Properties

Label 1512.2.c.d.757.5
Level $1512$
Weight $2$
Character 1512.757
Analytic conductor $12.073$
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] = [1512,2,Mod(757,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.757");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (of order \(2\), degree \(1\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 757.5
Root \(-0.156434 - 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 1512.757
Dual form 1512.2.c.d.757.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.437016 - 1.34500i) q^{2} +(-1.61803 + 1.17557i) q^{4} -4.29757i q^{5} -1.00000 q^{7} +(2.28825 + 1.66251i) q^{8} +O(q^{10})\) \(q+(-0.437016 - 1.34500i) q^{2} +(-1.61803 + 1.17557i) q^{4} -4.29757i q^{5} -1.00000 q^{7} +(2.28825 + 1.66251i) q^{8} +(-5.78022 + 1.87811i) q^{10} -1.60758i q^{11} +6.02967i q^{13} +(0.437016 + 1.34500i) q^{14} +(1.23607 - 3.80423i) q^{16} -3.16228 q^{17} +3.07768i q^{19} +(5.05210 + 6.95362i) q^{20} +(-2.16219 + 0.702537i) q^{22} -5.95080 q^{23} -13.4691 q^{25} +(8.10989 - 2.63506i) q^{26} +(1.61803 - 1.17557i) q^{28} +3.53972i q^{29} -2.08831 q^{31} -5.65685 q^{32} +(1.38197 + 4.25325i) q^{34} +4.29757i q^{35} +4.48276i q^{37} +(4.13948 - 1.34500i) q^{38} +(7.14475 - 9.83390i) q^{40} +6.69568 q^{41} -12.2811i q^{43} +(1.88982 + 2.60111i) q^{44} +(2.60059 + 8.00380i) q^{46} -3.82734 q^{47} +1.00000 q^{49} +(5.88622 + 18.1159i) q^{50} +(-7.08831 - 9.75621i) q^{52} +8.63711i q^{53} -6.90868 q^{55} +(-2.28825 - 1.66251i) q^{56} +(4.76091 - 1.54691i) q^{58} +1.55265i q^{59} +3.64886i q^{61} +(0.912623 + 2.80876i) q^{62} +(2.47214 + 7.60845i) q^{64} +25.9129 q^{65} +0.772361i q^{67} +(5.11667 - 3.71748i) q^{68} +(5.78022 - 1.87811i) q^{70} -7.36501 q^{71} +1.32739 q^{73} +(6.02930 - 1.95904i) q^{74} +(-3.61803 - 4.97980i) q^{76} +1.60758i q^{77} +12.7401 q^{79} +(-16.3489 - 5.31209i) q^{80} +(-2.92612 - 9.00567i) q^{82} -7.08446i q^{83} +13.5901i q^{85} +(-16.5180 + 5.36702i) q^{86} +(2.67261 - 3.67853i) q^{88} -9.73307 q^{89} -6.02967i q^{91} +(9.62859 - 6.99558i) q^{92} +(1.67261 + 5.14776i) q^{94} +13.2266 q^{95} -10.1803 q^{97} +(-0.437016 - 1.34500i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 16 q^{7} - 20 q^{10} - 16 q^{16} + 20 q^{22} - 32 q^{25} + 8 q^{28} + 40 q^{31} + 40 q^{34} + 40 q^{40} + 4 q^{46} + 16 q^{49} - 40 q^{52} - 72 q^{55} - 32 q^{64} + 20 q^{70} + 24 q^{73} - 40 q^{76} + 24 q^{79} - 28 q^{82} + 40 q^{88} + 24 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.437016 1.34500i −0.309017 0.951057i
\(3\) 0 0
\(4\) −1.61803 + 1.17557i −0.809017 + 0.587785i
\(5\) 4.29757i 1.92193i −0.276666 0.960966i \(-0.589230\pi\)
0.276666 0.960966i \(-0.410770\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.28825 + 1.66251i 0.809017 + 0.587785i
\(9\) 0 0
\(10\) −5.78022 + 1.87811i −1.82787 + 0.593910i
\(11\) 1.60758i 0.484703i −0.970189 0.242351i \(-0.922081\pi\)
0.970189 0.242351i \(-0.0779187\pi\)
\(12\) 0 0
\(13\) 6.02967i 1.67233i 0.548478 + 0.836165i \(0.315208\pi\)
−0.548478 + 0.836165i \(0.684792\pi\)
\(14\) 0.437016 + 1.34500i 0.116797 + 0.359466i
\(15\) 0 0
\(16\) 1.23607 3.80423i 0.309017 0.951057i
\(17\) −3.16228 −0.766965 −0.383482 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) 3.07768i 0.706069i 0.935610 + 0.353035i \(0.114850\pi\)
−0.935610 + 0.353035i \(0.885150\pi\)
\(20\) 5.05210 + 6.95362i 1.12968 + 1.55488i
\(21\) 0 0
\(22\) −2.16219 + 0.702537i −0.460980 + 0.149781i
\(23\) −5.95080 −1.24083 −0.620413 0.784275i \(-0.713035\pi\)
−0.620413 + 0.784275i \(0.713035\pi\)
\(24\) 0 0
\(25\) −13.4691 −2.69382
\(26\) 8.10989 2.63506i 1.59048 0.516778i
\(27\) 0 0
\(28\) 1.61803 1.17557i 0.305780 0.222162i
\(29\) 3.53972i 0.657310i 0.944450 + 0.328655i \(0.106595\pi\)
−0.944450 + 0.328655i \(0.893405\pi\)
\(30\) 0 0
\(31\) −2.08831 −0.375071 −0.187535 0.982258i \(-0.560050\pi\)
−0.187535 + 0.982258i \(0.560050\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) 1.38197 + 4.25325i 0.237005 + 0.729427i
\(35\) 4.29757i 0.726422i
\(36\) 0 0
\(37\) 4.48276i 0.736961i 0.929636 + 0.368480i \(0.120122\pi\)
−0.929636 + 0.368480i \(0.879878\pi\)
\(38\) 4.13948 1.34500i 0.671512 0.218187i
\(39\) 0 0
\(40\) 7.14475 9.83390i 1.12968 1.55488i
\(41\) 6.69568 1.04569 0.522845 0.852428i \(-0.324871\pi\)
0.522845 + 0.852428i \(0.324871\pi\)
\(42\) 0 0
\(43\) 12.2811i 1.87284i −0.350875 0.936422i \(-0.614116\pi\)
0.350875 0.936422i \(-0.385884\pi\)
\(44\) 1.88982 + 2.60111i 0.284901 + 0.392133i
\(45\) 0 0
\(46\) 2.60059 + 8.00380i 0.383437 + 1.18010i
\(47\) −3.82734 −0.558275 −0.279138 0.960251i \(-0.590049\pi\)
−0.279138 + 0.960251i \(0.590049\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.88622 + 18.1159i 0.832437 + 2.56198i
\(51\) 0 0
\(52\) −7.08831 9.75621i −0.982971 1.35294i
\(53\) 8.63711i 1.18640i 0.805056 + 0.593199i \(0.202135\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(54\) 0 0
\(55\) −6.90868 −0.931566
\(56\) −2.28825 1.66251i −0.305780 0.222162i
\(57\) 0 0
\(58\) 4.76091 1.54691i 0.625139 0.203120i
\(59\) 1.55265i 0.202137i 0.994879 + 0.101069i \(0.0322262\pi\)
−0.994879 + 0.101069i \(0.967774\pi\)
\(60\) 0 0
\(61\) 3.64886i 0.467189i 0.972334 + 0.233594i \(0.0750487\pi\)
−0.972334 + 0.233594i \(0.924951\pi\)
\(62\) 0.912623 + 2.80876i 0.115903 + 0.356713i
\(63\) 0 0
\(64\) 2.47214 + 7.60845i 0.309017 + 0.951057i
\(65\) 25.9129 3.21411
\(66\) 0 0
\(67\) 0.772361i 0.0943590i 0.998886 + 0.0471795i \(0.0150233\pi\)
