-0.755051\pi\)
0.718237 0.695798i \(-0.244949\pi\)
\(62\) 0 0
\(63\) 2.16604 + 1.51930i 0.272895 + 0.191414i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.81101 6.81101i 0.832098 0.832098i −0.155706 0.987803i \(-0.549765\pi\)
0.987803 + 0.155706i \(0.0497651\pi\)
\(68\) 0 0
\(69\) −6.27782 −0.755760
\(70\) 0 0
\(71\) 4.79796 0.569413 0.284706 0.958615i \(-0.408104\pi\)
0.284706 + 0.958615i \(0.408104\pi\)
\(72\) 0 0
\(73\) −9.72950 + 9.72950i −1.13875 + 1.13875i −0.150077 + 0.988674i \(0.547952\pi\)
−0.988674 + 0.150077i \(0.952048\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.67963 + 2.39461i −0.191412 + 0.272892i
\(78\) 0 0
\(79\) 12.5706i 1.41430i 0.707062 + 0.707152i \(0.250020\pi\)
−0.707062 + 0.707152i \(0.749980\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −0.227509 + 0.227509i −0.0249724 + 0.0249724i −0.719483 0.694510i \(-0.755621\pi\)
0.694510 + 0.719483i \(0.255621\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.85859 5.85859i −0.628107 0.628107i
\(88\) 0 0
\(89\) −8.93123 −0.946709 −0.473354 0.880872i \(-0.656957\pi\)
−0.473354 + 0.880872i \(0.656957\pi\)
\(90\) 0 0
\(91\) −2.27821 12.9822i −0.238822 1.36090i
\(92\) 0 0
\(93\) −3.24818 3.24818i −0.336820 0.336820i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.4114 12.4114i −1.26019 1.26019i −0.951002 0.309185i \(-0.899944\pi\)
−0.309185 0.951002i \(-0.600056\pi\)
\(98\) 0 0
\(99\) 1.10553i 0.111110i
\(100\) 0 0
\(101\) 4.91786i 0.489346i 0.969606 + 0.244673i \(0.0786806\pi\)
−0.969606 + 0.244673i \(0.921319\pi\)
\(102\) 0 0
\(103\) 1.91171 1.91171i 0.188366 0.188366i −0.606623 0.794989i \(-0.707476\pi\)
0.794989 + 0.606623i \(0.207476\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.3426 14.3426i 1.38655 1.38655i 0.554095 0.832453i \(-0.313064\pi\)
0.832453 0.554095i \(-0.186936\pi\)
\(108\) 0 0
\(109\) 9.15963i 0.877333i 0.898650 + 0.438667i \(0.144549\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(110\) 0 0
\(111\) 3.25767i 0.309205i
\(112\) 0 0
\(113\) 6.70424 + 6.70424i 0.630682 + 0.630682i 0.948239 0.317557i \(-0.102863\pi\)
−0.317557 + 0.948239i \(0.602863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.52265 + 3.52265i 0.325669 + 0.325669i
\(118\) 0 0
\(119\) 11.9347 2.09439i 1.09405 0.191993i
\(120\) 0 0
\(121\) −9.77781 −0.888892
\(122\) 0 0
\(123\) −2.22803 2.22803i −0.200895 0.200895i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.7414 + 14.7414i −1.30808 + 1.30808i −0.385288 + 0.922796i \(0.625898\pi\)
−0.922796 + 0.385288i \(0.874102\pi\)
\(128\) 0 0
\(129\) −3.82342 −0.336633
\(130\) 0 0
\(131\) 0.298477i 0.0260780i 0.999915 + 0.0130390i \(0.00415056\pi\)
−0.999915 + 0.0130390i \(0.995849\pi\)
\(132\) 0 0
\(133\) −1.39277 + 1.98565i −0.120769 + 0.172177i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.74069 4.74069i 0.405024 0.405024i −0.474975 0.879999i \(-0.657543\pi\)
0.879999 + 0.474975i \(0.157543\pi\)
\(138\) 0 0
\(139\) −2.35670 −0.199893 −0.0999465 0.994993i \(-0.531867\pi\)
−0.0999465 + 0.994993i \(0.531867\pi\)
\(140\) 0 0
\(141\) −6.66712 −0.561473
\(142\) 0 0
\(143\) −3.89438 + 3.89438i −0.325665 + 0.325665i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.33933 2.96865i −0.522859 0.244850i
\(148\) 0 0
\(149\) 6.95242i 0.569564i 0.958592 + 0.284782i \(0.0919213\pi\)
−0.958592 + 0.284782i \(0.908079\pi\)
\(150\) 0 0
\(151\) −1.15963 −0.0943691 −0.0471845 0.998886i \(-0.515025\pi\)
−0.0471845 + 0.998886i \(0.515025\pi\)
\(152\) 0 0
\(153\) −3.23842 + 3.23842i −0.261811 + 0.261811i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.0207600 0.0207600i −0.00165683 0.00165683i 0.706278 0.707935i \(-0.250373\pi\)
−0.707935 + 0.706278i \(0.750373\pi\)
\(158\) 0 0
\(159\) 9.07775 0.719913
\(160\) 0 0
\(161\) 16.3595 2.87090i 1.28931 0.226259i
\(162\) 0 0
\(163\) −3.58174 3.58174i −0.280543 0.280543i 0.552782 0.833326i \(-0.313566\pi\)
−0.833326 + 0.552782i \(0.813566\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.41192 + 2.41192i 0.186640 + 0.186640i 0.794242 0.607602i \(-0.207868\pi\)
−0.607602 + 0.794242i \(0.707868\pi\)
\(168\) 0 0
\(169\) 11.8181i 0.909085i
\(170\) 0 0
\(171\) 0.916719i 0.0701032i
\(172\) 0 0
\(173\) −10.8622 + 10.8622i −0.825838 + 0.825838i −0.986938 0.161100i \(-0.948496\pi\)
