Properties

Label 2432.2.c.h.1217.2
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $0$
Dimension $6$
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] = [2432,2,Mod(1217,2432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2432.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.4196177716\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.2
Root \(0.264658 - 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.h.1217.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24914i q^{3} -1.47068i q^{5} +1.47068 q^{7} -2.05863 q^{9} +O(q^{10})\) \(q-2.24914i q^{3} -1.47068i q^{5} +1.47068 q^{7} -2.05863 q^{9} +4.77846i q^{11} +4.49828i q^{13} -3.30777 q^{15} +1.71982 q^{17} -1.00000i q^{19} -3.30777i q^{21} +4.00000 q^{23} +2.83709 q^{25} -2.11727i q^{27} +7.80605i q^{29} +4.13187 q^{31} +10.7474 q^{33} -2.16291i q^{35} -3.30777i q^{37} +10.1173 q^{39} +9.80605 q^{41} -1.83709i q^{43} +3.02760i q^{45} +5.14486 q^{47} -4.83709 q^{49} -3.86813i q^{51} +12.8647i q^{53} +7.02760 q^{55} -2.24914 q^{57} -9.05863i q^{59} +8.91033i q^{61} -3.02760 q^{63} +6.61555 q^{65} +5.92332i q^{67} -8.99656i q^{69} -6.05520 q^{71} +0.335371 q^{73} -6.38101i q^{75} +7.02760i q^{77} +0.366407 q^{79} -10.9379 q^{81} -18.0552i q^{83} -2.52932i q^{85} +17.5569 q^{87} -2.36641 q^{89} +6.61555i q^{91} -9.29317i q^{93} -1.47068 q^{95} -7.05863 q^{97} -9.83709i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{17} + 24 q^{23} + 2 q^{25} + 4 q^{31} + 12 q^{33} + 64 q^{39} + 8 q^{41} - 14 q^{49} + 8 q^{55} + 4 q^{57} + 16 q^{63} + 8 q^{65} + 32 q^{71} - 48 q^{73} - 12 q^{79} + 6 q^{81} + 72 q^{87} - 8 q^{95} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 2.24914i − 1.29854i −0.760557 0.649271i \(-0.775074\pi\)
0.760557 0.649271i \(-0.224926\pi\)
\(4\) 0 0
\(5\) − 1.47068i − 0.657710i −0.944380 0.328855i \(-0.893337\pi\)
0.944380 0.328855i \(-0.106663\pi\)
\(6\) 0 0
\(7\) 1.47068 0.555866 0.277933 0.960600i \(-0.410351\pi\)
0.277933 + 0.960600i \(0.410351\pi\)
\(8\) 0 0
\(9\) −2.05863 −0.686211
\(10\) 0 0
\(11\) 4.77846i 1.44076i 0.693580 + 0.720380i \(0.256032\pi\)
−0.693580 + 0.720380i \(0.743968\pi\)
\(12\) 0 0
\(13\) 4.49828i 1.24760i 0.781585 + 0.623799i \(0.214412\pi\)
−0.781585 + 0.623799i \(0.785588\pi\)
\(14\) 0 0
\(15\) −3.30777 −0.854063
\(16\) 0 0
\(17\) 1.71982 0.417119 0.208559 0.978010i \(-0.433123\pi\)
0.208559 + 0.978010i \(0.433123\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) − 3.30777i − 0.721815i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 2.83709 0.567418
\(26\) 0 0
\(27\) − 2.11727i − 0.407468i
\(28\) 0 0
\(29\) 7.80605i 1.44955i 0.688987 + 0.724774i \(0.258056\pi\)
−0.688987 + 0.724774i \(0.741944\pi\)
\(30\) 0 0
\(31\) 4.13187 0.742107 0.371053 0.928612i \(-0.378997\pi\)
0.371053 + 0.928612i \(0.378997\pi\)
\(32\) 0 0
\(33\) 10.7474 1.87089
\(34\) 0 0
\(35\) − 2.16291i − 0.365598i
\(36\) 0 0
\(37\) − 3.30777i − 0.543795i −0.962326 0.271897i \(-0.912349\pi\)
0.962326 0.271897i \(-0.0876511\pi\)
\(38\) 0 0
\(39\) 10.1173 1.62006
\(40\) 0 0
\(41\) 9.80605 1.53145 0.765724 0.643169i \(-0.222381\pi\)
0.765724 + 0.643169i \(0.222381\pi\)
\(42\) 0 0
\(43\) − 1.83709i − 0.280154i −0.990141 0.140077i \(-0.955265\pi\)
0.990141 0.140077i \(-0.0447350\pi\)
\(44\) 0 0
\(45\) 3.02760i 0.451328i
\(46\) 0 0
\(47\) 5.14486 0.750456 0.375228 0.926933i \(-0.377564\pi\)
0.375228 + 0.926933i \(0.377564\pi\)
\(48\) 0 0
\(49\) −4.83709 −0.691013
\(50\) 0 0
\(51\) − 3.86813i − 0.541646i
\(52\) 0 0
\(53\) 12.8647i 1.76710i 0.468336 + 0.883550i \(0.344854\pi\)
−0.468336 + 0.883550i \(0.655146\pi\)
\(54\) 0 0