−0.998886 + 0.0471795i \(0.984977\pi\)
\(68\) 5.11667 3.71748i 0.620488 0.450811i
\(69\) 0 0
\(70\) 5.78022 1.87811i 0.690868 0.224477i
\(71\) −7.36501 −0.874066 −0.437033 0.899446i \(-0.643971\pi\)
−0.437033 + 0.899446i \(0.643971\pi\)
\(72\) 0 0
\(73\) 1.32739 0.155359 0.0776797 0.996978i \(-0.475249\pi\)
0.0776797 + 0.996978i \(0.475249\pi\)
\(74\) 6.02930 1.95904i 0.700891 0.227733i
\(75\) 0 0
\(76\) −3.61803 4.97980i −0.415017 0.571222i
\(77\) 1.60758i 0.183200i
\(78\) 0 0
\(79\) 12.7401 1.43337 0.716685 0.697397i \(-0.245659\pi\)
0.716685 + 0.697397i \(0.245659\pi\)
\(80\) −16.3489 5.31209i −1.82787 0.593910i
\(81\) 0 0
\(82\) −2.92612 9.00567i −0.323136 0.994510i
\(83\) 7.08446i 0.777620i −0.921318 0.388810i \(-0.872886\pi\)
0.921318 0.388810i \(-0.127114\pi\)
\(84\) 0 0
\(85\) 13.5901i 1.47405i
\(86\) −16.5180 + 5.36702i −1.78118 + 0.578741i
\(87\) 0 0
\(88\) 2.67261 3.67853i 0.284901 0.392133i
\(89\) −9.73307 −1.03170 −0.515852 0.856678i \(-0.672525\pi\)
−0.515852 + 0.856678i \(0.672525\pi\)
\(90\) 0 0
\(91\) 6.02967i 0.632081i
\(92\) 9.62859 6.99558i 1.00385 0.729340i
\(93\) 0 0
\(94\) 1.67261 + 5.14776i 0.172517 + 0.530951i
\(95\) 13.2266 1.35702
\(96\) 0 0
\(97\) −10.1803 −1.03366 −0.516828 0.856089i \(-0.672887\pi\)
−0.516828 + 0.856089i \(0.672887\pi\)
\(98\) −0.437016 1.34500i −0.0441453 0.135865i
\(99\) 0 0
\(100\) 21.7935 15.8339i 2.17935 1.58339i
\(101\) 1.02749i 0.102239i −0.998693 0.0511194i \(-0.983721\pi\)
0.998693 0.0511194i \(-0.0162789\pi\)
\(102\) 0 0
\(103\) −0.965118 −0.0950959 −0.0475479 0.998869i \(-0.515141\pi\)
−0.0475479 + 0.998869i \(0.515141\pi\)
\(104\) −10.0244 + 13.7974i −0.982971 + 1.35294i
\(105\) 0 0
\(106\) 11.6169 3.77455i 1.12833 0.366617i
\(107\) 16.0949i 1.55595i 0.628294 + 0.777976i \(0.283754\pi\)
−0.628294 + 0.777976i \(0.716246\pi\)
\(108\) 0 0
\(109\) 20.2686i 1.94138i 0.240327 + 0.970692i \(0.422745\pi\)
−0.240327 + 0.970692i \(0.577255\pi\)
\(110\) 3.01920 + 9.29215i 0.287870 + 0.885972i
\(111\) 0 0
\(112\) −1.23607 + 3.80423i −0.116797 + 0.359466i
\(113\) 0.951214 0.0894826 0.0447413 0.998999i \(-0.485754\pi\)
0.0447413 + 0.998999i \(0.485754\pi\)
\(114\) 0 0
\(115\) 25.5740i 2.38478i
\(116\) −4.16119 5.72739i −0.386357 0.531775i
\(117\) 0 0
\(118\) 2.08831 0.678531i 0.192244 0.0624639i
\(119\) 3.16228 0.289886
\(120\) 0 0
\(121\) 8.41570 0.765063
\(122\) 4.90770 1.59461i 0.444323 0.144369i
\(123\) 0 0
\(124\) 3.37895 2.45495i 0.303439 0.220461i
\(125\) 36.3966i 3.25541i
\(126\) 0 0
\(127\) −11.6132 −1.03050 −0.515250 0.857040i \(-0.672301\pi\)
−0.515250 + 0.857040i \(0.672301\pi\)
\(128\) 9.15298 6.65003i 0.809017 0.587785i
\(129\) 0 0
\(130\) −11.3244 34.8528i −0.993213 3.05680i
\(131\) 10.0120i 0.874752i −0.899279 0.437376i \(-0.855908\pi\)
0.899279 0.437376i \(-0.144092\pi\)
\(132\) 0 0
\(133\) 3.07768i 0.266869i
\(134\) 1.03882 0.337534i 0.0897407 0.0291585i
\(135\) 0 0
\(136\) −7.23607 5.25731i −0.620488 0.450811i
\(137\) 8.40383 0.717988 0.358994 0.933340i \(-0.383120\pi\)
0.358994 + 0.933340i \(0.383120\pi\)
\(138\) 0 0
\(139\) 6.61437i 0.561024i −0.959851 0.280512i \(-0.909496\pi\)
0.959851 0.280512i \(-0.0905042\pi\)
\(140\) −5.05210 6.95362i −0.426980 0.587688i
\(141\) 0 0
\(142\) 3.21863 + 9.90592i 0.270101 + 0.831286i
\(143\) 9.69316 0.810583
\(144\) 0 0
\(145\) 15.2122 1.26330
\(146\) −0.580091 1.78534i −0.0480087 0.147755i
\(147\) 0 0
\(148\) −5.26980 7.25325i −0.433175 0.596214i
\(149\) 16.9866i 1.39160i 0.718238 + 0.695798i \(0.244949\pi\)
−0.718238 + 0.695798i \(0.755051\pi\)
\(150\) 0 0
\(151\) −11.5256 −0.937937 −0.468968 0.883215i \(-0.655374\pi\)
−0.468968 + 0.883215i \(0.655374\pi\)
\(152\) −5.11667 + 7.04250i −0.415017 + 0.571222i
\(153\) 0 0
\(154\) 2.16219 0.702537i 0.174234 0.0566120i
\(155\) 8.97464i 0.720860i
\(156\) 0 0
\(157\) 10.3889i 0.829127i −0.910020 0.414563i \(-0.863934\pi\)
0.910020 0.414563i \(-0.136066\pi\)
\(158\) −5.56761 17.1354i −0.442936 1.36322i
\(159\) 0 0
\(160\) 24.3107i 1.92193i
\(161\) 5.95080 0.468988
\(162\) 0 0
\(163\) 11.8743i 0.930067i 0.885293 + 0.465034i \(0.153958\pi\)
−0.885293 + 0.465034i \(0.846042\pi\)
\(164\) −10.8338 + 7.87124i −0.845980 + 0.614641i
\(165\) 0 0
\(166\) −9.52858 + 3.09602i −0.739561 + 0.240298i
\(167\) −15.0596 −1.16535 −0.582673 0.812706i \(-0.697993\pi\)
−0.582673 + 0.812706i \(0.697993\pi\)
\(168\) 0 0
\(169\) −23.3570 −1.79669
\(170\) 18.2787 5.93910i 1.40191 0.455508i
\(171\) 0 0
\(172\) 14.4373 + 19.8712i 1.10083 + 1.51516i
\(173\) 23.7854i 1.80837i −0.427142 0.904185i \(-0.640479\pi\)
0.427142 0.904185i \(-0.359521\pi\)
\(174\) 0 0
\(175\) 13.4691 1.01817
\(176\) −6.11559 1.98707i −0.460980 0.149781i
\(177\) 0 0
\(178\) 4.25351 + 13.0910i 0.318814 + 0.981209i
\(179\) 22.1919i 1.65870i −0.558728 0.829351i \(-0.688710\pi\)
0.558728 0.829351i \(-0.311290\pi\)
\(180\) 0 0
\(181\) 8.60253i 0.639421i 0.947515 + 0.319711i \(0.103586\pi\)
−0.947515 + 0.319711i \(0.896414\pi\)
\(182\) −8.10989 + 2.63506i −0.601145 + 0.195324i
\(183\) 0 0
\(184\) −13.6169 9.89324i −1.00385 0.729340i
\(185\) 19.2650 1.41639
\(186\) 0 0
\(187\) 5.08361i 0.371750i
\(188\) 6.19277 4.49931i 0.451654 0.328146i
\(189\) 0 0
\(190\) −5.78022 17.7897i −0.419341 1.29060i
\(191\) 8.72989 0.631673 0.315836 0.948814i \(-0.397715\pi\)
0.315836 + 0.948814i \(0.397715\pi\)
\(192\) 0 0
\(193\) −12.8708 −0.926459 −0.463229 0.886238i \(-0.653309\pi\)
−0.463229 + 0.886238i \(0.653309\pi\)
\(194\) 4.44897 + 13.6925i 0.319418 + 0.983066i
\(195\) 0 0