0.161100 + 0.986938i \(0.448496\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.08538 + 3.08538i −0.231911 + 0.231911i
\(178\) 0 0
\(179\) 7.35054i 0.549405i −0.961529 0.274702i \(-0.911421\pi\)
0.961529 0.274702i \(-0.0885793\pi\)
\(180\) 0 0
\(181\) 23.1359i 1.71968i 0.510566 + 0.859839i \(0.329436\pi\)
−0.510566 + 0.859839i \(0.670564\pi\)
\(182\) 0 0
\(183\) −7.68534 7.68534i −0.568117 0.568117i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.58016 3.58016i −0.261807 0.261807i
\(188\) 0 0
\(189\) 2.60593 0.457309i 0.189553 0.0332644i
\(190\) 0 0
\(191\) 21.0326 1.52187 0.760934 0.648829i \(-0.224741\pi\)
0.760934 + 0.648829i \(0.224741\pi\)
\(192\) 0 0
\(193\) −8.47813 8.47813i −0.610269 0.610269i 0.332747 0.943016i \(-0.392025\pi\)
−0.943016 + 0.332747i \(0.892025\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.7407 + 12.7407i −0.907736 + 0.907736i −0.996089 0.0883529i \(-0.971840\pi\)
0.0883529 + 0.996089i \(0.471840\pi\)
\(198\) 0 0
\(199\) −22.7111 −1.60995 −0.804975 0.593309i \(-0.797821\pi\)
−0.804975 + 0.593309i \(0.797821\pi\)
\(200\) 0 0
\(201\) 9.63223i 0.679405i
\(202\) 0 0
\(203\) 17.9463 + 12.5879i 1.25958 + 0.883497i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.43909 + 4.43909i −0.308538 + 0.308538i
\(208\) 0 0
\(209\) 1.01346 0.0701023
\(210\) 0 0
\(211\) −2.55258 −0.175727 −0.0878634 0.996133i \(-0.528004\pi\)
−0.0878634 + 0.996133i \(0.528004\pi\)
\(212\) 0 0
\(213\) 3.39267 3.39267i 0.232462 0.232462i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.94994 + 6.97910i 0.675446 + 0.473772i
\(218\) 0 0
\(219\) 13.7596i 0.929787i
\(220\) 0 0
\(221\) 22.8156 1.53474
\(222\) 0 0
\(223\) 12.4422 12.4422i 0.833191 0.833191i −0.154761 0.987952i \(-0.549461\pi\)
0.987952 + 0.154761i \(0.0494609\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.1465 17.1465i −1.13806 1.13806i −0.988798 0.149257i \(-0.952312\pi\)
−0.149257 0.988798i \(-0.547688\pi\)
\(228\) 0 0
\(229\) 15.5306 1.02629 0.513145 0.858302i \(-0.328480\pi\)
0.513145 + 0.858302i \(0.328480\pi\)
\(230\) 0 0
\(231\) 0.505568 + 2.88093i 0.0332639 + 0.189551i
\(232\) 0 0
\(233\) 5.78962 + 5.78962i 0.379291 + 0.379291i 0.870846 0.491556i \(-0.163572\pi\)
−0.491556 + 0.870846i \(0.663572\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.88876 + 8.88876i 0.577387 + 0.577387i
\(238\) 0 0
\(239\) 14.4398i 0.934031i −0.884249 0.467015i \(-0.845329\pi\)
0.884249 0.467015i \(-0.154671\pi\)
\(240\) 0 0
\(241\) 10.0102i 0.644815i 0.946601 + 0.322407i \(0.104492\pi\)
−0.946601 + 0.322407i \(0.895508\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.22928 + 3.22928i −0.205474 + 0.205474i
\(248\) 0 0
\(249\) 0.321746i 0.0203898i
\(250\) 0 0
\(251\) 22.6809i 1.43160i −0.698303 0.715802i \(-0.746061\pi\)
0.698303 0.715802i \(-0.253939\pi\)
\(252\) 0 0
\(253\) −4.90753 4.90753i −0.308534 0.308534i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.8047 + 11.8047i 0.736356 + 0.736356i 0.971871 0.235515i \(-0.0756775\pi\)
−0.235515 + 0.971871i \(0.575678\pi\)
\(258\) 0 0
\(259\) 1.48976 + 8.48927i 0.0925695 + 0.527497i
\(260\) 0 0
\(261\) −8.28530 −0.512847
\(262\) 0 0
\(263\) 1.48454 + 1.48454i 0.0915408 + 0.0915408i 0.751394 0.659853i \(-0.229382\pi\)
−0.659853 + 0.751394i \(0.729382\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.31534 + 6.31534i −0.386492 + 0.386492i
\(268\) 0 0
\(269\) −30.7202 −1.87304 −0.936522 0.350609i \(-0.885975\pi\)
−0.936522 + 0.350609i \(0.885975\pi\)
\(270\) 0 0
\(271\) 5.29013i 0.321352i 0.987007 + 0.160676i \(0.0513675\pi\)
−0.987007 + 0.160676i \(0.948633\pi\)
\(272\) 0 0
\(273\) −10.7907 7.56883i −0.653084 0.458087i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.1817 + 11.1817i −0.671843 + 0.671843i −0.958141 0.286298i \(-0.907575\pi\)
0.286298 + 0.958141i \(0.407575\pi\)
\(278\) 0 0
\(279\) −4.59362 −0.275013
\(280\) 0 0
\(281\) 1.00864 0.0601706 0.0300853 0.999547i \(-0.490422\pi\)
0.0300853 + 0.999547i \(0.490422\pi\)
\(282\) 0 0
\(283\) 10.8457 10.8457i 0.644708 0.644708i −0.307001 0.951709i \(-0.599326\pi\)
0.951709 + 0.307001i \(0.0993256\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.82499 + 4.78719i 0.402867 + 0.282579i
\(288\) 0 0
\(289\) 3.97469i 0.233805i
\(290\) 0 0
\(291\) −17.5524 −1.02894
\(292\) 0 0
