Properties

Label 756.2.bo.b.169.9
Level $756$
Weight $2$
Character 756.169
Analytic conductor $6.037$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(85,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bo (of order \(9\), degree \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 169.9
Character \(\chi\) \(=\) 756.169
Dual form 756.2.bo.b.85.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.70918 + 0.280534i) q^{3} +(-0.389665 - 2.20990i) q^{5} +(-0.766044 - 0.642788i) q^{7} +(2.84260 + 0.958966i) q^{9} +O(q^{10})\) \(q+(1.70918 + 0.280534i) q^{3} +(-0.389665 - 2.20990i) q^{5} +(-0.766044 - 0.642788i) q^{7} +(2.84260 + 0.958966i) q^{9} +(0.724451 - 4.10856i) q^{11} +(-4.44929 - 1.61941i) q^{13} +(-0.0460568 - 3.88643i) q^{15} +(0.329574 + 0.570839i) q^{17} +(1.77783 - 3.07929i) q^{19} +(-1.12899 - 1.31354i) q^{21} +(-0.105065 + 0.0881600i) q^{23} +(-0.0333580 + 0.0121413i) q^{25} +(4.58950 + 2.43649i) q^{27} +(5.61090 - 2.04220i) q^{29} +(-6.28536 + 5.27404i) q^{31} +(2.39081 - 6.81905i) q^{33} +(-1.12200 + 1.94335i) q^{35} +(-1.43637 - 2.48786i) q^{37} +(-7.15035 - 4.01604i) q^{39} +(0.747037 + 0.271899i) q^{41} +(0.937735 - 5.31816i) q^{43} +(1.01156 - 6.65554i) q^{45} +(3.78429 + 3.17540i) q^{47} +(0.173648 + 0.984808i) q^{49} +(0.403162 + 1.06812i) q^{51} +2.53916 q^{53} -9.36181 q^{55} +(3.90248 - 4.76433i) q^{57} +(1.82277 + 10.3374i) q^{59} +(1.84891 + 1.55142i) q^{61} +(-1.56115 - 2.56180i) q^{63} +(-1.84500 + 10.4635i) q^{65} +(3.21501 + 1.17017i) q^{67} +(-0.204307 + 0.121207i) q^{69} +(1.20248 + 2.08276i) q^{71} +(3.09384 - 5.35868i) q^{73} +(-0.0604210 + 0.0113937i) q^{75} +(-3.19590 + 2.68167i) q^{77} +(-11.0346 + 4.01628i) q^{79} +(7.16077 + 5.45192i) q^{81} +(12.1830 - 4.43423i) q^{83} +(1.13307 - 0.950762i) q^{85} +(10.1629 - 1.91644i) q^{87} +(-5.93261 + 10.2756i) q^{89} +(2.36742 + 4.10049i) q^{91} +(-12.2224 + 7.25104i) q^{93} +(-7.49769 - 2.72893i) q^{95} +(1.18277 - 6.70779i) q^{97} +(5.99930 - 10.9843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9} + 9 q^{13} + 3 q^{15} + 3 q^{21} + 12 q^{23} + 27 q^{25} + 39 q^{27} + 30 q^{29} - 9 q^{31} - 6 q^{33} + 6 q^{35} - 18 q^{39} - 27 q^{41} + 9 q^{43} - 81 q^{45} + 9 q^{47} + 12 q^{53} - 21 q^{57} - 24 q^{59} - 3 q^{63} - 78 q^{65} - 9 q^{67} + 6 q^{69} - 30 q^{71} + 57 q^{75} + 9 q^{77} + 99 q^{81} + 78 q^{83} + 18 q^{85} + 21 q^{87} + 12 q^{89} - 51 q^{93} - 36 q^{95} + 36 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(1\)

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 0
\(3\) 1.70918 + 0.280534i 0.986796 + 0.161966i
\(4\) 0 0
\(5\) −0.389665 2.20990i −0.174264 0.988298i −0.938990 0.343943i \(-0.888237\pi\)
0.764727 0.644354i \(-0.222874\pi\)
\(6\) 0 0
\(7\) −0.766044 0.642788i −0.289538 0.242951i
\(8\) 0 0
\(9\) 2.84260 + 0.958966i 0.947534 + 0.319655i
\(10\) 0 0
\(11\) 0.724451 4.10856i 0.218430 1.23878i −0.656424 0.754392i \(-0.727932\pi\)
0.874854 0.484387i \(-0.160957\pi\)
\(12\) 0 0
\(13\) −4.44929 1.61941i −1.23401 0.449143i −0.359042 0.933322i \(-0.616896\pi\)
−0.874969 + 0.484178i \(0.839119\pi\)
\(14\) 0 0
\(15\) −0.0460568 3.88643i −0.0118918 1.00347i
\(16\) 0 0
\(17\) 0.329574 + 0.570839i 0.0799334 + 0.138449i 0.903221 0.429176i \(-0.141196\pi\)
−0.823288 + 0.567624i \(0.807863\pi\)
\(18\) 0 0
\(19\) 1.77783 3.07929i 0.407862 0.706438i −0.586788 0.809741i \(-0.699608\pi\)
0.994650 + 0.103303i \(0.0329411\pi\)
\(20\) 0 0
\(21\) −1.12899 1.31354i −0.246365 0.286638i
\(22\) 0 0
\(23\) −0.105065 + 0.0881600i −0.0219076 + 0.0183826i −0.653676 0.756775i \(-0.726774\pi\)
0.631768 + 0.775157i \(0.282329\pi\)
\(24\) 0 0
\(25\) −0.0333580 + 0.0121413i −0.00667160 + 0.00242827i
\(26\) 0 0
\(27\) 4.58950 + 2.43649i 0.883250 + 0.468903i
\(28\) 0 0
\(29\) 5.61090 2.04220i 1.04192 0.379227i 0.236311 0.971678i \(-0.424062\pi\)
0.805607 + 0.592451i \(0.201839\pi\)
\(30\) 0 0
\(31\) −6.28536 + 5.27404i −1.12888 + 0.947246i −0.999019 0.0442860i \(-0.985899\pi\)
−0.129865 + 0.991532i \(0.541454\pi\)
\(32\) 0 0
\(33\) 2.39081 6.81905i 0.416186 1.18704i
\(34\) 0 0
\(35\) −1.12200 + 1.94335i −0.189652 + 0.328487i
\(36\) 0 0
\(37\) −1.43637 2.48786i −0.236138 0.409002i 0.723465 0.690361i \(-0.242548\pi\)
−0.959603 + 0.281359i \(0.909215\pi\)
\(38\) 0 0
\(39\) −7.15035 4.01604i −1.14497 0.643081i
\(40\) 0 0
\(41\) 0.747037 + 0.271899i 0.116668 + 0.0424635i 0.399694 0.916649i \(-0.369116\pi\)
−0.283027 + 0.959112i \(0.591339\pi\)
\(42\) 0 0
\(43\) 0.937735 5.31816i 0.143003 0.811012i −0.825945 0.563750i \(-0.809358\pi\)
0.968949 0.247262i \(-0.0795308\pi\)
\(44\) 0 0
\(45\) 1.01156 6.65554i 0.150794 0.992150i
\(46\) 0 0
\(47\) 3.78429 + 3.17540i 0.551996 + 0.463179i 0.875616 0.483008i \(-0.160456\pi\)
−0.323621 + 0.946187i \(0.604900\pi\)
\(48\) 0 0
\(49\) 0.173648 + 0.984808i 0.0248069 + 0.140687i
\(50\) 0 0
\(51\) 0.403162 + 1.06812i 0.0564540 + 0.149567i
\(52\) 0 0
\(53\) 2.53916 0.348781 0.174391 0.984677i \(-0.444204\pi\)
0.174391 + 0.984677i \(0.444204\pi\)
\(54\) 0 0
\(55\) −9.36181 −1.26235
\(56\) 0 0
\(57\) 3.90248 4.76433i 0.516896 0.631050i
\(58\) 0 0
\(59\) 1.82277 + 10.3374i 0.237304 + 1.34582i 0.837706 + 0.546121i \(0.183896\pi\)
−0.600402 + 0.799698i \(0.704993\pi\)
\(60\) 0 0
\(61\) 1.84891 + 1.55142i 0.236729 + 0.198639i 0.753432 0.657525i \(-0.228397\pi\)
−0.516704 + 0.856164i \(0.672841\pi\)
\(62\) 0 0
\(63\) −1.56115 2.56180i −0.196686 0.322756i
\(64\) 0 0
\(65\) −1.84500 + 10.4635i −0.228844 + 1.29784i
\(66\) 0 0
\(67\) 3.21501 + 1.17017i 0.392776 + 0.142959i 0.530856 0.847462i \(-0.321871\pi\)