\(196\) −1.61803 + 1.17557i −0.115574 + 0.0839693i
\(197\) 22.4345i 1.59839i −0.601072 0.799195i \(-0.705259\pi\)
0.601072 0.799195i \(-0.294741\pi\)
\(198\) 0 0
\(199\) −14.9799 −1.06189 −0.530947 0.847405i \(-0.678164\pi\)
−0.530947 + 0.847405i \(0.678164\pi\)
\(200\) −30.8207 22.3925i −2.17935 1.58339i
\(201\) 0 0
\(202\) −1.38197 + 0.449028i −0.0972348 + 0.0315935i
\(203\) 3.53972i 0.248440i
\(204\) 0 0
\(205\) 28.7752i 2.00974i
\(206\) 0.421772 + 1.29808i 0.0293862 + 0.0904416i
\(207\) 0 0
\(208\) 22.9382 + 7.45309i 1.59048 + 0.516778i
\(209\) 4.94761 0.342234
\(210\) 0 0
\(211\) 8.50651i 0.585612i 0.956172 + 0.292806i \(0.0945890\pi\)
−0.956172 + 0.292806i \(0.905411\pi\)
\(212\) −10.1535 13.9751i −0.697347 0.959816i
\(213\) 0 0
\(214\) 21.6476 7.03373i 1.47980 0.480816i
\(215\) −52.7787 −3.59948
\(216\) 0 0
\(217\) 2.08831 0.141763
\(218\) 27.2613 8.85772i 1.84637 0.599921i
\(219\) 0 0
\(220\) 11.1785 8.12164i 0.753653 0.547561i
\(221\) 19.0675i 1.28262i
\(222\) 0 0
\(223\) 16.3214 1.09296 0.546479 0.837473i \(-0.315968\pi\)
0.546479 + 0.837473i \(0.315968\pi\)
\(224\) 5.65685 0.377964
\(225\) 0 0
\(226\) −0.415696 1.27938i −0.0276517 0.0851031i
\(227\) 10.8279i 0.718671i 0.933208 + 0.359336i \(0.116997\pi\)
−0.933208 + 0.359336i \(0.883003\pi\)
\(228\) 0 0
\(229\) 24.9251i 1.64710i −0.567245 0.823549i \(-0.691991\pi\)
0.567245 0.823549i \(-0.308009\pi\)
\(230\) 34.3969 11.1762i 2.26807 0.736939i
\(231\) 0 0
\(232\) −5.88481 + 8.09975i −0.386357 + 0.531775i
\(233\) 13.5950 0.890641 0.445321 0.895371i \(-0.353090\pi\)
0.445321 + 0.895371i \(0.353090\pi\)
\(234\) 0 0
\(235\) 16.4483i 1.07297i
\(236\) −1.82525 2.51224i −0.118813 0.163533i
\(237\) 0 0
\(238\) −1.38197 4.25325i −0.0895796 0.275698i
\(239\) −3.82998 −0.247741 −0.123870 0.992298i \(-0.539531\pi\)
−0.123870 + 0.992298i \(0.539531\pi\)
\(240\) 0 0
\(241\) 5.32509 0.343019 0.171509 0.985182i \(-0.445136\pi\)
0.171509 + 0.985182i \(0.445136\pi\)
\(242\) −3.67779 11.3191i −0.236418 0.727618i
\(243\) 0 0
\(244\) −4.28949 5.90398i −0.274607 0.377963i
\(245\) 4.29757i 0.274562i
\(246\) 0 0
\(247\) −18.5574 −1.18078
\(248\) −4.77856 3.47182i −0.303439 0.220461i
\(249\) 0 0
\(250\) 48.9534 15.9059i 3.09608 1.00598i
\(251\) 19.2193i 1.21311i −0.795040 0.606556i \(-0.792550\pi\)
0.795040 0.606556i \(-0.207450\pi\)
\(252\) 0 0
\(253\) 9.56636i 0.601432i
\(254\) 5.07513 + 15.6197i 0.318442 + 0.980064i
\(255\) 0 0
\(256\) −12.9443 9.40456i −0.809017 0.587785i
\(257\) −0.655642 −0.0408979 −0.0204489 0.999791i \(-0.506510\pi\)
−0.0204489 + 0.999791i \(0.506510\pi\)
\(258\) 0 0
\(259\) 4.48276i 0.278545i
\(260\) −41.9280 + 30.4625i −2.60027 + 1.88920i
\(261\) 0 0
\(262\) −13.4661 + 4.37540i −0.831939 + 0.270313i
\(263\) −25.8211 −1.59220 −0.796098 0.605168i \(-0.793106\pi\)
−0.796098 + 0.605168i \(0.793106\pi\)
\(264\) 0 0
\(265\) 37.1186 2.28018
\(266\) −4.13948 + 1.34500i −0.253808 + 0.0824671i
\(267\) 0 0
\(268\) −0.907965 1.24971i −0.0554628 0.0763380i
\(269\) 14.2078i 0.866266i 0.901330 + 0.433133i \(0.142592\pi\)
−0.901330 + 0.433133i \(0.857408\pi\)
\(270\) 0 0
\(271\) 3.52786 0.214302 0.107151 0.994243i \(-0.465827\pi\)
0.107151 + 0.994243i \(0.465827\pi\)
\(272\) −3.90879 + 12.0300i −0.237005 + 0.729427i
\(273\) 0 0
\(274\) −3.67261 11.3031i −0.221870 0.682847i
\(275\) 21.6526i 1.30570i
\(276\) 0 0
\(277\) 13.9396i 0.837548i 0.908091 + 0.418774i \(0.137540\pi\)
−0.908091 + 0.418774i \(0.862460\pi\)
\(278\) −8.89631 + 2.89059i −0.533565 + 0.173366i
\(279\) 0 0
\(280\) −7.14475 + 9.83390i −0.426980 + 0.587688i
\(281\) −20.4105 −1.21759 −0.608793 0.793329i \(-0.708346\pi\)
−0.608793 + 0.793329i \(0.708346\pi\)
\(282\) 0 0
\(283\) 29.6867i 1.76469i 0.470600 + 0.882347i \(0.344037\pi\)
−0.470600 + 0.882347i \(0.655963\pi\)
\(284\) 11.9168 8.65809i 0.707134 0.513763i
\(285\) 0 0
\(286\) −4.23607 13.0373i −0.250484 0.770910i
\(287\) −6.69568 −0.395233
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) −6.64798 20.4604i −0.390383 1.20147i
\(291\) 0 0
\(292\) −2.14776 + 1.56044i −0.125688 + 0.0913179i
\(293\) 16.4504i 0.961044i −0.876983 0.480522i \(-0.840447\pi\)
0.876983 0.480522i \(-0.159553\pi\)
\(294\) 0 0
\(295\) 6.67261 0.388494
\(296\) −7.45262 + 10.2577i −0.433175 + 0.596214i
\(297\) 0 0
\(298\) 22.8469 7.42341i 1.32349 0.430027i
\(299\) 35.8813i 2.07507i
\(300\) 0 0
\(301\) 12.2811i 0.707869i
\(302\) 5.03685 + 15.5018i 0.289838 + 0.892031i
\(303\) 0 0
\(304\) 11.7082 + 3.80423i 0.671512 + 0.218187i
\(305\) 15.6812 0.897905
\(306\) 0 0
\(307\) 19.8393i 1.13229i −0.824306 0.566145i \(-0.808434\pi\)
0.824306 0.566145i \(-0.191566\pi\)
\(308\) −1.88982 2.60111i −0.107683 0.148212i
\(309\) 0 0
\(310\) 12.0709 3.92206i 0.685579 0.222758i
\(311\) −32.3919 −1.83677 −0.918387 0.395683i \(-0.870508\pi\)
−0.918387 + 0.395683i \(0.870508\pi\)
\(312\) 0 0
\(313\) −6.02156 −0.340359 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(314\) −13.9731 + 4.54013i −0.788546 + 0.256214i
\(315\) 0 0
\(316\) −20.6139 + 14.9768i −1.15962 + 0.842514i
\(317\) 5.23318i 0.293925i −0.989142 0.146962i \(-0.953050\pi\)
0.989142 0.146962i \(-0.0469496\pi\)
\(318\) 0 0
\(319\) 5.69038 0.318600
\(320\) 32.6979 10.6242i 1.82787 0.593910i
\(321\) 0 0
\(322\) −2.60059 8.00380i −0.144925 0.446034i
\(323\) 9.73249i 0.541530i
\(324\) 0 0
\(325\) 81.2144i 4.50496i
\(326\) 15.9709 5.18926i 0.884547 0.287407i
\(327\) 0 0
\(328\) 15.3214 + 11.1316i 0.845980 + 0.614641i
\(329\) 3.82734 0.211008
\(330\) 0 0
