Properties

Label 1152.2.p.d.959.1
Level $1152$
Weight $2$
Character 1152.959
Analytic conductor $9.199$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,2,Mod(191,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.p (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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 16x^{14} + 178x^{12} - 1024x^{10} + 4267x^{8} - 7936x^{6} + 10594x^{4} - 2800x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 959.1
Root \(1.21704 - 0.702656i\) of defining polynomial
Character \(\chi\) \(=\) 1152.959
Dual form 1152.2.p.d.191.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.65068 + 0.524648i) q^{3} +(-1.01255 + 1.75379i) q^{5} +(-4.09406 + 2.36371i) q^{7} +(2.44949 - 1.73205i) q^{9} +O(q^{10})\) \(q+(-1.65068 + 0.524648i) q^{3} +(-1.01255 + 1.75379i) q^{5} +(-4.09406 + 2.36371i) q^{7} +(2.44949 - 1.73205i) q^{9} +(-2.93038 + 1.69185i) q^{11} +(-5.51787 - 3.18575i) q^{13} +(0.751275 - 3.42617i) q^{15} -4.87832i q^{17} +1.48393 q^{19} +(5.51787 - 6.04967i) q^{21} +(0.751275 - 1.30125i) q^{23} +(0.449490 + 0.778539i) q^{25} +(-3.13461 + 4.14418i) q^{27} +(3.49278 + 6.04967i) q^{29} +(5.93430 + 3.42617i) q^{31} +(3.94949 - 4.33013i) q^{33} -9.57348i q^{35} +2.86392i q^{37} +(10.7796 + 2.36371i) q^{39} +(4.62372 + 2.66951i) q^{41} +(2.76363 + 4.78674i) q^{43} +(0.557419 + 6.04967i) q^{45} +(-4.09406 - 7.09113i) q^{47} +(7.67423 - 13.2922i) q^{49} +(2.55940 + 8.05254i) q^{51} -4.96046 q^{53} -6.85234i q^{55} +(-2.44949 + 0.778539i) q^{57} +(-7.88242 - 4.55092i) q^{59} +(5.51787 - 3.18575i) q^{61} +(-5.93430 + 12.8810i) q^{63} +(11.1742 - 6.45145i) q^{65} +(5.48977 - 9.50857i) q^{67} +(-0.557419 + 2.54209i) q^{69} -6.68558 q^{71} +0.449490 q^{73} +(-1.15042 - 1.04930i) q^{75} +(7.99810 - 13.8531i) q^{77} +(-5.93430 + 3.42617i) q^{79} +(3.00000 - 8.48528i) q^{81} +(6.06499 - 3.50162i) q^{83} +(8.55552 + 4.93953i) q^{85} +(-8.93940 - 8.15359i) q^{87} +0.142865i q^{89} +30.1207 q^{91} +(-11.5932 - 2.54209i) q^{93} +(-1.50255 + 2.60249i) q^{95} +(0.724745 + 1.25529i) q^{97} +(-4.24755 + 9.21975i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{25} + 24 q^{33} - 24 q^{41} + 64 q^{49} + 120 q^{65} - 32 q^{73} + 48 q^{81} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.65068 + 0.524648i −0.953021 + 0.302905i
\(4\) 0 0
\(5\) −1.01255 + 1.75379i −0.452826 + 0.784317i −0.998560 0.0536411i \(-0.982917\pi\)
0.545735 + 0.837958i \(0.316251\pi\)
\(6\) 0 0
\(7\) −4.09406 + 2.36371i −1.54741 + 0.893398i −0.549072 + 0.835775i \(0.685019\pi\)
−0.998338 + 0.0576227i \(0.981648\pi\)
\(8\) 0 0
\(9\) 2.44949 1.73205i 0.816497 0.577350i
\(10\) 0 0
\(11\) −2.93038 + 1.69185i −0.883542 + 0.510113i −0.871825 0.489818i \(-0.837063\pi\)
−0.0117176 + 0.999931i \(0.503730\pi\)
\(12\) 0 0
\(13\) −5.51787 3.18575i −1.53038 0.883567i −0.999344 0.0362198i \(-0.988468\pi\)
−0.531039 0.847347i \(-0.678198\pi\)
\(14\) 0 0
\(15\) 0.751275 3.42617i 0.193978 0.884634i
\(16\) 0 0
\(17\) 4.87832i 1.18317i −0.806244 0.591583i \(-0.798503\pi\)
0.806244 0.591583i \(-0.201497\pi\)
\(18\) 0 0
\(19\) 1.48393 0.340436 0.170218 0.985406i \(-0.445553\pi\)
0.170218 + 0.985406i \(0.445553\pi\)
\(20\) 0 0
\(21\) 5.51787 6.04967i 1.20410 1.32015i
\(22\) 0 0
\(23\) 0.751275 1.30125i 0.156652 0.271328i −0.777008 0.629491i \(-0.783263\pi\)
0.933659 + 0.358163i \(0.116597\pi\)
\(24\) 0 0
\(25\) 0.449490 + 0.778539i 0.0898979 + 0.155708i
\(26\) 0 0
\(27\) −3.13461 + 4.14418i −0.603256 + 0.797548i
\(28\) 0 0
\(29\) 3.49278 + 6.04967i 0.648592 + 1.12339i 0.983459 + 0.181129i \(0.0579753\pi\)
−0.334867 + 0.942265i \(0.608691\pi\)
\(30\) 0 0
\(31\) 5.93430 + 3.42617i 1.06583 + 0.615358i 0.927040 0.374963i \(-0.122345\pi\)
0.138792 + 0.990321i \(0.455678\pi\)
\(32\) 0 0
\(33\) 3.94949 4.33013i 0.687518 0.753778i
\(34\) 0 0
\(35\) 9.57348i 1.61821i
\(36\) 0 0
\(37\) 2.86392i 0.470826i 0.971895 + 0.235413i \(0.0756442\pi\)
−0.971895 + 0.235413i \(0.924356\pi\)
\(38\) 0 0
\(39\) 10.7796 + 2.36371i 1.72612 + 0.378496i
\(40\) 0 0
\(41\) 4.62372 + 2.66951i 0.722104 + 0.416907i 0.815527 0.578719i \(-0.196447\pi\)
−0.0934223 + 0.995627i \(0.529781\pi\)
\(42\) 0 0
\(43\) 2.76363 + 4.78674i 0.421449 + 0.729971i 0.996081 0.0884405i \(-0.0281883\pi\)
−0.574632 + 0.818412i \(0.694855\pi\)
\(44\) 0 0
\(45\) 0.557419 + 6.04967i 0.0830950 + 0.901831i
\(46\) 0 0
\(47\) −4.09406 7.09113i −0.597180 1.03435i −0.993235 0.116120i \(-0.962954\pi\)
0.396055 0.918227i \(-0.370379\pi\)
\(48\) 0 0
\(49\) 7.67423 13.2922i 1.09632 1.89888i
\(50\) 0 0
\(51\) 2.55940 + 8.05254i 0.358387 + 1.12758i
\(52\) 0 0
\(53\) −4.96046 −0.681371 −0.340686 0.940177i \(-0.610659\pi\)
−0.340686 + 0.940177i \(0.610659\pi\)
\(54\) 0 0
\(55\) 6.85234i 0.923970i
\(56\) 0 0
\(57\) −2.44949 + 0.778539i −0.324443 + 0.103120i
\(58\) 0 0
\(59\) −7.88242 4.55092i −1.02620 0.592479i −0.110309 0.993897i \(-0.535184\pi\)
−0.915895 + 0.401418i \(0.868517\pi\)
\(60\) 0 0
\(61\) 5.51787 3.18575i 0.706491 0.407893i −0.103269 0.994653i \(-0.532930\pi\)
0.809761 + 0.586760i \(0.199597\pi\)
\(62\) 0 0
\(63\) −5.93430 + 12.8810i −0.747652 + 1.62285i
\(64\) 0 0
\(65\) 11.1742 6.45145i 1.38599 0.800204i
\(66\) 0 0
\(67\) 5.48977 9.50857i 0.670683 1.16166i −0.307028 0.951700i \(-0.599335\pi\)
0.977711 0.209956i \(-0.0673321\pi\)
\(68\) 0 0
\(69\) −0.557419 + 2.54209i −0.0671053 + 0.306032i
\(70\) 0 0
\(71\) −6.68558 −0.793432 −0.396716 0.917941i \(-0.629850\pi\)
