Properties

Label 504.2.cx.a.185.2
Level $504$
Weight $2$
Character 504.185
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
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] = [504,2,Mod(185,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.185");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cx (of order \(6\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 185.2
Character \(\chi\) \(=\) 504.185
Dual form 504.2.cx.a.425.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.69924 + 0.335557i) q^{3} -3.64567 q^{5} +(1.05524 + 2.42620i) q^{7} +(2.77480 - 1.14038i) q^{9} +O(q^{10})\) \(q+(-1.69924 + 0.335557i) q^{3} -3.64567 q^{5} +(1.05524 + 2.42620i) q^{7} +(2.77480 - 1.14038i) q^{9} -1.39835i q^{11} +(-2.97474 - 1.71747i) q^{13} +(6.19486 - 1.22333i) q^{15} +(-2.41670 + 4.18585i) q^{17} +(7.23869 - 4.17926i) q^{19} +(-2.60724 - 3.76860i) q^{21} -8.92741i q^{23} +8.29094 q^{25} +(-4.33238 + 2.86888i) q^{27} +(5.02182 - 2.89935i) q^{29} +(5.29782 - 3.05870i) q^{31} +(0.469228 + 2.37613i) q^{33} +(-3.84708 - 8.84514i) q^{35} +(-2.89032 - 5.00617i) q^{37} +(5.63110 + 1.92019i) q^{39} +(0.802159 - 1.38938i) q^{41} +(2.22777 + 3.85862i) q^{43} +(-10.1160 + 4.15746i) q^{45} +(1.51221 - 2.61923i) q^{47} +(-4.77292 + 5.12047i) q^{49} +(2.70196 - 7.92369i) q^{51} +(-4.40240 - 2.54173i) q^{53} +5.09795i q^{55} +(-10.8979 + 9.53053i) q^{57} +(3.21233 + 5.56392i) q^{59} +(-7.78670 - 4.49565i) q^{61} +(5.69489 + 5.52886i) q^{63} +(10.8449 + 6.26133i) q^{65} +(2.53496 + 4.39068i) q^{67} +(2.99565 + 15.1698i) q^{69} +4.20682i q^{71} +(-0.745839 - 0.430610i) q^{73} +(-14.0883 + 2.78208i) q^{75} +(3.39269 - 1.47561i) q^{77} +(-1.31221 + 2.27282i) q^{79} +(6.39906 - 6.32866i) q^{81} +(-3.82295 - 6.62154i) q^{83} +(8.81051 - 15.2603i) q^{85} +(-7.56036 + 6.61179i) q^{87} +(-4.44212 - 7.69398i) q^{89} +(1.02785 - 9.02968i) q^{91} +(-7.97587 + 6.97516i) q^{93} +(-26.3899 + 15.2362i) q^{95} +(9.63985 - 5.56557i) q^{97} +(-1.59466 - 3.88016i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 2 q^{9} + 8 q^{15} - 10 q^{21} + 48 q^{25} + 18 q^{27} + 18 q^{29} + 18 q^{31} + 12 q^{33} - 4 q^{39} - 6 q^{41} - 6 q^{43} - 18 q^{45} + 18 q^{47} - 12 q^{49} + 6 q^{51} - 12 q^{53} + 4 q^{57} + 18 q^{61} - 32 q^{63} - 36 q^{65} - 12 q^{77} + 6 q^{79} + 6 q^{81} - 54 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 54 q^{95} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.69924 + 0.335557i −0.981054 + 0.193734i
\(4\) 0 0
\(5\) −3.64567 −1.63039 −0.815197 0.579183i \(-0.803372\pi\)
−0.815197 + 0.579183i \(0.803372\pi\)
\(6\) 0 0
\(7\) 1.05524 + 2.42620i 0.398845 + 0.917018i
\(8\) 0 0
\(9\) 2.77480 1.14038i 0.924934 0.380127i
\(10\) 0 0
\(11\) 1.39835i 0.421620i −0.977527 0.210810i \(-0.932390\pi\)
0.977527 0.210810i \(-0.0676101\pi\)
\(12\) 0 0
\(13\) −2.97474 1.71747i −0.825046 0.476340i 0.0271077 0.999633i \(-0.491370\pi\)
−0.852153 + 0.523292i \(0.824704\pi\)
\(14\) 0 0
\(15\) 6.19486 1.22333i 1.59951 0.315863i
\(16\) 0 0
\(17\) −2.41670 + 4.18585i −0.586137 + 1.01522i 0.408596 + 0.912715i \(0.366018\pi\)
−0.994733 + 0.102503i \(0.967315\pi\)
\(18\) 0 0
\(19\) 7.23869 4.17926i 1.66067 0.958787i 0.688272 0.725453i \(-0.258370\pi\)
0.972397 0.233334i \(-0.0749637\pi\)
\(20\) 0 0
\(21\) −2.60724 3.76860i −0.568946 0.822375i
\(22\) 0 0
\(23\) 8.92741i 1.86149i −0.365665 0.930747i \(-0.619158\pi\)
0.365665 0.930747i \(-0.380842\pi\)
\(24\) 0 0
\(25\) 8.29094 1.65819
\(26\) 0 0
\(27\) −4.33238 + 2.86888i −0.833767 + 0.552116i
\(28\) 0 0
\(29\) 5.02182 2.89935i 0.932529 0.538396i 0.0449185 0.998991i \(-0.485697\pi\)
0.887611 + 0.460595i \(0.152364\pi\)
\(30\) 0 0
\(31\) 5.29782 3.05870i 0.951516 0.549358i 0.0579644 0.998319i \(-0.481539\pi\)
0.893552 + 0.448961i \(0.148206\pi\)
\(32\) 0 0
\(33\) 0.469228 + 2.37613i 0.0816820 + 0.413632i
\(34\) 0 0
\(35\) −3.84708 8.84514i −0.650274 1.49510i
\(36\) 0 0
\(37\) −2.89032 5.00617i −0.475165 0.823010i 0.524431 0.851453i \(-0.324278\pi\)
−0.999595 + 0.0284435i \(0.990945\pi\)
\(38\) 0 0
\(39\) 5.63110 + 1.92019i 0.901698 + 0.307476i
\(40\) 0 0
\(41\) 0.802159 1.38938i 0.125276 0.216985i −0.796565 0.604553i \(-0.793352\pi\)
0.921841 + 0.387569i \(0.126685\pi\)
\(42\) 0 0
\(43\) 2.22777 + 3.85862i 0.339732 + 0.588434i 0.984382 0.176044i \(-0.0563301\pi\)
−0.644650 + 0.764478i \(0.722997\pi\)
\(44\) 0 0
\(45\) −10.1160 + 4.15746i −1.50801 + 0.619757i
\(46\) 0 0
\(47\) 1.51221 2.61923i 0.220579 0.382053i −0.734405 0.678711i \(-0.762539\pi\)
0.954984 + 0.296658i \(0.0958721\pi\)
\(48\) 0 0
\(49\) −4.77292 + 5.12047i −0.681846 + 0.731496i
\(50\) 0 0
\(51\) 2.70196 7.92369i 0.378350 1.10954i
\(52\) 0 0
\(53\) −4.40240 2.54173i −0.604716 0.349133i 0.166178 0.986096i \(-0.446857\pi\)
−0.770895 + 0.636963i \(0.780191\pi\)
\(54\) 0 0
\(55\) 5.09795i 0.687407i
\(56\) 0 0
\(57\) −10.8979 + 9.53053i −1.44346 + 1.26235i
\(58\) 0 0
\(59\) 3.21233 + 5.56392i 0.418210 + 0.724360i 0.995759 0.0919952i \(-0.0293244\pi\)
−0.577550 + 0.816355i \(0.695991\pi\)
\(60\) 0 0
\(61\) −7.78670 4.49565i −0.996985 0.575609i −0.0896300 0.995975i \(-0.528568\pi\)
−0.907355 + 0.420366i \(0.861902\pi\)
\(62\) 0 0
\(63\) 5.69489 + 5.52886i 0.717489 + 0.696570i
\(64\) 0 0
\(65\) 10.8449 + 6.26133i 1.34515 + 0.776623i
\(66\) 0 0
\(67\) 2.53496 + 4.39068i 0.309694 + 0.536407i 0.978295 0.207215i \(-0.0664399\pi\)
−0.668601 + 0.743621i \(0.733107\pi\)
\(68\) 0 0
\(69\) 2.99565 + 15.1698i 0.360634 + 1.82623i