\(293\) −11.8047 + 11.8047i −0.689637 + 0.689637i −0.962152 0.272514i \(-0.912145\pi\)
0.272514 + 0.962152i \(0.412145\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.781726 0.781726i −0.0453603 0.0453603i
\(298\) 0 0
\(299\) 31.2747 1.80866
\(300\) 0 0
\(301\) 9.96355 1.74848i 0.574290 0.100781i
\(302\) 0 0
\(303\) 3.47746 + 3.47746i 0.199775 + 0.199775i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.0408 16.0408i −0.915498 0.915498i 0.0812002 0.996698i \(-0.474125\pi\)
−0.996698 + 0.0812002i \(0.974125\pi\)
\(308\) 0 0
\(309\) 2.70356i 0.153800i
\(310\) 0 0
\(311\) 24.0259i 1.36238i 0.732105 + 0.681191i \(0.238538\pi\)
−0.732105 + 0.681191i \(0.761462\pi\)
\(312\) 0 0
\(313\) 17.2483 17.2483i 0.974930 0.974930i −0.0247630 0.999693i \(-0.507883\pi\)
0.999693 + 0.0247630i \(0.00788310\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.2079 12.2079i 0.685663 0.685663i −0.275607 0.961270i \(-0.588879\pi\)
0.961270 + 0.275607i \(0.0888789\pi\)
\(318\) 0 0
\(319\) 9.15963i 0.512841i
\(320\) 0 0
\(321\) 20.2835i 1.13211i
\(322\) 0 0
\(323\) −2.96872 2.96872i −0.165184 0.165184i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.47683 + 6.47683i 0.358170 + 0.358170i
\(328\) 0 0
\(329\) 17.3740 3.04894i 0.957862 0.168093i
\(330\) 0 0
\(331\) −20.1746 −1.10890 −0.554448 0.832218i \(-0.687071\pi\)
−0.554448 + 0.832218i \(0.687071\pi\)
\(332\) 0 0
\(333\) −2.30352 2.30352i −0.126232 0.126232i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.7903 22.7903i 1.24147 1.24147i 0.282074 0.959393i \(-0.408978\pi\)
0.959393 0.282074i \(-0.0910224\pi\)
\(338\) 0 0
\(339\) 9.48123 0.514950
\(340\) 0 0
\(341\) 5.07837i 0.275009i
\(342\) 0 0
\(343\) 17.8774 + 4.83706i 0.965291 + 0.261176i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.9153 16.9153i 0.908061 0.908061i −0.0880550 0.996116i \(-0.528065\pi\)
0.996116 + 0.0880550i \(0.0280651\pi\)
\(348\) 0 0
\(349\) 8.04573 0.430678 0.215339 0.976539i \(-0.430914\pi\)
0.215339 + 0.976539i \(0.430914\pi\)
\(350\) 0 0
\(351\) 4.98178 0.265908
\(352\) 0 0
\(353\) −5.77285 + 5.77285i −0.307258 + 0.307258i −0.843845 0.536587i \(-0.819713\pi\)
0.536587 + 0.843845i \(0.319713\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6.95813 9.92004i 0.368263 0.525024i
\(358\) 0 0
\(359\) 33.1408i 1.74911i −0.484930 0.874553i \(-0.661155\pi\)
0.484930 0.874553i \(-0.338845\pi\)
\(360\) 0 0
\(361\) −18.1596 −0.955770
\(362\) 0 0
\(363\) −6.91396 + 6.91396i −0.362889 + 0.362889i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.56222 + 9.56222i 0.499144 + 0.499144i 0.911171 0.412028i \(-0.135179\pi\)
−0.412028 + 0.911171i \(0.635179\pi\)
\(368\) 0 0
\(369\) −3.15091 −0.164030
\(370\) 0 0
\(371\) −23.6560 + 4.15134i −1.22816 + 0.215527i
\(372\) 0 0
\(373\) −6.15963 6.15963i −0.318933 0.318933i 0.529424 0.848357i \(-0.322408\pi\)
−0.848357 + 0.529424i \(0.822408\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 29.1862 + 29.1862i 1.50317 + 1.50317i
\(378\) 0 0
\(379\) 7.42515i 0.381404i 0.981648 + 0.190702i \(0.0610765\pi\)
−0.981648 + 0.190702i \(0.938924\pi\)
\(380\) 0 0
\(381\) 20.8474i 1.06805i
\(382\) 0 0
\(383\) 21.9659 21.9659i 1.12240 1.12240i 0.131026 0.991379i \(-0.458173\pi\)
0.991379 0.131026i \(-0.0418270\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.70356 + 2.70356i −0.137430 + 0.137430i
\(388\) 0 0
\(389\) 9.95971i 0.504977i 0.967600 + 0.252489i \(0.0812490\pi\)
−0.967600 + 0.252489i \(0.918751\pi\)
\(390\) 0 0
\(391\) 28.7512i 1.45401i
\(392\) 0 0
\(393\) 0.211055 + 0.211055i 0.0106463 + 0.0106463i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.5244 + 13.5244i 0.678771 + 0.678771i 0.959722 0.280951i \(-0.0906498\pi\)
−0.280951 + 0.959722i \(0.590650\pi\)
\(398\) 0 0
\(399\) 0.419224 + 2.38890i 0.0209875 + 0.119595i
\(400\) 0 0
\(401\) 8.39215 0.419084 0.209542 0.977800i \(-0.432803\pi\)
0.209542 + 0.977800i \(0.432803\pi\)
\(402\) 0 0
\(403\) 16.1817 + 16.1817i 0.806068 + 0.806068i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.54661 2.54661i 0.126231 0.126231i
\(408\) 0 0
\(409\) −22.7124 −1.12305 −0.561527 0.827458i \(-0.689786\pi\)
−0.561527 + 0.827458i \(0.689786\pi\)
\(410\) 0 0
\(411\) 6.70434i 0.330701i
\(412\) 0 0