\(331\) 16.5740i 0.910988i 0.890239 + 0.455494i \(0.150537\pi\)
−0.890239 + 0.455494i \(0.849463\pi\)
\(332\) 8.32828 + 11.4629i 0.457074 + 0.629108i
\(333\) 0 0
\(334\) 6.58129 + 20.2551i 0.360112 + 1.10831i
\(335\) 3.31928 0.181352
\(336\) 0 0
\(337\) 1.70020 0.0926159 0.0463080 0.998927i \(-0.485254\pi\)
0.0463080 + 0.998927i \(0.485254\pi\)
\(338\) 10.2074 + 31.4150i 0.555207 + 1.70875i
\(339\) 0 0
\(340\) −15.9761 21.9893i −0.866428 1.19254i
\(341\) 3.35711i 0.181798i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 20.4174 28.1021i 1.10083 1.51516i
\(345\) 0 0
\(346\) −31.9913 + 10.3946i −1.71986 + 0.558817i
\(347\) 24.0032i 1.28856i 0.764790 + 0.644279i \(0.222843\pi\)
−0.764790 + 0.644279i \(0.777157\pi\)
\(348\) 0 0
\(349\) 21.3679i 1.14380i −0.820324 0.571898i \(-0.806207\pi\)
0.820324 0.571898i \(-0.193793\pi\)
\(350\) −5.88622 18.1159i −0.314632 0.968337i
\(351\) 0 0
\(352\) 9.09383i 0.484703i
\(353\) −25.1560 −1.33892 −0.669460 0.742849i \(-0.733474\pi\)
−0.669460 + 0.742849i \(0.733474\pi\)
\(354\) 0 0
\(355\) 31.6517i 1.67990i
\(356\) 15.7484 11.4419i 0.834666 0.606420i
\(357\) 0 0
\(358\) −29.8481 + 9.69822i −1.57752 + 0.512567i
\(359\) 18.1318 0.956957 0.478479 0.878099i \(-0.341188\pi\)
0.478479 + 0.878099i \(0.341188\pi\)
\(360\) 0 0
\(361\) 9.52786 0.501467
\(362\) 11.5704 3.75944i 0.608126 0.197592i
\(363\) 0 0
\(364\) 7.08831 + 9.75621i 0.371528 + 0.511365i
\(365\) 5.70456i 0.298590i
\(366\) 0 0
\(367\) −29.6249 −1.54641 −0.773203 0.634158i \(-0.781347\pi\)
−0.773203 + 0.634158i \(0.781347\pi\)
\(368\) −7.35559 + 22.6382i −0.383437 + 1.18010i
\(369\) 0 0
\(370\) −8.41910 25.9113i −0.437688 1.34707i
\(371\) 8.63711i 0.448416i
\(372\) 0 0
\(373\) 29.3696i 1.52070i 0.649514 + 0.760349i \(0.274972\pi\)
−0.649514 + 0.760349i \(0.725028\pi\)
\(374\) 6.83743 2.22162i 0.353555 0.114877i
\(375\) 0 0
\(376\) −8.75790 6.36298i −0.451654 0.328146i
\(377\) −21.3434 −1.09924
\(378\) 0 0
\(379\) 16.1375i 0.828930i −0.910065 0.414465i \(-0.863969\pi\)
0.910065 0.414465i \(-0.136031\pi\)
\(380\) −21.4010 + 15.5488i −1.09785 + 0.797634i
\(381\) 0 0
\(382\) −3.81510 11.7417i −0.195198 0.600757i
\(383\) −21.9227 −1.12020 −0.560099 0.828426i \(-0.689237\pi\)
−0.560099 + 0.828426i \(0.689237\pi\)
\(384\) 0 0
\(385\) 6.90868 0.352099
\(386\) 5.62474 + 17.3112i 0.286292 + 0.881115i
\(387\) 0 0
\(388\) 16.4721 11.9677i 0.836246 0.607568i
\(389\) 27.7725i 1.40812i 0.710140 + 0.704061i \(0.248632\pi\)
−0.710140 + 0.704061i \(0.751368\pi\)
\(390\) 0 0
\(391\) 18.8181 0.951671
\(392\) 2.28825 + 1.66251i 0.115574 + 0.0839693i
\(393\) 0 0
\(394\) −30.1743 + 9.80423i −1.52016 + 0.493930i
\(395\) 54.7514i 2.75484i
\(396\) 0 0
\(397\) 27.6193i 1.38617i 0.720855 + 0.693086i \(0.243749\pi\)
−0.720855 + 0.693086i \(0.756251\pi\)
\(398\) 6.54644 + 20.1479i 0.328143 + 1.00992i
\(399\) 0 0
\(400\) −16.6487 + 51.2396i −0.832437 + 2.56198i
\(401\) −10.4439 −0.521546 −0.260773 0.965400i \(-0.583977\pi\)
−0.260773 + 0.965400i \(0.583977\pi\)
\(402\) 0 0
\(403\) 12.5918i 0.627242i
\(404\) 1.20788 + 1.66251i 0.0600944 + 0.0827129i
\(405\) 0 0
\(406\) −4.76091 + 1.54691i −0.236280 + 0.0767721i
\(407\) 7.20638 0.357207
\(408\) 0 0
\(409\) 15.1424 0.748745 0.374373 0.927278i \(-0.377858\pi\)
0.374373 + 0.927278i \(0.377858\pi\)
\(410\) −38.7025 + 12.5752i −1.91138 + 0.621045i
\(411\) 0 0
\(412\) 1.56159 1.13456i 0.0769342 0.0558960i
\(413\) 1.55265i 0.0764007i
\(414\) 0 0
\(415\) −30.4460 −1.49453
\(416\) 34.1090i 1.67233i
\(417\) 0 0
\(418\) −2.16219 6.65453i −0.105756 0.325484i
\(419\) 19.9623i 0.975222i 0.873061 + 0.487611i \(0.162131\pi\)
−0.873061 + 0.487611i \(0.837869\pi\)
\(420\) 0 0
\(421\) 12.8339i 0.625486i 0.949838 + 0.312743i \(0.101248\pi\)
−0.949838 + 0.312743i \(0.898752\pi\)
\(422\) 11.4412 3.71748i 0.556950 0.180964i
\(423\) 0 0
\(424\) −14.3593 + 19.7638i −0.697347 + 0.959816i
\(425\) 42.5931 2.06607
\(426\) 0 0
\(427\) 3.64886i 0.176581i
\(428\) −18.9207 26.0421i −0.914566 1.25879i
\(429\) 0 0
\(430\) 23.0652 + 70.9872i 1.11230 + 3.42331i
\(431\) 2.66364 0.128303 0.0641514 0.997940i \(-0.479566\pi\)
0.0641514 + 0.997940i \(0.479566\pi\)
\(432\) 0 0
\(433\) −19.2122 −0.923280 −0.461640 0.887067i \(-0.652739\pi\)
−0.461640 + 0.887067i \(0.652739\pi\)
\(434\) −0.912623 2.80876i −0.0438073 0.134825i
\(435\) 0 0
\(436\) −23.8272 32.7954i −1.14112 1.57061i
\(437\) 18.3147i 0.876109i
\(438\) 0 0
\(439\) −35.1209 −1.67623 −0.838114 0.545495i \(-0.816342\pi\)
−0.838114 + 0.545495i \(0.816342\pi\)
\(440\) −15.8088 11.4857i −0.753653 0.547561i
\(441\) 0 0
\(442\) −25.6457 + 8.33280i −1.21984 + 0.396351i
\(443\) 8.40943i 0.399544i −0.979842 0.199772i \(-0.935980\pi\)
0.979842 0.199772i \(-0.0640202\pi\)
\(444\) 0 0
\(445\) 41.8286i 1.98286i
\(446\) −7.13269 21.9522i −0.337743 1.03947i
\(447\) 0 0
\(448\) −2.47214 7.60845i −0.116797 0.359466i
\(449\) 4.81067 0.227030 0.113515 0.993536i \(-0.463789\pi\)
0.113515 + 0.993536i \(0.463789\pi\)
\(450\) 0 0
\(451\) 10.7638i 0.506848i
\(452\) −1.53910 + 1.11822i −0.0723930 + 0.0525966i
\(453\) 0 0
\(454\) 14.5635 4.73195i 0.683497 0.222082i
\(455\) −25.9129 −1.21482
\(456\) 0 0
\(457\) −37.8291 −1.76957 −0.884785 0.465999i \(-0.845695\pi\)
−0.884785 + 0.465999i \(0.845695\pi\)
\(458\) −33.5242 + 10.8927i −1.56648 + 0.508981i
\(459\) 0 0
\(460\) −30.0640 41.3795i −1.40174 1.92933i
\(461\) 10.5113i 0.489558i −0.969579 0.244779i \(-0.921285\pi\)
0.969579 0.244779i \(-0.0787154\pi\)
\(462\) 0 0