\(70\) 0 0
\(71\) 4.20682i 0.499258i 0.968342 + 0.249629i \(0.0803087\pi\)
−0.968342 + 0.249629i \(0.919691\pi\)
\(72\) 0 0
\(73\) −0.745839 0.430610i −0.0872938 0.0503991i 0.455718 0.890124i \(-0.349383\pi\)
−0.543012 + 0.839725i \(0.682716\pi\)
\(74\) 0 0
\(75\) −14.0883 + 2.78208i −1.62677 + 0.321247i
\(76\) 0 0
\(77\) 3.39269 1.47561i 0.386633 0.168161i
\(78\) 0 0
\(79\) −1.31221 + 2.27282i −0.147636 + 0.255712i −0.930353 0.366665i \(-0.880500\pi\)
0.782718 + 0.622377i \(0.213833\pi\)
\(80\) 0 0
\(81\) 6.39906 6.32866i 0.711007 0.703185i
\(82\) 0 0
\(83\) −3.82295 6.62154i −0.419623 0.726808i 0.576278 0.817253i \(-0.304504\pi\)
−0.995901 + 0.0904451i \(0.971171\pi\)
\(84\) 0 0
\(85\) 8.81051 15.2603i 0.955634 1.65521i
\(86\) 0 0
\(87\) −7.56036 + 6.61179i −0.810556 + 0.708858i
\(88\) 0 0
\(89\) −4.44212 7.69398i −0.470864 0.815560i 0.528581 0.848883i \(-0.322724\pi\)
−0.999445 + 0.0333231i \(0.989391\pi\)
\(90\) 0 0
\(91\) 1.02785 9.02968i 0.107748 0.946568i
\(92\) 0 0
\(93\) −7.97587 + 6.97516i −0.827059 + 0.723291i
\(94\) 0 0
\(95\) −26.3899 + 15.2362i −2.70755 + 1.56320i
\(96\) 0 0
\(97\) 9.63985 5.56557i 0.978778 0.565098i 0.0768773 0.997041i \(-0.475505\pi\)
0.901901 + 0.431943i \(0.142172\pi\)
\(98\) 0 0
\(99\) −1.59466 3.88016i −0.160269 0.389971i
\(100\) 0 0
\(101\) 5.81086 0.578203 0.289101 0.957299i \(-0.406644\pi\)
0.289101 + 0.957299i \(0.406644\pi\)
\(102\) 0 0
\(103\) 12.4310i 1.22486i −0.790523 0.612432i \(-0.790191\pi\)
0.790523 0.612432i \(-0.209809\pi\)
\(104\) 0 0
\(105\) 9.50514 + 13.7391i 0.927606 + 1.34080i
\(106\) 0 0
\(107\) 10.1989 5.88835i 0.985968 0.569249i 0.0819012 0.996640i \(-0.473901\pi\)
0.904067 + 0.427392i \(0.140567\pi\)
\(108\) 0 0
\(109\) −4.47772 + 7.75563i −0.428887 + 0.742855i −0.996775 0.0802516i \(-0.974428\pi\)
0.567887 + 0.823106i \(0.307761\pi\)
\(110\) 0 0
\(111\) 6.59118 + 7.53680i 0.625607 + 0.715362i
\(112\) 0 0
\(113\) −10.2381 5.91096i −0.963117 0.556056i −0.0659864 0.997821i \(-0.521019\pi\)
−0.897131 + 0.441764i \(0.854353\pi\)
\(114\) 0 0
\(115\) 32.5464i 3.03497i
\(116\) 0 0
\(117\) −10.2129 1.37330i −0.944183 0.126962i
\(118\) 0 0
\(119\) −12.7059 1.44632i −1.16475 0.132584i
\(120\) 0 0
\(121\) 9.04460 0.822237
\(122\) 0 0
\(123\) −0.896841 + 2.63005i −0.0808654 + 0.237144i
\(124\) 0 0
\(125\) −11.9977 −1.07311
\(126\) 0 0
\(127\) −6.85247 −0.608059 −0.304029 0.952663i \(-0.598332\pi\)
−0.304029 + 0.952663i \(0.598332\pi\)
\(128\) 0 0
\(129\) −5.08030 5.80916i −0.447296 0.511468i
\(130\) 0 0
\(131\) −0.689055 −0.0602030 −0.0301015 0.999547i \(-0.509583\pi\)
−0.0301015 + 0.999547i \(0.509583\pi\)
\(132\) 0 0
\(133\) 17.7783 + 13.1524i 1.54157 + 1.14046i
\(134\) 0 0
\(135\) 15.7945 10.4590i 1.35937 0.900167i
\(136\) 0 0
\(137\) 3.47292i 0.296712i 0.988934 + 0.148356i \(0.0473981\pi\)
−0.988934 + 0.148356i \(0.952602\pi\)
\(138\) 0 0
\(139\) −7.69570 4.44311i −0.652741 0.376860i 0.136765 0.990604i \(-0.456330\pi\)
−0.789505 + 0.613744i \(0.789663\pi\)
\(140\) 0 0
\(141\) −1.69070 + 4.95811i −0.142383 + 0.417549i
\(142\) 0 0
\(143\) −2.40163 + 4.15975i −0.200835 + 0.347856i
\(144\) 0 0
\(145\) −18.3079 + 10.5701i −1.52039 + 0.877798i
\(146\) 0 0
\(147\) 6.39211 10.3025i 0.527212 0.849734i
\(148\) 0 0
\(149\) 0.567607i 0.0465002i −0.999730 0.0232501i \(-0.992599\pi\)
0.999730 0.0232501i \(-0.00740140\pi\)
\(150\) 0 0
\(151\) 0.206398 0.0167964 0.00839821 0.999965i \(-0.497327\pi\)
0.00839821 + 0.999965i \(0.497327\pi\)
\(152\) 0 0
\(153\) −1.93241 + 14.3709i −0.156226 + 1.16182i
\(154\) 0 0
\(155\) −19.3141 + 11.1510i −1.55135 + 0.895671i
\(156\) 0 0
\(157\) 7.69254 4.44129i 0.613931 0.354453i −0.160571 0.987024i \(-0.551334\pi\)
0.774502 + 0.632571i \(0.218000\pi\)
\(158\) 0 0
\(159\) 8.33361 + 2.84174i 0.660898 + 0.225364i
\(160\) 0 0
\(161\) 21.6597 9.42060i 1.70702 0.742447i
\(162\) 0 0
\(163\) −0.351149 0.608208i −0.0275041 0.0476386i 0.851946 0.523630i \(-0.175423\pi\)
−0.879450 + 0.475992i \(0.842089\pi\)
\(164\) 0 0
\(165\) −1.71065 8.66261i −0.133174 0.674383i
\(166\) 0 0
\(167\) 8.24845 14.2867i 0.638285 1.10554i −0.347524 0.937671i \(-0.612978\pi\)
0.985809 0.167870i \(-0.0536890\pi\)
\(168\) 0 0
\(169\) −0.600599 1.04027i −0.0461999 0.0800206i
\(170\) 0 0
\(171\) 15.3200 19.8515i 1.17155 1.51808i
\(172\) 0 0
\(173\) −5.14623 + 8.91353i −0.391261 + 0.677683i −0.992616 0.121298i \(-0.961294\pi\)
0.601355 + 0.798982i \(0.294628\pi\)
\(174\) 0 0
\(175\) 8.74896 + 20.1155i 0.661360 + 1.52059i
\(176\) 0 0
\(177\) −7.32551 8.37649i −0.550619 0.629615i
\(178\) 0 0
\(179\) 7.02136 + 4.05378i 0.524801 + 0.302994i 0.738897 0.673819i \(-0.235347\pi\)
−0.214096 + 0.976813i \(0.568680\pi\)
\(180\) 0 0
\(181\) 15.1524i 1.12627i −0.826365 0.563134i \(-0.809595\pi\)
0.826365 0.563134i \(-0.190405\pi\)
\(182\) 0 0
\(183\) 14.7400 + 5.02629i 1.08961 + 0.371554i
\(184\) 0 0
\(185\) 10.5371 + 18.2509i 0.774706 + 1.34183i
\(186\) 0 0
\(187\) 5.85331 + 3.37941i 0.428036 + 0.247127i
\(188\) 0 0
\(189\) −11.5322 7.48387i −0.838844 0.544371i
\(190\) 0 0
\(191\) 2.49394 + 1.43988i 0.180455 + 0.104186i 0.587506 0.809220i \(-0.300110\pi\)
−0.407051 + 0.913405i \(0.633443\pi\)
\(192\) 0 0
\(193\) −8.03659 13.9198i −0.578486 1.00197i −0.995653 0.0931378i \(-0.970310\pi\)
0.417167 0.908830i \(-0.363023\pi\)
\(194\) 0 0
\(195\) −20.5292 7.00038i −1.47012 0.501308i
\(196\) 0 0
\(197\) 6.37179i 0.453972i 0.973898 + 0.226986i \(0.0728871\pi\)
−0.973898 + 0.226986i \(0.927113\pi\)