\(413\) 6.62931 9.45126i 0.326207 0.465066i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.66644 + 1.66644i −0.0816060 + 0.0816060i
\(418\) 0 0
\(419\) −10.4037 −0.508253 −0.254126 0.967171i \(-0.581788\pi\)
−0.254126 + 0.967171i \(0.581788\pi\)
\(420\) 0 0
\(421\) −27.3523 −1.33307 −0.666534 0.745475i \(-0.732223\pi\)
−0.666534 + 0.745475i \(0.732223\pi\)
\(422\) 0 0
\(423\) −4.71436 + 4.71436i −0.229220 + 0.229220i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 23.5420 + 16.5129i 1.13928 + 0.799114i
\(428\) 0 0
\(429\) 5.50749i 0.265904i
\(430\) 0 0
\(431\) −1.64217 −0.0791008 −0.0395504 0.999218i \(-0.512593\pi\)
−0.0395504 + 0.999218i \(0.512593\pi\)
\(432\) 0 0
\(433\) 4.24765 4.24765i 0.204129 0.204129i −0.597637 0.801767i \(-0.703894\pi\)
0.801767 + 0.597637i \(0.203894\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.06939 4.06939i −0.194666 0.194666i
\(438\) 0 0
\(439\) 26.4776 1.26371 0.631853 0.775089i \(-0.282295\pi\)
0.631853 + 0.775089i \(0.282295\pi\)
\(440\) 0 0
\(441\) −6.58174 + 2.38343i −0.313416 + 0.113497i
\(442\) 0 0
\(443\) 6.74453 + 6.74453i 0.320442 + 0.320442i 0.848937 0.528494i \(-0.177243\pi\)
−0.528494 + 0.848937i \(0.677243\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.91610 + 4.91610i 0.232524 + 0.232524i
\(448\) 0 0
\(449\) 16.1785i 0.763508i 0.924264 + 0.381754i \(0.124680\pi\)
−0.924264 + 0.381754i \(0.875320\pi\)
\(450\) 0 0
\(451\) 3.48342i 0.164028i
\(452\) 0 0
\(453\) −0.819980 + 0.819980i −0.0385260 + 0.0385260i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.8923 + 17.8923i −0.836969 + 0.836969i −0.988459 0.151490i \(-0.951593\pi\)
0.151490 + 0.988459i \(0.451593\pi\)
\(458\) 0 0
\(459\) 4.57981i 0.213767i
\(460\) 0 0
\(461\) 33.5156i 1.56098i 0.625171 + 0.780488i \(0.285029\pi\)
−0.625171 + 0.780488i \(0.714971\pi\)
\(462\) 0 0
\(463\) −12.5303 12.5303i −0.582333 0.582333i 0.353211 0.935544i \(-0.385090\pi\)
−0.935544 + 0.353211i \(0.885090\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.9253 + 26.9253i 1.24595 + 1.24595i 0.957491 + 0.288462i \(0.0931438\pi\)
0.288462 + 0.957491i \(0.406856\pi\)
\(468\) 0 0
\(469\) 4.40491 + 25.1009i 0.203400 + 1.15905i
\(470\) 0 0
\(471\) −0.0293591 −0.00135280
\(472\) 0 0
\(473\) −2.98886 2.98886i −0.137428 0.137428i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.41894 6.41894i 0.293903 0.293903i
\(478\) 0 0
\(479\) 38.0334 1.73779 0.868896 0.494995i \(-0.164830\pi\)
0.868896 + 0.494995i \(0.164830\pi\)
\(480\) 0 0
\(481\) 16.2290i 0.739979i
\(482\) 0 0
\(483\) 9.53791 13.5980i 0.433990 0.618730i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.53280 + 2.53280i −0.114772 + 0.114772i −0.762160 0.647388i \(-0.775861\pi\)
0.647388 + 0.762160i \(0.275861\pi\)
\(488\) 0 0
\(489\) −5.06534 −0.229063
\(490\) 0 0
\(491\) 8.72506 0.393757 0.196878 0.980428i \(-0.436920\pi\)
0.196878 + 0.980428i \(0.436920\pi\)
\(492\) 0 0
\(493\) −26.8313 + 26.8313i −1.20842 + 1.20842i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.28955 + 10.3925i −0.326981 + 0.466169i
\(498\) 0 0
\(499\) 12.7452i 0.570554i −0.958445 0.285277i \(-0.907914\pi\)
0.958445 0.285277i \(-0.0920856\pi\)
\(500\) 0 0
\(501\) 3.41097 0.152391
\(502\) 0 0
\(503\) 15.9056 15.9056i 0.709194 0.709194i −0.257172 0.966366i \(-0.582791\pi\)
0.966366 + 0.257172i \(0.0827906\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −8.35666 8.35666i −0.371132 0.371132i
\(508\) 0 0
\(509\) 5.12085 0.226978 0.113489 0.993539i \(-0.463797\pi\)
0.113489 + 0.993539i \(0.463797\pi\)
\(510\) 0 0
\(511\) −6.29239 35.8565i −0.278359 1.58620i
\(512\) 0 0
\(513\) −0.648218 0.648218i −0.0286195 0.0286195i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.21186 5.21186i −0.229217 0.229217i
\(518\) 0 0
\(519\) 15.3615i 0.674294i
\(520\) 0 0
\(521\) 34.4071i 1.50740i 0.657217 + 0.753702i \(0.271734\pi\)
−0.657217 + 0.753702i \(0.728266\pi\)
\(522\) 0 0
\(523\) −9.25548 + 9.25548i −0.404714 + 0.404714i −0.879890 0.475177i \(-0.842384\pi\)
0.475177 + 0.879890i \(0.342384\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.8760 + 14.8760i −0.648011 + 0.648011i
\(528\) 0 0
\(529\) 16.4110i 0.713521i
\(530\) 0 0
\(531\) 4.36339i 0.189355i
\(532\) 0 0
\(533\) 11.0996 + 11.0996i 0.480775 + 0.480775i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.19761 5.19761i −0.224294 0.224294i