\(463\) 4.76163 0.221292 0.110646 0.993860i \(-0.464708\pi\)
0.110646 + 0.993860i \(0.464708\pi\)
\(464\) 13.4659 + 4.37534i 0.625139 + 0.203120i
\(465\) 0 0
\(466\) −5.94125 18.2853i −0.275223 0.847050i
\(467\) 19.7216i 0.912609i −0.889824 0.456305i \(-0.849173\pi\)
0.889824 0.456305i \(-0.150827\pi\)
\(468\) 0 0
\(469\) 0.772361i 0.0356643i
\(470\) 22.1229 7.18816i 1.02045 0.331565i
\(471\) 0 0
\(472\) −2.58129 + 3.55284i −0.118813 + 0.163533i
\(473\) −19.7428 −0.907773
\(474\) 0 0
\(475\) 41.4537i 1.90203i
\(476\) −5.11667 + 3.71748i −0.234522 + 0.170390i
\(477\) 0 0
\(478\) 1.67376 + 5.15131i 0.0765561 + 0.235615i
\(479\) −2.49558 −0.114026 −0.0570131 0.998373i \(-0.518158\pi\)
−0.0570131 + 0.998373i \(0.518158\pi\)
\(480\) 0 0
\(481\) −27.0296 −1.23244
\(482\) −2.32715 7.16222i −0.105999 0.326230i
\(483\) 0 0
\(484\) −13.6169 + 9.89324i −0.618949 + 0.449693i
\(485\) 43.7507i 1.98662i
\(486\) 0 0
\(487\) −10.0178 −0.453951 −0.226976 0.973900i \(-0.572884\pi\)
−0.226976 + 0.973900i \(0.572884\pi\)
\(488\) −6.06626 + 8.34949i −0.274607 + 0.377963i
\(489\) 0 0
\(490\) −5.78022 + 1.87811i −0.261124 + 0.0848442i
\(491\) 4.23469i 0.191109i 0.995424 + 0.0955544i \(0.0304624\pi\)
−0.995424 + 0.0955544i \(0.969538\pi\)
\(492\) 0 0
\(493\) 11.1936i 0.504134i
\(494\) 8.10989 + 24.9597i 0.364881 + 1.12299i
\(495\) 0 0
\(496\) −2.58129 + 7.94438i −0.115903 + 0.356713i
\(497\) 7.36501 0.330366
\(498\) 0 0
\(499\) 10.1176i 0.452925i 0.974020 + 0.226463i \(0.0727161\pi\)
−0.974020 + 0.226463i \(0.927284\pi\)
\(500\) −42.7868 58.8910i −1.91348 2.63369i
\(501\) 0 0
\(502\) −25.8499 + 8.39915i −1.15374 + 0.374873i
\(503\) −21.6693 −0.966186 −0.483093 0.875569i \(-0.660487\pi\)
−0.483093 + 0.875569i \(0.660487\pi\)
\(504\) 0 0
\(505\) −4.41570 −0.196496
\(506\) 12.8667 4.18065i 0.571996 0.185853i
\(507\) 0 0
\(508\) 18.7905 13.6521i 0.833692 0.605713i
\(509\) 13.8362i 0.613280i 0.951826 + 0.306640i \(0.0992048\pi\)
−0.951826 + 0.306640i \(0.900795\pi\)
\(510\) 0 0
\(511\) −1.32739 −0.0587203
\(512\) −6.99226 + 21.5200i −0.309017 + 0.951057i
\(513\) 0 0
\(514\) 0.286526 + 0.881837i 0.0126381 + 0.0388962i
\(515\) 4.14766i 0.182768i
\(516\) 0 0
\(517\) 6.15275i 0.270597i
\(518\) −6.02930 + 1.95904i −0.264912 + 0.0860751i
\(519\) 0 0
\(520\) 59.2952 + 43.0805i 2.60027 + 1.88920i
\(521\) 19.5067 0.854603 0.427302 0.904109i \(-0.359464\pi\)
0.427302 + 0.904109i \(0.359464\pi\)
\(522\) 0 0
\(523\) 0.616566i 0.0269606i −0.999909 0.0134803i \(-0.995709\pi\)
0.999909 0.0134803i \(-0.00429104\pi\)
\(524\) 11.7698 + 16.1997i 0.514166 + 0.707689i
\(525\) 0 0
\(526\) 11.2842 + 34.7293i 0.492015 + 1.51427i
\(527\) 6.60380 0.287666
\(528\) 0 0
\(529\) 12.4120 0.539651
\(530\) −16.2214 49.9244i −0.704613 2.16858i
\(531\) 0 0
\(532\) 3.61803 + 4.97980i 0.156862 + 0.215902i
\(533\) 40.3727i 1.74874i
\(534\) 0 0
\(535\) 69.1690 2.99044
\(536\) −1.28406 + 1.76735i −0.0554628 + 0.0763380i
\(537\) 0 0
\(538\) 19.1095 6.20905i 0.823868 0.267691i
\(539\) 1.60758i 0.0692433i
\(540\) 0 0
\(541\) 33.0638i 1.42152i −0.703432 0.710762i \(-0.748350\pi\)
0.703432 0.710762i \(-0.251650\pi\)
\(542\) −1.54173 4.74497i −0.0662231 0.203814i
\(543\) 0 0
\(544\) 17.8885 0.766965
\(545\) 87.1059 3.73121
\(546\) 0 0
\(547\) 45.6463i 1.95170i −0.218450 0.975848i \(-0.570100\pi\)
0.218450 0.975848i \(-0.429900\pi\)
\(548\) −13.5977 + 9.87930i −0.580864 + 0.422023i
\(549\) 0 0
\(550\) 29.1227 9.46255i 1.24180 0.403485i
\(551\) −10.8941 −0.464106
\(552\) 0 0
\(553\) −12.7401 −0.541763
\(554\) 18.7487 6.09182i 0.796555 0.258816i
\(555\) 0 0
\(556\) 7.77566 + 10.7023i 0.329762 + 0.453878i
\(557\) 0.0518694i 0.00219778i 0.999999 + 0.00109889i \(0.000349787\pi\)
−0.999999 + 0.00109889i \(0.999650\pi\)
\(558\) 0 0
\(559\) 74.0508 3.13201
\(560\) 16.3489 + 5.31209i 0.690868 + 0.224477i
\(561\) 0 0
\(562\) 8.91970 + 27.4520i 0.376255 + 1.15799i
\(563\) 10.9105i 0.459822i −0.973212 0.229911i \(-0.926156\pi\)
0.973212 0.229911i \(-0.0738436\pi\)
\(564\) 0 0
\(565\) 4.08791i 0.171980i
\(566\) 39.9286 12.9736i 1.67832 0.545320i
\(567\) 0 0
\(568\) −16.8529 12.2444i −0.707134 0.513763i
\(569\) 26.2746 1.10149 0.550745 0.834673i \(-0.314344\pi\)
0.550745 + 0.834673i \(0.314344\pi\)
\(570\) 0 0
\(571\) 24.4590i 1.02358i −0.859111 0.511789i \(-0.828983\pi\)
0.859111 0.511789i \(-0.171017\pi\)
\(572\) −15.6839 + 11.3950i −0.655775 + 0.476449i
\(573\) 0 0
\(574\) 2.92612 + 9.00567i 0.122134 + 0.375889i
\(575\) 80.1520 3.34257
\(576\) 0 0
\(577\) −10.2538 −0.426873 −0.213436 0.976957i \(-0.568466\pi\)
−0.213436 + 0.976957i \(0.568466\pi\)
\(578\) 3.05911 + 9.41498i 0.127242 + 0.391612i
\(579\) 0 0
\(580\) −24.6139 + 17.8830i −1.02204 + 0.742552i
\(581\) 7.08446i 0.293913i
\(582\) 0 0
\(583\) 13.8848 0.575050
\(584\) 3.03740 + 2.20680i 0.125688 + 0.0913179i
\(585\) 0 0
\(586\) −22.1258 + 7.18910i −0.914007 + 0.296979i
\(587\) 25.3720i 1.04722i 0.851959 + 0.523608i \(0.175414\pi\)
−0.851959 + 0.523608i \(0.824586\pi\)
\(588\) 0 0
\(589\) 6.42714i 0.264826i
\(590\) −2.91604 8.97464i −0.120051 0.369480i
\(591\) 0 0
\(592\) 17.0534 + 5.54099i 0.700891 + 0.227733i
\(593\) −7.14310 −0.293332 −0.146666 0.989186i \(-0.546854\pi\)
−0.146666 + 0.989186i \(0.546854\pi\)
\(594\) 0 0
\(595\) 13.5901i 0.557140i
\(596\) −19.9689 27.4849i −0.817959 1.12582i
\(597\) 0 0
\(598\) −48.2603 + 15.6807i −1.97351 + 0.641232i
\(599\) −3.00175 −0.122648 −0.0613242 0.998118i \(-0.519532\pi\)
−0.0613242 + 0.998118i \(0.519532\pi\)
\(600\) 0 0