\(198\) 0 0
\(199\) −5.73328 3.31011i −0.406422 0.234648i 0.282830 0.959170i \(-0.408727\pi\)
−0.689251 + 0.724523i \(0.742060\pi\)
\(200\) 0 0
\(201\) −5.78081 6.61017i −0.407747 0.466246i
\(202\) 0 0
\(203\) 12.3337 + 9.12444i 0.865653 + 0.640410i
\(204\) 0 0
\(205\) −2.92441 + 5.06522i −0.204250 + 0.353771i
\(206\) 0 0
\(207\) −10.1806 24.7718i −0.707604 1.72176i
\(208\) 0 0
\(209\) −5.84408 10.1223i −0.404244 0.700171i
\(210\) 0 0
\(211\) 2.13230 3.69326i 0.146794 0.254255i −0.783247 0.621711i \(-0.786438\pi\)
0.930041 + 0.367456i \(0.119771\pi\)
\(212\) 0 0
\(213\) −1.41163 7.14838i −0.0967232 0.489799i
\(214\) 0 0
\(215\) −8.12174 14.0673i −0.553898 0.959380i
\(216\) 0 0
\(217\) 13.0115 + 9.62591i 0.883279 + 0.653449i
\(218\) 0 0
\(219\) 1.41185 + 0.481437i 0.0954040 + 0.0325325i
\(220\) 0 0
\(221\) 14.3781 8.30123i 0.967179 0.558401i
\(222\) 0 0
\(223\) −0.338403 + 0.195377i −0.0226611 + 0.0130834i −0.511288 0.859410i \(-0.670831\pi\)
0.488627 + 0.872493i \(0.337498\pi\)
\(224\) 0 0
\(225\) 23.0057 9.45482i 1.53371 0.630322i
\(226\) 0 0
\(227\) −2.07487 −0.137714 −0.0688571 0.997627i \(-0.521935\pi\)
−0.0688571 + 0.997627i \(0.521935\pi\)
\(228\) 0 0
\(229\) 14.7152i 0.972411i −0.873845 0.486205i \(-0.838381\pi\)
0.873845 0.486205i \(-0.161619\pi\)
\(230\) 0 0
\(231\) −5.26983 + 3.64584i −0.346730 + 0.239879i
\(232\) 0 0
\(233\) −11.8713 + 6.85389i −0.777713 + 0.449013i −0.835619 0.549309i \(-0.814891\pi\)
0.0579059 + 0.998322i \(0.481558\pi\)
\(234\) 0 0
\(235\) −5.51303 + 9.54884i −0.359630 + 0.622898i
\(236\) 0 0
\(237\) 1.46710 4.30238i 0.0952984 0.279470i
\(238\) 0 0
\(239\) 8.08032 + 4.66517i 0.522672 + 0.301765i 0.738027 0.674771i \(-0.235758\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(240\) 0 0
\(241\) 17.1500i 1.10473i −0.833604 0.552363i \(-0.813726\pi\)
0.833604 0.552363i \(-0.186274\pi\)
\(242\) 0 0
\(243\) −8.74989 + 12.9011i −0.561306 + 0.827608i
\(244\) 0 0
\(245\) 17.4005 18.6676i 1.11168 1.19263i
\(246\) 0 0
\(247\) −28.7110 −1.82684
\(248\) 0 0
\(249\) 8.71799 + 9.96874i 0.552480 + 0.631743i
\(250\) 0 0
\(251\) 9.53124 0.601606 0.300803 0.953686i \(-0.402745\pi\)
0.300803 + 0.953686i \(0.402745\pi\)
\(252\) 0 0
\(253\) −12.4837 −0.784843
\(254\) 0 0
\(255\) −9.85045 + 28.8872i −0.616859 + 1.80899i
\(256\) 0 0
\(257\) 23.6663 1.47626 0.738131 0.674658i \(-0.235709\pi\)
0.738131 + 0.674658i \(0.235709\pi\)
\(258\) 0 0
\(259\) 9.09600 12.2952i 0.565198 0.763988i
\(260\) 0 0
\(261\) 10.6282 13.7719i 0.657869 0.852460i
\(262\) 0 0
\(263\) 18.8037i 1.15949i −0.814799 0.579744i \(-0.803153\pi\)
0.814799 0.579744i \(-0.196847\pi\)
\(264\) 0 0
\(265\) 16.0497 + 9.26631i 0.985926 + 0.569225i
\(266\) 0 0
\(267\) 10.1300 + 11.5833i 0.619944 + 0.708886i
\(268\) 0 0
\(269\) −1.16840 + 2.02372i −0.0712383 + 0.123388i −0.899444 0.437035i \(-0.856028\pi\)
0.828206 + 0.560424i \(0.189362\pi\)
\(270\) 0 0
\(271\) −19.9051 + 11.4922i −1.20915 + 0.698103i −0.962574 0.271021i \(-0.912639\pi\)
−0.246576 + 0.969123i \(0.579306\pi\)
\(272\) 0 0
\(273\) 1.28342 + 15.6885i 0.0776759 + 0.949509i
\(274\) 0 0
\(275\) 11.5937i 0.699125i
\(276\) 0 0
\(277\) −30.9918 −1.86211 −0.931057 0.364873i \(-0.881113\pi\)
−0.931057 + 0.364873i \(0.881113\pi\)
\(278\) 0 0
\(279\) 11.2123 14.5288i 0.671264 0.869817i
\(280\) 0 0
\(281\) −7.29924 + 4.21422i −0.435436 + 0.251399i −0.701660 0.712512i \(-0.747557\pi\)
0.266224 + 0.963911i \(0.414224\pi\)
\(282\) 0 0
\(283\) 18.6948 10.7935i 1.11129 0.641604i 0.172128 0.985075i \(-0.444936\pi\)
0.939163 + 0.343471i \(0.111603\pi\)
\(284\) 0 0
\(285\) 39.7300 34.7452i 2.35340 2.05813i
\(286\) 0 0
\(287\) 4.21739 + 0.480065i 0.248945 + 0.0283373i
\(288\) 0 0
\(289\) −3.18091 5.50950i −0.187112 0.324088i
\(290\) 0 0
\(291\) −14.5128 + 12.6919i −0.850756 + 0.744014i
\(292\) 0 0
\(293\) −12.7083 + 22.0114i −0.742427 + 1.28592i 0.208960 + 0.977924i \(0.432992\pi\)
−0.951387 + 0.307998i \(0.900341\pi\)
\(294\) 0 0
\(295\) −11.7111 20.2842i −0.681847 1.18099i
\(296\) 0 0
\(297\) 4.01171 + 6.05821i 0.232783 + 0.351533i
\(298\) 0 0
\(299\) −15.3326 + 26.5568i −0.886704 + 1.53582i
\(300\) 0 0
\(301\) −7.01095 + 9.47682i −0.404104 + 0.546235i
\(302\) 0 0
\(303\) −9.87403 + 1.94988i −0.567248 + 0.112017i
\(304\) 0 0
\(305\) 28.3878 + 16.3897i 1.62548 + 0.938471i
\(306\) 0 0
\(307\) 9.44895i 0.539280i 0.962961 + 0.269640i \(0.0869047\pi\)
−0.962961 + 0.269640i \(0.913095\pi\)
\(308\) 0 0
\(309\) 4.17131 + 21.1232i 0.237298 + 1.20166i
\(310\) 0 0
\(311\) −4.73567 8.20242i −0.268535 0.465117i 0.699949 0.714193i \(-0.253206\pi\)
−0.968484 + 0.249077i \(0.919873\pi\)
\(312\) 0 0
\(313\) 5.90654 + 3.41015i 0.333858 + 0.192753i 0.657552 0.753409i \(-0.271592\pi\)
−0.323695 + 0.946162i \(0.604925\pi\)
\(314\) 0 0
\(315\) −20.7617 20.1564i −1.16979 1.13568i
\(316\) 0 0
\(317\) 24.8540 + 14.3494i 1.39594 + 0.805945i 0.993964 0.109706i \(-0.0349910\pi\)
0.401974 + 0.915651i \(0.368324\pi\)
\(318\) 0 0
\(319\) −4.05432 7.02229i −0.226998 0.393173i
\(320\) 0 0
\(321\) −15.3545 + 13.4280i −0.857005 + 0.749479i
\(322\) 0 0
\(323\) 40.4001i 2.24792i
\(324\) 0 0
\(325\) −24.6634 14.2394i −1.36808 0.789862i
\(326\) 0 0
\(327\) 5.00624 14.6812i 0.276846 0.811871i
\(328\) 0 0
\(329\) 7.95052 + 0.905007i 0.438327 + 0.0498947i
\(330\) 0 0
\(331\) −0.600666 + 1.04038i −0.0330156 + 0.0571847i −0.882061 0.471135i \(-0.843844\pi\)
0.849045 + 0.528320i \(0.177178\pi\)
\(332\) 0 0
\(333\) −13.7290 10.5951i −0.752344 0.580607i