\(538\) 0 0
\(539\) −2.63495 7.27629i −0.113495 0.313412i
\(540\) 0 0
\(541\) 13.3304 0.573118 0.286559 0.958063i \(-0.407489\pi\)
0.286559 + 0.958063i \(0.407489\pi\)
\(542\) 0 0
\(543\) 16.3595 + 16.3595i 0.702055 + 0.702055i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.0071 10.0071i 0.427872 0.427872i −0.460031 0.887903i \(-0.652162\pi\)
0.887903 + 0.460031i \(0.152162\pi\)
\(548\) 0 0
\(549\) −10.8687 −0.463865
\(550\) 0 0
\(551\) 7.59529i 0.323570i
\(552\) 0 0
\(553\) −27.2284 19.0986i −1.15787 0.812153i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.8441 + 17.8441i −0.756078 + 0.756078i −0.975606 0.219528i \(-0.929548\pi\)
0.219528 + 0.975606i \(0.429548\pi\)
\(558\) 0 0
\(559\) 19.0474 0.805620
\(560\) 0 0
\(561\) −5.06311 −0.213765
\(562\) 0 0
\(563\) −4.36339 + 4.36339i −0.183895 + 0.183895i −0.793051 0.609156i \(-0.791508\pi\)
0.609156 + 0.793051i \(0.291508\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.51930 2.16604i 0.0638048 0.0909650i
\(568\) 0 0
\(569\) 35.9854i 1.50859i −0.656537 0.754294i \(-0.727980\pi\)
0.656537 0.754294i \(-0.272020\pi\)
\(570\) 0 0
\(571\) 17.5414 0.734087 0.367043 0.930204i \(-0.380370\pi\)
0.367043 + 0.930204i \(0.380370\pi\)
\(572\) 0 0
\(573\) 14.8723 14.8723i 0.621300 0.621300i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.45573 2.45573i −0.102234 0.102234i 0.654140 0.756374i \(-0.273031\pi\)
−0.756374 + 0.654140i \(0.773031\pi\)
\(578\) 0 0
\(579\) −11.9899 −0.498283
\(580\) 0 0
\(581\) −0.147138 0.838448i −0.00610430 0.0347847i
\(582\) 0 0
\(583\) 7.09631 + 7.09631i 0.293899 + 0.293899i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.1108 24.1108i −0.995161 0.995161i 0.00482767 0.999988i \(-0.498463\pi\)
−0.999988 + 0.00482767i \(0.998463\pi\)
\(588\) 0 0
\(589\) 4.21105i 0.173513i
\(590\) 0 0
\(591\) 18.0181i 0.741164i
\(592\) 0 0
\(593\) 10.8752 10.8752i 0.446592 0.446592i −0.447628 0.894220i \(-0.647731\pi\)
0.894220 + 0.447628i \(0.147731\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −16.0592 + 16.0592i −0.657259 + 0.657259i
\(598\) 0 0
\(599\) 20.8319i 0.851169i −0.904919 0.425584i \(-0.860069\pi\)
0.904919 0.425584i \(-0.139931\pi\)
\(600\) 0 0
\(601\) 11.0253i 0.449730i 0.974390 + 0.224865i \(0.0721940\pi\)
−0.974390 + 0.224865i \(0.927806\pi\)
\(602\) 0 0
\(603\) −6.81101 6.81101i −0.277366 0.277366i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.6440 + 17.6440i 0.716149 + 0.716149i 0.967814 0.251665i \(-0.0809783\pi\)
−0.251665 + 0.967814i \(0.580978\pi\)
\(608\) 0 0
\(609\) 21.5909 3.78895i 0.874908 0.153536i
\(610\) 0 0
\(611\) 33.2141 1.34370
\(612\) 0 0
\(613\) 5.97854 + 5.97854i 0.241471 + 0.241471i 0.817458 0.575988i \(-0.195382\pi\)
−0.575988 + 0.817458i \(0.695382\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.90416 1.90416i 0.0766586 0.0766586i −0.667738 0.744396i \(-0.732737\pi\)
0.744396 + 0.667738i \(0.232737\pi\)
\(618\) 0 0
\(619\) −38.2048 −1.53558 −0.767790 0.640702i \(-0.778643\pi\)
−0.767790 + 0.640702i \(0.778643\pi\)
\(620\) 0 0
\(621\) 6.27782i 0.251920i
\(622\) 0 0
\(623\) 13.5693 19.3454i 0.543641 0.775056i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.716623 0.716623i 0.0286192 0.0286192i
\(628\) 0 0
\(629\) −14.9195 −0.594881
\(630\) 0 0
\(631\) 46.8028 1.86319 0.931595 0.363499i \(-0.118418\pi\)
0.931595 + 0.363499i \(0.118418\pi\)
\(632\) 0 0
\(633\) −1.80495 + 1.80495i −0.0717402 + 0.0717402i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 31.5811 + 14.7892i 1.25129 + 0.585967i
\(638\) 0 0
\(639\) 4.79796i 0.189804i
\(640\) 0 0
\(641\) 18.4235 0.727683 0.363842 0.931461i \(-0.381465\pi\)
0.363842 + 0.931461i \(0.381465\pi\)
\(642\) 0 0
\(643\) 5.26164 5.26164i 0.207499 0.207499i −0.595705 0.803204i \(-0.703127\pi\)
0.803204 + 0.595705i \(0.203127\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.421944 + 0.421944i 0.0165883 + 0.0165883i 0.715352 0.698764i \(-0.246266\pi\)
−0.698764 + 0.715352i \(0.746266\pi\)
\(648\) 0 0
\(649\) −4.82385 −0.189352
\(650\) 0 0
\(651\) 11.9706 2.10070i 0.469166 0.0823331i
\(652\) 0 0
\(653\) −13.2593 13.2593i −0.518877 0.518877i 0.398355 0.917232i \(-0.369581\pi\)
−0.917232 + 0.398355i \(0.869581\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.72950 + 9.72950i 0.379584 + 0.379584i