−0.396716 + 0.917941i \(0.629850\pi\)
\(72\) 0 0
\(73\) 0.449490 0.0526088 0.0263044 0.999654i \(-0.491626\pi\)
0.0263044 + 0.999654i \(0.491626\pi\)
\(74\) 0 0
\(75\) −1.15042 1.04930i −0.132839 0.121162i
\(76\) 0 0
\(77\) 7.99810 13.8531i 0.911468 1.57871i
\(78\) 0 0
\(79\) −5.93430 + 3.42617i −0.667661 + 0.385474i −0.795190 0.606361i \(-0.792629\pi\)
0.127529 + 0.991835i \(0.459296\pi\)
\(80\) 0 0
\(81\) 3.00000 8.48528i 0.333333 0.942809i
\(82\) 0 0
\(83\) 6.06499 3.50162i 0.665719 0.384353i −0.128734 0.991679i \(-0.541091\pi\)
0.794452 + 0.607326i \(0.207758\pi\)
\(84\) 0 0
\(85\) 8.55552 + 4.93953i 0.927977 + 0.535768i
\(86\) 0 0
\(87\) −8.93940 8.15359i −0.958404 0.874156i
\(88\) 0 0
\(89\) 0.142865i 0.0151436i 0.999971 + 0.00757181i \(0.00241020\pi\)
−0.999971 + 0.00757181i \(0.997590\pi\)
\(90\) 0 0
\(91\) 30.1207 3.15751
\(92\) 0 0
\(93\) −11.5932 2.54209i −1.20216 0.263603i
\(94\) 0 0
\(95\) −1.50255 + 2.60249i −0.154158 + 0.267010i
\(96\) 0 0
\(97\) 0.724745 + 1.25529i 0.0735867 + 0.127456i 0.900471 0.434916i \(-0.143222\pi\)
−0.826884 + 0.562372i \(0.809889\pi\)
\(98\) 0 0
\(99\) −4.24755 + 9.21975i −0.426895 + 0.926619i
\(100\) 0 0
\(101\) −3.03765 5.26136i −0.302257 0.523525i 0.674390 0.738376i \(-0.264407\pi\)
−0.976647 + 0.214851i \(0.931073\pi\)
\(102\) 0 0
\(103\) 4.09406 + 2.36371i 0.403400 + 0.232903i 0.687950 0.725758i \(-0.258511\pi\)
−0.284550 + 0.958661i \(0.591844\pi\)
\(104\) 0 0
\(105\) 5.02270 + 15.8028i 0.490166 + 1.54219i
\(106\) 0 0
\(107\) 11.9079i 1.15118i −0.817739 0.575590i \(-0.804773\pi\)
0.817739 0.575590i \(-0.195227\pi\)
\(108\) 0 0
\(109\) 4.15122i 0.397615i 0.980039 + 0.198808i \(0.0637069\pi\)
−0.980039 + 0.198808i \(0.936293\pi\)
\(110\) 0 0
\(111\) −1.50255 4.72742i −0.142616 0.448707i
\(112\) 0 0
\(113\) −6.39898 3.69445i −0.601965 0.347545i 0.167849 0.985813i \(-0.446318\pi\)
−0.769814 + 0.638268i \(0.779651\pi\)
\(114\) 0 0
\(115\) 1.52140 + 2.63515i 0.141872 + 0.245729i
\(116\) 0 0
\(117\) −19.0339 + 1.75379i −1.75968 + 0.162138i
\(118\) 0 0
\(119\) 11.5309 + 19.9721i 1.05704 + 1.83084i
\(120\) 0 0
\(121\) 0.224745 0.389270i 0.0204314 0.0353881i
\(122\) 0 0
\(123\) −9.03284 1.98068i −0.814464 0.178592i
\(124\) 0 0
\(125\) −11.9460 −1.06848
\(126\) 0 0
\(127\) 9.45483i 0.838981i −0.907760 0.419490i \(-0.862209\pi\)
0.907760 0.419490i \(-0.137791\pi\)
\(128\) 0 0
\(129\) −7.07321 6.45145i −0.622762 0.568018i
\(130\) 0 0
\(131\) −10.5168 6.07186i −0.918854 0.530501i −0.0355850 0.999367i \(-0.511329\pi\)
−0.883269 + 0.468866i \(0.844663\pi\)
\(132\) 0 0
\(133\) −6.07529 + 3.50757i −0.526795 + 0.304145i
\(134\) 0 0
\(135\) −4.09406 9.69362i −0.352361 0.834294i
\(136\) 0 0
\(137\) −9.27526 + 5.35507i −0.792439 + 0.457515i −0.840820 0.541314i \(-0.817927\pi\)
0.0483818 + 0.998829i \(0.484594\pi\)
\(138\) 0 0
\(139\) 3.50559 6.07186i 0.297340 0.515008i −0.678186 0.734890i \(-0.737234\pi\)
0.975527 + 0.219882i \(0.0705671\pi\)
\(140\) 0 0
\(141\) 10.4783 + 9.55724i 0.882435 + 0.804865i
\(142\) 0 0
\(143\) 21.5593 1.80288
\(144\) 0 0
\(145\) −14.1464 −1.17480
\(146\) 0 0
\(147\) −5.69400 + 25.9674i −0.469634 + 2.14175i
\(148\) 0 0
\(149\) 3.49278 6.04967i 0.286139 0.495608i −0.686745 0.726898i \(-0.740961\pi\)
0.972885 + 0.231290i \(0.0742946\pi\)
\(150\) 0 0
\(151\) 2.25382 1.30125i 0.183414 0.105894i −0.405482 0.914103i \(-0.632896\pi\)
0.588896 + 0.808209i \(0.299563\pi\)
\(152\) 0 0
\(153\) −8.44949 11.9494i −0.683101 0.966050i
\(154\) 0 0
\(155\) −12.0175 + 6.93833i −0.965272 + 0.557300i
\(156\) 0 0
\(157\) 12.9586 + 7.48163i 1.03421 + 0.597099i 0.918187 0.396148i \(-0.129653\pi\)
0.116019 + 0.993247i \(0.462987\pi\)
\(158\) 0 0
\(159\) 8.18813 2.60249i 0.649361 0.206391i
\(160\) 0 0
\(161\) 7.10318i 0.559809i
\(162\) 0 0
\(163\) −3.63487 −0.284705 −0.142352 0.989816i \(-0.545467\pi\)
−0.142352 + 0.989816i \(0.545467\pi\)
\(164\) 0 0
\(165\) 3.59506 + 11.3110i 0.279875 + 0.880562i
\(166\) 0 0
\(167\) 2.59151 4.48863i 0.200537 0.347341i −0.748164 0.663513i \(-0.769065\pi\)
0.948702 + 0.316173i \(0.102398\pi\)
\(168\) 0 0
\(169\) 13.7980 + 23.8988i 1.06138 + 1.83837i
\(170\) 0 0
\(171\) 3.63487 2.57024i 0.277965 0.196551i
\(172\) 0 0
\(173\) −1.01255 1.75379i −0.0769827 0.133338i 0.824964 0.565185i \(-0.191195\pi\)
−0.901947 + 0.431847i \(0.857862\pi\)
\(174\) 0 0
\(175\) −3.68048 2.12493i −0.278218 0.160629i
\(176\) 0 0
\(177\) 15.3990 + 3.37662i 1.15746 + 0.253802i
\(178\) 0 0
\(179\) 16.5767i 1.23900i −0.784996 0.619501i \(-0.787335\pi\)
0.784996 0.619501i \(-0.212665\pi\)
\(180\) 0 0
\(181\) 22.6220i 1.68148i −0.541436 0.840742i \(-0.682119\pi\)
0.541436 0.840742i \(-0.317881\pi\)
\(182\) 0 0
\(183\) −7.43685 + 8.15359i −0.549748 + 0.602731i
\(184\) 0 0
\(185\) −5.02270 2.89986i −0.369277 0.213202i
\(186\) 0 0
\(187\) 8.25340 + 14.2953i 0.603548 + 1.04538i
\(188\) 0 0
\(189\) 3.03765 24.3758i 0.220956 1.77308i
\(190\) 0 0
\(191\) −2.59151 4.48863i −0.187515 0.324786i 0.756906 0.653524i \(-0.226710\pi\)
−0.944421 + 0.328738i \(0.893377\pi\)
\(192\) 0 0
\(193\) −0.376276 + 0.651729i −0.0270849 + 0.0469124i −0.879250 0.476360i \(-0.841956\pi\)
0.852165 + 0.523273i \(0.175289\pi\)
\(194\) 0 0
\(195\) −15.0604 + 16.5118i −1.07849 + 1.18244i
\(196\) 0 0
\(197\) 18.0213 1.28396 0.641982 0.766719i \(-0.278112\pi\)
0.641982 + 0.766719i \(0.278112\pi\)
\(198\) 0 0
\(199\) 2.12493i 0.150632i 0.997160 + 0.0753160i \(0.0239965\pi\)