−0.138079 + 0.990421i \(0.544093\pi\)
\(68\) 0 0
\(69\) −0.204307 + 0.121207i −0.0245957 + 0.0145916i
\(70\) 0 0
\(71\) 1.20248 + 2.08276i 0.142709 + 0.247179i 0.928516 0.371293i \(-0.121085\pi\)
−0.785807 + 0.618472i \(0.787752\pi\)
\(72\) 0 0
\(73\) 3.09384 5.35868i 0.362106 0.627186i −0.626201 0.779661i \(-0.715391\pi\)
0.988307 + 0.152475i \(0.0487245\pi\)
\(74\) 0 0
\(75\) −0.0604210 + 0.0113937i −0.00697681 + 0.00131563i
\(76\) 0 0
\(77\) −3.19590 + 2.68167i −0.364206 + 0.305605i
\(78\) 0 0
\(79\) −11.0346 + 4.01628i −1.24149 + 0.451866i −0.877519 0.479543i \(-0.840803\pi\)
−0.363974 + 0.931409i \(0.618580\pi\)
\(80\) 0 0
\(81\) 7.16077 + 5.45192i 0.795641 + 0.605768i
\(82\) 0 0
\(83\) 12.1830 4.43423i 1.33725 0.486720i 0.428306 0.903634i \(-0.359110\pi\)
0.908946 + 0.416913i \(0.136888\pi\)
\(84\) 0 0
\(85\) 1.13307 0.950762i 0.122899 0.103125i
\(86\) 0 0
\(87\) 10.1629 1.91644i 1.08958 0.205464i
\(88\) 0 0
\(89\) −5.93261 + 10.2756i −0.628856 + 1.08921i 0.358926 + 0.933366i \(0.383143\pi\)
−0.987782 + 0.155844i \(0.950190\pi\)
\(90\) 0 0
\(91\) 2.36742 + 4.10049i 0.248173 + 0.429848i
\(92\) 0 0
\(93\) −12.2224 + 7.25104i −1.26740 + 0.751898i
\(94\) 0 0
\(95\) −7.49769 2.72893i −0.769246 0.279983i
\(96\) 0 0
\(97\) 1.18277 6.70779i 0.120092 0.681073i −0.864011 0.503473i \(-0.832055\pi\)
0.984103 0.177601i \(-0.0568335\pi\)
\(98\) 0 0
\(99\) 5.99930 10.9843i 0.602952 1.10396i
\(100\) 0 0
\(101\) 7.06952 + 5.93203i 0.703444 + 0.590260i 0.922751 0.385396i \(-0.125935\pi\)
−0.219307 + 0.975656i \(0.570380\pi\)
\(102\) 0 0
\(103\) 2.57504 + 14.6038i 0.253726 + 1.43895i 0.799321 + 0.600905i \(0.205193\pi\)
−0.545594 + 0.838049i \(0.683696\pi\)
\(104\) 0 0
\(105\) −2.46287 + 3.00679i −0.240352 + 0.293432i
\(106\) 0 0
\(107\) 6.34284 0.613186 0.306593 0.951841i \(-0.400811\pi\)
0.306593 + 0.951841i \(0.400811\pi\)
\(108\) 0 0
\(109\) −2.19105 −0.209864 −0.104932 0.994479i \(-0.533463\pi\)
−0.104932 + 0.994479i \(0.533463\pi\)
\(110\) 0 0
\(111\) −1.75709 4.65516i −0.166775 0.441848i
\(112\) 0 0
\(113\) −1.61960 9.18519i −0.152359 0.864070i −0.961161 0.275989i \(-0.910995\pi\)
0.808802 0.588081i \(-0.200116\pi\)
\(114\) 0 0
\(115\) 0.235765 + 0.197830i 0.0219852 + 0.0184478i
\(116\) 0 0
\(117\) −11.0946 8.87005i −1.02570 0.820037i
\(118\) 0 0
\(119\) 0.114460 0.649134i 0.0104925 0.0595060i
\(120\) 0 0
\(121\) −6.01885 2.19068i −0.547168 0.199153i
\(122\) 0 0
\(123\) 1.20054 + 0.674294i 0.108249 + 0.0607990i
\(124\) 0 0
\(125\) −5.57015 9.64778i −0.498209 0.862924i
\(126\) 0 0
\(127\) 6.18166 10.7069i 0.548533 0.950087i −0.449842 0.893108i \(-0.648520\pi\)
0.998375 0.0569793i \(-0.0181469\pi\)
\(128\) 0 0
\(129\) 3.09468 8.82664i 0.272472 0.777142i
\(130\) 0 0
\(131\) 3.96590 3.32779i 0.346503 0.290750i −0.452881 0.891571i \(-0.649604\pi\)
0.799384 + 0.600821i \(0.205159\pi\)
\(132\) 0 0
\(133\) −3.34123 + 1.21611i −0.289721 + 0.105450i
\(134\) 0 0
\(135\) 3.59604 11.0918i 0.309498 0.954626i
\(136\) 0 0
\(137\) −3.18055 + 1.15763i −0.271733 + 0.0989027i −0.474293 0.880367i \(-0.657296\pi\)
0.202560 + 0.979270i \(0.435074\pi\)
\(138\) 0 0
\(139\) −16.0388 + 13.4581i −1.36039 + 1.14150i −0.384528 + 0.923113i \(0.625636\pi\)
−0.975862 + 0.218389i \(0.929920\pi\)
\(140\) 0 0
\(141\) 5.57723 + 6.48895i 0.469688 + 0.546468i
\(142\) 0 0
\(143\) −9.87674 + 17.1070i −0.825934 + 1.43056i
\(144\) 0 0
\(145\) −6.69943 11.6038i −0.556357 0.963639i
\(146\) 0 0
\(147\) 0.0205245 + 1.73193i 0.00169283 + 0.142847i
\(148\) 0 0
\(149\) −6.26273 2.27945i −0.513063 0.186740i 0.0724971 0.997369i \(-0.476903\pi\)
−0.585560 + 0.810629i \(0.699125\pi\)
\(150\) 0 0
\(151\) −3.30480 + 18.7425i −0.268941 + 1.52524i 0.488632 + 0.872490i \(0.337496\pi\)
−0.757573 + 0.652750i \(0.773615\pi\)
\(152\) 0 0
\(153\) 0.389433 + 1.93872i 0.0314838 + 0.156736i
\(154\) 0 0
\(155\) 14.1043 + 11.8349i 1.13288 + 0.950602i
\(156\) 0 0
\(157\) −3.48569 19.7683i −0.278188 1.57769i −0.728648 0.684888i \(-0.759851\pi\)
0.450459 0.892797i \(-0.351260\pi\)
\(158\) 0 0
\(159\) 4.33989 + 0.712321i 0.344176 + 0.0564907i
\(160\) 0 0
\(161\) 0.137153 0.0108091
\(162\) 0 0
\(163\) −19.5218 −1.52907 −0.764533 0.644585i \(-0.777030\pi\)
−0.764533 + 0.644585i \(0.777030\pi\)
\(164\) 0 0
\(165\) −16.0010 2.62630i −1.24568 0.204457i
\(166\) 0 0
\(167\) 2.87283 + 16.2926i 0.222306 + 1.26076i 0.867769 + 0.496968i \(0.165553\pi\)
−0.645463 + 0.763792i \(0.723335\pi\)
\(168\) 0 0
\(169\) 7.21512 + 6.05421i 0.555009 + 0.465708i
\(170\) 0 0
\(171\) 8.00660 7.04832i 0.612280 0.538999i
\(172\) 0 0
\(173\) −3.61178 + 20.4834i −0.274599 + 1.55733i 0.465635 + 0.884977i \(0.345826\pi\)
−0.740233 + 0.672350i \(0.765285\pi\)
\(174\) 0 0
\(175\) 0.0333580 + 0.0121413i 0.00252163 + 0.000917798i
\(176\) 0 0
\(177\) 0.215444 + 18.1799i 0.0161937 + 1.36648i
\(178\) 0 0
\(179\) 11.6046 + 20.0997i 0.867366 + 1.50232i 0.864678 + 0.502326i \(0.167522\pi\)
0.00268829 + 0.999996i \(0.499144\pi\)
\(180\) 0 0
\(181\) −2.37943 + 4.12129i −0.176861 + 0.306333i −0.940804 0.338951i \(-0.889928\pi\)
0.763943 + 0.645284i \(0.223261\pi\)
\(182\) 0 0
\(183\) 2.72490 + 3.17034i 0.201430 + 0.234358i
\(184\) 0 0
\(185\) −4.93823 + 4.14367i −0.363066 + 0.304648i
\(186\) 0 0
\(187\) 2.58409 0.940531i 0.188967 0.0687785i
\(188\) 0 0
\(189\) −1.94961 4.81653i −0.141814 0.350351i
\(190\) 0 0
\(191\) 10.3233 3.75738i 0.746968 0.271874i 0.0596388 0.998220i \(-0.481005\pi\)
0.687329 + 0.726346i \(0.258783\pi\)
\(192\) 0 0