\(601\) 21.2516 0.866871 0.433435 0.901185i \(-0.357301\pi\)
0.433435 + 0.901185i \(0.357301\pi\)
\(602\) 16.5180 5.36702i 0.673223 0.218743i
\(603\) 0 0
\(604\) 18.6487 13.5491i 0.758807 0.551305i
\(605\) 36.1671i 1.47040i
\(606\) 0 0
\(607\) 24.0614 0.976623 0.488312 0.872669i \(-0.337613\pi\)
0.488312 + 0.872669i \(0.337613\pi\)
\(608\) 17.4100i 0.706069i
\(609\) 0 0
\(610\) −6.85295 21.0912i −0.277468 0.853958i
\(611\) 23.0776i 0.933620i
\(612\) 0 0
\(613\) 20.9197i 0.844939i 0.906377 + 0.422469i \(0.138837\pi\)
−0.906377 + 0.422469i \(0.861163\pi\)
\(614\) −26.6838 + 8.67010i −1.07687 + 0.349897i
\(615\) 0 0
\(616\) −2.67261 + 3.67853i −0.107683 + 0.148212i
\(617\) −3.95122 −0.159070 −0.0795350 0.996832i \(-0.525344\pi\)
−0.0795350 + 0.996832i \(0.525344\pi\)
\(618\) 0 0
\(619\) 23.8651i 0.959220i −0.877482 0.479610i \(-0.840778\pi\)
0.877482 0.479610i \(-0.159222\pi\)
\(620\) −10.5503 14.5213i −0.423711 0.583188i
\(621\) 0 0
\(622\) 14.1558 + 43.5670i 0.567594 + 1.74688i
\(623\) 9.73307 0.389947
\(624\) 0 0
\(625\) 89.0716 3.56286
\(626\) 2.63152 + 8.09898i 0.105177 + 0.323700i
\(627\) 0 0
\(628\) 12.2129 + 16.8096i 0.487348 + 0.670778i
\(629\) 14.1757i 0.565223i
\(630\) 0 0
\(631\) 36.3725 1.44797 0.723983 0.689818i \(-0.242310\pi\)
0.723983 + 0.689818i \(0.242310\pi\)
\(632\) 29.1524 + 21.1805i 1.15962 + 0.842514i
\(633\) 0 0
\(634\) −7.03861 + 2.28698i −0.279539 + 0.0908277i
\(635\) 49.9083i 1.98055i
\(636\) 0 0
\(637\) 6.02967i 0.238904i
\(638\) −2.48679 7.65354i −0.0984528 0.303006i
\(639\) 0 0
\(640\) −28.5790 39.3356i −1.12968 1.55488i
\(641\) −0.876669 −0.0346264 −0.0173132 0.999850i \(-0.505511\pi\)
−0.0173132 + 0.999850i \(0.505511\pi\)
\(642\) 0 0
\(643\) 15.2486i 0.601347i −0.953727 0.300674i \(-0.902789\pi\)
0.953727 0.300674i \(-0.0972114\pi\)
\(644\) −9.62859 + 6.99558i −0.379420 + 0.275664i
\(645\) 0 0
\(646\) −13.0902 + 4.25325i −0.515026 + 0.167342i
\(647\) 9.53027 0.374673 0.187337 0.982296i \(-0.440014\pi\)
0.187337 + 0.982296i \(0.440014\pi\)
\(648\) 0 0
\(649\) 2.49600 0.0979765
\(650\) −109.233 + 35.4920i −4.28447 + 1.39211i
\(651\) 0 0
\(652\) −13.9591 19.2130i −0.546680 0.752440i
\(653\) 9.29496i 0.363740i 0.983323 + 0.181870i \(0.0582150\pi\)
−0.983323 + 0.181870i \(0.941785\pi\)
\(654\) 0 0
\(655\) −43.0273 −1.68121
\(656\) 8.27631 25.4719i 0.323136 0.994510i
\(657\) 0 0
\(658\) −1.67261 5.14776i −0.0652051 0.200681i
\(659\) 36.1879i 1.40968i 0.709366 + 0.704841i \(0.248982\pi\)
−0.709366 + 0.704841i \(0.751018\pi\)
\(660\) 0 0
\(661\) 19.8091i 0.770483i −0.922816 0.385242i \(-0.874118\pi\)
0.922816 0.385242i \(-0.125882\pi\)
\(662\) 22.2919 7.24309i 0.866401 0.281511i
\(663\) 0 0
\(664\) 11.7780 16.2110i 0.457074 0.629108i
\(665\) −13.2266 −0.512904
\(666\) 0 0
\(667\) 21.0642i 0.815607i
\(668\) 24.3669 17.7036i 0.942785 0.684974i
\(669\) 0 0
\(670\) −1.45058 4.46442i −0.0560407 0.172476i
\(671\) 5.86582 0.226448
\(672\) 0 0
\(673\) −0.186375 −0.00718421 −0.00359211 0.999994i \(-0.501143\pi\)
−0.00359211 + 0.999994i \(0.501143\pi\)
\(674\) −0.743016 2.28677i −0.0286199 0.0880830i
\(675\) 0 0
\(676\) 37.7923 27.4577i 1.45355 1.05607i
\(677\) 3.31885i 0.127554i 0.997964 + 0.0637770i \(0.0203146\pi\)
−0.997964 + 0.0637770i \(0.979685\pi\)
\(678\) 0 0
\(679\) 10.1803 0.390686
\(680\) −22.5937 + 31.0975i −0.866428 + 1.19254i
\(681\) 0 0
\(682\) 4.51531 1.46711i 0.172900 0.0561786i
\(683\) 17.9992i 0.688720i 0.938838 + 0.344360i \(0.111904\pi\)
−0.938838 + 0.344360i \(0.888096\pi\)
\(684\) 0 0
\(685\) 36.1161i 1.37992i
\(686\) 0.437016 + 1.34500i 0.0166853 + 0.0513522i
\(687\) 0 0
\(688\) −46.7199 15.1802i −1.78118 0.578741i
\(689\) −52.0789 −1.98405
\(690\) 0 0
\(691\) 3.67153i 0.139671i −0.997559 0.0698357i \(-0.977752\pi\)
0.997559 0.0698357i \(-0.0222475\pi\)
\(692\) 27.9614 + 38.4856i 1.06293 + 1.46300i
\(693\) 0 0
\(694\) 32.2842 10.4898i 1.22549 0.398187i
\(695\) −28.4257 −1.07825
\(696\) 0 0
\(697\) −21.1736 −0.802007
\(698\) −28.7397 + 9.33811i −1.08782 + 0.353453i
\(699\) 0 0
\(700\) −21.7935 + 15.8339i −0.823717 + 0.598465i
\(701\) 7.06843i 0.266971i 0.991051 + 0.133485i \(0.0426169\pi\)
−0.991051 + 0.133485i \(0.957383\pi\)
\(702\) 0 0
\(703\) −13.7965 −0.520345
\(704\) 12.2312 3.97415i 0.460980 0.149781i
\(705\) 0 0
\(706\) 10.9936 + 33.8348i 0.413749 + 1.27339i
\(707\) 1.02749i 0.0386426i
\(708\) 0 0
\(709\) 20.4607i 0.768417i 0.923246 + 0.384209i \(0.125526\pi\)
−0.923246 + 0.384209i \(0.874474\pi\)
\(710\) 42.5714 13.8323i 1.59768 0.519116i
\(711\) 0 0
\(712\) −22.2717 16.1813i −0.834666 0.606420i
\(713\) 12.4271 0.465398
\(714\) 0 0
\(715\) 41.6571i 1.55789i
\(716\) 26.0882 + 35.9073i 0.974961 + 1.34192i
\(717\) 0 0
\(718\) −7.92387 24.3872i −0.295716 0.910120i
\(719\) 40.5191 1.51111 0.755554 0.655087i \(-0.227368\pi\)
0.755554 + 0.655087i \(0.227368\pi\)
\(720\) 0 0
\(721\) 0.965118 0.0359429
\(722\) −4.16383 12.8149i −0.154962 0.476923i
\(723\) 0 0
\(724\) −10.1129 13.9192i −0.375842 0.517302i
\(725\) 47.6769i 1.77068i
\(726\) 0 0
\(727\) 6.87078 0.254823 0.127412 0.991850i \(-0.459333\pi\)
0.127412 + 0.991850i \(0.459333\pi\)
\(728\) 10.0244 13.7974i 0.371528 0.511365i
\(729\) 0 0
\(730\) −7.67261 + 2.49298i −0.283976 + 0.0922694i
\(731\) 38.8361i 1.43641i
\(732\) 0 0
\(733\) 18.5847i 0.686442i 0.939255 + 0.343221i \(0.111518\pi\)
−0.939255 + 0.343221i \(0.888482\pi\)
\(734\) 12.9465 + 39.8454i 0.477866 + 1.47072i
\(735\) 0 0
\(736\) 33.6628 1.24083
\(737\) 1.24163 0.0457360
\(738\) 0 0