\(55\) 7.02760 0.947601
\(56\) 0 0
\(57\) −2.24914 −0.297906
\(58\) 0 0
\(59\) − 9.05863i − 1.17933i −0.807647 0.589667i \(-0.799259\pi\)
0.807647 0.589667i \(-0.200741\pi\)
\(60\) 0 0
\(61\) 8.91033i 1.14085i 0.821349 + 0.570426i \(0.193222\pi\)
−0.821349 + 0.570426i \(0.806778\pi\)
\(62\) 0 0
\(63\) −3.02760 −0.381441
\(64\) 0 0
\(65\) 6.61555 0.820558
\(66\) 0 0
\(67\) 5.92332i 0.723649i 0.932246 + 0.361824i \(0.117846\pi\)
−0.932246 + 0.361824i \(0.882154\pi\)
\(68\) 0 0
\(69\) − 8.99656i − 1.08306i
\(70\) 0 0
\(71\) −6.05520 −0.718619 −0.359310 0.933218i \(-0.616988\pi\)
−0.359310 + 0.933218i \(0.616988\pi\)
\(72\) 0 0
\(73\) 0.335371 0.0392522 0.0196261 0.999807i \(-0.493752\pi\)
0.0196261 + 0.999807i \(0.493752\pi\)
\(74\) 0 0
\(75\) − 6.38101i − 0.736816i
\(76\) 0 0
\(77\) 7.02760i 0.800869i
\(78\) 0 0
\(79\) 0.366407 0.0412240 0.0206120 0.999788i \(-0.493439\pi\)
0.0206120 + 0.999788i \(0.493439\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) 0 0
\(83\) − 18.0552i − 1.98182i −0.134544 0.990908i \(-0.542957\pi\)
0.134544 0.990908i \(-0.457043\pi\)
\(84\) 0 0
\(85\) − 2.52932i − 0.274343i
\(86\) 0 0
\(87\) 17.5569 1.88230
\(88\) 0 0
\(89\) −2.36641 −0.250839 −0.125419 0.992104i \(-0.540028\pi\)
−0.125419 + 0.992104i \(0.540028\pi\)
\(90\) 0 0
\(91\) 6.61555i 0.693498i
\(92\) 0 0
\(93\) − 9.29317i − 0.963656i
\(94\) 0 0
\(95\) −1.47068 −0.150889
\(96\) 0 0
\(97\) −7.05863 −0.716696 −0.358348 0.933588i \(-0.616660\pi\)
−0.358348 + 0.933588i \(0.616660\pi\)
\(98\) 0 0
\(99\) − 9.83709i − 0.988665i
\(100\) 0 0
\(101\) 6.94137i 0.690692i 0.938475 + 0.345346i \(0.112238\pi\)
−0.938475 + 0.345346i \(0.887762\pi\)
\(102\) 0 0
\(103\) 0.824101 0.0812010 0.0406005 0.999175i \(-0.487073\pi\)
0.0406005 + 0.999175i \(0.487073\pi\)
\(104\) 0 0
\(105\) −4.86469 −0.474745
\(106\) 0 0
\(107\) − 11.4396i − 1.10591i −0.833210 0.552956i \(-0.813500\pi\)
0.833210 0.552956i \(-0.186500\pi\)
\(108\) 0 0
\(109\) − 4.13187i − 0.395762i −0.980226 0.197881i \(-0.936594\pi\)
0.980226 0.197881i \(-0.0634059\pi\)
\(110\) 0 0
\(111\) −7.43965 −0.706140
\(112\) 0 0
\(113\) −7.92332 −0.745363 −0.372682 0.927959i \(-0.621562\pi\)
−0.372682 + 0.927959i \(0.621562\pi\)
\(114\) 0 0
\(115\) − 5.88273i − 0.548568i
\(116\) 0 0
\(117\) − 9.26031i − 0.856116i
\(118\) 0 0
\(119\) 2.52932 0.231862
\(120\) 0 0
\(121\) −11.8337 −1.07579
\(122\) 0 0
\(123\) − 22.0552i − 1.98865i
\(124\) 0 0
\(125\) − 11.5259i − 1.03091i
\(126\) 0 0
\(127\) −1.19051 −0.105640 −0.0528202 0.998604i \(-0.516821\pi\)
−0.0528202 + 0.998604i \(0.516821\pi\)
\(128\) 0 0
\(129\) −4.13187 −0.363791
\(130\) 0 0
\(131\) 2.39744i 0.209466i 0.994500 + 0.104733i \(0.0333987\pi\)
−0.994500 + 0.104733i \(0.966601\pi\)
\(132\) 0 0
\(133\) − 1.47068i − 0.127524i
\(134\) 0 0
\(135\) −3.11383 −0.267996
\(136\) 0 0
\(137\) 9.39400 0.802584 0.401292 0.915950i \(-0.368561\pi\)
0.401292 + 0.915950i \(0.368561\pi\)
\(138\) 0 0
\(139\) 12.2181i 1.03633i 0.855282 + 0.518163i \(0.173384\pi\)
−0.855282 + 0.518163i \(0.826616\pi\)
\(140\) 0 0
\(141\) − 11.5715i − 0.974498i
\(142\) 0 0
\(143\) −21.4948 −1.79749
\(144\) 0 0
\(145\) 11.4802 0.953382
\(146\) 0 0
\(147\) 10.8793i 0.897309i
\(148\) 0 0
\(149\) − 16.8190i − 1.37787i −0.724823 0.688935i \(-0.758079\pi\)
0.724823 0.688935i \(-0.241921\pi\)
\(150\) 0 0
\(151\) −7.17590 −0.583966 −0.291983 0.956423i \(-0.594315\pi\)
−0.291983 + 0.956423i \(0.594315\pi\)
\(152\) 0 0
\(153\) −3.54049 −0.286231
\(154\) 0 0
\(155\) − 6.07668i − 0.488091i
\(156\) 0 0
\(157\) 15.6742i 1.25094i 0.780250 + 0.625468i \(0.215092\pi\)
−0.780250 + 0.625468i \(0.784908\pi\)
\(158\) 0 0