\(334\) 0 0
\(335\) −9.24163 16.0070i −0.504924 0.874554i
\(336\) 0 0
\(337\) 3.34451 5.79285i 0.182187 0.315557i −0.760438 0.649410i \(-0.775016\pi\)
0.942625 + 0.333854i \(0.108349\pi\)
\(338\) 0 0
\(339\) 19.3804 + 6.60865i 1.05260 + 0.358933i
\(340\) 0 0
\(341\) −4.27714 7.40823i −0.231620 0.401178i
\(342\) 0 0
\(343\) −17.4599 6.17672i −0.942746 0.333512i
\(344\) 0 0
\(345\) −10.9212 55.3040i −0.587976 2.97747i
\(346\) 0 0
\(347\) −1.14577 + 0.661512i −0.0615083 + 0.0355118i −0.530439 0.847723i \(-0.677973\pi\)
0.468930 + 0.883235i \(0.344639\pi\)
\(348\) 0 0
\(349\) −3.25327 + 1.87828i −0.174144 + 0.100542i −0.584538 0.811366i \(-0.698724\pi\)
0.410395 + 0.911908i \(0.365391\pi\)
\(350\) 0 0
\(351\) 17.8149 1.09345i 0.950891 0.0583640i
\(352\) 0 0
\(353\) 11.7059 0.623040 0.311520 0.950240i \(-0.399162\pi\)
0.311520 + 0.950240i \(0.399162\pi\)
\(354\) 0 0
\(355\) 15.3367i 0.813988i
\(356\) 0 0
\(357\) 22.0757 1.80593i 1.16837 0.0955802i
\(358\) 0 0
\(359\) 23.0847 13.3280i 1.21836 0.703423i 0.253797 0.967258i \(-0.418321\pi\)
0.964568 + 0.263835i \(0.0849873\pi\)
\(360\) 0 0
\(361\) 25.4324 44.0502i 1.33855 2.31843i
\(362\) 0 0
\(363\) −15.3689 + 3.03498i −0.806659 + 0.159295i
\(364\) 0 0
\(365\) 2.71909 + 1.56986i 0.142323 + 0.0821705i
\(366\) 0 0
\(367\) 27.0815i 1.41364i 0.707392 + 0.706821i \(0.249872\pi\)
−0.707392 + 0.706821i \(0.750128\pi\)
\(368\) 0 0
\(369\) 0.641411 4.77002i 0.0333905 0.248317i
\(370\) 0 0
\(371\) 1.52114 13.3633i 0.0789736 0.693786i
\(372\) 0 0
\(373\) 8.60898 0.445756 0.222878 0.974846i \(-0.428455\pi\)
0.222878 + 0.974846i \(0.428455\pi\)
\(374\) 0 0
\(375\) 20.3869 4.02591i 1.05278 0.207897i
\(376\) 0 0
\(377\) −19.9182 −1.02584
\(378\) 0 0
\(379\) 2.49028 0.127917 0.0639585 0.997953i \(-0.479627\pi\)
0.0639585 + 0.997953i \(0.479627\pi\)
\(380\) 0 0
\(381\) 11.6440 2.29940i 0.596539 0.117802i
\(382\) 0 0
\(383\) 20.2597 1.03522 0.517612 0.855616i \(-0.326821\pi\)
0.517612 + 0.855616i \(0.326821\pi\)
\(384\) 0 0
\(385\) −12.3687 + 5.37958i −0.630365 + 0.274169i
\(386\) 0 0
\(387\) 10.5819 + 8.16640i 0.537910 + 0.415121i
\(388\) 0 0
\(389\) 3.60894i 0.182980i 0.995806 + 0.0914902i \(0.0291630\pi\)
−0.995806 + 0.0914902i \(0.970837\pi\)
\(390\) 0 0
\(391\) 37.3688 + 21.5749i 1.88982 + 1.09109i
\(392\) 0 0
\(393\) 1.17087 0.231217i 0.0590624 0.0116634i
\(394\) 0 0
\(395\) 4.78390 8.28597i 0.240704 0.416912i
\(396\) 0 0
\(397\) −16.3076 + 9.41521i −0.818456 + 0.472536i −0.849884 0.526970i \(-0.823328\pi\)
0.0314275 + 0.999506i \(0.489995\pi\)
\(398\) 0 0
\(399\) −34.6229 16.3834i −1.73331 0.820194i
\(400\) 0 0
\(401\) 14.3026i 0.714239i −0.934059 0.357119i \(-0.883759\pi\)
0.934059 0.357119i \(-0.116241\pi\)
\(402\) 0 0
\(403\) −21.0129 −1.04673
\(404\) 0 0
\(405\) −23.3289 + 23.0722i −1.15922 + 1.14647i
\(406\) 0 0
\(407\) −7.00041 + 4.04169i −0.346997 + 0.200339i
\(408\) 0 0
\(409\) −30.3685 + 17.5332i −1.50162 + 0.866963i −0.501626 + 0.865085i \(0.667265\pi\)
−0.999998 + 0.00187827i \(0.999402\pi\)
\(410\) 0 0
\(411\) −1.16536 5.90131i −0.0574831 0.291090i
\(412\) 0 0
\(413\) −10.1094 + 13.6650i −0.497451 + 0.672413i
\(414\) 0 0
\(415\) 13.9372 + 24.1400i 0.684151 + 1.18498i
\(416\) 0 0
\(417\) 14.5677 + 4.96755i 0.713384 + 0.243262i
\(418\) 0 0
\(419\) −5.83034 + 10.0984i −0.284831 + 0.493341i −0.972568 0.232618i \(-0.925271\pi\)
0.687737 + 0.725959i \(0.258604\pi\)
\(420\) 0 0
\(421\) 1.42981 + 2.47650i 0.0696845 + 0.120697i 0.898762 0.438436i \(-0.144467\pi\)
−0.829078 + 0.559133i \(0.811134\pi\)
\(422\) 0 0
\(423\) 1.20917 8.99233i 0.0587920 0.437222i
\(424\) 0 0
\(425\) −20.0367 + 34.7046i −0.971925 + 1.68342i
\(426\) 0 0
\(427\) 2.69050 23.6361i 0.130202 1.14383i
\(428\) 0 0
\(429\) 2.68511 7.87428i 0.129638 0.380174i
\(430\) 0 0
\(431\) −0.720513 0.415989i −0.0347059 0.0200375i 0.482547 0.875870i \(-0.339712\pi\)
−0.517253 + 0.855833i \(0.673045\pi\)
\(432\) 0 0
\(433\) 41.2560i 1.98264i 0.131479 + 0.991319i \(0.458027\pi\)
−0.131479 + 0.991319i \(0.541973\pi\)
\(434\) 0 0
\(435\) 27.5626 24.1044i 1.32153 1.15572i
\(436\) 0 0
\(437\) −37.3099 64.6227i −1.78478 3.09132i
\(438\) 0 0
\(439\) −3.72695 2.15175i −0.177877 0.102698i 0.408418 0.912795i \(-0.366081\pi\)
−0.586295 + 0.810098i \(0.699414\pi\)
\(440\) 0 0
\(441\) −7.40463 + 19.6512i −0.352601 + 0.935774i
\(442\) 0 0
\(443\) 8.98795 + 5.18919i 0.427030 + 0.246546i 0.698081 0.716019i \(-0.254038\pi\)
−0.271050 + 0.962565i \(0.587371\pi\)
\(444\) 0 0
\(445\) 16.1945 + 28.0497i 0.767694 + 1.32968i
\(446\) 0 0
\(447\) 0.190464 + 0.964498i 0.00900866 + 0.0456192i
\(448\) 0 0
\(449\) 29.5119i 1.39275i 0.717677 + 0.696376i \(0.245205\pi\)
−0.717677 + 0.696376i \(0.754795\pi\)
\(450\) 0 0
\(451\) −1.94285 1.12170i −0.0914850 0.0528189i
\(452\) 0 0
\(453\) −0.350719 + 0.0692583i −0.0164782 + 0.00325404i
\(454\) 0 0
\(455\) −3.74720 + 32.9193i −0.175671 + 1.54328i
\(456\) 0 0
\(457\) −1.11805 + 1.93652i −0.0523001 + 0.0905864i −0.890990 0.454023i \(-0.849989\pi\)
0.838690 + 0.544609i \(0.183322\pi\)
\(458\) 0 0
\(459\) −1.53863 25.0679i −0.0718169 1.17007i
\(460\) 0 0
\(461\) −5.62018 9.73443i −0.261758 0.453378i 0.704951 0.709256i \(-0.250969\pi\)
−0.966709 + 0.255878i \(0.917635\pi\)
\(462\) 0 0
\(463\) −2.58576 + 4.47867i −0.120170 + 0.208141i −0.919835 0.392306i \(-0.871677\pi\)
0.799664 + 0.600447i \(0.205011\pi\)
\(464\) 0 0
\(465\) 29.0774 25.4292i 1.34843 1.17925i
\(466\) 0 0