−0.997160 + 0.0753160i \(0.976003\pi\)
\(200\) 0 0
\(201\) −4.07321 + 18.5758i −0.287302 + 1.31024i
\(202\) 0 0
\(203\) −28.5993 16.5118i −2.00728 1.15890i
\(204\) 0 0
\(205\) −9.36349 + 5.40602i −0.653975 + 0.377572i
\(206\) 0 0
\(207\) −0.413584 4.48863i −0.0287461 0.311982i
\(208\) 0 0
\(209\) −4.34847 + 2.51059i −0.300790 + 0.173661i
\(210\) 0 0
\(211\) −3.33884 + 5.78304i −0.229855 + 0.398121i −0.957765 0.287552i \(-0.907159\pi\)
0.727910 + 0.685673i \(0.240492\pi\)
\(212\) 0 0
\(213\) 11.0357 3.50757i 0.756157 0.240335i
\(214\) 0 0
\(215\) −11.1932 −0.763372
\(216\) 0 0
\(217\) −32.3939 −2.19904
\(218\) 0 0
\(219\) −0.741964 + 0.235824i −0.0501373 + 0.0159355i
\(220\) 0 0
\(221\) −15.5411 + 26.9179i −1.04541 + 1.81070i
\(222\) 0 0
\(223\) 15.9627 9.21605i 1.06894 0.617152i 0.141048 0.990003i \(-0.454953\pi\)
0.927891 + 0.372850i \(0.121620\pi\)
\(224\) 0 0
\(225\) 2.44949 + 1.12848i 0.163299 + 0.0752323i
\(226\) 0 0
\(227\) 17.2862 9.98022i 1.14733 0.662410i 0.199093 0.979981i \(-0.436200\pi\)
0.948235 + 0.317570i \(0.102867\pi\)
\(228\) 0 0
\(229\) −12.9586 7.48163i −0.856326 0.494400i 0.00645448 0.999979i \(-0.497945\pi\)
−0.862780 + 0.505579i \(0.831279\pi\)
\(230\) 0 0
\(231\) −5.93430 + 27.0633i −0.390448 + 1.78063i
\(232\) 0 0
\(233\) 9.75663i 0.639178i 0.947556 + 0.319589i \(0.103545\pi\)
−0.947556 + 0.319589i \(0.896455\pi\)
\(234\) 0 0
\(235\) 16.5818 1.08167
\(236\) 0 0
\(237\) 7.99810 8.76893i 0.519533 0.569603i
\(238\) 0 0
\(239\) −12.6199 + 21.8583i −0.816312 + 1.41389i 0.0920698 + 0.995753i \(0.470652\pi\)
−0.908382 + 0.418142i \(0.862682\pi\)
\(240\) 0 0
\(241\) 4.94949 + 8.57277i 0.318825 + 0.552221i 0.980243 0.197797i \(-0.0633786\pi\)
−0.661418 + 0.750017i \(0.730045\pi\)
\(242\) 0 0
\(243\) −0.500258 + 15.5804i −0.0320915 + 0.999485i
\(244\) 0 0
\(245\) 15.5411 + 26.9179i 0.992883 + 1.71972i
\(246\) 0 0
\(247\) −8.18813 4.72742i −0.520998 0.300798i
\(248\) 0 0
\(249\) −8.17423 + 8.96204i −0.518021 + 0.567946i
\(250\) 0 0
\(251\) 18.2037i 1.14901i 0.818503 + 0.574503i \(0.194804\pi\)
−0.818503 + 0.574503i \(0.805196\pi\)
\(252\) 0 0
\(253\) 5.08419i 0.319640i
\(254\) 0 0
\(255\) −16.7139 3.66495i −1.04667 0.229508i
\(256\) 0 0
\(257\) −5.05051 2.91591i −0.315042 0.181890i 0.334138 0.942524i \(-0.391555\pi\)
−0.649181 + 0.760634i \(0.724888\pi\)
\(258\) 0 0
\(259\) −6.76947 11.7251i −0.420635 0.728560i
\(260\) 0 0
\(261\) 19.0339 + 8.76893i 1.17817 + 0.542783i
\(262\) 0 0
\(263\) −14.1224 24.4608i −0.870826 1.50832i −0.861143 0.508362i \(-0.830251\pi\)
−0.00968287 0.999953i \(-0.503082\pi\)
\(264\) 0 0
\(265\) 5.02270 8.69958i 0.308542 0.534411i
\(266\) 0 0
\(267\) −0.0749536 0.235824i −0.00458708 0.0144322i
\(268\) 0 0
\(269\) 0.910261 0.0554996 0.0277498 0.999615i \(-0.491166\pi\)
0.0277498 + 0.999615i \(0.491166\pi\)
\(270\) 0 0
\(271\) 18.9097i 1.14868i 0.818617 + 0.574340i \(0.194741\pi\)
−0.818617 + 0.574340i \(0.805259\pi\)
\(272\) 0 0
\(273\) −49.7196 + 15.8028i −3.00917 + 0.956426i
\(274\) 0 0
\(275\) −2.63435 1.52094i −0.158857 0.0917163i
\(276\) 0 0
\(277\) −9.11294 + 5.26136i −0.547543 + 0.316124i −0.748131 0.663551i \(-0.769048\pi\)
0.200587 + 0.979676i \(0.435715\pi\)
\(278\) 0 0
\(279\) 20.4703 1.88614i 1.22553 0.112920i
\(280\) 0 0
\(281\) −19.0732 + 11.0119i −1.13781 + 0.656916i −0.945888 0.324493i \(-0.894806\pi\)
−0.191925 + 0.981410i \(0.561473\pi\)
\(282\) 0 0
\(283\) 10.1083 17.5081i 0.600877 1.04075i −0.391812 0.920045i \(-0.628152\pi\)
0.992689 0.120704i \(-0.0385151\pi\)
\(284\) 0 0
\(285\) 1.11484 5.08419i 0.0660372 0.301161i
\(286\) 0 0
\(287\) −25.2398 −1.48986
\(288\) 0 0
\(289\) −6.79796 −0.399880
\(290\) 0 0
\(291\) −1.85491 1.69185i −0.108737 0.0991783i
\(292\) 0 0
\(293\) 1.01255 1.75379i 0.0591537 0.102457i −0.834932 0.550353i \(-0.814493\pi\)
0.894086 + 0.447896i \(0.147826\pi\)
\(294\) 0 0
\(295\) 15.9627 9.21605i 0.929382 0.536579i
\(296\) 0 0
\(297\) 2.17423 17.4473i 0.126162 1.01240i
\(298\) 0 0
\(299\) −8.29088 + 4.78674i −0.479474 + 0.276824i
\(300\) 0 0
\(301\) −22.6289 13.0648i −1.30431 0.753043i
\(302\) 0 0
\(303\) 7.77454 + 7.09113i 0.446636 + 0.407374i
\(304\) 0 0
\(305\) 12.9029i 0.738818i
\(306\) 0 0
\(307\) 2.96786 0.169384 0.0846922 0.996407i \(-0.473009\pi\)
0.0846922 + 0.996407i \(0.473009\pi\)
\(308\) 0 0
\(309\) −7.99810 1.75379i −0.454996 0.0997694i
\(310\) 0 0
\(311\) 15.6250 27.0633i 0.886011 1.53462i 0.0414609 0.999140i \(-0.486799\pi\)
0.844550 0.535476i \(-0.179868\pi\)
\(312\) 0 0
\(313\) −1.94949 3.37662i −0.110192 0.190858i 0.805656 0.592384i \(-0.201813\pi\)
−0.915847 + 0.401526i \(0.868480\pi\)
\(314\) 0 0
\(315\) −16.5818 23.4501i −0.934276 1.32127i
\(316\) 0 0
\(317\) −8.45323 14.6414i −0.474781 0.822345i 0.524802 0.851224i \(-0.324139\pi\)
−0.999583 + 0.0288797i \(0.990806\pi\)
\(318\) 0 0
\(319\) −20.4703 11.8185i −1.14612 0.661711i
\(320\) 0 0
\(321\) 6.24745 + 19.6561i 0.348699 + 1.09710i
\(322\) 0 0
\(323\) 7.23907i 0.402792i
\(324\) 0 0
\(325\) 5.72784i 0.317723i
\(326\) 0 0
\(327\) −2.17793 6.85234i −0.120440 0.378935i
\(328\) 0 0
\(329\) 33.5227 + 19.3543i 1.84817 + 1.06704i
\(330\) 0 0
\(331\) 16.8778 + 29.2332i 0.927687 + 1.60680i 0.787182 + 0.616721i \(0.211539\pi\)
0.140505 + 0.990080i \(0.455127\pi\)
\(332\) 0 0
\(333\) 4.96046 + 7.01514i 0.271831 + 0.384428i
\(334\) 0 0
\(335\) 11.1173 + 19.2558i 0.607405 + 1.05206i
\(336\) 0 0