\(159\) 28.9345 2.29465
\(160\) 0 0
\(161\) 5.88273 0.463624
\(162\) 0 0
\(163\) 9.32238i 0.730185i 0.930971 + 0.365093i \(0.118963\pi\)
−0.930971 + 0.365093i \(0.881037\pi\)
\(164\) 0 0
\(165\) − 15.8061i − 1.23050i
\(166\) 0 0
\(167\) 20.3043 1.57120 0.785598 0.618737i \(-0.212355\pi\)
0.785598 + 0.618737i \(0.212355\pi\)
\(168\) 0 0
\(169\) −7.23453 −0.556503
\(170\) 0 0
\(171\) 2.05863i 0.157428i
\(172\) 0 0
\(173\) − 0.560352i − 0.0426028i −0.999773 0.0213014i \(-0.993219\pi\)
0.999773 0.0213014i \(-0.00678096\pi\)
\(174\) 0 0
\(175\) 4.17246 0.315408
\(176\) 0 0
\(177\) −20.3741 −1.53141
\(178\) 0 0
\(179\) 4.36641i 0.326361i 0.986596 + 0.163180i \(0.0521752\pi\)
−0.986596 + 0.163180i \(0.947825\pi\)
\(180\) 0 0
\(181\) − 5.05863i − 0.376005i −0.982169 0.188003i \(-0.939799\pi\)
0.982169 0.188003i \(-0.0602013\pi\)
\(182\) 0 0
\(183\) 20.0406 1.48144
\(184\) 0 0
\(185\) −4.86469 −0.357659
\(186\) 0 0
\(187\) 8.21811i 0.600967i
\(188\) 0 0
\(189\) − 3.11383i − 0.226498i
\(190\) 0 0
\(191\) 2.46725 0.178524 0.0892618 0.996008i \(-0.471549\pi\)
0.0892618 + 0.996008i \(0.471549\pi\)
\(192\) 0 0
\(193\) 12.2491 0.881712 0.440856 0.897578i \(-0.354675\pi\)
0.440856 + 0.897578i \(0.354675\pi\)
\(194\) 0 0
\(195\) − 14.8793i − 1.06553i
\(196\) 0 0
\(197\) 1.23109i 0.0877119i 0.999038 + 0.0438559i \(0.0139643\pi\)
−0.999038 + 0.0438559i \(0.986036\pi\)
\(198\) 0 0
\(199\) 27.0276 1.91594 0.957968 0.286876i \(-0.0926167\pi\)
0.957968 + 0.286876i \(0.0926167\pi\)
\(200\) 0 0
\(201\) 13.3224 0.939688
\(202\) 0 0
\(203\) 11.4802i 0.805755i
\(204\) 0 0
\(205\) − 14.4216i − 1.00725i
\(206\) 0 0
\(207\) −8.23453 −0.572340
\(208\) 0 0
\(209\) 4.77846 0.330533
\(210\) 0 0
\(211\) 0.325819i 0.0224303i 0.999937 + 0.0112152i \(0.00356997\pi\)
−0.999937 + 0.0112152i \(0.996430\pi\)
\(212\) 0 0
\(213\) 13.6190i 0.933157i
\(214\) 0 0
\(215\) −2.70178 −0.184260
\(216\) 0 0
\(217\) 6.07668 0.412512
\(218\) 0 0
\(219\) − 0.754297i − 0.0509707i
\(220\) 0 0
\(221\) 7.73625i 0.520397i
\(222\) 0 0
\(223\) −23.4182 −1.56820 −0.784098 0.620637i \(-0.786874\pi\)
−0.784098 + 0.620637i \(0.786874\pi\)
\(224\) 0 0
\(225\) −5.84053 −0.389369
\(226\) 0 0
\(227\) − 23.8466i − 1.58276i −0.611326 0.791379i \(-0.709364\pi\)
0.611326 0.791379i \(-0.290636\pi\)
\(228\) 0 0
\(229\) 13.9690i 0.923095i 0.887115 + 0.461548i \(0.152706\pi\)
−0.887115 + 0.461548i \(0.847294\pi\)
\(230\) 0 0
\(231\) 15.8061 1.03996
\(232\) 0 0
\(233\) 12.8337 0.840761 0.420380 0.907348i \(-0.361897\pi\)
0.420380 + 0.907348i \(0.361897\pi\)
\(234\) 0 0
\(235\) − 7.56647i − 0.493582i
\(236\) 0 0
\(237\) − 0.824101i − 0.0535311i
\(238\) 0 0
\(239\) −24.2587 −1.56916 −0.784582 0.620025i \(-0.787123\pi\)
−0.784582 + 0.620025i \(0.787123\pi\)
\(240\) 0 0
\(241\) −26.0698 −1.67930 −0.839652 0.543125i \(-0.817241\pi\)
−0.839652 + 0.543125i \(0.817241\pi\)
\(242\) 0 0
\(243\) 18.2491i 1.17068i
\(244\) 0 0
\(245\) 7.11383i 0.454486i
\(246\) 0 0
\(247\) 4.49828 0.286219
\(248\) 0 0
\(249\) −40.6087 −2.57347
\(250\) 0 0
\(251\) − 23.3319i − 1.47270i −0.676602 0.736349i \(-0.736548\pi\)
0.676602 0.736349i \(-0.263452\pi\)
\(252\) 0 0
\(253\) 19.1138i 1.20168i
\(254\) 0 0
\(255\) −5.68879 −0.356246
\(256\) 0 0
\(257\) 2.82410 0.176163 0.0880813 0.996113i \(-0.471926\pi\)
0.0880813 + 0.996113i \(0.471926\pi\)
\(258\) 0 0
\(259\) − 4.86469i − 0.302277i
\(260\) 0 0
\(261\) − 16.0698i − 0.994696i
\(262\) 0 0
\(263\) 7.08967 0.437168 0.218584 0.975818i \(-0.429856\pi\)
0.218584 + 0.975818i \(0.429856\pi\)
\(264\) 0 0
\(265\) 18.9199 1.16224
\(266\) 0 0
\(267\) 5.32238i 0.325724i
\(268\) 0 0