\(467\) 14.9873 + 25.9587i 0.693528 + 1.20123i 0.970674 + 0.240398i \(0.0772780\pi\)
−0.277146 + 0.960828i \(0.589389\pi\)
\(468\) 0 0
\(469\) −7.97767 + 10.7836i −0.368375 + 0.497938i
\(470\) 0 0
\(471\) −11.5811 + 10.1281i −0.533630 + 0.466677i
\(472\) 0 0
\(473\) 5.39572 3.11522i 0.248095 0.143238i
\(474\) 0 0
\(475\) 60.0155 34.6500i 2.75370 1.58985i
\(476\) 0 0
\(477\) −15.1143 2.03238i −0.692038 0.0930563i
\(478\) 0 0
\(479\) 35.2436 1.61032 0.805159 0.593058i \(-0.202080\pi\)
0.805159 + 0.593058i \(0.202080\pi\)
\(480\) 0 0
\(481\) 19.8561i 0.905361i
\(482\) 0 0
\(483\) −33.6438 + 23.2759i −1.53085 + 1.05909i
\(484\) 0 0
\(485\) −35.1437 + 20.2903i −1.59580 + 0.921333i
\(486\) 0 0
\(487\) −12.3984 + 21.4747i −0.561826 + 0.973111i 0.435511 + 0.900183i \(0.356568\pi\)
−0.997337 + 0.0729280i \(0.976766\pi\)
\(488\) 0 0
\(489\) 0.800774 + 0.915659i 0.0362122 + 0.0414075i
\(490\) 0 0
\(491\) −17.4696 10.0860i −0.788390 0.455177i 0.0510056 0.998698i \(-0.483757\pi\)
−0.839395 + 0.543521i \(0.817091\pi\)
\(492\) 0 0
\(493\) 28.0275i 1.26229i
\(494\) 0 0
\(495\) 5.81360 + 14.1458i 0.261302 + 0.635806i
\(496\) 0 0
\(497\) −10.2066 + 4.43922i −0.457829 + 0.199126i
\(498\) 0 0
\(499\) −27.5961 −1.23537 −0.617687 0.786424i \(-0.711930\pi\)
−0.617687 + 0.786424i \(0.711930\pi\)
\(500\) 0 0
\(501\) −9.22205 + 27.0444i −0.412011 + 1.20825i
\(502\) 0 0
\(503\) −27.9512 −1.24628 −0.623142 0.782109i \(-0.714144\pi\)
−0.623142 + 0.782109i \(0.714144\pi\)
\(504\) 0 0
\(505\) −21.1845 −0.942699
\(506\) 0 0
\(507\) 1.36963 + 1.56612i 0.0608273 + 0.0695540i
\(508\) 0 0
\(509\) −43.1118 −1.91090 −0.955448 0.295160i \(-0.904627\pi\)
−0.955448 + 0.295160i \(0.904627\pi\)
\(510\) 0 0
\(511\) 0.257706 2.26396i 0.0114002 0.100151i
\(512\) 0 0
\(513\) −19.3710 + 38.8731i −0.855249 + 1.71629i
\(514\) 0 0
\(515\) 45.3194i 1.99701i
\(516\) 0 0
\(517\) −3.66261 2.11461i −0.161081 0.0930003i
\(518\) 0 0
\(519\) 5.75366 16.8730i 0.252558 0.740645i
\(520\) 0 0
\(521\) 3.06052 5.30098i 0.134084 0.232240i −0.791163 0.611605i \(-0.790524\pi\)
0.925247 + 0.379365i \(0.123857\pi\)
\(522\) 0 0
\(523\) 13.1647 7.60065i 0.575652 0.332353i −0.183751 0.982973i \(-0.558824\pi\)
0.759404 + 0.650620i \(0.225491\pi\)
\(524\) 0 0
\(525\) −21.6164 31.2452i −0.943419 1.36365i
\(526\) 0 0
\(527\) 29.5678i 1.28800i
\(528\) 0 0
\(529\) −56.6986 −2.46516
\(530\) 0 0
\(531\) 15.2586 + 11.7755i 0.662165 + 0.511013i
\(532\) 0 0
\(533\) −4.77243 + 2.75537i −0.206717 + 0.119348i
\(534\) 0 0
\(535\) −37.1820 + 21.4670i −1.60752 + 0.928100i
\(536\) 0 0
\(537\) −13.2912 4.53227i −0.573558 0.195582i
\(538\) 0 0
\(539\) 7.16024 + 6.67424i 0.308413 + 0.287480i
\(540\) 0 0
\(541\) 12.9995 + 22.5159i 0.558894 + 0.968033i 0.997589 + 0.0693968i \(0.0221074\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(542\) 0 0
\(543\) 5.08449 + 25.7475i 0.218196 + 1.10493i
\(544\) 0 0
\(545\) 16.3243 28.2745i 0.699256 1.21115i
\(546\) 0 0
\(547\) −15.7120 27.2139i −0.671794 1.16358i −0.977395 0.211422i \(-0.932190\pi\)
0.305600 0.952160i \(-0.401143\pi\)
\(548\) 0 0
\(549\) −26.7333 3.59475i −1.14095 0.153420i
\(550\) 0 0
\(551\) 24.2343 41.9750i 1.03241 1.78819i
\(552\) 0 0
\(553\) −6.89903 0.785316i −0.293377 0.0333950i
\(554\) 0 0
\(555\) −24.0293 27.4767i −1.01999 1.16632i
\(556\) 0 0
\(557\) 33.8867 + 19.5645i 1.43582 + 0.828974i 0.997556 0.0698707i \(-0.0222587\pi\)
0.438268 + 0.898844i \(0.355592\pi\)
\(558\) 0 0
\(559\) 15.3045i 0.647313i
\(560\) 0 0
\(561\) −11.0801 3.77829i −0.467804 0.159520i
\(562\) 0 0
\(563\) 14.6402 + 25.3576i 0.617012 + 1.06870i 0.990028 + 0.140871i \(0.0449902\pi\)
−0.373016 + 0.927825i \(0.621676\pi\)
\(564\) 0 0
\(565\) 37.3247 + 21.5494i 1.57026 + 0.906591i
\(566\) 0 0
\(567\) 22.1072 + 8.84715i 0.928415 + 0.371545i
\(568\) 0 0
\(569\) 28.3754 + 16.3825i 1.18956 + 0.686791i 0.958206 0.286079i \(-0.0923521\pi\)
0.231351 + 0.972870i \(0.425685\pi\)
\(570\) 0 0
\(571\) −10.5170 18.2160i −0.440123 0.762315i 0.557576 0.830126i \(-0.311732\pi\)
−0.997698 + 0.0678114i \(0.978398\pi\)
\(572\) 0 0
\(573\) −4.72095 1.60983i −0.197220 0.0672516i
\(574\) 0 0
\(575\) 74.0166i 3.08671i
\(576\) 0 0
\(577\) 14.9926 + 8.65597i 0.624149 + 0.360353i 0.778483 0.627666i \(-0.215990\pi\)
−0.154334 + 0.988019i \(0.549323\pi\)
\(578\) 0 0
\(579\) 18.3269 + 20.9563i 0.761641 + 0.870912i
\(580\) 0 0
\(581\) 12.0311 16.2626i 0.499132 0.674686i
\(582\) 0 0
\(583\) −3.55424 + 6.15612i −0.147201 + 0.254960i
\(584\) 0 0
\(585\) 37.2329 + 5.00660i 1.53939 + 0.206997i
\(586\) 0 0
\(587\) −8.95712 15.5142i −0.369700 0.640339i 0.619819 0.784745i \(-0.287206\pi\)
−0.989518 + 0.144406i \(0.953873\pi\)
\(588\) 0 0
\(589\) 25.5662 44.2819i 1.05343 1.82460i
\(590\) 0 0
\(591\) −2.13810 10.8272i −0.0879497 0.445371i
\(592\) 0 0
\(593\) −17.6799 30.6225i −0.726028 1.25752i −0.958550 0.284926i \(-0.908031\pi\)
0.232522 0.972591i \(-0.425302\pi\)
\(594\) 0 0
\(595\) 46.3217 + 5.27280i 1.89901 + 0.216164i
\(596\) 0 0
\(597\) 10.8529 + 3.70082i 0.444181 + 0.151464i
\(598\) 0 0
\(599\) −0.773338 + 0.446487i −0.0315977 + 0.0182430i −0.515716 0.856760i \(-0.672474\pi\)
0.484118 + 0.875003i \(0.339141\pi\)
\(600\) 0 0
\(601\) 23.7713 13.7244i 0.969653 0.559829i 0.0705225 0.997510i \(-0.477533\pi\)
0.899130 + 0.437681i \(0.144200\pi\)
\(602\) 0 0
\(603\) 12.0411 + 9.29245i 0.490350 + 0.378418i
\(604\) 0 0
\(605\) −32.9737 −1.34057
\(606\) 0 0
\(607\) 20.3891i 0.827568i −0.910375 0.413784i \(-0.864207\pi\)