\(193\) 3.18305 2.67089i 0.229121 0.192255i −0.520999 0.853557i \(-0.674440\pi\)
0.750119 + 0.661302i \(0.229996\pi\)
\(194\) 0 0
\(195\) −6.08881 + 17.3665i −0.436029 + 1.24364i
\(196\) 0 0
\(197\) 8.33244 14.4322i 0.593662 1.02825i −0.400072 0.916484i \(-0.631015\pi\)
0.993734 0.111769i \(-0.0356516\pi\)
\(198\) 0 0
\(199\) 0.979653 + 1.69681i 0.0694457 + 0.120284i 0.898657 0.438651i \(-0.144544\pi\)
−0.829212 + 0.558935i \(0.811210\pi\)
\(200\) 0 0
\(201\) 5.16677 + 2.90195i 0.364436 + 0.204688i
\(202\) 0 0
\(203\) −5.61090 2.04220i −0.393808 0.143334i
\(204\) 0 0
\(205\) 0.309776 1.75683i 0.0216357 0.122702i
\(206\) 0 0
\(207\) −0.383200 + 0.149850i −0.0266343 + 0.0104153i
\(208\) 0 0
\(209\) −11.3635 9.53512i −0.786031 0.659558i
\(210\) 0 0
\(211\) 4.11602 + 23.3431i 0.283359 + 1.60701i 0.711090 + 0.703101i \(0.248202\pi\)
−0.427731 + 0.903906i \(0.640687\pi\)
\(212\) 0 0
\(213\) 1.47098 + 3.89716i 0.100790 + 0.267029i
\(214\) 0 0
\(215\) −12.1180 −0.826441
\(216\) 0 0
\(217\) 8.20495 0.556988
\(218\) 0 0
\(219\) 6.79122 8.29103i 0.458908 0.560256i
\(220\) 0 0
\(221\) −0.541949 3.07354i −0.0364554 0.206749i
\(222\) 0 0
\(223\) 21.6027 + 18.1268i 1.44662 + 1.21386i 0.934999 + 0.354649i \(0.115400\pi\)
0.511623 + 0.859210i \(0.329045\pi\)
\(224\) 0 0
\(225\) −0.106467 + 0.00252376i −0.00709778 + 0.000168250i
\(226\) 0 0
\(227\) 3.23644 18.3547i 0.214810 1.21825i −0.666426 0.745571i \(-0.732177\pi\)
0.881236 0.472676i \(-0.156712\pi\)
\(228\) 0 0
\(229\) −22.8729 8.32504i −1.51148 0.550134i −0.552478 0.833528i \(-0.686318\pi\)
−0.959004 + 0.283394i \(0.908540\pi\)
\(230\) 0 0
\(231\) −6.21466 + 3.68691i −0.408895 + 0.242581i
\(232\) 0 0
\(233\) 7.89990 + 13.6830i 0.517540 + 0.896405i 0.999792 + 0.0203732i \(0.00648543\pi\)
−0.482253 + 0.876032i \(0.660181\pi\)
\(234\) 0 0
\(235\) 5.54270 9.60025i 0.361566 0.626251i
\(236\) 0 0
\(237\) −19.9869 + 3.76896i −1.29829 + 0.244820i
\(238\) 0 0
\(239\) 20.3071 17.0397i 1.31356 1.10221i 0.325930 0.945394i \(-0.394323\pi\)
0.987628 0.156812i \(-0.0501217\pi\)
\(240\) 0 0
\(241\) 2.19158 0.797671i 0.141172 0.0513825i −0.270468 0.962729i \(-0.587178\pi\)
0.411640 + 0.911347i \(0.364956\pi\)
\(242\) 0 0
\(243\) 10.7096 + 11.3271i 0.687022 + 0.726637i
\(244\) 0 0
\(245\) 2.10866 0.767490i 0.134718 0.0490332i
\(246\) 0 0
\(247\) −12.8967 + 10.8216i −0.820598 + 0.688564i
\(248\) 0 0
\(249\) 22.0668 4.16118i 1.39843 0.263704i
\(250\) 0 0
\(251\) −12.8194 + 22.2038i −0.809152 + 1.40149i 0.104300 + 0.994546i \(0.466740\pi\)
−0.913452 + 0.406946i \(0.866594\pi\)
\(252\) 0 0
\(253\) 0.286097 + 0.495534i 0.0179867 + 0.0311539i
\(254\) 0 0
\(255\) 2.20335 1.30716i 0.137979 0.0818575i
\(256\) 0 0
\(257\) −26.7053 9.71993i −1.66583 0.606312i −0.674567 0.738214i \(-0.735670\pi\)
−0.991263 + 0.131901i \(0.957892\pi\)
\(258\) 0 0
\(259\) −0.498846 + 2.82909i −0.0309968 + 0.175791i
\(260\) 0 0
\(261\) 17.9079 0.424502i 1.10847 0.0262760i
\(262\) 0 0
\(263\) 0.0336811 + 0.0282618i 0.00207687 + 0.00174270i 0.643825 0.765172i \(-0.277346\pi\)
−0.641749 + 0.766915i \(0.721791\pi\)
\(264\) 0 0
\(265\) −0.989424 5.61130i −0.0607798 0.344699i
\(266\) 0 0
\(267\) −13.0226 + 15.8985i −0.796968 + 0.972975i
\(268\) 0 0
\(269\) 1.31036 0.0798941 0.0399471 0.999202i \(-0.487281\pi\)
0.0399471 + 0.999202i \(0.487281\pi\)
\(270\) 0 0
\(271\) −16.0504 −0.974992 −0.487496 0.873125i \(-0.662090\pi\)
−0.487496 + 0.873125i \(0.662090\pi\)
\(272\) 0 0
\(273\) 2.89602 + 7.67262i 0.175275 + 0.464368i
\(274\) 0 0
\(275\) 0.0257172 + 0.145849i 0.00155080 + 0.00879505i
\(276\) 0 0
\(277\) 13.0158 + 10.9216i 0.782046 + 0.656214i 0.943763 0.330623i \(-0.107259\pi\)
−0.161717 + 0.986837i \(0.551703\pi\)
\(278\) 0 0
\(279\) −22.9244 + 8.96456i −1.37245 + 0.536694i
\(280\) 0 0
\(281\) 4.41685 25.0492i 0.263487 1.49431i −0.509822 0.860280i \(-0.670289\pi\)
0.773309 0.634029i \(-0.218600\pi\)
\(282\) 0 0
\(283\) −13.0614 4.75398i −0.776422 0.282595i −0.0767423 0.997051i \(-0.524452\pi\)
−0.699680 + 0.714456i \(0.746674\pi\)
\(284\) 0 0
\(285\) −12.0493 6.76760i −0.713742 0.400878i
\(286\) 0 0
\(287\) −0.397490 0.688473i −0.0234631 0.0406393i
\(288\) 0 0
\(289\) 8.28276 14.3462i 0.487221 0.843892i
\(290\) 0 0
\(291\) 3.90332 11.1330i 0.228817 0.652630i
\(292\) 0 0
\(293\) 11.0500 9.27202i 0.645546 0.541677i −0.260170 0.965563i \(-0.583779\pi\)
0.905716 + 0.423885i \(0.139334\pi\)
\(294\) 0 0
\(295\) 22.1344 8.05627i 1.28872 0.469054i
\(296\) 0 0
\(297\) 13.3353 17.0911i 0.773795 0.991728i
\(298\) 0 0
\(299\) 0.610232 0.222106i 0.0352906 0.0128447i
\(300\) 0 0
\(301\) −4.13680 + 3.47118i −0.238441 + 0.200076i
\(302\) 0 0
\(303\) 10.4190 + 12.1222i 0.598554 + 0.696400i
\(304\) 0 0
\(305\) 2.70803 4.69044i 0.155061 0.268574i
\(306\) 0 0
\(307\) 3.52983 + 6.11384i 0.201458 + 0.348935i 0.948998 0.315281i \(-0.102099\pi\)
−0.747540 + 0.664216i \(0.768765\pi\)
\(308\) 0 0
\(309\) 0.304359 + 25.6829i 0.0173144 + 1.46105i
\(310\) 0 0
\(311\) −16.9115 6.15529i −0.958965 0.349035i −0.185337 0.982675i \(-0.559338\pi\)
−0.773628 + 0.633640i \(0.781560\pi\)
\(312\) 0 0
\(313\) −5.08431 + 28.8346i −0.287382 + 1.62983i 0.409268 + 0.912414i \(0.365784\pi\)
−0.696650 + 0.717411i \(0.745327\pi\)
\(314\) 0 0
\(315\) −5.05300 + 4.44823i −0.284704 + 0.250629i
\(316\) 0 0
\(317\) −18.3830 15.4252i −1.03249 0.866365i −0.0413480 0.999145i \(-0.513165\pi\)
−0.991146 + 0.132780i \(0.957610\pi\)
\(318\) 0 0
\(319\) −4.32569 24.5322i −0.242192 1.37354i
\(320\) 0 0
\(321\) 10.8411 + 1.77938i 0.605089 + 0.0993153i
\(322\) 0 0
\(323\) 2.34371 0.130407