\(269\) − 20.8647i − 1.27214i −0.771630 0.636071i \(-0.780558\pi\)
0.771630 0.636071i \(-0.219442\pi\)
\(270\) 0 0
\(271\) −0.0620710 −0.00377055 −0.00188527 0.999998i \(-0.500600\pi\)
−0.00188527 + 0.999998i \(0.500600\pi\)
\(272\) 0 0
\(273\) 14.8793 0.900536
\(274\) 0 0
\(275\) 13.5569i 0.817513i
\(276\) 0 0
\(277\) − 7.82248i − 0.470007i −0.971994 0.235004i \(-0.924490\pi\)
0.971994 0.235004i \(-0.0755102\pi\)
\(278\) 0 0
\(279\) −8.50601 −0.509242
\(280\) 0 0
\(281\) −20.6448 −1.23156 −0.615782 0.787917i \(-0.711160\pi\)
−0.615782 + 0.787917i \(0.711160\pi\)
\(282\) 0 0
\(283\) − 21.3871i − 1.27133i −0.771964 0.635666i \(-0.780725\pi\)
0.771964 0.635666i \(-0.219275\pi\)
\(284\) 0 0
\(285\) 3.30777i 0.195936i
\(286\) 0 0
\(287\) 14.4216 0.851280
\(288\) 0 0
\(289\) −14.0422 −0.826012
\(290\) 0 0
\(291\) 15.8759i 0.930659i
\(292\) 0 0
\(293\) 24.6087i 1.43765i 0.695189 + 0.718827i \(0.255321\pi\)
−0.695189 + 0.718827i \(0.744679\pi\)
\(294\) 0 0
\(295\) −13.3224 −0.775659
\(296\) 0 0
\(297\) 10.1173 0.587063
\(298\) 0 0
\(299\) 17.9931i 1.04057i
\(300\) 0 0
\(301\) − 2.70178i − 0.155728i
\(302\) 0 0
\(303\) 15.6121 0.896892
\(304\) 0 0
\(305\) 13.1043 0.750349
\(306\) 0 0
\(307\) − 10.4216i − 0.594792i −0.954754 0.297396i \(-0.903882\pi\)
0.954754 0.297396i \(-0.0961182\pi\)
\(308\) 0 0
\(309\) − 1.85352i − 0.105443i
\(310\) 0 0
\(311\) 7.91377 0.448749 0.224374 0.974503i \(-0.427966\pi\)
0.224374 + 0.974503i \(0.427966\pi\)
\(312\) 0 0
\(313\) 17.7846 1.00524 0.502622 0.864506i \(-0.332369\pi\)
0.502622 + 0.864506i \(0.332369\pi\)
\(314\) 0 0
\(315\) 4.45264i 0.250878i
\(316\) 0 0
\(317\) − 30.0552i − 1.68807i −0.536290 0.844034i \(-0.680175\pi\)
0.536290 0.844034i \(-0.319825\pi\)
\(318\) 0 0
\(319\) −37.3009 −2.08845
\(320\) 0 0
\(321\) −25.7294 −1.43607
\(322\) 0 0
\(323\) − 1.71982i − 0.0956936i
\(324\) 0 0
\(325\) 12.7620i 0.707910i
\(326\) 0 0
\(327\) −9.29317 −0.513913
\(328\) 0 0
\(329\) 7.56647 0.417153
\(330\) 0 0
\(331\) 18.8095i 1.03386i 0.856027 + 0.516932i \(0.172926\pi\)
−0.856027 + 0.516932i \(0.827074\pi\)
\(332\) 0 0
\(333\) 6.80949i 0.373158i
\(334\) 0 0
\(335\) 8.71133 0.475951
\(336\) 0 0
\(337\) 30.2017 1.64519 0.822595 0.568628i \(-0.192525\pi\)
0.822595 + 0.568628i \(0.192525\pi\)
\(338\) 0 0
\(339\) 17.8207i 0.967886i
\(340\) 0 0
\(341\) 19.7440i 1.06920i
\(342\) 0 0
\(343\) −17.4086 −0.939977
\(344\) 0 0
\(345\) −13.2311 −0.712338
\(346\) 0 0
\(347\) − 9.36802i − 0.502902i −0.967870 0.251451i \(-0.919092\pi\)
0.967870 0.251451i \(-0.0809077\pi\)
\(348\) 0 0
\(349\) − 21.3173i − 1.14109i −0.821266 0.570545i \(-0.806732\pi\)
0.821266 0.570545i \(-0.193268\pi\)
\(350\) 0 0
\(351\) 9.52406 0.508357
\(352\) 0 0
\(353\) 18.4362 0.981260 0.490630 0.871368i \(-0.336767\pi\)
0.490630 + 0.871368i \(0.336767\pi\)
\(354\) 0 0
\(355\) 8.90528i 0.472643i
\(356\) 0 0
\(357\) − 5.68879i − 0.301083i
\(358\) 0 0
\(359\) 37.1932 1.96298 0.981491 0.191510i \(-0.0613386\pi\)
0.981491 + 0.191510i \(0.0613386\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 26.6155i 1.39695i
\(364\) 0 0
\(365\) − 0.493225i − 0.0258166i
\(366\) 0 0
\(367\) −11.6121 −0.606147 −0.303074 0.952967i \(-0.598013\pi\)
−0.303074 + 0.952967i \(0.598013\pi\)
\(368\) 0 0
\(369\) −20.1871 −1.05090
\(370\) 0 0
\(371\) 18.9199i 0.982271i
\(372\) 0 0
\(373\) 24.6087i 1.27419i 0.770786 + 0.637094i \(0.219864\pi\)
−0.770786 + 0.637094i \(0.780136\pi\)
\(374\) 0 0
\(375\) −25.9233 −1.33867
\(376\) 0 0
\(377\) −35.1138 −1.80845
\(378\) 0 0
\(379\) 1.05863i 0.0543783i 0.999630 + 0.0271892i \(0.00865565\pi\)
−0.999630 + 0.0271892i \(0.991344\pi\)
\(380\) 0 0