\(739\) 4.69519i 0.172715i 0.996264 + 0.0863576i \(0.0275228\pi\)
−0.996264 + 0.0863576i \(0.972477\pi\)
\(740\) −31.1714 + 22.6473i −1.14588 + 0.832532i
\(741\) 0 0
\(742\) −11.6169 + 3.77455i −0.426469 + 0.138568i
\(743\) 15.5867 0.571821 0.285911 0.958256i \(-0.407704\pi\)
0.285911 + 0.958256i \(0.407704\pi\)
\(744\) 0 0
\(745\) 73.0011 2.67455
\(746\) 39.5020 12.8350i 1.44627 0.469922i
\(747\) 0 0
\(748\) −5.97614 8.22545i −0.218509 0.300752i
\(749\) 16.0949i 0.588095i
\(750\) 0 0
\(751\) −18.7401 −0.683835 −0.341917 0.939730i \(-0.611076\pi\)
−0.341917 + 0.939730i \(0.611076\pi\)
\(752\) −4.73085 + 14.5601i −0.172517 + 0.530951i
\(753\) 0 0
\(754\) 9.32739 + 28.7068i 0.339684 + 1.04544i
\(755\) 49.5319i 1.80265i
\(756\) 0 0
\(757\) 24.9095i 0.905352i 0.891675 + 0.452676i \(0.149531\pi\)
−0.891675 + 0.452676i \(0.850469\pi\)
\(758\) −21.7049 + 7.05236i −0.788359 + 0.256153i
\(759\) 0 0
\(760\) 30.2656 + 21.9893i 1.09785 + 0.797634i
\(761\) −4.70991 −0.170734 −0.0853670 0.996350i \(-0.527206\pi\)
−0.0853670 + 0.996350i \(0.527206\pi\)
\(762\) 0 0
\(763\) 20.2686i 0.733774i
\(764\) −14.1253 + 10.2626i −0.511034 + 0.371288i
\(765\) 0 0
\(766\) 9.58057 + 29.4860i 0.346160 + 1.06537i
\(767\) −9.36195 −0.338040
\(768\) 0 0
\(769\) 13.8989 0.501208 0.250604 0.968090i \(-0.419371\pi\)
0.250604 + 0.968090i \(0.419371\pi\)
\(770\) −3.01920 9.29215i −0.108805 0.334866i
\(771\) 0 0
\(772\) 20.8254 15.1305i 0.749521 0.544559i
\(773\) 14.8995i 0.535899i −0.963433 0.267949i \(-0.913654\pi\)
0.963433 0.267949i \(-0.0863460\pi\)
\(774\) 0 0
\(775\) 28.1276 1.01037
\(776\) −23.2951 16.9249i −0.836246 0.607568i
\(777\) 0 0
\(778\) 37.3539 12.1370i 1.33920 0.435134i
\(779\) 20.6072i 0.738329i
\(780\) 0 0
\(781\) 11.8398i 0.423662i
\(782\) −8.22380 25.3102i −0.294082 0.905092i
\(783\) 0 0
\(784\) 1.23607 3.80423i 0.0441453 0.135865i
\(785\) −44.6472 −1.59353
\(786\) 0 0
\(787\) 10.1839i 0.363018i 0.983389 + 0.181509i \(0.0580981\pi\)
−0.983389 + 0.181509i \(0.941902\pi\)
\(788\) 26.3733 + 36.2997i 0.939510 + 1.29312i
\(789\) 0 0
\(790\) −73.6404 + 23.9272i −2.62001 + 0.851292i
\(791\) −0.951214 −0.0338213
\(792\) 0 0
\(793\) −22.0014 −0.781293
\(794\) 37.1478 12.0701i 1.31833 0.428351i
\(795\) 0 0
\(796\) 24.2379 17.6099i 0.859091 0.624166i
\(797\) 14.0210i 0.496648i −0.968677 0.248324i \(-0.920120\pi\)
0.968677 0.248324i \(-0.0798798\pi\)
\(798\) 0 0
\(799\) 12.1031 0.428177
\(800\) 76.1928 2.69382
\(801\) 0 0
\(802\) 4.56417 + 14.0471i 0.161166 + 0.496019i
\(803\) 2.13388i 0.0753031i
\(804\) 0 0
\(805\) 25.5740i 0.901364i
\(806\) −16.9359 + 5.50282i −0.596543 + 0.193828i
\(807\) 0 0
\(808\) 1.70820 2.35114i 0.0600944 0.0827129i
\(809\) −28.0399 −0.985831 −0.492916 0.870077i \(-0.664069\pi\)
−0.492916 + 0.870077i \(0.664069\pi\)
\(810\) 0 0
\(811\) 13.2362i 0.464785i −0.972622 0.232393i \(-0.925345\pi\)
0.972622 0.232393i \(-0.0746554\pi\)
\(812\) 4.16119 + 5.72739i 0.146029 + 0.200992i
\(813\) 0 0
\(814\) −3.14930 9.69256i −0.110383 0.339724i
\(815\) 51.0307 1.78753
\(816\) 0 0
\(817\) 37.7972 1.32236
\(818\) −6.61749 20.3665i −0.231375 0.712099i
\(819\) 0 0
\(820\) 33.8272 + 46.5592i 1.18130 + 1.62592i
\(821\) 13.8343i 0.482821i 0.970423 + 0.241411i \(0.0776100\pi\)
−0.970423 + 0.241411i \(0.922390\pi\)
\(822\) 0 0
\(823\) 29.4862 1.02782 0.513912 0.857843i \(-0.328196\pi\)
0.513912 + 0.857843i \(0.328196\pi\)
\(824\) −2.20843 1.60452i −0.0769342 0.0558960i
\(825\) 0 0
\(826\) −2.08831 + 0.678531i −0.0726614 + 0.0236091i
\(827\) 28.9797i 1.00772i −0.863784 0.503861i \(-0.831912\pi\)
0.863784 0.503861i \(-0.168088\pi\)
\(828\) 0 0
\(829\) 32.5800i 1.13155i 0.824560 + 0.565774i \(0.191423\pi\)
−0.824560 + 0.565774i \(0.808577\pi\)
\(830\) 13.3054 + 40.9497i 0.461836 + 1.42139i
\(831\) 0 0
\(832\) −45.8765 + 14.9062i −1.59048 + 0.516778i
\(833\) −3.16228 −0.109566
\(834\) 0 0
\(835\) 64.7197i 2.23972i
\(836\) −8.00541 + 5.81627i −0.276873 + 0.201160i
\(837\) 0 0
\(838\) 26.8492 8.72384i 0.927491 0.301360i
\(839\) −19.8215 −0.684313 −0.342157 0.939643i \(-0.611157\pi\)
−0.342157 + 0.939643i \(0.611157\pi\)
\(840\) 0 0
\(841\) 16.4704 0.567944
\(842\) 17.2616 5.60862i 0.594872 0.193286i
\(843\) 0 0
\(844\) −10.0000 13.7638i −0.344214 0.473770i
\(845\) 100.378i 3.45311i
\(846\) 0 0
\(847\) −8.41570 −0.289167
\(848\) 32.8575 + 10.6760i 1.12833 + 0.366617i
\(849\) 0 0
\(850\) −18.6139 57.2876i −0.638450 1.96495i
\(851\) 26.6760i 0.914441i
\(852\) 0 0
\(853\) 17.4764i 0.598381i −0.954193 0.299190i \(-0.903283\pi\)
0.954193 0.299190i \(-0.0967166\pi\)
\(854\) −4.90770 + 1.59461i −0.167938 + 0.0545664i
\(855\) 0 0
\(856\) −26.7579 + 36.8291i −0.914566 + 1.25879i
\(857\) −26.0767 −0.890764 −0.445382 0.895341i \(-0.646932\pi\)
−0.445382 + 0.895341i \(0.646932\pi\)
\(858\) 0 0
\(859\) 40.6339i 1.38641i −0.720740 0.693205i \(-0.756198\pi\)
0.720740 0.693205i \(-0.243802\pi\)
\(860\) 85.3978 62.0451i 2.91204 2.11572i
\(861\) 0 0
\(862\) −1.16405 3.58258i −0.0396477 0.122023i
\(863\) −41.4216 −1.41001 −0.705004 0.709204i \(-0.749055\pi\)
−0.705004 + 0.709204i \(0.749055\pi\)
\(864\) 0 0
\(865\) −102.219 −3.47556
\(866\) 8.39604 + 25.8404i 0.285309 + 0.878091i
\(867\) 0 0
\(868\) −3.37895 + 2.45495i −0.114689 + 0.0833264i
\(869\) 20.4806i 0.694758i
\(870\) 0 0
\(871\) −4.65709 −0.157799
\(872\) −33.6968 + 46.3796i −1.14112 + 1.57061i
\(873\) 0 0
\(874\) −24.6332 + 8.00380i −0.833229 + 0.270733i
\(875\) 36.3966i 1.23043i
\(876\) 0 0