\(337\) −0.0505103 + 0.0874863i −0.00275147 + 0.00476568i −0.867398 0.497615i \(-0.834209\pi\)
0.864646 + 0.502381i \(0.167542\pi\)
\(338\) 0 0
\(339\) 12.5010 + 2.74115i 0.678959 + 0.148879i
\(340\) 0 0
\(341\) −23.1863 −1.25561
\(342\) 0 0
\(343\) 39.4667i 2.13100i
\(344\) 0 0
\(345\) −3.89388 3.55159i −0.209639 0.191211i
\(346\) 0 0
\(347\) 9.60805 + 5.54721i 0.515787 + 0.297790i 0.735209 0.677840i \(-0.237084\pi\)
−0.219422 + 0.975630i \(0.570417\pi\)
\(348\) 0 0
\(349\) 17.9190 10.3455i 0.959183 0.553784i 0.0632614 0.997997i \(-0.479850\pi\)
0.895921 + 0.444213i \(0.146516\pi\)
\(350\) 0 0
\(351\) 30.4987 12.8810i 1.62790 0.687537i
\(352\) 0 0
\(353\) −4.62372 + 2.66951i −0.246096 + 0.142084i −0.617975 0.786197i \(-0.712047\pi\)
0.371879 + 0.928281i \(0.378714\pi\)
\(354\) 0 0
\(355\) 6.76947 11.7251i 0.359286 0.622302i
\(356\) 0 0
\(357\) −29.5122 26.9179i −1.56195 1.42465i
\(358\) 0 0
\(359\) 29.7474 1.57001 0.785004 0.619491i \(-0.212661\pi\)
0.785004 + 0.619491i \(0.212661\pi\)
\(360\) 0 0
\(361\) −16.7980 −0.884103
\(362\) 0 0
\(363\) −0.166753 + 0.760471i −0.00875224 + 0.0399144i
\(364\) 0 0
\(365\) −0.455130 + 0.788309i −0.0238226 + 0.0412620i
\(366\) 0 0
\(367\) −10.4419 + 6.02866i −0.545065 + 0.314694i −0.747129 0.664679i \(-0.768568\pi\)
0.202064 + 0.979372i \(0.435235\pi\)
\(368\) 0 0
\(369\) 15.9495 1.46959i 0.830297 0.0765039i
\(370\) 0 0
\(371\) 20.3084 11.7251i 1.05436 0.608735i
\(372\) 0 0
\(373\) 10.4783 + 6.04967i 0.542547 + 0.313240i 0.746111 0.665822i \(-0.231919\pi\)
−0.203563 + 0.979062i \(0.565252\pi\)
\(374\) 0 0
\(375\) 19.7190 6.26745i 1.01829 0.323649i
\(376\) 0 0
\(377\) 44.5084i 2.29230i
\(378\) 0 0
\(379\) 7.60324 0.390552 0.195276 0.980748i \(-0.437440\pi\)
0.195276 + 0.980748i \(0.437440\pi\)
\(380\) 0 0
\(381\) 4.96046 + 15.6069i 0.254132 + 0.799566i
\(382\) 0 0
\(383\) 12.2822 21.2734i 0.627591 1.08702i −0.360443 0.932781i \(-0.617375\pi\)
0.988034 0.154238i \(-0.0492921\pi\)
\(384\) 0 0
\(385\) 16.1969 + 28.0539i 0.825472 + 1.42976i
\(386\) 0 0
\(387\) 15.0604 + 6.93833i 0.765561 + 0.352695i
\(388\) 0 0
\(389\) −12.9586 22.4449i −0.657025 1.13800i −0.981382 0.192066i \(-0.938481\pi\)
0.324357 0.945935i \(-0.394852\pi\)
\(390\) 0 0
\(391\) −6.34789 3.66495i −0.321026 0.185345i
\(392\) 0 0
\(393\) 20.5454 + 4.50510i 1.03638 + 0.227252i
\(394\) 0 0
\(395\) 13.8767i 0.698211i
\(396\) 0 0
\(397\) 23.9094i 1.19998i 0.800009 + 0.599988i \(0.204828\pi\)
−0.800009 + 0.599988i \(0.795172\pi\)
\(398\) 0 0
\(399\) 8.18813 8.97727i 0.409919 0.449426i
\(400\) 0 0
\(401\) −16.6237 9.59771i −0.830149 0.479287i 0.0237546 0.999718i \(-0.492438\pi\)
−0.853904 + 0.520431i \(0.825771\pi\)
\(402\) 0 0
\(403\) −21.8298 37.8104i −1.08742 1.88347i
\(404\) 0 0
\(405\) 11.8437 + 13.8531i 0.588519 + 0.688367i
\(406\) 0 0
\(407\) −4.84534 8.39237i −0.240174 0.415994i
\(408\) 0 0
\(409\) 17.3990 30.1359i 0.860324 1.49013i −0.0112920 0.999936i \(-0.503594\pi\)
0.871616 0.490189i \(-0.163072\pi\)
\(410\) 0 0
\(411\) 12.5010 13.7057i 0.616627 0.676055i
\(412\) 0 0
\(413\) 43.0282 2.11728
\(414\) 0 0
\(415\) 14.1823i 0.696179i
\(416\) 0 0
\(417\) −2.60102 + 11.8619i −0.127373 + 0.580880i
\(418\) 0 0
\(419\) 30.8252 + 17.7969i 1.50591 + 0.869437i 0.999976 + 0.00686390i \(0.00218486\pi\)
0.505933 + 0.862573i \(0.331148\pi\)
\(420\) 0 0
\(421\) 12.7080 7.33697i 0.619350 0.357582i −0.157266 0.987556i \(-0.550268\pi\)
0.776616 + 0.629974i \(0.216935\pi\)
\(422\) 0 0
\(423\) −22.3106 10.2785i −1.08478 0.499758i
\(424\) 0 0
\(425\) 3.79796 2.19275i 0.184228 0.106364i
\(426\) 0 0
\(427\) −15.0604 + 26.0853i −0.728821 + 1.26236i
\(428\) 0 0
\(429\) −35.5875 + 11.3110i −1.71818 + 0.546101i
\(430\) 0 0
\(431\) 3.00510 0.144750 0.0723752 0.997377i \(-0.476942\pi\)
0.0723752 + 0.997377i \(0.476942\pi\)
\(432\) 0 0
\(433\) −26.2474 −1.26137 −0.630686 0.776038i \(-0.717226\pi\)
−0.630686 + 0.776038i \(0.717226\pi\)
\(434\) 0 0
\(435\) 23.3512 7.42189i 1.11961 0.355852i
\(436\) 0 0
\(437\) 1.11484 1.93095i 0.0533299 0.0923701i
\(438\) 0 0
\(439\) −10.4419 + 6.02866i −0.498367 + 0.287732i −0.728039 0.685536i \(-0.759568\pi\)
0.229672 + 0.973268i \(0.426235\pi\)
\(440\) 0 0
\(441\) −4.22474 45.8512i −0.201178 2.18339i
\(442\) 0 0
\(443\) −31.7339 + 18.3216i −1.50772 + 0.870484i −0.507764 + 0.861496i \(0.669528\pi\)
−0.999960 + 0.00898805i \(0.997139\pi\)
\(444\) 0 0
\(445\) −0.250554 0.144657i −0.0118774 0.00685742i
\(446\) 0 0
\(447\) −2.59151 + 11.8185i −0.122574 + 0.558998i
\(448\) 0 0
\(449\) 37.8980i 1.78852i 0.447549 + 0.894259i \(0.352297\pi\)
−0.447549 + 0.894259i \(0.647703\pi\)
\(450\) 0 0
\(451\) −18.0657 −0.850680
\(452\) 0 0
\(453\) −3.03765 + 3.33040i −0.142721 + 0.156476i
\(454\) 0 0
\(455\) −30.4987 + 52.8253i −1.42980 + 2.47649i
\(456\) 0 0
\(457\) 12.0732 + 20.9114i 0.564761 + 0.978195i 0.997072 + 0.0764703i \(0.0243650\pi\)
−0.432311 + 0.901725i \(0.642302\pi\)
\(458\) 0 0
\(459\) 20.2166 + 15.2916i 0.943631 + 0.713751i
\(460\) 0 0
\(461\) −5.51787 9.55724i −0.256993 0.445125i 0.708442 0.705769i \(-0.249398\pi\)
−0.965435 + 0.260644i \(0.916065\pi\)
\(462\) 0 0
\(463\) −5.93430 3.42617i −0.275790 0.159228i 0.355726 0.934590i \(-0.384234\pi\)
−0.631516 + 0.775363i \(0.717567\pi\)
\(464\) 0 0
\(465\) 16.1969 17.7579i 0.751115 0.823505i
\(466\) 0 0
\(467\) 20.7739i 0.961302i −0.876912 0.480651i \(-0.840400\pi\)
0.876912 0.480651i \(-0.159600\pi\)
\(468\) 0 0