\(381\) 2.67762i 0.137179i
\(382\) 0 0
\(383\) −18.4837 −0.944472 −0.472236 0.881472i \(-0.656553\pi\)
−0.472236 + 0.881472i \(0.656553\pi\)
\(384\) 0 0
\(385\) 10.3354 0.526739
\(386\) 0 0
\(387\) 3.78189i 0.192245i
\(388\) 0 0
\(389\) 2.46725i 0.125094i 0.998042 + 0.0625472i \(0.0199224\pi\)
−0.998042 + 0.0625472i \(0.980078\pi\)
\(390\) 0 0
\(391\) 6.87930 0.347901
\(392\) 0 0
\(393\) 5.39218 0.272000
\(394\) 0 0
\(395\) − 0.538868i − 0.0271134i
\(396\) 0 0
\(397\) − 6.91721i − 0.347165i −0.984819 0.173582i \(-0.944466\pi\)
0.984819 0.173582i \(-0.0555343\pi\)
\(398\) 0 0
\(399\) −3.30777 −0.165596
\(400\) 0 0
\(401\) 20.8578 1.04159 0.520795 0.853682i \(-0.325636\pi\)
0.520795 + 0.853682i \(0.325636\pi\)
\(402\) 0 0
\(403\) 18.5863i 0.925851i
\(404\) 0 0
\(405\) 16.0862i 0.799331i
\(406\) 0 0
\(407\) 15.8061 0.783477
\(408\) 0 0
\(409\) 24.9820 1.23528 0.617639 0.786462i \(-0.288089\pi\)
0.617639 + 0.786462i \(0.288089\pi\)
\(410\) 0 0
\(411\) − 21.1284i − 1.04219i
\(412\) 0 0
\(413\) − 13.3224i − 0.655552i
\(414\) 0 0
\(415\) −26.5535 −1.30346
\(416\) 0 0
\(417\) 27.4802 1.34571
\(418\) 0 0
\(419\) − 31.6742i − 1.54738i −0.633561 0.773692i \(-0.718408\pi\)
0.633561 0.773692i \(-0.281592\pi\)
\(420\) 0 0
\(421\) 38.5941i 1.88096i 0.339850 + 0.940480i \(0.389624\pi\)
−0.339850 + 0.940480i \(0.610376\pi\)
\(422\) 0 0
\(423\) −10.5914 −0.514971
\(424\) 0 0
\(425\) 4.87930 0.236681
\(426\) 0 0
\(427\) 13.1043i 0.634160i
\(428\) 0 0
\(429\) 48.3449i 2.33411i
\(430\) 0 0
\(431\) −37.4948 −1.80606 −0.903032 0.429574i \(-0.858664\pi\)
−0.903032 + 0.429574i \(0.858664\pi\)
\(432\) 0 0
\(433\) 13.4182 0.644836 0.322418 0.946597i \(-0.395504\pi\)
0.322418 + 0.946597i \(0.395504\pi\)
\(434\) 0 0
\(435\) − 25.8207i − 1.23801i
\(436\) 0 0
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −36.3043 −1.73271 −0.866356 0.499427i \(-0.833544\pi\)
−0.866356 + 0.499427i \(0.833544\pi\)
\(440\) 0 0
\(441\) 9.95779 0.474181
\(442\) 0 0
\(443\) 27.9215i 1.32659i 0.748358 + 0.663295i \(0.230843\pi\)
−0.748358 + 0.663295i \(0.769157\pi\)
\(444\) 0 0
\(445\) 3.48024i 0.164979i
\(446\) 0 0
\(447\) −37.8284 −1.78922
\(448\) 0 0
\(449\) −12.2086 −0.576157 −0.288079 0.957607i \(-0.593016\pi\)
−0.288079 + 0.957607i \(0.593016\pi\)
\(450\) 0 0
\(451\) 46.8578i 2.20645i
\(452\) 0 0
\(453\) 16.1396i 0.758305i
\(454\) 0 0
\(455\) 9.72938 0.456120
\(456\) 0 0
\(457\) 4.42666 0.207070 0.103535 0.994626i \(-0.466985\pi\)
0.103535 + 0.994626i \(0.466985\pi\)
\(458\) 0 0
\(459\) − 3.64133i − 0.169963i
\(460\) 0 0
\(461\) − 6.96553i − 0.324417i −0.986757 0.162208i \(-0.948138\pi\)
0.986757 0.162208i \(-0.0518617\pi\)
\(462\) 0 0
\(463\) 30.7501 1.42908 0.714539 0.699595i \(-0.246636\pi\)
0.714539 + 0.699595i \(0.246636\pi\)
\(464\) 0 0
\(465\) −13.6673 −0.633806
\(466\) 0 0
\(467\) 5.16635i 0.239070i 0.992830 + 0.119535i \(0.0381404\pi\)
−0.992830 + 0.119535i \(0.961860\pi\)
\(468\) 0 0
\(469\) 8.71133i 0.402252i
\(470\) 0 0
\(471\) 35.2534 1.62439
\(472\) 0 0
\(473\) 8.77846 0.403634
\(474\) 0 0
\(475\) − 2.83709i − 0.130175i
\(476\) 0 0
\(477\) − 26.4837i − 1.21260i
\(478\) 0 0
\(479\) 23.3776 1.06815 0.534074 0.845437i \(-0.320660\pi\)
0.534074 + 0.845437i \(0.320660\pi\)
\(480\) 0 0
\(481\) 14.8793 0.678437
\(482\) 0 0
\(483\) − 13.2311i − 0.602036i
\(484\) 0 0
\(485\) 10.3810i 0.471378i
\(486\) 0 0
\(487\) −41.9018 −1.89875 −0.949377 0.314140i \(-0.898284\pi\)
−0.949377 + 0.314140i \(0.898284\pi\)
\(488\) 0 0
\(489\) 20.9673 0.948176
\(490\) 0 0
\(491\) 4.90528i 0.221372i 0.993855 + 0.110686i \(0.0353048\pi\)
−0.993855 + 0.110686i \(0.964695\pi\)
\(492\) 0 0