\(877\) 33.0093i 1.11464i −0.830296 0.557322i \(-0.811829\pi\)
0.830296 0.557322i \(-0.188171\pi\)
\(878\) 15.3484 + 47.2375i 0.517983 + 1.59419i
\(879\) 0 0
\(880\) −8.53960 + 26.2822i −0.287870 + 0.885972i
\(881\) 18.0587 0.608414 0.304207 0.952606i \(-0.401609\pi\)
0.304207 + 0.952606i \(0.401609\pi\)
\(882\) 0 0
\(883\) 46.6771i 1.57081i 0.618983 + 0.785404i \(0.287545\pi\)
−0.618983 + 0.785404i \(0.712455\pi\)
\(884\) 22.4152 + 30.8519i 0.753904 + 1.03766i
\(885\) 0 0
\(886\) −11.3107 + 3.67506i −0.379989 + 0.123466i
\(887\) 29.2001 0.980444 0.490222 0.871598i \(-0.336916\pi\)
0.490222 + 0.871598i \(0.336916\pi\)
\(888\) 0 0
\(889\) 11.6132 0.389493
\(890\) 56.2593 18.2798i 1.88582 0.612739i
\(891\) 0 0
\(892\) −26.4085 + 19.1869i −0.884222 + 0.642425i
\(893\) 11.7793i 0.394181i
\(894\) 0 0
\(895\) −95.3714 −3.18791
\(896\) −9.15298 + 6.65003i −0.305780 + 0.222162i
\(897\) 0 0
\(898\) −2.10234 6.47034i −0.0701560 0.215918i
\(899\) 7.39202i 0.246538i
\(900\) 0 0
\(901\) 27.3129i 0.909926i
\(902\) −14.4773 + 4.70396i −0.482042 + 0.156625i
\(903\) 0 0
\(904\) 2.17661 + 1.58140i 0.0723930 + 0.0525966i
\(905\) 36.9700 1.22892
\(906\) 0 0
\(907\) 9.06592i 0.301029i −0.988608 0.150514i \(-0.951907\pi\)
0.988608 0.150514i \(-0.0480930\pi\)
\(908\) −12.7289 17.5199i −0.422424 0.581417i
\(909\) 0 0
\(910\) 11.3244 + 34.8528i 0.375399 + 1.15536i
\(911\) 14.3571 0.475673 0.237837 0.971305i \(-0.423562\pi\)
0.237837 + 0.971305i \(0.423562\pi\)
\(912\) 0 0
\(913\) −11.3888 −0.376915
\(914\) 16.5319 + 50.8800i 0.546827 + 1.68296i
\(915\) 0 0
\(916\) 29.3012 + 40.3297i 0.968140 + 1.33253i
\(917\) 10.0120i 0.330625i
\(918\) 0 0
\(919\) 49.8788 1.64535 0.822675 0.568513i \(-0.192481\pi\)
0.822675 + 0.568513i \(0.192481\pi\)
\(920\) −42.5169 + 58.5195i −1.40174 + 1.92933i
\(921\) 0 0
\(922\) −14.1376 + 4.59359i −0.465597 + 0.151282i
\(923\) 44.4086i 1.46173i
\(924\) 0 0
\(925\) 60.3788i 1.98524i
\(926\) −2.08091 6.40437i −0.0683829 0.210461i
\(927\) 0 0
\(928\) 20.0237i 0.657310i
\(929\) 39.3069 1.28962 0.644809 0.764343i \(-0.276937\pi\)
0.644809 + 0.764343i \(0.276937\pi\)
\(930\) 0 0
\(931\) 3.07768i 0.100867i
\(932\) −21.9973 + 15.9819i −0.720544 + 0.523506i
\(933\) 0 0
\(934\) −26.5256 + 8.61868i −0.867943 + 0.282012i
\(935\) 21.8472 0.714478
\(936\) 0 0
\(937\) 37.3504 1.22018 0.610092 0.792331i \(-0.291133\pi\)
0.610092 + 0.792331i \(0.291133\pi\)
\(938\) −1.03882 + 0.337534i −0.0339188 + 0.0110209i
\(939\) 0 0
\(940\) −19.3361 26.6139i −0.630674 0.868049i
\(941\) 38.7781i 1.26413i −0.774915 0.632065i \(-0.782208\pi\)
0.774915 0.632065i \(-0.217792\pi\)
\(942\) 0 0
\(943\) −39.8446 −1.29752
\(944\) 5.90662 + 1.91918i 0.192244 + 0.0624639i
\(945\) 0 0
\(946\) 8.62790 + 26.5539i 0.280517 + 0.863343i
\(947\) 3.75328i 0.121965i −0.998139 0.0609826i \(-0.980577\pi\)
0.998139 0.0609826i \(-0.0194234\pi\)
\(948\) 0 0
\(949\) 8.00373i 0.259812i
\(950\) −55.7551 + 18.1159i −1.80893 + 0.587758i
\(951\) 0 0
\(952\) 7.23607 + 5.25731i 0.234522 + 0.170390i
\(953\) −46.5443 −1.50772 −0.753859 0.657036i \(-0.771810\pi\)
−0.753859 + 0.657036i \(0.771810\pi\)
\(954\) 0 0
\(955\) 37.5173i 1.21403i
\(956\) 6.19704 4.50241i 0.200426 0.145618i
\(957\) 0 0
\(958\) 1.09061 + 3.35655i 0.0352360 + 0.108445i
\(959\) −8.40383 −0.271374
\(960\) 0 0
\(961\) −26.6390 −0.859322
\(962\) 11.8124 + 36.3547i 0.380845 + 1.17212i
\(963\) 0 0
\(964\) −8.61617 + 6.26001i −0.277508 + 0.201621i
\(965\) 55.3131i 1.78059i
\(966\) 0 0
\(967\) −16.6487 −0.535388 −0.267694 0.963504i \(-0.586262\pi\)
−0.267694 + 0.963504i \(0.586262\pi\)
\(968\) 19.2572 + 13.9912i 0.618949 + 0.449693i
\(969\) 0 0
\(970\) 58.8446 19.1198i 1.88939 0.613899i
\(971\) 5.74652i 0.184415i −0.995740 0.0922073i \(-0.970608\pi\)
0.995740 0.0922073i \(-0.0293922\pi\)
\(972\) 0 0
\(973\) 6.61437i 0.212047i
\(974\) 4.37795 + 13.4739i 0.140279 + 0.431733i
\(975\) 0 0
\(976\) 13.8811 + 4.51024i 0.444323 + 0.144369i
\(977\) −27.6236 −0.883758 −0.441879 0.897075i \(-0.645688\pi\)
−0.441879 + 0.897075i \(0.645688\pi\)
\(978\) 0 0
\(979\) 15.6467i 0.500070i
\(980\) 5.05210 + 6.95362i 0.161383 + 0.222125i
\(981\) 0 0
\(982\) 5.69565 1.85063i 0.181755 0.0590559i
\(983\) −2.19812 −0.0701091 −0.0350545 0.999385i \(-0.511160\pi\)
−0.0350545 + 0.999385i \(0.511160\pi\)
\(984\) 0 0
\(985\) −96.4138 −3.07200
\(986\) −15.0553 + 4.89177i −0.479460 + 0.155786i
\(987\) 0 0
\(988\) 30.0265 21.8156i 0.955272 0.694045i
\(989\) 73.0821i 2.32388i
\(990\) 0 0
\(991\) 2.62346 0.0833369 0.0416684 0.999131i \(-0.486733\pi\)
0.0416684 + 0.999131i \(0.486733\pi\)
\(992\) 11.8132 0.375071
\(993\) 0 0
\(994\) −3.21863 9.90592i −0.102089 0.314197i
\(995\) 64.3770i 2.04089i
\(996\) 0 0
\(997\) 29.1361i 0.922749i 0.887205 + 0.461375i \(0.152644\pi\)
−0.887205 + 0.461375i \(0.847356\pi\)
\(998\) 13.6081 4.42154i 0.430757 0.139962i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1512.2.c.d.757.5 16
3.2 odd 2 inner 1512.2.c.d.757.12 yes 16
4.3 odd 2 6048.2.c.d.3025.2 16
8.3 odd 2 6048.2.c.d.3025.15 16
8.5 even 2 inner 1512.2.c.d.757.8 yes 16
12.11 even 2 6048.2.c.d.3025.16 16
24.5 odd 2 inner 1512.2.c.d.757.9 yes 16
24.11 even 2 6048.2.c.d.3025.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.5 16 1.1 even 1 trivial
1512.2.c.d.757.8 yes 16 8.5 even 2 inner
1512.2.c.d.757.9 yes 16 24.5 odd 2 inner
1512.2.c.d.757.12 yes 16 3.2 odd 2 inner
6048.2.c.d.3025.1 16 24.11 even 2
6048.2.c.d.3025.2 16 4.3 odd 2
6048.2.c.d.3025.15 16 8.3 odd 2
6048.2.c.d.3025.16 16 12.11 even 2