\(469\) 51.9049i 2.39675i
\(470\) 0 0
\(471\) −25.3157 5.55110i −1.16648 0.255781i
\(472\) 0 0
\(473\) −16.1969 9.35131i −0.744736 0.429974i
\(474\) 0 0
\(475\) 0.667010 + 1.15530i 0.0306045 + 0.0530086i
\(476\) 0 0
\(477\) −12.1506 + 8.59176i −0.556337 + 0.393390i
\(478\) 0 0
\(479\) −4.43175 7.67602i −0.202492 0.350726i 0.746839 0.665005i \(-0.231571\pi\)
−0.949331 + 0.314279i \(0.898237\pi\)
\(480\) 0 0
\(481\) 9.12372 15.8028i 0.416006 0.720544i
\(482\) 0 0
\(483\) −3.72666 11.7251i −0.169569 0.533509i
\(484\) 0 0
\(485\) −2.93536 −0.133288
\(486\) 0 0
\(487\) 18.9097i 0.856879i −0.903571 0.428439i \(-0.859064\pi\)
0.903571 0.428439i \(-0.140936\pi\)
\(488\) 0 0
\(489\) 6.00000 1.90702i 0.271329 0.0862386i
\(490\) 0 0
\(491\) 10.1083 + 5.83604i 0.456182 + 0.263377i 0.710437 0.703760i \(-0.248497\pi\)
−0.254256 + 0.967137i \(0.581830\pi\)
\(492\) 0 0
\(493\) 29.5122 17.0389i 1.32916 0.767392i
\(494\) 0 0
\(495\) −11.8686 16.7847i −0.533454 0.754418i
\(496\) 0 0
\(497\) 27.3712 15.8028i 1.22776 0.708850i
\(498\) 0 0
\(499\) −14.0767 + 24.3815i −0.630159 + 1.09147i 0.357360 + 0.933967i \(0.383677\pi\)
−0.987519 + 0.157500i \(0.949656\pi\)
\(500\) 0 0
\(501\) −1.92281 + 8.76893i −0.0859048 + 0.391767i
\(502\) 0 0
\(503\) −33.4279 −1.49048 −0.745238 0.666799i \(-0.767664\pi\)
−0.745238 + 0.666799i \(0.767664\pi\)
\(504\) 0 0
\(505\) 12.3031 0.547479
\(506\) 0 0
\(507\) −35.3144 32.2102i −1.56837 1.43050i
\(508\) 0 0
\(509\) 11.5932 20.0800i 0.513858 0.890028i −0.486013 0.873952i \(-0.661549\pi\)
0.999871 0.0160766i \(-0.00511756\pi\)
\(510\) 0 0
\(511\) −1.84024 + 1.06246i −0.0814074 + 0.0470006i
\(512\) 0 0
\(513\) −4.65153 + 6.14966i −0.205370 + 0.271514i
\(514\) 0 0
\(515\) −8.29088 + 4.78674i −0.365340 + 0.210929i
\(516\) 0 0
\(517\) 23.9943 + 13.8531i 1.05527 + 0.609260i
\(518\) 0 0
\(519\) 2.59151 + 2.36371i 0.113755 + 0.103755i
\(520\) 0 0
\(521\) 27.0771i 1.18627i −0.805103 0.593135i \(-0.797890\pi\)
0.805103 0.593135i \(-0.202110\pi\)
\(522\) 0 0
\(523\) −36.9820 −1.61711 −0.808554 0.588421i \(-0.799750\pi\)
−0.808554 + 0.588421i \(0.799750\pi\)
\(524\) 0 0
\(525\) 7.19013 + 1.57662i 0.313803 + 0.0688092i
\(526\) 0 0
\(527\) 16.7139 28.9494i 0.728071 1.26106i
\(528\) 0 0
\(529\) 10.3712 + 17.9634i 0.450921 + 0.781017i
\(530\) 0 0
\(531\) −27.1903 + 2.50533i −1.17996 + 0.108722i
\(532\) 0 0
\(533\) −17.0088 29.4600i −0.736731 1.27606i
\(534\) 0 0
\(535\) 20.8839 + 12.0573i 0.902890 + 0.521284i
\(536\) 0 0
\(537\) 8.69694 + 27.3629i 0.375301 + 1.18079i
\(538\) 0 0
\(539\) 51.9348i 2.23699i
\(540\) 0 0
\(541\) 43.9568i 1.88985i −0.327287 0.944925i \(-0.606134\pi\)
0.327287 0.944925i \(-0.393866\pi\)
\(542\) 0 0
\(543\) 11.8686 + 37.3418i 0.509331 + 1.60249i
\(544\) 0 0
\(545\) −7.28036 4.20332i −0.311856 0.180050i
\(546\) 0 0
\(547\) 6.23174 + 10.7937i 0.266450 + 0.461505i 0.967942 0.251172i \(-0.0808160\pi\)
−0.701493 + 0.712677i \(0.747483\pi\)
\(548\) 0 0
\(549\) 7.99810 17.3607i 0.341351 0.740936i
\(550\) 0 0
\(551\) 5.18303 + 8.97727i 0.220804 + 0.382444i
\(552\) 0 0
\(553\) 16.1969 28.0539i 0.688764 1.19297i
\(554\) 0 0
\(555\) 9.81228 + 2.15159i 0.416508 + 0.0913299i
\(556\) 0 0
\(557\) 13.0608 0.553406 0.276703 0.960956i \(-0.410758\pi\)
0.276703 + 0.960956i \(0.410758\pi\)
\(558\) 0 0
\(559\) 35.2168i 1.48951i
\(560\) 0 0
\(561\) −21.1237 19.2669i −0.891844 0.813447i
\(562\) 0 0
\(563\) 9.19959 + 5.31139i 0.387717 + 0.223848i 0.681170 0.732125i \(-0.261471\pi\)
−0.293454 + 0.955973i \(0.594805\pi\)
\(564\) 0 0
\(565\) 12.9586 7.48163i 0.545171 0.314754i
\(566\) 0 0
\(567\) 7.77454 + 41.8304i 0.326500 + 1.75671i
\(568\) 0 0
\(569\) 15.7020 9.06558i 0.658264 0.380049i −0.133351 0.991069i \(-0.542574\pi\)
0.791615 + 0.611020i \(0.209241\pi\)
\(570\) 0 0
\(571\) 8.12412 14.0714i 0.339984 0.588870i −0.644445 0.764650i \(-0.722912\pi\)
0.984429 + 0.175781i \(0.0562450\pi\)
\(572\) 0 0
\(573\) 6.63271 + 6.04967i 0.277085 + 0.252728i
\(574\) 0 0
\(575\) 1.35076 0.0563306
\(576\) 0 0
\(577\) −6.44949 −0.268496 −0.134248 0.990948i \(-0.542862\pi\)
−0.134248 + 0.990948i \(0.542862\pi\)
\(578\) 0 0
\(579\) 0.279183 1.27321i 0.0116024 0.0529127i
\(580\) 0 0
\(581\) −16.5536 + 28.6717i −0.686760 + 1.18950i
\(582\) 0 0
\(583\) 14.5360 8.39237i 0.602020 0.347576i
\(584\) 0 0
\(585\) 16.1969 35.1571i 0.669661 1.45357i
\(586\) 0 0
\(587\) 1.20474 0.695560i 0.0497251 0.0287088i −0.474931 0.880023i \(-0.657527\pi\)
0.524656 + 0.851314i \(0.324194\pi\)
\(588\) 0 0
\(589\) 8.80607 + 5.08419i 0.362848 + 0.209490i
\(590\) 0 0
\(591\) −29.7474 + 9.45483i −1.22364 + 0.388920i
\(592\) 0 0
\(593\) 2.04989i 0.0841788i −0.999114 0.0420894i \(-0.986599\pi\)
0.999114 0.0420894i \(-0.0134014\pi\)
\(594\) 0 0
\(595\) −46.7025 −1.91461
\(596\) 0 0
\(597\) −1.11484 3.50757i −0.0456272 0.143555i
\(598\) 0 0
\(599\) 2.59151 4.48863i 0.105886 0.183401i −0.808214 0.588889i \(-0.799565\pi\)
0.914100 + 0.405489i \(0.132899\pi\)
\(600\) 0 0
\(601\) −16.7247 28.9681i −0.682217 1.18163i −0.974303 0.225242i \(-0.927683\pi\)
0.292086 0.956392i \(-0.405651\pi\)
\(602\) 0 0
\(603\) −3.02218 32.7997i −0.123073 1.33571i
\(604\) 0 0
\(605\) 0.455130 + 0.788309i 0.0185037 + 0.0320493i
\(606\) 0 0
\(607\) 32.3389 + 18.6709i 1.31260 + 0.757828i 0.982525 0.186128i \(-0.0595940\pi\)
0.330071 + 0.943956i \(0.392927\pi\)
\(608\) 0 0
\(609\) 55.8712 + 12.2512i 2.26401 + 0.496442i
\(610\) 0 0