\(493\) 13.4250i 0.604633i
\(494\) 0 0
\(495\) −14.4672 −0.650254
\(496\) 0 0
\(497\) −8.90528 −0.399456
\(498\) 0 0
\(499\) − 5.77502i − 0.258525i −0.991610 0.129263i \(-0.958739\pi\)
0.991610 0.129263i \(-0.0412610\pi\)
\(500\) 0 0
\(501\) − 45.6673i − 2.04026i
\(502\) 0 0
\(503\) −31.3776 −1.39906 −0.699529 0.714605i \(-0.746607\pi\)
−0.699529 + 0.714605i \(0.746607\pi\)
\(504\) 0 0
\(505\) 10.2086 0.454275
\(506\) 0 0
\(507\) 16.2715i 0.722642i
\(508\) 0 0
\(509\) 17.2603i 0.765050i 0.923945 + 0.382525i \(0.124945\pi\)
−0.923945 + 0.382525i \(0.875055\pi\)
\(510\) 0 0
\(511\) 0.493225 0.0218190
\(512\) 0 0
\(513\) −2.11727 −0.0934796
\(514\) 0 0
\(515\) − 1.21199i − 0.0534067i
\(516\) 0 0
\(517\) 24.5845i 1.08123i
\(518\) 0 0
\(519\) −1.26031 −0.0553215
\(520\) 0 0
\(521\) 18.8939 0.827757 0.413878 0.910332i \(-0.364174\pi\)
0.413878 + 0.910332i \(0.364174\pi\)
\(522\) 0 0
\(523\) 40.2017i 1.75790i 0.476917 + 0.878948i \(0.341754\pi\)
−0.476917 + 0.878948i \(0.658246\pi\)
\(524\) 0 0
\(525\) − 9.38445i − 0.409571i
\(526\) 0 0
\(527\) 7.10610 0.309546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 18.6484i 0.809272i
\(532\) 0 0
\(533\) 44.1104i 1.91063i
\(534\) 0 0
\(535\) −16.8241 −0.727369
\(536\) 0 0
\(537\) 9.82066 0.423793
\(538\) 0 0
\(539\) − 23.1138i − 0.995583i
\(540\) 0 0
\(541\) − 22.4052i − 0.963274i −0.876371 0.481637i \(-0.840042\pi\)
0.876371 0.481637i \(-0.159958\pi\)
\(542\) 0 0
\(543\) −11.3776 −0.488259
\(544\) 0 0
\(545\) −6.07668 −0.260296
\(546\) 0 0
\(547\) − 37.9931i − 1.62447i −0.583331 0.812234i \(-0.698251\pi\)
0.583331 0.812234i \(-0.301749\pi\)
\(548\) 0 0
\(549\) − 18.3431i − 0.782865i
\(550\) 0 0
\(551\) 7.80605 0.332549
\(552\) 0 0
\(553\) 0.538868 0.0229150
\(554\) 0 0
\(555\) 10.9414i 0.464435i
\(556\) 0 0
\(557\) 16.1775i 0.685463i 0.939433 + 0.342732i \(0.111352\pi\)
−0.939433 + 0.342732i \(0.888648\pi\)
\(558\) 0 0
\(559\) 8.26375 0.349519
\(560\) 0 0
\(561\) 18.4837 0.780381
\(562\) 0 0
\(563\) − 9.59750i − 0.404486i −0.979335 0.202243i \(-0.935177\pi\)
0.979335 0.202243i \(-0.0648232\pi\)
\(564\) 0 0
\(565\) 11.6527i 0.490233i
\(566\) 0 0
\(567\) −16.0862 −0.675558
\(568\) 0 0
\(569\) −23.9018 −1.00202 −0.501009 0.865442i \(-0.667037\pi\)
−0.501009 + 0.865442i \(0.667037\pi\)
\(570\) 0 0
\(571\) 36.1725i 1.51377i 0.653548 + 0.756885i \(0.273280\pi\)
−0.653548 + 0.756885i \(0.726720\pi\)
\(572\) 0 0
\(573\) − 5.54918i − 0.231820i
\(574\) 0 0
\(575\) 11.3484 0.473259
\(576\) 0 0
\(577\) −12.5699 −0.523292 −0.261646 0.965164i \(-0.584265\pi\)
−0.261646 + 0.965164i \(0.584265\pi\)
\(578\) 0 0
\(579\) − 27.5500i − 1.14494i
\(580\) 0 0
\(581\) − 26.5535i − 1.10162i
\(582\) 0 0
\(583\) −61.4734 −2.54597
\(584\) 0 0
\(585\) −13.6190 −0.563076
\(586\) 0 0
\(587\) 0.107714i 0.00444585i 0.999998 + 0.00222292i \(0.000707579\pi\)
−0.999998 + 0.00222292i \(0.999292\pi\)
\(588\) 0 0
\(589\) − 4.13187i − 0.170251i
\(590\) 0 0
\(591\) 2.76891 0.113898
\(592\) 0 0
\(593\) 25.7846 1.05885 0.529423 0.848358i \(-0.322409\pi\)
0.529423 + 0.848358i \(0.322409\pi\)
\(594\) 0 0
\(595\) − 3.71982i − 0.152498i
\(596\) 0 0
\(597\) − 60.7889i − 2.48792i
\(598\) 0 0
\(599\) 23.6336 0.965642 0.482821 0.875719i \(-0.339612\pi\)
0.482821 + 0.875719i \(0.339612\pi\)
\(600\) 0 0
\(601\) −35.3415 −1.44161 −0.720805 0.693138i \(-0.756228\pi\)
−0.720805 + 0.693138i \(0.756228\pi\)
\(602\) 0 0
\(603\) − 12.1939i − 0.496576i
\(604\) 0 0
\(605\) 17.4036i 0.707555i
\(606\) 0 0
\(607\) 28.8432 1.17071 0.585355 0.810777i \(-0.300955\pi\)
0.585355 + 0.810777i \(0.300955\pi\)
\(608\) 0 0
\(609\) 25.8207 1.04631
\(610\) 0 0