\(658\) 0 0
\(659\) 38.1322i 1.48542i 0.669614 + 0.742710i \(0.266460\pi\)
−0.669614 + 0.742710i \(0.733540\pi\)
\(660\) 0 0
\(661\) 48.1006i 1.87090i −0.353462 0.935449i \(-0.614996\pi\)
0.353462 0.935449i \(-0.385004\pi\)
\(662\) 0 0
\(663\) 16.1331 16.1331i 0.626557 0.626557i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −36.7792 + 36.7792i −1.42409 + 1.42409i
\(668\) 0 0
\(669\) 17.5959i 0.680297i
\(670\) 0 0
\(671\) 12.0157i 0.463859i
\(672\) 0 0
\(673\) 3.10360 + 3.10360i 0.119635 + 0.119635i 0.764390 0.644755i \(-0.223040\pi\)
−0.644755 + 0.764390i \(0.723040\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.28037 6.28037i −0.241374 0.241374i 0.576044 0.817418i \(-0.304596\pi\)
−0.817418 + 0.576044i \(0.804596\pi\)
\(678\) 0 0
\(679\) 45.7403 8.02687i 1.75535 0.308043i
\(680\) 0 0
\(681\) −24.2489 −0.929218
\(682\) 0 0
\(683\) 7.16030 + 7.16030i 0.273981 + 0.273981i 0.830701 0.556719i \(-0.187940\pi\)
−0.556719 + 0.830701i \(0.687940\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.9818 10.9818i 0.418981 0.418981i
\(688\) 0 0
\(689\) −45.2233 −1.72287
\(690\) 0 0
\(691\) 2.60364i 0.0990470i −0.998773 0.0495235i \(-0.984230\pi\)
0.998773 0.0495235i \(-0.0157703\pi\)
\(692\) 0 0
\(693\) 2.39461 + 1.67963i 0.0909638 + 0.0638039i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10.2040 + 10.2040i −0.386503 + 0.386503i
\(698\) 0 0
\(699\) 8.18776 0.309690
\(700\) 0 0
\(701\) −33.1974 −1.25385 −0.626925 0.779080i \(-0.715687\pi\)
−0.626925 + 0.779080i \(0.715687\pi\)
\(702\) 0 0
\(703\) 2.11168 2.11168i 0.0796436 0.0796436i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.6523 7.47173i −0.400620 0.281003i
\(708\) 0 0
\(709\) 29.2613i 1.09893i −0.835516 0.549466i \(-0.814831\pi\)
0.835516 0.549466i \(-0.185169\pi\)
\(710\) 0 0
\(711\) 12.5706 0.471434
\(712\) 0 0
\(713\) −20.3915 + 20.3915i −0.763666 + 0.763666i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.2105 10.2105i −0.381316 0.381316i
\(718\) 0 0
\(719\) 21.3340 0.795622 0.397811 0.917467i \(-0.369770\pi\)
0.397811 + 0.917467i \(0.369770\pi\)
\(720\) 0 0
\(721\) 1.23636 + 7.04530i 0.0460446 + 0.262381i
\(722\) 0 0
\(723\) 7.07829 + 7.07829i 0.263245 + 0.263245i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −5.80881 5.80881i −0.215437 0.215437i 0.591135 0.806572i \(-0.298680\pi\)
−0.806572 + 0.591135i \(0.798680\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 17.5105i 0.647651i
\(732\) 0 0
\(733\) −0.239203 + 0.239203i −0.00883517 + 0.00883517i −0.711511 0.702675i \(-0.751989\pi\)
0.702675 + 0.711511i \(0.251989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.52976 7.52976i 0.277362 0.277362i
\(738\) 0 0
\(739\) 22.4536i 0.825968i 0.910738 + 0.412984i \(0.135513\pi\)
−0.910738 + 0.412984i \(0.864487\pi\)
\(740\) 0 0
\(741\) 4.56689i 0.167769i
\(742\) 0 0
\(743\) 9.25547 + 9.25547i 0.339550 + 0.339550i 0.856198 0.516648i \(-0.172820\pi\)
−0.516648 + 0.856198i \(0.672820\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.227509 + 0.227509i 0.00832412 + 0.00832412i
\(748\) 0 0
\(749\) 9.27581 + 52.8573i 0.338931 + 1.93136i
\(750\) 0 0
\(751\) −5.77477 −0.210724 −0.105362 0.994434i \(-0.533600\pi\)
−0.105362 + 0.994434i \(0.533600\pi\)
\(752\) 0 0
\(753\) −16.0378 16.0378i −0.584450 0.584450i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −13.5904 + 13.5904i −0.493951 + 0.493951i −0.909549 0.415598i \(-0.863572\pi\)
0.415598 + 0.909549i \(0.363572\pi\)
\(758\) 0 0
\(759\) −6.94030 −0.251917
\(760\) 0 0
\(761\) 35.7986i 1.29770i −0.760918 0.648849i \(-0.775251\pi\)
0.760918 0.648849i \(-0.224749\pi\)
\(762\) 0 0
\(763\) −19.8401 13.9163i −0.718259 0.503802i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.3707 15.3707i 0.555003 0.555003i
\(768\) 0 0
\(769\) −9.22523 −0.332670 −0.166335 0.986069i \(-0.553193\pi\)
−0.166335 + 0.986069i \(0.553193\pi\)
\(770\) 0 0
\(771\) 16.6944 0.601232
\(772\) 0 0
\(773\) 13.7901 13.7901i 0.495995 0.495995i −0.414194 0.910189i \(-0.635936\pi\)
0.910189 + 0.414194i \(0.135936\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.05624 + 4.94940i 0.253141 + 0.177559i
\(778\) 0 0
\(779\) 2.88850i 0.103491i
\(780\) 0 0
\(781\) 5.30427 0.189802
\(782\) 0 0