\(324\) 0 0
\(325\) 0.168081 0.00932347
\(326\) 0 0
\(327\) −3.74489 0.614662i −0.207093 0.0339909i
\(328\) 0 0
\(329\) −0.857829 4.86499i −0.0472936 0.268216i
\(330\) 0 0
\(331\) −20.4863 17.1901i −1.12603 0.944852i −0.127137 0.991885i \(-0.540579\pi\)
−0.998893 + 0.0470331i \(0.985023\pi\)
\(332\) 0 0
\(333\) −1.69725 8.44943i −0.0930086 0.463026i
\(334\) 0 0
\(335\) 1.33318 7.56083i 0.0728393 0.413092i
\(336\) 0 0
\(337\) −4.09662 1.49105i −0.223157 0.0812225i 0.228022 0.973656i \(-0.426774\pi\)
−0.451179 + 0.892434i \(0.648996\pi\)
\(338\) 0 0
\(339\) −0.191430 16.1535i −0.0103970 0.877338i
\(340\) 0 0
\(341\) 17.1153 + 29.6446i 0.926846 + 1.60534i
\(342\) 0 0
\(343\) 0.500000 0.866025i 0.0269975 0.0467610i
\(344\) 0 0
\(345\) 0.347467 + 0.404268i 0.0187070 + 0.0217650i
\(346\) 0 0
\(347\) 21.1934 17.7833i 1.13772 0.954660i 0.138358 0.990382i \(-0.455818\pi\)
0.999362 + 0.0357223i \(0.0113732\pi\)
\(348\) 0 0
\(349\) 29.7165 10.8159i 1.59069 0.578963i 0.613195 0.789932i \(-0.289884\pi\)
0.977493 + 0.210969i \(0.0676618\pi\)
\(350\) 0 0
\(351\) −16.4743 18.2729i −0.879335 0.975337i
\(352\) 0 0
\(353\) 5.18178 1.88601i 0.275798 0.100382i −0.200418 0.979710i \(-0.564230\pi\)
0.476217 + 0.879328i \(0.342008\pi\)
\(354\) 0 0
\(355\) 4.13414 3.46895i 0.219417 0.184113i
\(356\) 0 0
\(357\) 0.377737 1.07738i 0.0199919 0.0570209i
\(358\) 0 0
\(359\) 5.32457 9.22243i 0.281020 0.486741i −0.690616 0.723222i \(-0.742661\pi\)
0.971636 + 0.236480i \(0.0759939\pi\)
\(360\) 0 0
\(361\) 3.17864 + 5.50557i 0.167297 + 0.289767i
\(362\) 0 0
\(363\) −9.67275 5.43276i −0.507688 0.285146i
\(364\) 0 0
\(365\) −13.0477 4.74898i −0.682948 0.248573i
\(366\) 0 0
\(367\) 3.36599 19.0895i 0.175703 0.996462i −0.761626 0.648017i \(-0.775599\pi\)
0.937329 0.348445i \(-0.113290\pi\)
\(368\) 0 0
\(369\) 1.86279 + 1.48928i 0.0969727 + 0.0775290i
\(370\) 0 0
\(371\) −1.94511 1.63214i −0.100985 0.0847367i
\(372\) 0 0
\(373\) 2.48984 + 14.1206i 0.128919 + 0.731137i 0.978903 + 0.204328i \(0.0655008\pi\)
−0.849983 + 0.526809i \(0.823388\pi\)
\(374\) 0 0
\(375\) −6.81387 18.0524i −0.351867 0.932223i
\(376\) 0 0
\(377\) −28.2717 −1.45607
\(378\) 0 0
\(379\) 16.5576 0.850507 0.425253 0.905074i \(-0.360185\pi\)
0.425253 + 0.905074i \(0.360185\pi\)
\(380\) 0 0
\(381\) 13.5692 16.5659i 0.695173 0.848699i
\(382\) 0 0
\(383\) −2.44240 13.8516i −0.124801 0.707782i −0.981426 0.191842i \(-0.938554\pi\)
0.856625 0.515940i \(-0.172557\pi\)
\(384\) 0 0
\(385\) 7.17156 + 6.01766i 0.365497 + 0.306688i
\(386\) 0 0
\(387\) 7.76554 14.2182i 0.394745 0.722750i
\(388\) 0 0
\(389\) 4.29093 24.3351i 0.217559 1.23384i −0.658852 0.752272i \(-0.728958\pi\)
0.876411 0.481564i \(-0.159931\pi\)
\(390\) 0 0
\(391\) −0.0849518 0.0309199i −0.00429620 0.00156369i
\(392\) 0 0
\(393\) 7.71201 4.57522i 0.389019 0.230790i
\(394\) 0 0
\(395\) 13.1754 + 22.8204i 0.662925 + 1.14822i
\(396\) 0 0
\(397\) 7.37799 12.7791i 0.370291 0.641362i −0.619319 0.785139i \(-0.712591\pi\)
0.989610 + 0.143777i \(0.0459247\pi\)
\(398\) 0 0
\(399\) −6.05192 + 1.14122i −0.302975 + 0.0571325i
\(400\) 0 0
\(401\) 7.69453 6.45648i 0.384246 0.322421i −0.430120 0.902772i \(-0.641529\pi\)
0.814367 + 0.580351i \(0.197084\pi\)
\(402\) 0 0
\(403\) 36.5062 13.2872i 1.81850 0.661881i
\(404\) 0 0
\(405\) 9.25789 17.9490i 0.460028 0.891893i
\(406\) 0 0
\(407\) −11.2621 + 4.09908i −0.558243 + 0.203184i
\(408\) 0 0
\(409\) 10.0627 8.44360i 0.497568 0.417509i −0.359161 0.933275i \(-0.616937\pi\)
0.856729 + 0.515766i \(0.172493\pi\)
\(410\) 0 0
\(411\) −5.76090 + 1.08634i −0.284164 + 0.0535853i
\(412\) 0 0
\(413\) 5.24845 9.09059i 0.258259 0.447319i
\(414\) 0 0
\(415\) −14.5465 25.1952i −0.714059 1.23679i
\(416\) 0 0
\(417\) −31.1886 + 18.5030i −1.52731 + 0.906093i
\(418\) 0 0
\(419\) −7.15156 2.60295i −0.349376 0.127163i 0.161370 0.986894i \(-0.448409\pi\)
−0.510747 + 0.859731i \(0.670631\pi\)
\(420\) 0 0
\(421\) −0.0575341 + 0.326292i −0.00280404 + 0.0159025i −0.986178 0.165691i \(-0.947014\pi\)
0.983374 + 0.181594i \(0.0581256\pi\)
\(422\) 0 0
\(423\) 7.71213 + 12.6554i 0.374977 + 0.615326i
\(424\) 0 0
\(425\) −0.0179247 0.0150406i −0.000869475 0.000729576i
\(426\) 0 0
\(427\) −0.419114 2.37691i −0.0202823 0.115027i
\(428\) 0 0
\(429\) −21.6802 + 26.4682i −1.04673 + 1.27790i
\(430\) 0 0
\(431\) −14.1982 −0.683903 −0.341952 0.939718i \(-0.611088\pi\)
−0.341952 + 0.939718i \(0.611088\pi\)
\(432\) 0 0
\(433\) 1.20852 0.0580778 0.0290389 0.999578i \(-0.490755\pi\)
0.0290389 + 0.999578i \(0.490755\pi\)
\(434\) 0 0
\(435\) −8.19530 21.7123i −0.392934 1.04103i
\(436\) 0 0
\(437\) 0.0846827 + 0.480259i 0.00405092 + 0.0229739i
\(438\) 0 0
\(439\) −9.81113 8.23251i −0.468259 0.392916i 0.377900 0.925847i \(-0.376646\pi\)
−0.846159 + 0.532930i \(0.821091\pi\)
\(440\) 0 0
\(441\) −0.450784 + 2.96594i −0.0214659 + 0.141235i
\(442\) 0 0
\(443\) −6.39038 + 36.2416i −0.303616 + 1.72189i 0.326333 + 0.945255i \(0.394187\pi\)
−0.629949 + 0.776637i \(0.716924\pi\)
\(444\) 0 0
\(445\) 25.0198 + 9.10645i 1.18605 + 0.431687i
\(446\) 0 0
\(447\) −10.0647 5.65290i −0.476043 0.267373i
\(448\) 0 0
\(449\) −11.1010 19.2274i −0.523887 0.907399i −0.999613 0.0278059i \(-0.991148\pi\)
0.475726 0.879593i \(-0.342185\pi\)
\(450\) 0 0
\(451\) 1.65831 2.87227i 0.0780866 0.135250i
\(452\) 0 0
\(453\) −10.9064 + 31.1072i −0.512427 + 1.46154i
\(454\) 0 0
\(455\) 8.13917 6.82958i 0.381570 0.320175i
\(456\) 0 0
\(457\) 33.9936 12.3727i 1.59015 0.578769i 0.612774 0.790258i \(-0.290054\pi\)
0.977380 + 0.211489i \(0.0678313\pi\)