\(611\) 23.1430i 0.936267i
\(612\) 0 0
\(613\) − 27.7604i − 1.12123i −0.828076 0.560616i \(-0.810564\pi\)
0.828076 0.560616i \(-0.189436\pi\)
\(614\) 0 0
\(615\) −32.4362 −1.30795
\(616\) 0 0
\(617\) −32.5370 −1.30989 −0.654946 0.755676i \(-0.727309\pi\)
−0.654946 + 0.755676i \(0.727309\pi\)
\(618\) 0 0
\(619\) 6.20855i 0.249543i 0.992186 + 0.124771i \(0.0398198\pi\)
−0.992186 + 0.124771i \(0.960180\pi\)
\(620\) 0 0
\(621\) − 8.46907i − 0.339852i
\(622\) 0 0
\(623\) −3.48024 −0.139433
\(624\) 0 0
\(625\) −2.76547 −0.110619
\(626\) 0 0
\(627\) − 10.7474i − 0.429211i
\(628\) 0 0
\(629\) − 5.68879i − 0.226827i
\(630\) 0 0
\(631\) 2.67256 0.106393 0.0531965 0.998584i \(-0.483059\pi\)
0.0531965 + 0.998584i \(0.483059\pi\)
\(632\) 0 0
\(633\) 0.732814 0.0291267
\(634\) 0 0
\(635\) 1.75086i 0.0694807i
\(636\) 0 0
\(637\) − 21.7586i − 0.862107i
\(638\) 0 0
\(639\) 12.4654 0.493125
\(640\) 0 0
\(641\) −24.6639 −0.974164 −0.487082 0.873356i \(-0.661939\pi\)
−0.487082 + 0.873356i \(0.661939\pi\)
\(642\) 0 0
\(643\) 25.5405i 1.00722i 0.863932 + 0.503609i \(0.167995\pi\)
−0.863932 + 0.503609i \(0.832005\pi\)
\(644\) 0 0
\(645\) 6.07668i 0.239269i
\(646\) 0 0
\(647\) −21.7535 −0.855220 −0.427610 0.903963i \(-0.640644\pi\)
−0.427610 + 0.903963i \(0.640644\pi\)
\(648\) 0 0
\(649\) 43.2863 1.69914
\(650\) 0 0
\(651\) − 13.6673i − 0.535664i
\(652\) 0 0
\(653\) 9.37940i 0.367044i 0.983016 + 0.183522i \(0.0587499\pi\)
−0.983016 + 0.183522i \(0.941250\pi\)
\(654\) 0 0
\(655\) 3.52588 0.137767
\(656\) 0 0
\(657\) −0.690407 −0.0269353
\(658\) 0 0
\(659\) 39.8138i 1.55092i 0.631394 + 0.775462i \(0.282483\pi\)
−0.631394 + 0.775462i \(0.717517\pi\)
\(660\) 0 0
\(661\) − 19.1138i − 0.743442i −0.928345 0.371721i \(-0.878768\pi\)
0.928345 0.371721i \(-0.121232\pi\)
\(662\) 0 0
\(663\) 17.3999 0.675757
\(664\) 0 0
\(665\) −2.16291 −0.0838740
\(666\) 0 0
\(667\) 31.2242i 1.20901i
\(668\) 0 0
\(669\) 52.6707i 2.03637i
\(670\) 0 0
\(671\) −42.5776 −1.64369
\(672\) 0 0
\(673\) −21.1138 −0.813878 −0.406939 0.913455i \(-0.633404\pi\)
−0.406939 + 0.913455i \(0.633404\pi\)
\(674\) 0 0
\(675\) − 6.00688i − 0.231205i
\(676\) 0 0
\(677\) − 36.3855i − 1.39841i −0.714922 0.699204i \(-0.753538\pi\)
0.714922 0.699204i \(-0.246462\pi\)
\(678\) 0 0
\(679\) −10.3810 −0.398387
\(680\) 0 0
\(681\) −53.6344 −2.05528
\(682\) 0 0
\(683\) − 12.6087i − 0.482457i −0.970468 0.241229i \(-0.922450\pi\)
0.970468 0.241229i \(-0.0775504\pi\)
\(684\) 0 0
\(685\) − 13.8156i − 0.527867i
\(686\) 0 0
\(687\) 31.4182 1.19868
\(688\) 0 0
\(689\) −57.8690 −2.20463
\(690\) 0 0
\(691\) 16.8405i 0.640644i 0.947309 + 0.320322i \(0.103791\pi\)
−0.947309 + 0.320322i \(0.896209\pi\)
\(692\) 0 0
\(693\) − 14.4672i − 0.549565i
\(694\) 0 0
\(695\) 17.9690 0.681602
\(696\) 0 0
\(697\) 16.8647 0.638796
\(698\) 0 0
\(699\) − 28.8647i − 1.09176i
\(700\) 0 0
\(701\) − 33.9311i − 1.28156i −0.767725 0.640779i \(-0.778611\pi\)
0.767725 0.640779i \(-0.221389\pi\)
\(702\) 0 0
\(703\) −3.30777 −0.124755
\(704\) 0 0
\(705\) −17.0180 −0.640937
\(706\) 0 0
\(707\) 10.2086i 0.383932i
\(708\) 0 0
\(709\) − 30.8984i − 1.16041i −0.814469 0.580207i \(-0.802972\pi\)
0.814469 0.580207i \(-0.197028\pi\)
\(710\) 0 0
\(711\) −0.754297 −0.0282884
\(712\) 0 0
\(713\) 16.5275 0.618960
\(714\) 0 0
\(715\) 31.6121i 1.18223i
\(716\) 0 0
\(717\) 54.5612i 2.03763i
\(718\) 0 0
\(719\) −26.8122 −0.999925 −0.499963 0.866047i \(-0.666653\pi\)
−0.499963 + 0.866047i \(0.666653\pi\)
\(720\) 0 0
\(721\) 1.21199 0.0451369
\(722\) 0 0
\(723\) 58.6347i 2.18065i
\(724\) 0 0
\(725\) 22.1465i 0.822500i
\(726\) 0 0
\(727\) 30.1897 1.11968 0.559838 0.828602i \(-0.310863\pi\)