\(611\) 52.1706i 2.11060i
\(612\) 0 0
\(613\) 1.57662i 0.0636790i −0.999493 0.0318395i \(-0.989863\pi\)
0.999493 0.0318395i \(-0.0101365\pi\)
\(614\) 0 0
\(615\) 12.6199 13.8361i 0.508883 0.557927i
\(616\) 0 0
\(617\) 10.3763 + 5.99075i 0.417733 + 0.241178i 0.694107 0.719872i \(-0.255799\pi\)
−0.276374 + 0.961050i \(0.589133\pi\)
\(618\) 0 0
\(619\) −16.5443 28.6555i −0.664971 1.15176i −0.979293 0.202447i \(-0.935111\pi\)
0.314323 0.949316i \(-0.398223\pi\)
\(620\) 0 0
\(621\) 3.03765 + 7.19231i 0.121897 + 0.288618i
\(622\) 0 0
\(623\) −0.337690 0.584897i −0.0135293 0.0234334i
\(624\) 0 0
\(625\) 9.84847 17.0580i 0.393939 0.682322i
\(626\) 0 0
\(627\) 5.86076 6.42559i 0.234056 0.256614i
\(628\) 0 0
\(629\) 13.9711 0.557065
\(630\) 0 0
\(631\) 46.3190i 1.84393i −0.387271 0.921966i \(-0.626582\pi\)
0.387271 0.921966i \(-0.373418\pi\)
\(632\) 0 0
\(633\) 2.47730 11.2977i 0.0984637 0.449041i
\(634\) 0 0
\(635\) 16.5818 + 9.57348i 0.658027 + 0.379912i
\(636\) 0 0
\(637\) −84.6909 + 48.8963i −3.35558 + 1.93734i
\(638\) 0 0
\(639\) −16.3763 + 11.5798i −0.647835 + 0.458088i
\(640\) 0 0
\(641\) 29.2980 16.9152i 1.15720 0.668110i 0.206568 0.978432i \(-0.433770\pi\)
0.950631 + 0.310323i \(0.100437\pi\)
\(642\) 0 0
\(643\) −7.71567 + 13.3639i −0.304276 + 0.527022i −0.977100 0.212781i \(-0.931748\pi\)
0.672824 + 0.739803i \(0.265081\pi\)
\(644\) 0 0
\(645\) 18.4764 5.87250i 0.727509 0.231229i
\(646\) 0 0
\(647\) 49.1288 1.93145 0.965725 0.259566i \(-0.0835795\pi\)
0.965725 + 0.259566i \(0.0835795\pi\)
\(648\) 0 0
\(649\) 30.7980 1.20893
\(650\) 0 0
\(651\) 53.4719 16.9954i 2.09573 0.666101i
\(652\) 0 0
\(653\) −14.0734 + 24.3758i −0.550735 + 0.953900i 0.447487 + 0.894290i \(0.352319\pi\)
−0.998222 + 0.0596098i \(0.981014\pi\)
\(654\) 0 0
\(655\) 21.2975 12.2961i 0.832162 0.480449i
\(656\) 0 0
\(657\) 1.10102 0.778539i 0.0429549 0.0303737i
\(658\) 0 0
\(659\) −2.33832 + 1.35003i −0.0910881 + 0.0525897i −0.544852 0.838532i \(-0.683414\pi\)
0.453764 + 0.891122i \(0.350081\pi\)
\(660\) 0 0
\(661\) −33.9152 19.5810i −1.31915 0.761611i −0.335557 0.942020i \(-0.608925\pi\)
−0.983592 + 0.180409i \(0.942258\pi\)
\(662\) 0 0
\(663\) 11.5309 52.5865i 0.447824 2.04229i
\(664\) 0 0
\(665\) 14.2064i 0.550899i
\(666\) 0 0
\(667\) 10.4961 0.406412
\(668\) 0 0
\(669\) −21.5141 + 23.5875i −0.831782 + 0.911946i
\(670\) 0 0
\(671\) −10.7796 + 18.6709i −0.416143 + 0.720781i
\(672\) 0 0
\(673\) 2.07321 + 3.59091i 0.0799165 + 0.138419i 0.903214 0.429191i \(-0.141201\pi\)
−0.823297 + 0.567611i \(0.807868\pi\)
\(674\) 0 0
\(675\) −4.63538 0.577648i −0.178416 0.0222337i
\(676\) 0 0
\(677\) 14.0734 + 24.3758i 0.540885 + 0.936839i 0.998854 + 0.0478713i \(0.0152437\pi\)
−0.457969 + 0.888968i \(0.651423\pi\)
\(678\) 0 0
\(679\) −5.93430 3.42617i −0.227738 0.131484i
\(680\) 0 0
\(681\) −23.2980 + 25.5433i −0.892780 + 0.978823i
\(682\) 0 0
\(683\) 3.51353i 0.134442i 0.997738 + 0.0672208i \(0.0214132\pi\)
−0.997738 + 0.0672208i \(0.978587\pi\)
\(684\) 0 0
\(685\) 21.6891i 0.828697i
\(686\) 0 0
\(687\) 25.3157 + 5.55110i 0.965852 + 0.211788i
\(688\) 0 0
\(689\) 27.3712 + 15.8028i 1.04276 + 0.602037i
\(690\) 0 0
\(691\) −6.47344 11.2123i −0.246261 0.426537i 0.716224 0.697870i \(-0.245869\pi\)
−0.962486 + 0.271333i \(0.912536\pi\)
\(692\) 0 0
\(693\) −4.40304 47.7862i −0.167258 1.81525i
\(694\) 0 0
\(695\) 7.09916 + 12.2961i 0.269287 + 0.466418i
\(696\) 0 0
\(697\) 13.0227 22.5560i 0.493270 0.854369i
\(698\) 0 0
\(699\) −5.11879 16.1051i −0.193611 0.609150i
\(700\) 0 0
\(701\) −33.3118 −1.25817 −0.629085 0.777336i \(-0.716570\pi\)
−0.629085 + 0.777336i \(0.716570\pi\)
\(702\) 0 0
\(703\) 4.24985i 0.160286i
\(704\) 0 0
\(705\) −27.3712 + 8.69958i −1.03086 + 0.327645i
\(706\) 0 0
\(707\) 24.8726 + 14.3602i 0.935432 + 0.540072i
\(708\) 0 0
\(709\) −7.99810 + 4.61771i −0.300375 + 0.173422i −0.642611 0.766192i \(-0.722149\pi\)
0.342236 + 0.939614i \(0.388816\pi\)
\(710\) 0 0
\(711\) −8.60171 + 18.6709i −0.322589 + 0.700213i
\(712\) 0 0
\(713\) 8.91658 5.14799i 0.333929 0.192794i
\(714\) 0 0
\(715\) −21.8298 + 37.8104i −0.816389 + 1.41403i
\(716\) 0 0
\(717\) 9.36349 42.7020i 0.349686 1.59474i
\(718\) 0 0
\(719\) −32.7525 −1.22146 −0.610731 0.791838i \(-0.709124\pi\)
−0.610731 + 0.791838i \(0.709124\pi\)
\(720\) 0 0
\(721\) −22.3485 −0.832300
\(722\) 0 0
\(723\) −12.6677 11.5542i −0.471117 0.429704i
\(724\) 0 0
\(725\) −3.13993 + 5.43853i −0.116614 + 0.201982i
\(726\) 0 0
\(727\) −25.9910 + 15.0059i −0.963954 + 0.556539i −0.897388 0.441243i \(-0.854538\pi\)
−0.0665663 + 0.997782i \(0.521204\pi\)
\(728\) 0 0
\(729\) −7.34847 25.9808i −0.272166 0.962250i
\(730\) 0 0
\(731\) 23.3512 13.4818i 0.863676 0.498644i
\(732\) 0 0
\(733\) −4.40304 2.54209i −0.162630 0.0938944i 0.416476 0.909147i \(-0.363265\pi\)
−0.579106 + 0.815252i \(0.696598\pi\)
\(734\) 0 0
\(735\) −39.7758 36.2793i −1.46715 1.33818i
\(736\) 0 0
\(737\) 37.1516i 1.36850i
\(738\) 0 0
\(739\) −12.8719 −0.473502 −0.236751 0.971570i \(-0.576083\pi\)
−0.236751 + 0.971570i \(0.576083\pi\)
\(740\) 0 0
\(741\) 15.9962 + 3.50757i 0.587635 + 0.128854i
\(742\) 0 0
\(743\) −4.09406 + 7.09113i −0.150197 + 0.260148i −0.931300 0.364254i \(-0.881324\pi\)
0.781103 + 0.624402i \(0.214657\pi\)
\(744\) 0 0
\(745\) 7.07321 + 12.2512i 0.259143 + 0.448848i
\(746\) 0 0
\(747\) 8.79114 19.0820i 0.321651 0.698176i
\(748\) 0 0
\(749\) 28.1468 + 48.7517i 1.02846 + 1.78135i
\(750\) 0 0