\(458\) 0 0
\(459\) 0.121735 + 3.42287i 0.00568212 + 0.159766i
\(460\) 0 0
\(461\) 18.0994 6.58763i 0.842971 0.306817i 0.115800 0.993273i \(-0.463057\pi\)
0.727171 + 0.686456i \(0.240835\pi\)
\(462\) 0 0
\(463\) 6.48139 5.43853i 0.301216 0.252750i −0.479634 0.877468i \(-0.659231\pi\)
0.780850 + 0.624718i \(0.214786\pi\)
\(464\) 0 0
\(465\) 20.7867 + 24.1847i 0.963960 + 1.12154i
\(466\) 0 0
\(467\) 1.73757 3.00957i 0.0804054 0.139266i −0.823019 0.568014i \(-0.807712\pi\)
0.903424 + 0.428748i \(0.141045\pi\)
\(468\) 0 0
\(469\) −1.71067 2.96297i −0.0789915 0.136817i
\(470\) 0 0
\(471\) −0.411994 34.7655i −0.0189837 1.60191i
\(472\) 0 0
\(473\) −21.1707 7.70549i −0.973428 0.354299i
\(474\) 0 0
\(475\) −0.0219182 + 0.124304i −0.00100568 + 0.00570347i
\(476\) 0 0
\(477\) 7.21783 + 2.43497i 0.330482 + 0.111490i
\(478\) 0 0
\(479\) 31.3025 + 26.2659i 1.43025 + 1.20012i 0.945567 + 0.325428i \(0.105508\pi\)
0.484680 + 0.874692i \(0.338936\pi\)
\(480\) 0 0
\(481\) 2.36195 + 13.3953i 0.107696 + 0.610773i
\(482\) 0 0
\(483\) 0.234419 + 0.0384759i 0.0106664 + 0.00175071i
\(484\) 0 0
\(485\) −15.2844 −0.694031
\(486\) 0 0
\(487\) 12.7621 0.578307 0.289153 0.957283i \(-0.406626\pi\)
0.289153 + 0.957283i \(0.406626\pi\)
\(488\) 0 0
\(489\) −33.3663 5.47652i −1.50888 0.247657i
\(490\) 0 0
\(491\) −1.65692 9.39688i −0.0747759 0.424075i −0.999098 0.0424622i \(-0.986480\pi\)
0.924322 0.381613i \(-0.124631\pi\)
\(492\) 0 0
\(493\) 3.01497 + 2.52986i 0.135788 + 0.113939i
\(494\) 0 0
\(495\) −26.6119 8.97766i −1.19612 0.403516i
\(496\) 0 0
\(497\) 0.417619 2.36843i 0.0187328 0.106239i
\(498\) 0 0
\(499\) 38.1274 + 13.8772i 1.70682 + 0.621230i 0.996572 0.0827269i \(-0.0263629\pi\)
0.710243 + 0.703957i \(0.248585\pi\)
\(500\) 0 0
\(501\) 0.339556 + 28.6529i 0.0151703 + 1.28012i
\(502\) 0 0
\(503\) −14.5715 25.2386i −0.649711 1.12533i −0.983192 0.182575i \(-0.941557\pi\)
0.333481 0.942757i \(-0.391777\pi\)
\(504\) 0 0
\(505\) 10.3545 17.9345i 0.460767 0.798073i
\(506\) 0 0
\(507\) 10.6335 + 12.3718i 0.472252 + 0.549452i
\(508\) 0 0
\(509\) −12.0178 + 10.0841i −0.532679 + 0.446971i −0.869025 0.494768i \(-0.835253\pi\)
0.336346 + 0.941738i \(0.390809\pi\)
\(510\) 0 0
\(511\) −5.81451 + 2.11631i −0.257219 + 0.0936199i
\(512\) 0 0
\(513\) 15.6620 9.80074i 0.691495 0.432713i
\(514\) 0 0
\(515\) 31.2695 11.3812i 1.37790 0.501514i
\(516\) 0 0
\(517\) 15.7879 13.2476i 0.694349 0.582628i
\(518\) 0 0
\(519\) −11.9195 + 33.9967i −0.523207 + 1.49229i
\(520\) 0 0
\(521\) −14.7459 + 25.5406i −0.646028 + 1.11895i 0.338035 + 0.941134i \(0.390238\pi\)
−0.984063 + 0.177820i \(0.943096\pi\)
\(522\) 0 0
\(523\) 5.86587 + 10.1600i 0.256497 + 0.444265i 0.965301 0.261140i \(-0.0840984\pi\)
−0.708804 + 0.705405i \(0.750765\pi\)
\(524\) 0 0
\(525\) 0.0536089 + 0.0301098i 0.00233968 + 0.00131410i
\(526\) 0 0
\(527\) −5.08212 1.84974i −0.221381 0.0805759i
\(528\) 0 0
\(529\) −3.99064 + 22.6321i −0.173506 + 0.984002i
\(530\) 0 0
\(531\) −4.73184 + 31.1332i −0.205344 + 1.35106i
\(532\) 0 0
\(533\) −2.88347 2.41952i −0.124897 0.104801i
\(534\) 0 0
\(535\) −2.47158 14.0170i −0.106856 0.606010i
\(536\) 0 0
\(537\) 14.1957 + 37.6095i 0.612589 + 1.62297i
\(538\) 0 0
\(539\) 4.17195 0.179698
\(540\) 0 0
\(541\) 5.55402 0.238786 0.119393 0.992847i \(-0.461905\pi\)
0.119393 + 0.992847i \(0.461905\pi\)
\(542\) 0 0
\(543\) −5.22303 + 6.37652i −0.224142 + 0.273643i
\(544\) 0 0
\(545\) 0.853774 + 4.84199i 0.0365716 + 0.207408i
\(546\) 0 0
\(547\) 16.6714 + 13.9890i 0.712818 + 0.598126i 0.925388 0.379020i \(-0.123739\pi\)
−0.212570 + 0.977146i \(0.568183\pi\)
\(548\) 0 0
\(549\) 3.76796 + 6.18311i 0.160812 + 0.263889i
\(550\) 0 0
\(551\) 3.68669 20.9083i 0.157058 0.890723i
\(552\) 0 0
\(553\) 11.0346 + 4.01628i 0.469240 + 0.170789i
\(554\) 0 0
\(555\) −9.60277 + 5.69694i −0.407615 + 0.241822i
\(556\) 0 0
\(557\) −20.0541 34.7348i −0.849721 1.47176i −0.881457 0.472264i \(-0.843437\pi\)
0.0317365 0.999496i \(-0.489896\pi\)
\(558\) 0 0
\(559\) −12.7845 + 22.1435i −0.540728 + 0.936569i
\(560\) 0 0
\(561\) 4.68053 0.882615i 0.197612 0.0372640i
\(562\) 0 0
\(563\) −4.62638 + 3.88200i −0.194979 + 0.163607i −0.735051 0.678012i \(-0.762842\pi\)
0.540072 + 0.841619i \(0.318397\pi\)
\(564\) 0 0
\(565\) −19.6672 + 7.15829i −0.827407 + 0.301152i
\(566\) 0 0
\(567\) −1.98104 8.77926i −0.0831960 0.368694i
\(568\) 0 0
\(569\) −42.4698 + 15.4577i −1.78043 + 0.648022i −0.780694 + 0.624914i \(0.785134\pi\)
−0.999733 + 0.0231086i \(0.992644\pi\)
\(570\) 0 0
\(571\) −20.7441 + 17.4063i −0.868112 + 0.728432i −0.963700 0.266989i \(-0.913971\pi\)
0.0955878 + 0.995421i \(0.469527\pi\)
\(572\) 0 0
\(573\) 18.6985 3.52600i 0.781140 0.147301i
\(574\) 0 0
\(575\) 0.00243438 0.00421647i 0.000101521 0.000175839i
\(576\) 0 0
\(577\) −17.4393 30.2058i −0.726010 1.25749i −0.958557 0.284900i \(-0.908039\pi\)
0.232548 0.972585i \(-0.425294\pi\)
\(578\) 0 0
\(579\) 6.18968 3.67209i 0.257234 0.152607i
\(580\) 0 0
\(581\) −12.1830 4.43423i −0.505434 0.183963i
\(582\) 0 0
\(583\) 1.83950 10.4323i 0.0761843 0.432063i
\(584\) 0 0
\(585\) −15.2788 + 27.9743i −0.631699 + 1.15660i
\(586\) 0 0
\(587\) −8.37653 7.02874i −0.345736 0.290107i 0.453339 0.891338i \(-0.350233\pi\)
−0.799075 + 0.601231i \(0.794677\pi\)
\(588\) 0 0
\(589\) 5.06601 + 28.7308i 0.208741 + 1.18383i
\(590\) 0 0
\(591\) 18.2904 22.3297i 0.752365 0.918523i
\(592\) 0 0
\(593\) 3.70184 0.152016 0.0760082 0.997107i \(-0.475782\pi\)
0.0760082 + 0.997107i \(0.475782\pi\)
\(594\) 0 0
\(595\) −1.47912 −0.0606381
\(596\) 0 0