0.910375 0.413784i \(-0.135793\pi\)
\(608\) 0 0
\(609\) −24.0196 11.3659i −0.973322 0.460570i
\(610\) 0 0
\(611\) −8.99688 + 5.19435i −0.363975 + 0.210141i
\(612\) 0 0
\(613\) 16.6541 28.8457i 0.672652 1.16507i −0.304497 0.952513i \(-0.598488\pi\)
0.977149 0.212554i \(-0.0681783\pi\)
\(614\) 0 0
\(615\) 3.26959 9.58832i 0.131843 0.386638i
\(616\) 0 0
\(617\) −27.6812 15.9817i −1.11440 0.643401i −0.174437 0.984668i \(-0.555810\pi\)
−0.939966 + 0.341268i \(0.889144\pi\)
\(618\) 0 0
\(619\) 4.39913i 0.176816i −0.996084 0.0884079i \(-0.971822\pi\)
0.996084 0.0884079i \(-0.0281779\pi\)
\(620\) 0 0
\(621\) 25.6117 + 38.6769i 1.02776 + 1.55205i
\(622\) 0 0
\(623\) 13.9796 18.8965i 0.560082 0.757072i
\(624\) 0 0
\(625\) 2.28497 0.0913990
\(626\) 0 0
\(627\) 13.3271 + 15.2391i 0.532232 + 0.608590i
\(628\) 0 0
\(629\) 27.9401 1.11405
\(630\) 0 0
\(631\) −18.0749 −0.719549 −0.359775 0.933039i \(-0.617146\pi\)
−0.359775 + 0.933039i \(0.617146\pi\)
\(632\) 0 0
\(633\) −2.38399 + 6.99123i −0.0947551 + 0.277876i
\(634\) 0 0
\(635\) 24.9819 0.991376
\(636\) 0 0
\(637\) 22.9925 7.03475i 0.910995 0.278727i
\(638\) 0 0
\(639\) 4.79738 + 11.6731i 0.189781 + 0.461781i
\(640\) 0 0
\(641\) 4.10040i 0.161956i −0.996716 0.0809779i \(-0.974196\pi\)
0.996716 0.0809779i \(-0.0258043\pi\)
\(642\) 0 0
\(643\) 7.12890 + 4.11587i 0.281136 + 0.162314i 0.633938 0.773384i \(-0.281438\pi\)
−0.352801 + 0.935698i \(0.614771\pi\)
\(644\) 0 0
\(645\) 18.5211 + 21.1783i 0.729268 + 0.833895i
\(646\) 0 0
\(647\) 7.07138 12.2480i 0.278004 0.481518i −0.692884 0.721049i \(-0.743660\pi\)
0.970889 + 0.239531i \(0.0769937\pi\)
\(648\) 0 0
\(649\) 7.78033 4.49198i 0.305405 0.176325i
\(650\) 0 0
\(651\) −25.3397 11.9906i −0.993139 0.469948i
\(652\) 0 0
\(653\) 23.0473i 0.901911i 0.892546 + 0.450956i \(0.148917\pi\)
−0.892546 + 0.450956i \(0.851083\pi\)
\(654\) 0 0
\(655\) 2.51207 0.0981547
\(656\) 0 0
\(657\) −2.56062 0.344319i −0.0998991 0.0134332i
\(658\) 0 0
\(659\) 7.09924 4.09875i 0.276547 0.159665i −0.355312 0.934748i \(-0.615625\pi\)
0.631859 + 0.775083i \(0.282292\pi\)
\(660\) 0 0
\(661\) 26.3464 15.2111i 1.02476 0.591643i 0.109277 0.994011i \(-0.465146\pi\)
0.915478 + 0.402369i \(0.131813\pi\)
\(662\) 0 0
\(663\) −21.6463 + 18.9304i −0.840674 + 0.735197i
\(664\) 0 0
\(665\) −64.8139 47.9493i −2.51338 1.85939i
\(666\) 0 0
\(667\) −25.8837 44.8319i −1.00222 1.73590i
\(668\) 0 0
\(669\) 0.509466 0.445545i 0.0196971 0.0172258i
\(670\) 0 0
\(671\) −6.28652 + 10.8886i −0.242688 + 0.420349i
\(672\) 0 0
\(673\) 2.44406 + 4.23323i 0.0942115 + 0.163179i 0.909279 0.416187i \(-0.136634\pi\)
−0.815068 + 0.579366i \(0.803300\pi\)
\(674\) 0 0
\(675\) −35.9195 + 23.7857i −1.38254 + 0.915512i
\(676\) 0 0
\(677\) 10.7121 18.5539i 0.411699 0.713083i −0.583377 0.812202i \(-0.698269\pi\)
0.995076 + 0.0991185i \(0.0316023\pi\)
\(678\) 0 0
\(679\) 23.6756 + 17.5152i 0.908586 + 0.672171i
\(680\) 0 0
\(681\) 3.52570 0.696238i 0.135105 0.0266799i
\(682\) 0 0
\(683\) −3.49101 2.01553i −0.133580 0.0771222i 0.431721 0.902007i \(-0.357906\pi\)
−0.565301 + 0.824885i \(0.691240\pi\)
\(684\) 0 0
\(685\) 12.6611i 0.483757i
\(686\) 0 0
\(687\) 4.93780 + 25.0047i 0.188389 + 0.953988i
\(688\) 0 0
\(689\) 8.73068 + 15.1220i 0.332612 + 0.576101i
\(690\) 0 0
\(691\) −30.0272 17.3362i −1.14229 0.659500i −0.195291 0.980745i \(-0.562565\pi\)
−0.946996 + 0.321246i \(0.895899\pi\)
\(692\) 0 0
\(693\) 7.73130 7.96348i 0.293688 0.302507i
\(694\) 0 0
\(695\) 28.0560 + 16.1981i 1.06422 + 0.614431i
\(696\) 0 0
\(697\) 3.87716 + 6.71544i 0.146858 + 0.254365i
\(698\) 0 0
\(699\) 17.8722 15.6299i 0.675990 0.591176i
\(700\) 0 0
\(701\) 31.1575i 1.17680i 0.808569 + 0.588401i \(0.200242\pi\)
−0.808569 + 0.588401i \(0.799758\pi\)
\(702\) 0 0
\(703\) −41.8442 24.1587i −1.57818 0.911164i
\(704\) 0 0
\(705\) 6.16375 18.0757i 0.232140 0.680769i
\(706\) 0 0
\(707\) 6.13188 + 14.0983i 0.230613 + 0.530222i
\(708\) 0 0
\(709\) −14.1925 + 24.5821i −0.533009 + 0.923199i 0.466248 + 0.884654i \(0.345606\pi\)
−0.999257 + 0.0385448i \(0.987728\pi\)
\(710\) 0 0
\(711\) −1.04925 + 7.80306i −0.0393501 + 0.292637i
\(712\) 0 0
\(713\) −27.3062 47.2958i −1.02263 1.77124i
\(714\) 0 0
\(715\) 8.75557 15.1651i 0.327440 0.567142i
\(716\) 0 0
\(717\) −15.2958 5.21582i −0.571232 0.194788i
\(718\) 0 0
\(719\) 14.8969 + 25.8022i 0.555561 + 0.962259i 0.997860 + 0.0653917i \(0.0208297\pi\)
−0.442299 + 0.896868i \(0.645837\pi\)
\(720\) 0 0
\(721\) 30.1602 13.1178i 1.12322 0.488531i
\(722\) 0 0
\(723\) 5.75479 + 29.1418i 0.214023 + 1.08380i
\(724\) 0 0
\(725\) 41.6356 24.0383i 1.54631 0.892762i
\(726\) 0 0
\(727\) 36.8231 21.2598i 1.36569 0.788484i 0.375320 0.926895i \(-0.377533\pi\)
0.990375 + 0.138411i \(0.0441996\pi\)
\(728\) 0 0
\(729\) 10.5391 24.8582i 0.390336 0.920673i
\(730\) 0 0
\(731\) −21.5355 −0.796519
\(732\) 0 0
\(733\) 36.9920i 1.36633i 0.730263 + 0.683166i \(0.239398\pi\)
−0.730263 + 0.683166i \(0.760602\pi\)
\(734\) 0 0
\(735\) −23.3035 + 37.5595i −0.859564 + 1.38540i
\(736\) 0 0
\(737\) 6.13972 3.54477i 0.226160 0.130573i
\(738\) 0 0
\(739\) 17.3127 29.9864i 0.636857 1.10307i −0.349261 0.937025i \(-0.613567\pi\)
0.986118 0.166044i \(-0.0530993\pi\)
\(740\) 0 0
\(741\) 48.7867 9.63417i 1.79222 0.353920i
\(742\) 0 0
\(743\) −21.1788 12.2276i −0.776976 0.448587i 0.0583814 0.998294i \(-0.481406\pi\)
−0.835358 + 0.549707i \(0.814739\pi\)
\(744\) 0 0
\(745\) 2.06931i 0.0758136i
\(746\) 0 0
\(747\) −18.1590 14.0139i −0.664403 0.512740i