\(751\) 11.4550 + 6.61356i 0.418000 + 0.241332i 0.694221 0.719762i \(-0.255749\pi\)
−0.276221 + 0.961094i \(0.589082\pi\)
\(752\) 0 0
\(753\) −9.55051 30.0484i −0.348040 1.09503i
\(754\) 0 0
\(755\) 5.27030i 0.191806i
\(756\) 0 0
\(757\) 48.3973i 1.75903i −0.475870 0.879516i \(-0.657867\pi\)
0.475870 0.879516i \(-0.342133\pi\)
\(758\) 0 0
\(759\) −2.66741 8.39237i −0.0968208 0.304624i
\(760\) 0 0
\(761\) 20.9722 + 12.1083i 0.760241 + 0.438926i 0.829382 0.558681i \(-0.188693\pi\)
−0.0691410 + 0.997607i \(0.522026\pi\)
\(762\) 0 0
\(763\) −9.81228 16.9954i −0.355228 0.615274i
\(764\) 0 0
\(765\) 29.5122 2.71926i 1.06702 0.0983152i
\(766\) 0 0
\(767\) 28.9961 + 50.2228i 1.04699 + 1.81344i
\(768\) 0 0
\(769\) 13.3990 23.2077i 0.483180 0.836892i −0.516634 0.856206i \(-0.672815\pi\)
0.999813 + 0.0193149i \(0.00614850\pi\)
\(770\) 0 0
\(771\) 9.86660 + 2.16350i 0.355337 + 0.0779166i
\(772\) 0 0
\(773\) 0.910261 0.0327398 0.0163699 0.999866i \(-0.494789\pi\)
0.0163699 + 0.999866i \(0.494789\pi\)
\(774\) 0 0
\(775\) 6.16011i 0.221278i
\(776\) 0 0
\(777\) 17.3258 + 15.8028i 0.621558 + 0.566921i
\(778\) 0 0
\(779\) 6.86127 + 3.96136i 0.245831 + 0.141930i
\(780\) 0 0
\(781\) 19.5913 11.3110i 0.701031 0.404740i
\(782\) 0 0
\(783\) −36.0194 4.48863i −1.28723 0.160411i
\(784\) 0 0
\(785\) −26.2423 + 15.1510i −0.936629 + 0.540763i
\(786\) 0 0
\(787\) −6.47344 + 11.2123i −0.230753 + 0.399677i −0.958030 0.286668i \(-0.907452\pi\)
0.727277 + 0.686344i \(0.240786\pi\)
\(788\) 0 0
\(789\) 36.1449 + 32.9676i 1.28679 + 1.17368i
\(790\) 0 0
\(791\) 34.9304 1.24198
\(792\) 0 0
\(793\) −40.5959 −1.44160
\(794\) 0 0
\(795\) −3.72666 + 16.9954i −0.132171 + 0.602764i
\(796\) 0 0
\(797\) 12.5034 21.6566i 0.442894 0.767115i −0.555009 0.831845i \(-0.687285\pi\)
0.997903 + 0.0647294i \(0.0206184\pi\)
\(798\) 0 0
\(799\) −34.5927 + 19.9721i −1.22380 + 0.706563i
\(800\) 0 0
\(801\) 0.247449 + 0.349945i 0.00874317 + 0.0123647i
\(802\) 0 0
\(803\) −1.31718 + 0.760471i −0.0464821 + 0.0268365i
\(804\) 0 0
\(805\) −12.4575 7.19231i −0.439067 0.253496i
\(806\) 0 0
\(807\) −1.50255 + 0.477566i −0.0528922 + 0.0168111i
\(808\) 0 0
\(809\) 34.7839i 1.22294i −0.791269 0.611468i \(-0.790579\pi\)
0.791269 0.611468i \(-0.209421\pi\)
\(810\) 0 0
\(811\) −50.6708 −1.77929 −0.889647 0.456650i \(-0.849049\pi\)
−0.889647 + 0.456650i \(0.849049\pi\)
\(812\) 0 0
\(813\) −9.92091 31.2138i −0.347942 1.09472i
\(814\) 0 0
\(815\) 3.68048 6.37478i 0.128922 0.223299i
\(816\) 0 0
\(817\) 4.10102 + 7.10318i 0.143477 + 0.248509i
\(818\) 0 0
\(819\) 73.7803 52.1706i 2.57809 1.82299i
\(820\) 0 0
\(821\) 8.45323 + 14.6414i 0.295020 + 0.510989i 0.974990 0.222251i \(-0.0713404\pi\)
−0.679970 + 0.733240i \(0.738007\pi\)
\(822\) 0 0
\(823\) −26.8182 15.4835i −0.934824 0.539721i −0.0464898 0.998919i \(-0.514804\pi\)
−0.888334 + 0.459198i \(0.848137\pi\)
\(824\) 0 0
\(825\) 5.14643 + 1.12848i 0.179176 + 0.0392888i
\(826\) 0 0
\(827\) 11.4362i 0.397677i 0.980032 + 0.198839i \(0.0637170\pi\)
−0.980032 + 0.198839i \(0.936283\pi\)
\(828\) 0 0
\(829\) 1.57662i 0.0547582i 0.999625 + 0.0273791i \(0.00871613\pi\)
−0.999625 + 0.0273791i \(0.991284\pi\)
\(830\) 0 0
\(831\) 12.2822 13.4659i 0.426064 0.467127i
\(832\) 0 0
\(833\) −64.8434 37.4373i −2.24669 1.29713i
\(834\) 0 0
\(835\) 5.24807 + 9.08992i 0.181617 + 0.314570i
\(836\) 0 0
\(837\) −32.8004 + 13.8531i −1.13375 + 0.478834i
\(838\) 0 0
\(839\) −20.4703 35.4556i −0.706714 1.22406i −0.966069 0.258282i \(-0.916844\pi\)
0.259356 0.965782i \(-0.416490\pi\)
\(840\) 0 0
\(841\) −9.89898 + 17.1455i −0.341344 + 0.591225i
\(842\) 0 0
\(843\) 25.7064 28.1839i 0.885375 0.970704i
\(844\) 0 0
\(845\) −55.8844 −1.92248
\(846\) 0 0
\(847\) 2.12493i 0.0730133i
\(848\) 0 0
\(849\) −7.50000 + 34.2036i −0.257399 + 1.17386i
\(850\) 0 0
\(851\) 3.72666 + 2.15159i 0.127748 + 0.0737556i
\(852\) 0 0
\(853\) −19.2844 + 11.1339i −0.660285 + 0.381216i −0.792386 0.610020i \(-0.791161\pi\)
0.132100 + 0.991236i \(0.457828\pi\)
\(854\) 0 0
\(855\) 0.827169 + 8.97727i 0.0282886 + 0.307016i
\(856\) 0 0
\(857\) −46.3207 + 26.7432i −1.58228 + 0.913532i −0.587758 + 0.809037i \(0.699989\pi\)
−0.994525 + 0.104495i \(0.966677\pi\)
\(858\) 0 0
\(859\) 17.6197 30.5183i 0.601178 1.04127i −0.391465 0.920193i \(-0.628032\pi\)
0.992643 0.121078i \(-0.0386351\pi\)
\(860\) 0 0
\(861\) 41.6628 13.2420i 1.41986 0.451285i
\(862\) 0 0
\(863\) 3.00510 0.102295 0.0511474 0.998691i \(-0.483712\pi\)
0.0511474 + 0.998691i \(0.483712\pi\)
\(864\) 0 0
\(865\) 4.10102 0.139439
\(866\) 0 0
\(867\) 11.2213 3.56653i 0.381094 0.121126i
\(868\) 0 0
\(869\) 11.5932 20.0800i 0.393271 0.681166i
\(870\) 0 0
\(871\) −60.5838 + 34.9781i −2.05280 + 1.18519i
\(872\) 0 0
\(873\) 3.94949 + 1.81954i 0.133670 + 0.0615820i
\(874\) 0 0
\(875\) 48.9077 28.2369i 1.65338 0.954581i
\(876\) 0 0
\(877\) 39.7400 + 22.9439i 1.34192 + 0.774760i 0.987089 0.160170i \(-0.0512042\pi\)
0.354834 + 0.934929i \(0.384538\pi\)
\(878\) 0 0
\(879\) −0.751275 + 3.42617i −0.0253399 + 0.115562i
\(880\) 0 0
\(881\) 24.0416i 0.809983i −0.914320 0.404992i \(-0.867274\pi\)
0.914320 0.404992i \(-0.132726\pi\)
\(882\) 0 0
\(883\) −24.9270 −0.838859 −0.419429 0.907788i \(-0.637770\pi\)
−0.419429 + 0.907788i \(0.637770\pi\)
\(884\) 0 0
\(885\) −21.5141 + 23.5875i −0.723188 + 0.792886i
\(886\) 0 0
\(887\) −11.1173 + 19.2558i −0.373283 + 0.646546i −0.990069 0.140586i \(-0.955101\pi\)