\(597\) 1.19839 + 3.17498i 0.0490469 + 0.129943i
\(598\) 0 0
\(599\) −3.77569 21.4130i −0.154270 0.874911i −0.959450 0.281880i \(-0.909042\pi\)
0.805179 0.593031i \(-0.202069\pi\)
\(600\) 0 0
\(601\) 4.44306 + 3.72817i 0.181236 + 0.152075i 0.728892 0.684628i \(-0.240035\pi\)
−0.547656 + 0.836703i \(0.684480\pi\)
\(602\) 0 0
\(603\) 8.01685 + 6.40941i 0.326471 + 0.261011i
\(604\) 0 0
\(605\) −2.49585 + 14.1547i −0.101471 + 0.575470i
\(606\) 0 0
\(607\) −19.4016 7.06161i −0.787487 0.286622i −0.0831961 0.996533i \(-0.526513\pi\)
−0.704291 + 0.709911i \(0.748735\pi\)
\(608\) 0 0
\(609\) −9.01714 5.06454i −0.365393 0.205225i
\(610\) 0 0
\(611\) −11.6951 20.2566i −0.473135 0.819493i
\(612\) 0 0
\(613\) −6.26954 + 10.8592i −0.253224 + 0.438597i −0.964412 0.264405i \(-0.914824\pi\)
0.711187 + 0.703002i \(0.248158\pi\)
\(614\) 0 0
\(615\) 1.02231 2.91583i 0.0412236 0.117578i
\(616\) 0 0
\(617\) 4.11956 3.45672i 0.165847 0.139162i −0.556087 0.831124i \(-0.687698\pi\)
0.721934 + 0.691962i \(0.243253\pi\)
\(618\) 0 0
\(619\) −40.0011 + 14.5592i −1.60778 + 0.585184i −0.980999 0.194011i \(-0.937850\pi\)
−0.626781 + 0.779195i \(0.715628\pi\)
\(620\) 0 0
\(621\) −0.696997 + 0.148620i −0.0279695 + 0.00596393i
\(622\) 0 0
\(623\) 11.1497 4.05815i 0.446702 0.162586i
\(624\) 0 0
\(625\) −19.2861 + 16.1830i −0.771445 + 0.647319i
\(626\) 0 0
\(627\) −16.7474 19.4851i −0.668826 0.778160i
\(628\) 0 0
\(629\) 0.946780 1.63987i 0.0377506 0.0653859i
\(630\) 0 0
\(631\) −1.94269 3.36484i −0.0773372 0.133952i 0.824763 0.565478i \(-0.191308\pi\)
−0.902100 + 0.431527i \(0.857975\pi\)
\(632\) 0 0
\(633\) 0.486497 + 41.0523i 0.0193365 + 1.63168i
\(634\) 0 0
\(635\) −26.0700 9.48872i −1.03456 0.376548i
\(636\) 0 0
\(637\) 0.822196 4.66290i 0.0325766 0.184751i
\(638\) 0 0
\(639\) 1.42089 + 7.07361i 0.0562093 + 0.279828i
\(640\) 0 0
\(641\) 4.65667 + 3.90741i 0.183927 + 0.154333i 0.730103 0.683338i \(-0.239472\pi\)
−0.546175 + 0.837671i \(0.683917\pi\)
\(642\) 0 0
\(643\) 1.98712 + 11.2695i 0.0783643 + 0.444426i 0.998592 + 0.0530431i \(0.0168921\pi\)
−0.920228 + 0.391383i \(0.871997\pi\)
\(644\) 0 0
\(645\) −20.7119 3.39951i −0.815529 0.133856i
\(646\) 0 0
\(647\) −1.11757 −0.0439360 −0.0219680 0.999759i \(-0.506993\pi\)
−0.0219680 + 0.999759i \(0.506993\pi\)
\(648\) 0 0
\(649\) 43.7925 1.71901
\(650\) 0 0
\(651\) 14.0238 + 2.30177i 0.549634 + 0.0902133i
\(652\) 0 0
\(653\) 8.40011 + 47.6394i 0.328722 + 1.86427i 0.482116 + 0.876107i \(0.339868\pi\)
−0.153394 + 0.988165i \(0.549021\pi\)
\(654\) 0 0
\(655\) −8.89946 7.46753i −0.347731 0.291781i
\(656\) 0 0
\(657\) 13.9333 12.2657i 0.543591 0.478531i
\(658\) 0 0
\(659\) 2.76692 15.6920i 0.107784 0.611273i −0.882288 0.470710i \(-0.843998\pi\)
0.990072 0.140563i \(-0.0448912\pi\)
\(660\) 0 0
\(661\) −42.9510 15.6329i −1.67060 0.608048i −0.678625 0.734485i \(-0.737424\pi\)
−0.991975 + 0.126437i \(0.959646\pi\)
\(662\) 0 0
\(663\) −0.0640561 5.40528i −0.00248773 0.209924i
\(664\) 0 0
\(665\) 3.98944 + 6.90991i 0.154704 + 0.267955i
\(666\) 0 0
\(667\) −0.409469 + 0.709220i −0.0158547 + 0.0274611i
\(668\) 0 0
\(669\) 31.8377 + 37.0423i 1.23092 + 1.43214i
\(670\) 0 0
\(671\) 7.71355 6.47244i 0.297778 0.249866i
\(672\) 0 0
\(673\) −28.2025 + 10.2649i −1.08712 + 0.395681i −0.822555 0.568686i \(-0.807452\pi\)
−0.264570 + 0.964367i \(0.585230\pi\)
\(674\) 0 0
\(675\) −0.182679 0.0255539i −0.00703131 0.000983571i
\(676\) 0 0
\(677\) 31.8213 11.5820i 1.22299 0.445133i 0.351801 0.936075i \(-0.385569\pi\)
0.871192 + 0.490942i \(0.163347\pi\)
\(678\) 0 0
\(679\) −5.21774 + 4.37820i −0.200238 + 0.168020i
\(680\) 0 0
\(681\) 10.6808 30.4637i 0.409288 1.16737i
\(682\) 0 0
\(683\) −6.13893 + 10.6329i −0.234900 + 0.406858i −0.959244 0.282581i \(-0.908809\pi\)
0.724344 + 0.689439i \(0.242143\pi\)
\(684\) 0 0
\(685\) 3.79759 + 6.57762i 0.145098 + 0.251318i
\(686\) 0 0
\(687\) −36.7584 20.6456i −1.40242 0.787679i
\(688\) 0 0
\(689\) −11.2975 4.11195i −0.430400 0.156653i
\(690\) 0 0
\(691\) 3.28717 18.6425i 0.125050 0.709194i −0.856228 0.516598i \(-0.827198\pi\)
0.981278 0.192596i \(-0.0616907\pi\)
\(692\) 0 0
\(693\) −11.6563 + 4.55818i −0.442786 + 0.173151i
\(694\) 0 0
\(695\) 35.9908 + 30.1999i 1.36521 + 1.14555i
\(696\) 0 0
\(697\) 0.0909933 + 0.516048i 0.00344662 + 0.0195467i
\(698\) 0 0
\(699\) 9.66382 + 25.6030i 0.365519 + 0.968393i
\(700\) 0 0
\(701\) 37.9263 1.43246 0.716228 0.697867i \(-0.245867\pi\)
0.716228 + 0.697867i \(0.245867\pi\)
\(702\) 0 0
\(703\) −10.2145 −0.385246
\(704\) 0 0
\(705\) 12.1667 14.8536i 0.458224 0.559421i
\(706\) 0 0
\(707\) −1.60253 9.08840i −0.0602694 0.341805i
\(708\) 0 0
\(709\) 17.1424 + 14.3842i 0.643797 + 0.540210i 0.905182 0.425025i \(-0.139735\pi\)
−0.261385 + 0.965235i \(0.584179\pi\)
\(710\) 0 0
\(711\) −35.2185 + 0.834843i −1.32080 + 0.0313091i
\(712\) 0 0
\(713\) 0.195412 1.10823i 0.00731822 0.0415037i
\(714\) 0 0
\(715\) 41.6534 + 15.1606i 1.55775 + 0.566974i
\(716\) 0 0
\(717\) 39.4888 23.4271i 1.47473 0.874901i
\(718\) 0 0
\(719\) −0.290549 0.503245i −0.0108356 0.0187679i 0.860557 0.509355i \(-0.170116\pi\)
−0.871392 + 0.490587i \(0.836782\pi\)
\(720\) 0 0
\(721\) 7.41454 12.8424i 0.276132 0.478274i
\(722\) 0 0
\(723\) 3.96959 0.748551i 0.147630 0.0278389i
\(724\) 0 0
\(725\) −0.162373 + 0.136247i −0.00603040 + 0.00506011i
\(726\) 0 0
\(727\) −21.4772 + 7.81705i −0.796544 + 0.289918i −0.708054 0.706159i \(-0.750427\pi\)
−0.0884907 + 0.996077i \(0.528204\pi\)
\(728\) 0 0
\(729\) 15.1270 + 22.3646i 0.560260 + 0.828317i
\(730\) 0 0