\(748\) 0 0
\(749\) 25.0487 + 18.5310i 0.915260 + 0.677109i
\(750\) 0 0
\(751\) −4.19662 −0.153137 −0.0765685 0.997064i \(-0.524396\pi\)
−0.0765685 + 0.997064i \(0.524396\pi\)
\(752\) 0 0
\(753\) −16.1958 + 3.19827i −0.590208 + 0.116552i
\(754\) 0 0
\(755\) −0.752460 −0.0273848
\(756\) 0 0
\(757\) 19.3020 0.701545 0.350772 0.936461i \(-0.385919\pi\)
0.350772 + 0.936461i \(0.385919\pi\)
\(758\) 0 0
\(759\) 21.2127 4.18899i 0.769973 0.152051i
\(760\) 0 0
\(761\) −14.7261 −0.533821 −0.266910 0.963721i \(-0.586003\pi\)
−0.266910 + 0.963721i \(0.586003\pi\)
\(762\) 0 0
\(763\) −23.5418 2.67976i −0.852271 0.0970139i
\(764\) 0 0
\(765\) 7.04494 52.3915i 0.254710 1.89422i
\(766\) 0 0
\(767\) 22.0683i 0.796840i
\(768\) 0 0
\(769\) −43.6719 25.2140i −1.57485 0.909240i −0.995561 0.0941181i \(-0.969997\pi\)
−0.579289 0.815122i \(-0.696670\pi\)
\(770\) 0 0
\(771\) −40.2146 + 7.94138i −1.44829 + 0.286002i
\(772\) 0 0
\(773\) 9.54706 16.5360i 0.343384 0.594758i −0.641675 0.766977i \(-0.721760\pi\)
0.985059 + 0.172218i \(0.0550935\pi\)
\(774\) 0 0
\(775\) 43.9239 25.3595i 1.57779 0.910939i
\(776\) 0 0
\(777\) −11.3305 + 23.9447i −0.406480 + 0.859012i
\(778\) 0 0
\(779\) 13.4097i 0.480453i
\(780\) 0 0
\(781\) 5.88263 0.210497
\(782\) 0 0
\(783\) −13.4386 + 26.9681i −0.480255 + 0.963761i
\(784\) 0 0
\(785\) −28.0445 + 16.1915i −1.00095 + 0.577899i
\(786\) 0 0
\(787\) 43.0297 24.8432i 1.53384 0.885566i 0.534665 0.845064i \(-0.320438\pi\)
0.999179 0.0405015i \(-0.0128956\pi\)
\(788\) 0 0
\(789\) 6.30972 + 31.9520i 0.224632 + 1.13752i
\(790\) 0 0
\(791\) 3.53751 31.0772i 0.125779 1.10498i
\(792\) 0 0
\(793\) 15.4423 + 26.7468i 0.548372 + 0.949808i
\(794\) 0 0
\(795\) −30.3816 10.3600i −1.07753 0.367433i
\(796\) 0 0
\(797\) −5.17665 + 8.96621i −0.183366 + 0.317600i −0.943025 0.332723i \(-0.892033\pi\)
0.759659 + 0.650322i \(0.225366\pi\)
\(798\) 0 0
\(799\) 7.30913 + 12.6598i 0.258578 + 0.447871i
\(800\) 0 0
\(801\) −21.1001 16.2836i −0.745534 0.575351i
\(802\) 0 0
\(803\) −0.602146 + 1.04295i −0.0212493 + 0.0368048i
\(804\) 0 0
\(805\) −78.9642 + 34.3444i −2.78312 + 1.21048i
\(806\) 0 0
\(807\) 1.30631 3.83084i 0.0459841 0.134852i
\(808\) 0 0
\(809\) −41.5520 23.9901i −1.46089 0.843446i −0.461838 0.886964i \(-0.652810\pi\)
−0.999053 + 0.0435184i \(0.986143\pi\)
\(810\) 0 0
\(811\) 4.79426i 0.168349i 0.996451 + 0.0841745i \(0.0268253\pi\)
−0.996451 + 0.0841745i \(0.973175\pi\)
\(812\) 0 0
\(813\) 29.9672 26.2073i 1.05100 0.919130i
\(814\) 0 0
\(815\) 1.28018 + 2.21733i 0.0448426 + 0.0776697i
\(816\) 0 0
\(817\) 32.2523 + 18.6209i 1.12837 + 0.651462i
\(818\) 0 0
\(819\) −7.44520 26.2277i −0.260156 0.916471i
\(820\) 0 0
\(821\) −24.3923 14.0829i −0.851296 0.491496i 0.00979179 0.999952i \(-0.496883\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(822\) 0 0
\(823\) 28.6428 + 49.6107i 0.998424 + 1.72932i 0.547813 + 0.836601i \(0.315460\pi\)
0.450611 + 0.892720i \(0.351206\pi\)
\(824\) 0 0
\(825\) 3.89034 + 19.7004i 0.135444 + 0.685879i
\(826\) 0 0
\(827\) 27.9805i 0.972976i −0.873687 0.486488i \(-0.838278\pi\)
0.873687 0.486488i \(-0.161722\pi\)
\(828\) 0 0
\(829\) 44.6428 + 25.7745i 1.55051 + 0.895186i 0.998100 + 0.0616119i \(0.0196241\pi\)
0.552408 + 0.833574i \(0.313709\pi\)
\(830\) 0 0
\(831\) 52.6623 10.3995i 1.82684 0.360755i
\(832\) 0 0
\(833\) −9.89881 32.3534i −0.342973 1.12098i
\(834\) 0 0
\(835\) −30.0712 + 52.0848i −1.04066 + 1.80247i
\(836\) 0 0
\(837\) −14.1771 + 28.4502i −0.490033 + 0.983384i
\(838\) 0 0
\(839\) 4.64286 + 8.04167i 0.160289 + 0.277629i 0.934972 0.354720i \(-0.115424\pi\)
−0.774683 + 0.632350i \(0.782091\pi\)
\(840\) 0 0
\(841\) 2.31247 4.00531i 0.0797403 0.138114i
\(842\) 0 0
\(843\) 10.9890 9.61027i 0.378482 0.330995i
\(844\) 0 0
\(845\) 2.18959 + 3.79248i 0.0753241 + 0.130465i
\(846\) 0 0
\(847\) 9.54426 + 21.9440i 0.327945 + 0.754006i
\(848\) 0 0
\(849\) −28.1451 + 24.6138i −0.965936 + 0.844743i
\(850\) 0 0
\(851\) −44.6922 + 25.8030i −1.53203 + 0.884516i
\(852\) 0 0
\(853\) 21.5591 12.4471i 0.738168 0.426181i −0.0832349 0.996530i \(-0.526525\pi\)
0.821403 + 0.570349i \(0.193192\pi\)
\(854\) 0 0
\(855\) −55.8517 + 72.3720i −1.91009 + 2.47507i
\(856\) 0 0
\(857\) 33.4238 1.14174 0.570868 0.821042i \(-0.306607\pi\)
0.570868 + 0.821042i \(0.306607\pi\)
\(858\) 0 0
\(859\) 33.7864i 1.15278i 0.817175 + 0.576389i \(0.195539\pi\)
−0.817175 + 0.576389i \(0.804461\pi\)
\(860\) 0 0
\(861\) −7.32743 + 0.599431i −0.249718 + 0.0204285i
\(862\) 0 0
\(863\) 9.74739 5.62766i 0.331805 0.191568i −0.324837 0.945770i \(-0.605309\pi\)
0.656642 + 0.754202i \(0.271976\pi\)
\(864\) 0 0
\(865\) 18.7615 32.4958i 0.637910 1.10489i
\(866\) 0 0
\(867\) 7.25386 + 8.29456i 0.246354 + 0.281698i
\(868\) 0 0
\(869\) 3.17821 + 1.83494i 0.107813 + 0.0622461i
\(870\) 0 0
\(871\) 17.4149i 0.590080i
\(872\) 0 0
\(873\) 20.4018 26.4365i 0.690497 0.894738i
\(874\) 0 0
\(875\) −12.6605 29.1088i −0.428003 0.984058i
\(876\) 0 0
\(877\) −41.7801 −1.41081 −0.705407 0.708803i \(-0.749236\pi\)
−0.705407 + 0.708803i \(0.749236\pi\)
\(878\) 0 0
\(879\) 14.2083 41.6670i 0.479235 1.40539i
\(880\) 0 0
\(881\) 19.2845 0.649710 0.324855 0.945764i \(-0.394685\pi\)
0.324855 + 0.945764i \(0.394685\pi\)
\(882\) 0 0
\(883\) 55.1130 1.85470 0.927350 0.374195i \(-0.122081\pi\)
0.927350 + 0.374195i \(0.122081\pi\)
\(884\) 0 0
\(885\) 26.7064 + 30.5379i 0.897727 + 1.02652i
\(886\) 0 0
\(887\) 24.9546 0.837892 0.418946 0.908011i \(-0.362400\pi\)