0.616785 + 0.787132i \(0.288435\pi\)
\(888\) 0 0
\(889\) 22.3485 + 38.7087i 0.749544 + 1.29825i
\(890\) 0 0
\(891\) 5.56473 + 29.9406i 0.186425 + 1.00305i
\(892\) 0 0
\(893\) −6.07529 10.5227i −0.203302 0.352129i
\(894\) 0 0
\(895\) 29.0720 + 16.7847i 0.971771 + 0.561052i
\(896\) 0 0
\(897\) 11.1742 12.2512i 0.373097 0.409055i
\(898\) 0 0
\(899\) 47.8674i 1.59647i
\(900\) 0 0
\(901\) 24.1987i 0.806174i
\(902\) 0 0
\(903\) 44.2075 + 9.69362i 1.47113 + 0.322583i
\(904\) 0 0
\(905\) 39.6742 + 22.9059i 1.31882 + 0.761419i
\(906\) 0 0
\(907\) 2.09662 + 3.63144i 0.0696170 + 0.120580i 0.898733 0.438497i \(-0.144489\pi\)
−0.829116 + 0.559077i \(0.811156\pi\)
\(908\) 0 0
\(909\) −16.5536 7.62628i −0.549049 0.252948i
\(910\) 0 0
\(911\) −22.3106 38.6430i −0.739182 1.28030i −0.952864 0.303397i \(-0.901879\pi\)
0.213683 0.976903i \(-0.431454\pi\)
\(912\) 0 0
\(913\) −11.8485 + 20.5222i −0.392127 + 0.679184i
\(914\) 0 0
\(915\) −6.76947 21.2986i −0.223792 0.704108i
\(916\) 0 0
\(917\) 57.4084 1.89579
\(918\) 0 0
\(919\) 14.6598i 0.483583i −0.970328 0.241791i \(-0.922265\pi\)
0.970328 0.241791i \(-0.0777350\pi\)
\(920\) 0 0
\(921\) −4.89898 + 1.55708i −0.161427 + 0.0513075i
\(922\) 0 0
\(923\) 36.8902 + 21.2986i 1.21425 + 0.701050i
\(924\) 0 0
\(925\) −2.22967 + 1.28730i −0.0733112 + 0.0423263i
\(926\) 0 0
\(927\) 14.1224 1.30125i 0.463841 0.0427385i
\(928\) 0 0
\(929\) 47.9722 27.6968i 1.57392 0.908701i 0.578235 0.815870i \(-0.303742\pi\)
0.995682 0.0928307i \(-0.0295915\pi\)
\(930\) 0 0
\(931\) 11.3880 19.7246i 0.373227 0.646448i
\(932\) 0 0
\(933\) −11.5932 + 52.8704i −0.379543 + 1.73090i
\(934\) 0 0
\(935\) −33.4279 −1.09321
\(936\) 0 0
\(937\) −37.1464 −1.21352 −0.606760 0.794885i \(-0.707531\pi\)
−0.606760 + 0.794885i \(0.707531\pi\)
\(938\) 0 0
\(939\) 4.98952 + 4.55092i 0.162827 + 0.148514i
\(940\) 0 0
\(941\) −5.97300 + 10.3455i −0.194714 + 0.337255i −0.946807 0.321802i \(-0.895711\pi\)
0.752092 + 0.659058i \(0.229045\pi\)
\(942\) 0 0
\(943\) 6.94737 4.01107i 0.226238 0.130618i
\(944\) 0 0
\(945\) 39.6742 + 30.0091i 1.29060 + 0.976196i
\(946\) 0 0
\(947\) −6.15679 + 3.55462i −0.200069 + 0.115510i −0.596687 0.802474i \(-0.703517\pi\)
0.396619 + 0.917983i \(0.370183\pi\)
\(948\) 0 0
\(949\) −2.48023 1.43196i −0.0805116 0.0464834i
\(950\) 0 0
\(951\) 21.6352 + 19.7333i 0.701569 + 0.639898i
\(952\) 0 0
\(953\) 27.5699i 0.893078i 0.894764 + 0.446539i \(0.147344\pi\)
−0.894764 + 0.446539i \(0.852656\pi\)
\(954\) 0 0
\(955\) 10.4961 0.339647
\(956\) 0 0
\(957\) 39.9905 + 8.76893i 1.29271 + 0.283459i
\(958\) 0 0
\(959\) 25.3157 43.8480i 0.817485 1.41593i
\(960\) 0 0
\(961\) 7.97730 + 13.8171i 0.257332 + 0.445712i
\(962\) 0 0
\(963\) −20.6251 29.1683i −0.664634 0.939934i
\(964\) 0 0
\(965\) −0.761995 1.31981i −0.0245295 0.0424863i
\(966\) 0 0
\(967\) 34.1792 + 19.7333i 1.09913 + 0.634582i 0.935992 0.352022i \(-0.114506\pi\)
0.163136 + 0.986604i \(0.447839\pi\)
\(968\) 0 0
\(969\) 3.79796 + 11.9494i 0.122008 + 0.383869i
\(970\) 0 0
\(971\) 15.7394i 0.505102i −0.967584 0.252551i \(-0.918730\pi\)
0.967584 0.252551i \(-0.0812696\pi\)
\(972\) 0 0
\(973\) 33.1448i 1.06257i
\(974\) 0 0
\(975\) 3.00510 + 9.45483i 0.0962402 + 0.302797i
\(976\) 0 0
\(977\) −9.94949 5.74434i −0.318312 0.183778i 0.332328 0.943164i \(-0.392166\pi\)
−0.650640 + 0.759386i \(0.725499\pi\)
\(978\) 0 0
\(979\) −0.241706 0.418647i −0.00772496 0.0133800i
\(980\) 0 0
\(981\) 7.19013 + 10.1684i 0.229563 + 0.324651i
\(982\) 0 0
\(983\) 4.43175 + 7.67602i 0.141351 + 0.244827i 0.928006 0.372566i \(-0.121522\pi\)
−0.786655 + 0.617393i \(0.788189\pi\)
\(984\) 0 0
\(985\) −18.2474 + 31.6055i −0.581412 + 1.00704i
\(986\) 0 0
\(987\) −65.4895 14.3602i −2.08455 0.457091i
\(988\) 0 0
\(989\) 8.30497 0.264083
\(990\) 0 0
\(991\) 42.0692i 1.33637i −0.743994 0.668186i \(-0.767071\pi\)
0.743994 0.668186i \(-0.232929\pi\)
\(992\) 0 0
\(993\) −43.1969 39.3997i −1.37081 1.25031i
\(994\) 0 0
\(995\) −3.72666 2.15159i −0.118143 0.0682100i
\(996\) 0 0
\(997\) 35.0301 20.2246i 1.10941 0.640520i 0.170736 0.985317i \(-0.445386\pi\)
0.938677 + 0.344797i \(0.112052\pi\)
\(998\) 0 0
\(999\) −11.8686 8.97727i −0.375506 0.284028i
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 1152.2.p.d.959.1 yes 16
3.2 odd 2 3456.2.p.d.2879.5 16
4.3 odd 2 inner 1152.2.p.d.959.7 yes 16
8.3 odd 2 inner 1152.2.p.d.959.2 yes 16
8.5 even 2 inner 1152.2.p.d.959.8 yes 16
9.2 odd 6 inner 1152.2.p.d.191.2 yes 16
9.7 even 3 3456.2.p.d.575.4 16
12.11 even 2 3456.2.p.d.2879.6 16
24.5 odd 2 3456.2.p.d.2879.3 16
24.11 even 2 3456.2.p.d.2879.4 16
36.7 odd 6 3456.2.p.d.575.3 16
36.11 even 6 inner 1152.2.p.d.191.8 yes 16
72.11 even 6 inner 1152.2.p.d.191.1 16
72.29 odd 6 inner 1152.2.p.d.191.7 yes 16
72.43 odd 6 3456.2.p.d.575.5 16
72.61 even 6 3456.2.p.d.575.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.p.d.191.1 16 72.11 even 6 inner
1152.2.p.d.191.2 yes 16 9.2 odd 6 inner
1152.2.p.d.191.7 yes 16 72.29 odd 6 inner
1152.2.p.d.191.8 yes 16 36.11 even 6 inner
1152.2.p.d.959.1 yes 16 1.1 even 1 trivial
1152.2.p.d.959.2 yes 16 8.3 odd 2 inner
1152.2.p.d.959.7 yes 16 4.3 odd 2 inner
1152.2.p.d.959.8 yes 16 8.5 even 2 inner
3456.2.p.d.575.3 16 36.7 odd 6
3456.2.p.d.575.4 16 9.7 even 3
3456.2.p.d.575.5 16 72.43 odd 6
3456.2.p.d.575.6 16 72.61 even 6
3456.2.p.d.2879.3 16 24.5 odd 2
3456.2.p.d.2879.4 16 24.11 even 2
3456.2.p.d.2879.5 16 3.2 odd 2
3456.2.p.d.2879.6 16 12.11 even 2