\(731\) 3.34487 1.21743i 0.123714 0.0450283i
\(732\) 0 0
\(733\) 20.7441 17.4064i 0.766202 0.642920i −0.173531 0.984828i \(-0.555518\pi\)
0.939733 + 0.341908i \(0.111073\pi\)
\(734\) 0 0
\(735\) 3.81939 0.720229i 0.140880 0.0265661i
\(736\) 0 0
\(737\) 7.13683 12.3614i 0.262889 0.455336i
\(738\) 0 0
\(739\) 5.31515 + 9.20610i 0.195521 + 0.338652i 0.947071 0.321024i \(-0.104027\pi\)
−0.751550 + 0.659676i \(0.770694\pi\)
\(740\) 0 0
\(741\) −25.0787 + 14.8782i −0.921287 + 0.546563i
\(742\) 0 0
\(743\) −46.0289 16.7531i −1.68863 0.614613i −0.694182 0.719799i \(-0.744234\pi\)
−0.994453 + 0.105187i \(0.966456\pi\)
\(744\) 0 0
\(745\) −2.59699 + 14.7282i −0.0951462 + 0.539601i
\(746\) 0 0
\(747\) 38.8836 0.921721i 1.42267 0.0337240i
\(748\) 0 0
\(749\) −4.85890 4.07710i −0.177540 0.148974i
\(750\) 0 0
\(751\) 6.13951 + 34.8189i 0.224034 + 1.27056i 0.864524 + 0.502592i \(0.167620\pi\)
−0.640490 + 0.767967i \(0.721269\pi\)
\(752\) 0 0
\(753\) −28.1396 + 34.3541i −1.02546 + 1.25193i
\(754\) 0 0
\(755\) 42.7067 1.55426
\(756\) 0 0
\(757\) 11.5606 0.420176 0.210088 0.977682i \(-0.432625\pi\)
0.210088 + 0.977682i \(0.432625\pi\)
\(758\) 0 0
\(759\) 0.349977 + 0.927217i 0.0127034 + 0.0336558i
\(760\) 0 0
\(761\) 9.25585 + 52.4925i 0.335524 + 1.90285i 0.421995 + 0.906598i \(0.361330\pi\)
−0.0864711 + 0.996254i \(0.527559\pi\)
\(762\) 0 0
\(763\) 1.67844 + 1.40838i 0.0607635 + 0.0509867i
\(764\) 0 0
\(765\) 4.13263 1.61606i 0.149415 0.0584287i
\(766\) 0 0
\(767\) 8.63051 48.9461i 0.311630 1.76734i
\(768\) 0 0
\(769\) 3.67996 + 1.33939i 0.132703 + 0.0482998i 0.407518 0.913197i \(-0.366394\pi\)
−0.274815 + 0.961497i \(0.588617\pi\)
\(770\) 0 0
\(771\) −42.9174 24.1049i −1.54563 0.868115i
\(772\) 0 0
\(773\) −1.93752 3.35588i −0.0696877 0.120703i 0.829076 0.559136i \(-0.188867\pi\)
−0.898764 + 0.438433i \(0.855534\pi\)
\(774\) 0 0
\(775\) 0.145633 0.252244i 0.00523130 0.00906088i
\(776\) 0 0
\(777\) −1.64627 + 4.69549i −0.0590597 + 0.168450i
\(778\) 0 0
\(779\) 2.16536 1.81695i 0.0775821 0.0650991i
\(780\) 0 0
\(781\) 9.42831 3.43163i 0.337372 0.122793i
\(782\) 0 0
\(783\) 30.7270 + 4.29823i 1.09809 + 0.153606i
\(784\) 0 0
\(785\) −42.3278 + 15.4061i −1.51074 + 0.549866i
\(786\) 0 0
\(787\) −39.1587 + 32.8581i −1.39586 + 1.17126i −0.432954 + 0.901416i \(0.642529\pi\)
−0.962903 + 0.269848i \(0.913027\pi\)
\(788\) 0 0
\(789\) 0.0496388 + 0.0577533i 0.00176719 + 0.00205607i
\(790\) 0 0
\(791\) −4.66344 + 8.07732i −0.165813 + 0.287196i
\(792\) 0 0
\(793\) −5.71395 9.89686i −0.202908 0.351448i
\(794\) 0 0
\(795\) −0.116946 9.86830i −0.00414764 0.349992i
\(796\) 0 0
\(797\) 20.0884 + 7.31159i 0.711568 + 0.258990i 0.672342 0.740241i \(-0.265288\pi\)
0.0392264 + 0.999230i \(0.487511\pi\)
\(798\) 0 0
\(799\) −0.565436 + 3.20675i −0.0200037 + 0.113447i
\(800\) 0 0
\(801\) −26.7180 + 23.5202i −0.944034 + 0.831047i
\(802\) 0 0
\(803\) −19.7751 16.5933i −0.697850 0.585565i
\(804\) 0 0
\(805\) −0.0534436 0.303094i −0.00188364 0.0106826i
\(806\) 0 0
\(807\) 2.23964 + 0.367600i 0.0788392 + 0.0129401i
\(808\) 0 0
\(809\) −8.80808 −0.309676 −0.154838 0.987940i \(-0.549485\pi\)
−0.154838 + 0.987940i \(0.549485\pi\)
\(810\) 0 0
\(811\) −18.4884 −0.649215 −0.324608 0.945849i \(-0.605232\pi\)
−0.324608 + 0.945849i \(0.605232\pi\)
\(812\) 0 0
\(813\) −27.4330 4.50267i −0.962118 0.157916i
\(814\) 0 0
\(815\) 7.60697 + 43.1413i 0.266460 + 1.51117i
\(816\) 0 0
\(817\) −14.7090 12.3423i −0.514604 0.431804i
\(818\) 0 0
\(819\) 2.79740 + 13.9263i 0.0977490 + 0.486625i
\(820\) 0 0
\(821\) −5.08814 + 28.8563i −0.177577 + 1.00709i 0.757550 + 0.652778i \(0.226396\pi\)
−0.935127 + 0.354313i \(0.884715\pi\)
\(822\) 0 0
\(823\) −13.5921 4.94710i −0.473789 0.172445i 0.0940788 0.995565i \(-0.470009\pi\)
−0.567868 + 0.823120i \(0.692232\pi\)
\(824\) 0 0
\(825\) 0.00303966 + 0.256498i 0.000105828 + 0.00893010i
\(826\) 0 0
\(827\) 9.54113 + 16.5257i 0.331778 + 0.574656i 0.982860 0.184351i \(-0.0590183\pi\)
−0.651083 + 0.759007i \(0.725685\pi\)
\(828\) 0 0
\(829\) 2.91554 5.04986i 0.101261 0.175389i −0.810944 0.585124i \(-0.801046\pi\)
0.912204 + 0.409735i \(0.134379\pi\)
\(830\) 0 0
\(831\) 19.1826 + 22.3183i 0.665435 + 0.774215i
\(832\) 0 0
\(833\) −0.504937 + 0.423692i −0.0174950 + 0.0146801i
\(834\) 0 0
\(835\) 34.8856 12.6973i 1.20727 0.439409i
\(836\) 0 0
\(837\) −41.6968 + 8.89099i −1.44125 + 0.307317i
\(838\) 0 0
\(839\) −18.0046 + 6.55312i −0.621586 + 0.226239i −0.633565 0.773689i \(-0.718409\pi\)
0.0119788 + 0.999928i \(0.496187\pi\)
\(840\) 0 0
\(841\) 5.09631 4.27631i 0.175735 0.147459i
\(842\) 0 0
\(843\) 14.5763 41.5745i 0.502035 1.43190i
\(844\) 0 0
\(845\) 10.5677 18.3038i 0.363540 0.629670i
\(846\) 0 0
\(847\) 3.20256 + 5.54700i 0.110041 + 0.190597i
\(848\) 0 0
\(849\) −20.9907 11.7896i −0.720400 0.404617i
\(850\) 0 0
\(851\) 0.370242 + 0.134757i 0.0126917 + 0.00461941i
\(852\) 0 0
\(853\) 4.07927 23.1347i 0.139672 0.792117i −0.831820 0.555045i \(-0.812701\pi\)
0.971492 0.237072i \(-0.0761878\pi\)
\(854\) 0 0
\(855\) −18.6960 14.9473i −0.639389 0.511187i
\(856\) 0 0
\(857\) −23.4793 19.7014i −0.802036 0.672988i 0.146657 0.989187i \(-0.453149\pi\)
−0.948693 + 0.316199i \(0.897593\pi\)
\(858\) 0 0
\(859\) −5.69176 32.2796i −0.194200 1.10137i −0.913553 0.406720i \(-0.866672\pi\)
0.719352 0.694645i \(-0.244439\pi\)
\(860\) 0 0
\(861\) −0.486243 1.28823i −0.0165711 0.0439029i
\(862\) 0 0
\(863\) −26.5644 −0.904263 −0.452132 0.891951i \(-0.649336\pi\)
−0.452132 + 0.891951i \(0.649336\pi\)
\(864\) 0 0
\(865\) 46.6737 1.58696
\(866\) 0 0