0.418946 + 0.908011i \(0.362400\pi\)
\(888\) 0 0
\(889\) −7.23103 16.6255i −0.242521 0.557601i
\(890\) 0 0
\(891\) −8.84972 8.94816i −0.296477 0.299775i
\(892\) 0 0
\(893\) 25.2797i 0.845952i
\(894\) 0 0
\(895\) −25.5976 14.7788i −0.855633 0.494000i
\(896\) 0 0
\(897\) 17.1423 50.2711i 0.572365 1.67850i
\(898\) 0 0
\(899\) 17.7365 30.7205i 0.591544 1.02458i
\(900\) 0 0
\(901\) 21.2786 12.2852i 0.708893 0.409279i
\(902\) 0 0
\(903\) 8.73324 18.4559i 0.290624 0.614174i
\(904\) 0 0
\(905\) 55.2407i 1.83626i
\(906\) 0 0
\(907\) 58.8709 1.95478 0.977388 0.211452i \(-0.0678192\pi\)
0.977388 + 0.211452i \(0.0678192\pi\)
\(908\) 0 0
\(909\) 16.1240 6.62660i 0.534799 0.219790i
\(910\) 0 0
\(911\) −36.6533 + 21.1618i −1.21438 + 0.701121i −0.963710 0.266952i \(-0.913983\pi\)
−0.250668 + 0.968073i \(0.580650\pi\)
\(912\) 0 0
\(913\) −9.25926 + 5.34584i −0.306437 + 0.176921i
\(914\) 0 0
\(915\) −53.7372 18.3242i −1.77650 0.605780i
\(916\) 0 0
\(917\) −0.727121 1.67179i −0.0240117 0.0552073i
\(918\) 0 0
\(919\) 16.9879 + 29.4239i 0.560379 + 0.970605i 0.997463 + 0.0711842i \(0.0226778\pi\)
−0.437084 + 0.899421i \(0.643989\pi\)
\(920\) 0 0
\(921\) −3.17066 16.0560i −0.104477 0.529063i
\(922\) 0 0
\(923\) 7.22509 12.5142i 0.237817 0.411911i
\(924\) 0 0
\(925\) −23.9634 41.5059i −0.787913 1.36470i
\(926\) 0 0
\(927\) −14.1761 34.4936i −0.465604 1.13292i
\(928\) 0 0
\(929\) −6.41930 + 11.1185i −0.210610 + 0.364788i −0.951906 0.306391i \(-0.900878\pi\)
0.741295 + 0.671179i \(0.234212\pi\)
\(930\) 0 0
\(931\) −13.1499 + 57.0127i −0.430970 + 1.86852i
\(932\) 0 0
\(933\) 10.7994 + 12.3488i 0.353556 + 0.404280i
\(934\) 0 0
\(935\) −21.3393 12.3202i −0.697868 0.402914i
\(936\) 0 0
\(937\) 28.2886i 0.924149i 0.886841 + 0.462075i \(0.152895\pi\)
−0.886841 + 0.462075i \(0.847105\pi\)
\(938\) 0 0
\(939\) −11.1809 3.81266i −0.364875 0.124421i
\(940\) 0 0
\(941\) 12.8901 + 22.3263i 0.420205 + 0.727817i 0.995959 0.0898059i \(-0.0286247\pi\)
−0.575754 + 0.817623i \(0.695291\pi\)
\(942\) 0 0
\(943\) −12.4036 7.16120i −0.403915 0.233201i
\(944\) 0 0
\(945\) 42.0427 + 27.2837i 1.36765 + 0.887540i
\(946\) 0 0
\(947\) −52.3308 30.2132i −1.70052 0.981798i −0.945227 0.326415i \(-0.894159\pi\)
−0.755297 0.655383i \(-0.772507\pi\)
\(948\) 0 0
\(949\) 1.47912 + 2.56191i 0.0480143 + 0.0831631i
\(950\) 0 0
\(951\) −47.0478 16.0432i −1.52563 0.520235i
\(952\) 0 0
\(953\) 3.83769i 0.124315i −0.998066 0.0621575i \(-0.980202\pi\)
0.998066 0.0621575i \(-0.0197981\pi\)
\(954\) 0 0
\(955\) −9.09208 5.24932i −0.294213 0.169864i
\(956\) 0 0
\(957\) 9.24562 + 10.5721i 0.298869 + 0.341746i
\(958\) 0 0
\(959\) −8.42601 + 3.66478i −0.272090 + 0.118342i
\(960\) 0 0
\(961\) 3.21124 5.56203i 0.103588 0.179420i
\(962\) 0 0
\(963\) 21.5851 27.9697i 0.695569 0.901311i
\(964\) 0 0
\(965\) 29.2988 + 50.7470i 0.943161 + 1.63360i
\(966\) 0 0
\(967\) −3.68243 + 6.37816i −0.118419 + 0.205108i −0.919141 0.393928i \(-0.871116\pi\)
0.800722 + 0.599036i \(0.204449\pi\)
\(968\) 0 0
\(969\) −13.5565 68.6493i −0.435499 2.20533i
\(970\) 0 0
\(971\) 26.6276 + 46.1204i 0.854521 + 1.48007i 0.877089 + 0.480328i \(0.159482\pi\)
−0.0225686 + 0.999745i \(0.507184\pi\)
\(972\) 0 0
\(973\) 2.65905 23.3599i 0.0852454 0.748884i
\(974\) 0 0
\(975\) 46.6871 + 15.9202i 1.49518 + 0.509853i
\(976\) 0 0
\(977\) −50.5473 + 29.1835i −1.61715 + 0.933663i −0.629500 + 0.777001i \(0.716740\pi\)
−0.987652 + 0.156662i \(0.949927\pi\)
\(978\) 0 0
\(979\) −10.7589 + 6.21166i −0.343856 + 0.198525i
\(980\) 0 0
\(981\) −3.58041 + 26.6266i −0.114314 + 0.850123i
\(982\) 0 0
\(983\) −56.9508 −1.81645 −0.908224 0.418485i \(-0.862561\pi\)
−0.908224 + 0.418485i \(0.862561\pi\)
\(984\) 0 0
\(985\) 23.2295i 0.740153i
\(986\) 0 0
\(987\) −13.8135 + 1.13003i −0.439688 + 0.0359693i
\(988\) 0 0
\(989\) 34.4475 19.8883i 1.09537 0.632410i
\(990\) 0 0
\(991\) −7.39894 + 12.8153i −0.235035 + 0.407093i −0.959283 0.282447i \(-0.908854\pi\)
0.724248 + 0.689540i \(0.242187\pi\)
\(992\) 0 0
\(993\) 0.671565 1.96942i 0.0213115 0.0624975i
\(994\) 0 0
\(995\) 20.9017 + 12.0676i 0.662628 + 0.382568i
\(996\) 0 0
\(997\) 39.0440i 1.23654i −0.785967 0.618268i \(-0.787835\pi\)
0.785967 0.618268i \(-0.212165\pi\)
\(998\) 0 0
\(999\) 26.8841 + 13.3967i 0.850574 + 0.423852i
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 504.2.cx.a.185.2 yes 48
3.2 odd 2 1512.2.cx.a.17.23 48
4.3 odd 2 1008.2.df.e.689.23 48
7.5 odd 6 504.2.bs.a.257.9 48
9.2 odd 6 504.2.bs.a.353.9 yes 48
9.7 even 3 1512.2.bs.a.521.23 48
12.11 even 2 3024.2.df.e.17.23 48
21.5 even 6 1512.2.bs.a.1097.23 48
28.19 even 6 1008.2.ca.e.257.16 48
36.7 odd 6 3024.2.ca.e.2033.23 48
36.11 even 6 1008.2.ca.e.353.16 48
63.47 even 6 inner 504.2.cx.a.425.2 yes 48
63.61 odd 6 1512.2.cx.a.89.23 48
84.47 odd 6 3024.2.ca.e.2609.23 48
252.47 odd 6 1008.2.df.e.929.23 48
252.187 even 6 3024.2.df.e.1601.23 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.9 48 7.5 odd 6
504.2.bs.a.353.9 yes 48 9.2 odd 6
504.2.cx.a.185.2 yes 48 1.1 even 1 trivial
504.2.cx.a.425.2 yes 48 63.47 even 6 inner
1008.2.ca.e.257.16 48 28.19 even 6
1008.2.ca.e.353.16 48 36.11 even 6
1008.2.df.e.689.23 48 4.3 odd 2
1008.2.df.e.929.23 48 252.47 odd 6
1512.2.bs.a.521.23 48 9.7 even 3
1512.2.bs.a.1097.23 48 21.5 even 6
1512.2.cx.a.17.23 48 3.2 odd 2
1512.2.cx.a.89.23 48 63.61 odd 6
3024.2.ca.e.2033.23 48 36.7 odd 6
3024.2.ca.e.2609.23 48 84.47 odd 6
3024.2.df.e.17.23 48 12.11 even 2
3024.2.df.e.1601.23 48 252.187 even 6