\(783\) −5.85859 + 5.85859i −0.209369 + 0.209369i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −23.5876 23.5876i −0.840806 0.840806i 0.148158 0.988964i \(-0.452666\pi\)
−0.988964 + 0.148158i \(0.952666\pi\)
\(788\) 0 0
\(789\) 2.09946 0.0747428
\(790\) 0 0
\(791\) −24.7074 + 4.33585i −0.878494 + 0.154165i
\(792\) 0 0
\(793\) 38.2867 + 38.2867i 1.35960 + 1.35960i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.38379 8.38379i −0.296969 0.296969i 0.542856 0.839826i \(-0.317343\pi\)
−0.839826 + 0.542856i \(0.817343\pi\)
\(798\) 0 0
\(799\) 30.5342i 1.08022i
\(800\) 0 0
\(801\) 8.93123i 0.315570i
\(802\) 0 0
\(803\) −10.7562 + 10.7562i −0.379579 + 0.379579i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −21.7225 + 21.7225i −0.764667 + 0.764667i
\(808\) 0 0
\(809\) 5.45910i 0.191932i 0.995385 + 0.0959659i \(0.0305940\pi\)
−0.995385 + 0.0959659i \(0.969406\pi\)
\(810\) 0 0
\(811\) 24.7871i 0.870393i −0.900335 0.435197i \(-0.856679\pi\)
0.900335 0.435197i \(-0.143321\pi\)
\(812\) 0 0
\(813\) 3.74069 + 3.74069i 0.131192 + 0.131192i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.47841 2.47841i −0.0867085 0.0867085i
\(818\) 0 0
\(819\) −12.9822 + 2.27821i −0.453633 + 0.0796072i
\(820\) 0 0
\(821\) 27.6418 0.964706 0.482353 0.875977i \(-0.339782\pi\)
0.482353 + 0.875977i \(0.339782\pi\)
\(822\) 0 0
\(823\) −8.76364 8.76364i −0.305481 0.305481i 0.537673 0.843154i \(-0.319304\pi\)
−0.843154 + 0.537673i \(0.819304\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.39400 3.39400i 0.118021 0.118021i −0.645630 0.763651i \(-0.723405\pi\)
0.763651 + 0.645630i \(0.223405\pi\)
\(828\) 0 0
\(829\) 1.04845 0.0364140 0.0182070 0.999834i \(-0.494204\pi\)
0.0182070 + 0.999834i \(0.494204\pi\)
\(830\) 0 0
\(831\) 15.8133i 0.548557i
\(832\) 0 0
\(833\) −13.5959 + 29.0330i −0.471069 + 1.00593i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.24818 + 3.24818i −0.112273 + 0.112273i
\(838\) 0 0
\(839\) −51.4166 −1.77510 −0.887550 0.460712i \(-0.847594\pi\)
−0.887550 + 0.460712i \(0.847594\pi\)
\(840\) 0 0
\(841\) −39.6462 −1.36711
\(842\) 0 0
\(843\) 0.713218 0.713218i 0.0245645 0.0245645i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.8555 21.1791i 0.510440 0.727722i
\(848\) 0 0
\(849\) 15.3381i 0.526402i
\(850\) 0 0
\(851\) −20.4511 −0.701054
\(852\) 0 0
\(853\) −21.3132 + 21.3132i −0.729750 + 0.729750i −0.970570 0.240820i \(-0.922584\pi\)
0.240820 + 0.970570i \(0.422584\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.7450 18.7450i −0.640317 0.640317i 0.310317 0.950633i \(-0.399565\pi\)
−0.950633 + 0.310317i \(0.899565\pi\)
\(858\) 0 0
\(859\) 39.5452 1.34926 0.674632 0.738154i \(-0.264302\pi\)
0.674632 + 0.738154i \(0.264302\pi\)
\(860\) 0 0
\(861\) 8.21105 1.44094i 0.279832 0.0491072i
\(862\) 0 0
\(863\) −24.4872 24.4872i −0.833554 0.833554i 0.154447 0.988001i \(-0.450641\pi\)
−0.988001 + 0.154447i \(0.950641\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.81053 + 2.81053i 0.0954506 + 0.0954506i
\(868\) 0 0
\(869\) 13.8971i 0.471428i
\(870\) 0 0
\(871\) 47.9856i 1.62593i
\(872\) 0 0
\(873\) −12.4114 + 12.4114i −0.420062 + 0.420062i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.3269 31.3269i 1.05784 1.05784i 0.0596143 0.998221i \(-0.481013\pi\)
0.998221 0.0596143i \(-0.0189871\pi\)
\(878\) 0 0
\(879\) 16.6944i 0.563086i
\(880\) 0 0
\(881\) 20.8419i 0.702181i 0.936341 + 0.351091i \(0.114189\pi\)
−0.936341 + 0.351091i \(0.885811\pi\)
\(882\) 0 0
\(883\) −9.72563 9.72563i −0.327294 0.327294i 0.524263 0.851556i \(-0.324341\pi\)
−0.851556 + 0.524263i \(0.824341\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.0130 + 19.0130i 0.638395 + 0.638395i 0.950160 0.311764i \(-0.100920\pi\)
−0.311764 + 0.950160i \(0.600920\pi\)
\(888\) 0 0
\(889\) −9.53373 54.3269i −0.319751 1.82207i
\(890\) 0 0
\(891\) −1.10553 −0.0370366
\(892\) 0 0
\(893\) −4.32175 4.32175i −0.144622 0.144622i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 22.1145 22.1145i 0.738383 0.738383i
\(898\) 0 0
\(899\) −38.0595 −1.26936
\(900\) 0 0
\(901\) 41.5744i 1.38504i
\(902\) 0 0
\(903\) 5.80893 8.28166i 0.193309 0.275596i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.5146 11.5146i 0.382335 0.382335i −0.489608 0.871943i \(-0.662860\pi\)
0.871943 + 0.489608i \(0.162860\pi\)
\(908\) 0 0
\(909\) 4.91786 0.163115