\(867\) 18.1813 22.1966i 0.617470 0.753836i
\(868\) 0 0
\(869\) 8.50709 + 48.2461i 0.288583 + 1.63664i
\(870\) 0 0
\(871\) −12.4095 10.4128i −0.420481 0.352826i
\(872\) 0 0
\(873\) 9.79468 17.9334i 0.331500 0.606952i
\(874\) 0 0
\(875\) −1.93449 + 10.9711i −0.0653978 + 0.370889i
\(876\) 0 0
\(877\) −0.284980 0.103724i −0.00962307 0.00350251i 0.337204 0.941432i \(-0.390519\pi\)
−0.346827 + 0.937929i \(0.612741\pi\)
\(878\) 0 0
\(879\) 21.4875 12.7477i 0.724756 0.429969i
\(880\) 0 0
\(881\) −8.69674 15.0632i −0.293001 0.507492i 0.681517 0.731802i \(-0.261320\pi\)
−0.974518 + 0.224310i \(0.927987\pi\)
\(882\) 0 0
\(883\) −23.7684 + 41.1681i −0.799871 + 1.38542i 0.119829 + 0.992795i \(0.461765\pi\)
−0.919700 + 0.392622i \(0.871568\pi\)
\(884\) 0 0
\(885\) 40.0918 7.56018i 1.34767 0.254133i
\(886\) 0 0
\(887\) 21.2988 17.8718i 0.715142 0.600076i −0.210895 0.977509i \(-0.567638\pi\)
0.926037 + 0.377433i \(0.123193\pi\)
\(888\) 0 0
\(889\) −11.6177 + 4.22850i −0.389646 + 0.141819i
\(890\) 0 0
\(891\) 27.5872 25.4708i 0.924205 0.853305i
\(892\) 0 0
\(893\) 16.5058 6.00762i 0.552345 0.201037i
\(894\) 0 0
\(895\) 39.8965 33.4771i 1.33359 1.11902i
\(896\) 0 0
\(897\) 1.10530 0.208429i 0.0369051 0.00695925i
\(898\) 0 0
\(899\) −24.4959 + 42.4281i −0.816983 + 1.41506i
\(900\) 0 0
\(901\) 0.836843 + 1.44945i 0.0278793 + 0.0482883i
\(902\) 0 0
\(903\) −8.04432 + 4.77237i −0.267698 + 0.158815i
\(904\) 0 0
\(905\) 10.0348 + 3.65237i 0.333568 + 0.121409i
\(906\) 0 0
\(907\) 0.141543 0.802731i 0.00469986 0.0266542i −0.982368 0.186959i \(-0.940137\pi\)
0.987068 + 0.160305i \(0.0512478\pi\)
\(908\) 0 0
\(909\) 14.4072 + 23.6418i 0.477857 + 0.784150i
\(910\) 0 0
\(911\) 7.71690 + 6.47525i 0.255672 + 0.214535i 0.761610 0.648035i \(-0.224409\pi\)
−0.505938 + 0.862570i \(0.668854\pi\)
\(912\) 0 0
\(913\) −9.39238 53.2668i −0.310842 1.76287i
\(914\) 0 0
\(915\) 5.94434 7.25712i 0.196514 0.239913i
\(916\) 0 0
\(917\) −5.17712 −0.170964
\(918\) 0 0
\(919\) −37.4451 −1.23520 −0.617601 0.786492i \(-0.711895\pi\)
−0.617601 + 0.786492i \(0.711895\pi\)
\(920\) 0 0
\(921\) 4.31798 + 11.4399i 0.142282 + 0.376958i
\(922\) 0 0
\(923\) −1.97736 11.2141i −0.0650854 0.369118i
\(924\) 0 0
\(925\) 0.0781204 + 0.0655508i 0.00256858 + 0.00215530i
\(926\) 0 0
\(927\) −6.68472 + 43.9821i −0.219555 + 1.44456i
\(928\) 0 0
\(929\) −1.21351 + 6.88214i −0.0398139 + 0.225796i −0.998222 0.0596054i \(-0.981016\pi\)
0.958408 + 0.285401i \(0.0921269\pi\)
\(930\) 0 0
\(931\) 3.34123 + 1.21611i 0.109504 + 0.0398563i
\(932\) 0 0
\(933\) −27.1781 15.2648i −0.889771 0.499746i
\(934\) 0 0
\(935\) −3.08541 5.34409i −0.100904 0.174770i
\(936\) 0 0
\(937\) 2.76071 4.78169i 0.0901885 0.156211i −0.817402 0.576068i \(-0.804586\pi\)
0.907590 + 0.419857i \(0.137920\pi\)
\(938\) 0 0
\(939\) −16.7791 + 47.8572i −0.547564 + 1.56176i
\(940\) 0 0
\(941\) 26.4546 22.1980i 0.862394 0.723635i −0.100088 0.994979i \(-0.531912\pi\)
0.962483 + 0.271344i \(0.0874680\pi\)
\(942\) 0 0
\(943\) −0.102458 + 0.0372917i −0.00333649 + 0.00121438i
\(944\) 0 0
\(945\) −9.88437 + 6.18529i −0.321538 + 0.201207i
\(946\) 0 0
\(947\) −54.3306 + 19.7747i −1.76551 + 0.642592i −1.00000 0.000833325i \(-0.999735\pi\)
−0.765509 + 0.643426i \(0.777513\pi\)
\(948\) 0 0
\(949\) −22.4433 + 18.8321i −0.728539 + 0.611317i
\(950\) 0 0
\(951\) −27.0926 31.5215i −0.878539 1.02215i
\(952\) 0 0
\(953\) 18.0011 31.1788i 0.583112 1.00998i −0.411996 0.911186i \(-0.635168\pi\)
0.995108 0.0987942i \(-0.0314986\pi\)
\(954\) 0 0
\(955\) −12.3261 21.3494i −0.398862 0.690849i
\(956\) 0 0
\(957\) −0.511279 43.1435i −0.0165273 1.39463i
\(958\) 0 0
\(959\) 3.18055 + 1.15763i 0.102705 + 0.0373817i
\(960\) 0 0
\(961\) 6.30712 35.7694i 0.203455 1.15385i
\(962\) 0 0
\(963\) 18.0302 + 6.08257i 0.581014 + 0.196008i
\(964\) 0 0
\(965\) −7.14273 5.99346i −0.229933 0.192936i
\(966\) 0 0
\(967\) −1.27673 7.24071i −0.0410569 0.232846i 0.957373 0.288853i \(-0.0932740\pi\)
−0.998430 + 0.0560077i \(0.982163\pi\)
\(968\) 0 0
\(969\) 4.00582 + 0.657488i 0.128685 + 0.0211216i
\(970\) 0 0
\(971\) 24.6929 0.792432 0.396216 0.918157i \(-0.370323\pi\)
0.396216 + 0.918157i \(0.370323\pi\)
\(972\) 0 0
\(973\) 20.9371 0.671213
\(974\) 0 0
\(975\) 0.287281 + 0.0471525i 0.00920037 + 0.00151009i
\(976\) 0 0
\(977\) 7.94661 + 45.0674i 0.254234 + 1.44184i 0.798030 + 0.602618i \(0.205876\pi\)
−0.543796 + 0.839218i \(0.683013\pi\)
\(978\) 0 0
\(979\) 37.9200 + 31.8187i 1.21193 + 1.01693i
\(980\) 0 0
\(981\) −6.22827 2.10114i −0.198853 0.0670841i
\(982\) 0 0
\(983\) −6.28709 + 35.6558i −0.200527 + 1.13724i 0.703798 + 0.710400i \(0.251486\pi\)
−0.904325 + 0.426845i \(0.859625\pi\)
\(984\) 0 0
\(985\) −35.1406 12.7901i −1.11967 0.407528i
\(986\) 0 0
\(987\) −0.101392 8.55580i −0.00322734 0.272334i
\(988\) 0 0
\(989\) 0.370326 + 0.641423i 0.0117757 + 0.0203961i
\(990\) 0 0
\(991\) −20.3445 + 35.2378i −0.646266 + 1.11937i 0.337742 + 0.941239i \(0.390337\pi\)
−0.984008 + 0.178126i \(0.942996\pi\)
\(992\) 0 0
\(993\) −30.1925 35.1281i −0.958129 1.11476i
\(994\) 0 0
\(995\) 3.36804 2.82612i 0.106774 0.0895941i
\(996\) 0 0
\(997\) −51.0089 + 18.5657i −1.61547 + 0.587982i −0.982511 0.186205i \(-0.940381\pi\)
−0.632957 + 0.774187i \(0.718159\pi\)
\(998\) 0 0
\(999\) −0.530554 14.9178i −0.0167860 0.471977i
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 756.2.bo.b.169.9 yes 54
27.4 even 9 inner 756.2.bo.b.85.9 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.bo.b.85.9 54 27.4 even 9 inner
756.2.bo.b.169.9 yes 54 1.1 even 1 trivial