\(910\) 0 0
\(911\) −34.5480 −1.14463 −0.572313 0.820036i \(-0.693954\pi\)
−0.572313 + 0.820036i \(0.693954\pi\)
\(912\) 0 0
\(913\) −0.251517 + 0.251517i −0.00832401 + 0.00832401i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.646511 0.453477i −0.0213497 0.0149751i
\(918\) 0 0
\(919\) 36.6646i 1.20945i 0.796433 + 0.604727i \(0.206718\pi\)
−0.796433 + 0.604727i \(0.793282\pi\)
\(920\) 0 0
\(921\) −22.6851 −0.747501
\(922\) 0 0
\(923\) −16.9015 + 16.9015i −0.556320 + 0.556320i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.91171 1.91171i −0.0627887 0.0627887i
\(928\) 0 0
\(929\) 55.6250 1.82500 0.912498 0.409081i \(-0.134151\pi\)
0.912498 + 0.409081i \(0.134151\pi\)
\(930\) 0 0
\(931\) −2.18494 6.03360i −0.0716084 0.197743i
\(932\) 0 0
\(933\) 16.9889 + 16.9889i 0.556190 + 0.556190i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.414213 + 0.414213i 0.0135317 + 0.0135317i 0.713840 0.700309i \(-0.246954\pi\)
−0.700309 + 0.713840i \(0.746954\pi\)
\(938\) 0 0
\(939\) 24.3928i 0.796027i
\(940\) 0 0
\(941\) 7.32435i 0.238767i −0.992848 0.119383i \(-0.961908\pi\)
0.992848 0.119383i \(-0.0380918\pi\)
\(942\) 0 0
\(943\) −13.9872 + 13.9872i −0.455485 + 0.455485i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0774 32.0774i 1.04238 1.04238i 0.0433144 0.999061i \(-0.486208\pi\)
0.999061 0.0433144i \(-0.0137917\pi\)
\(948\) 0 0
\(949\) 68.5472i 2.22514i
\(950\) 0 0
\(951\) 17.2646i 0.559842i
\(952\) 0 0
\(953\) 28.2117 + 28.2117i 0.913868 + 0.913868i 0.996574 0.0827063i \(-0.0263564\pi\)
−0.0827063 + 0.996574i \(0.526356\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.47683 6.47683i −0.209366 0.209366i
\(958\) 0 0
\(959\) 3.06596 + 17.4710i 0.0990050 + 0.564169i
\(960\) 0 0
\(961\) 9.89869 0.319312
\(962\) 0 0
\(963\) −14.3426 14.3426i −0.462183 0.462183i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 29.9303 29.9303i 0.962495 0.962495i −0.0368265 0.999322i \(-0.511725\pi\)
0.999322 + 0.0368265i \(0.0117249\pi\)
\(968\) 0 0
\(969\) −4.19840 −0.134872
\(970\) 0 0
\(971\) 36.0470i 1.15680i −0.815752 0.578401i \(-0.803677\pi\)
0.815752 0.578401i \(-0.196323\pi\)
\(972\) 0 0
\(973\) 3.58055 5.10471i 0.114787 0.163649i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.64977 + 4.64977i −0.148759 + 0.148759i −0.777564 0.628804i \(-0.783545\pi\)
0.628804 + 0.777564i \(0.283545\pi\)
\(978\) 0 0
\(979\) −9.87372 −0.315566
\(980\) 0 0
\(981\) 9.15963 0.292444
\(982\) 0 0
\(983\) −21.0934 + 21.0934i −0.672775 + 0.672775i −0.958355 0.285580i \(-0.907814\pi\)
0.285580 + 0.958355i \(0.407814\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 10.1294 14.4412i 0.322422 0.459669i
\(988\) 0 0
\(989\) 24.0027i 0.763242i
\(990\) 0 0
\(991\) −21.2260 −0.674267 −0.337134 0.941457i \(-0.609457\pi\)
−0.337134 + 0.941457i \(0.609457\pi\)
\(992\) 0 0
\(993\) −14.2656 + 14.2656i −0.452705 + 0.452705i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.71103 9.71103i −0.307551 0.307551i 0.536408 0.843959i \(-0.319781\pi\)
−0.843959 + 0.536408i \(0.819781\pi\)
\(998\) 0 0
\(999\) −3.25767 −0.103068
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.1693.6 16
5.2 odd 4 inner 2100.2.x.d.1357.1 16
5.3 odd 4 420.2.x.a.97.8 yes 16
5.4 even 2 420.2.x.a.13.1 16
7.6 odd 2 inner 2100.2.x.d.1693.1 16
15.8 even 4 1260.2.ba.b.937.1 16
15.14 odd 2 1260.2.ba.b.433.8 16
20.3 even 4 1680.2.cz.c.97.4 16
20.19 odd 2 1680.2.cz.c.433.5 16
35.13 even 4 420.2.x.a.97.1 yes 16
35.27 even 4 inner 2100.2.x.d.1357.6 16
35.34 odd 2 420.2.x.a.13.8 yes 16
105.83 odd 4 1260.2.ba.b.937.8 16
105.104 even 2 1260.2.ba.b.433.1 16
140.83 odd 4 1680.2.cz.c.97.5 16
140.139 even 2 1680.2.cz.c.433.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.x.a.13.1 16 5.4 even 2
420.2.x.a.13.8 yes 16 35.34 odd 2
420.2.x.a.97.1 yes 16 35.13 even 4
420.2.x.a.97.8 yes 16 5.3 odd 4
1260.2.ba.b.433.1 16 105.104 even 2
1260.2.ba.b.433.8 16 15.14 odd 2
1260.2.ba.b.937.1 16 15.8 even 4
1260.2.ba.b.937.8 16 105.83 odd 4
1680.2.cz.c.97.4 16 20.3 even 4
1680.2.cz.c.97.5 16 140.83 odd 4
1680.2.cz.c.433.4 16 140.139 even 2
1680.2.cz.c.433.5 16 20.19 odd 2
2100.2.x.d.1357.1 16 5.2 odd 4 inner
2100.2.x.d.1357.6 16 35.27 even 4 inner
2100.2.x.d.1693.1 16 7.6 odd 2 inner
2100.2.x.d.1693.6 16 1.1 even 1 trivial