0.559838 + 0.828602i \(0.310863\pi\)
\(728\) 0 0
\(729\) 8.23109 0.304855
\(730\) 0 0
\(731\) − 3.15947i − 0.116857i
\(732\) 0 0
\(733\) 37.8207i 1.39694i 0.715640 + 0.698469i \(0.246135\pi\)
−0.715640 + 0.698469i \(0.753865\pi\)
\(734\) 0 0
\(735\) 16.0000 0.590169
\(736\) 0 0
\(737\) −28.3043 −1.04260
\(738\) 0 0
\(739\) − 39.3803i − 1.44863i −0.689471 0.724313i \(-0.742157\pi\)
0.689471 0.724313i \(-0.257843\pi\)
\(740\) 0 0
\(741\) − 10.1173i − 0.371667i
\(742\) 0 0
\(743\) 13.9854 0.513074 0.256537 0.966534i \(-0.417418\pi\)
0.256537 + 0.966534i \(0.417418\pi\)
\(744\) 0 0
\(745\) −24.7355 −0.906238
\(746\) 0 0
\(747\) 37.1690i 1.35994i
\(748\) 0 0
\(749\) − 16.8241i − 0.614739i
\(750\) 0 0
\(751\) −25.0777 −0.915100 −0.457550 0.889184i \(-0.651273\pi\)
−0.457550 + 0.889184i \(0.651273\pi\)
\(752\) 0 0
\(753\) −52.4768 −1.91236
\(754\) 0 0
\(755\) 10.5535i 0.384080i
\(756\) 0 0
\(757\) − 24.6137i − 0.894601i −0.894384 0.447301i \(-0.852385\pi\)
0.894384 0.447301i \(-0.147615\pi\)
\(758\) 0 0
\(759\) 42.9897 1.56043
\(760\) 0 0
\(761\) 37.8010 1.37029 0.685143 0.728409i \(-0.259740\pi\)
0.685143 + 0.728409i \(0.259740\pi\)
\(762\) 0 0
\(763\) − 6.07668i − 0.219991i
\(764\) 0 0
\(765\) 5.20693i 0.188257i
\(766\) 0 0
\(767\) 40.7483 1.47134
\(768\) 0 0
\(769\) 9.98357 0.360017 0.180008 0.983665i \(-0.442388\pi\)
0.180008 + 0.983665i \(0.442388\pi\)
\(770\) 0 0
\(771\) − 6.35180i − 0.228754i
\(772\) 0 0
\(773\) − 12.5896i − 0.452815i −0.974033 0.226408i \(-0.927302\pi\)
0.974033 0.226408i \(-0.0726981\pi\)
\(774\) 0 0
\(775\) 11.7225 0.421085
\(776\) 0 0
\(777\) −10.9414 −0.392519
\(778\) 0 0
\(779\) − 9.80605i − 0.351338i
\(780\) 0 0
\(781\) − 28.9345i − 1.03536i
\(782\) 0 0
\(783\) 16.5275 0.590645
\(784\) 0 0
\(785\) 23.0518 0.822753
\(786\) 0 0
\(787\) 29.2603i 1.04302i 0.853246 + 0.521509i \(0.174631\pi\)
−0.853246 + 0.521509i \(0.825369\pi\)
\(788\) 0 0
\(789\) − 15.9457i − 0.567681i
\(790\) 0 0
\(791\) −11.6527 −0.414322
\(792\) 0 0
\(793\) −40.0812 −1.42332
\(794\) 0 0
\(795\) − 42.5535i − 1.50922i
\(796\) 0 0
\(797\) 46.8578i 1.65979i 0.557920 + 0.829894i \(0.311599\pi\)
−0.557920 + 0.829894i \(0.688401\pi\)
\(798\) 0 0
\(799\) 8.84826 0.313029
\(800\) 0 0
\(801\) 4.87156 0.172128
\(802\) 0 0
\(803\) 1.60256i 0.0565530i
\(804\) 0 0
\(805\) − 8.65164i − 0.304930i
\(806\) 0 0
\(807\) −46.9276 −1.65193
\(808\) 0 0
\(809\) 22.6681 0.796967 0.398483 0.917176i \(-0.369537\pi\)
0.398483 + 0.917176i \(0.369537\pi\)
\(810\) 0 0
\(811\) 32.7620i 1.15043i 0.818002 + 0.575215i \(0.195082\pi\)
−0.818002 + 0.575215i \(0.804918\pi\)
\(812\) 0 0
\(813\) 0.139606i 0.00489621i
\(814\) 0 0
\(815\) 13.7103 0.480250
\(816\) 0 0
\(817\) −1.83709 −0.0642717
\(818\) 0 0
\(819\) − 13.6190i − 0.475886i
\(820\) 0 0
\(821\) − 3.20006i − 0.111683i −0.998440 0.0558414i \(-0.982216\pi\)
0.998440 0.0558414i \(-0.0177841\pi\)
\(822\) 0 0
\(823\) −24.4741 −0.853114 −0.426557 0.904461i \(-0.640274\pi\)
−0.426557 + 0.904461i \(0.640274\pi\)
\(824\) 0 0
\(825\) 30.4914 1.06157
\(826\) 0 0
\(827\) − 24.5681i − 0.854316i −0.904177 0.427158i \(-0.859515\pi\)
0.904177 0.427158i \(-0.140485\pi\)
\(828\) 0 0
\(829\) − 11.2380i − 0.390311i −0.980772 0.195155i \(-0.937479\pi\)
0.980772 0.195155i \(-0.0625211\pi\)
\(830\) 0 0
\(831\) −17.5939 −0.610324
\(832\) 0 0
\(833\) −8.31894 −0.288234
\(834\) 0 0
\(835\) − 29.8613i − 1.03339i
\(836\) 0 0
\(837\) − 8.74828i − 0.302385i
\(838\) 0 0
\(839\) 1.94480 0.0671421 0.0335711 0.999436i \(-0.489312\pi\)
0.0335711 + 0.999436i \(0.489312\pi\)
\(840\) 0 0
\(841\) −31.9345 −1.10119
\(842\) 0 0
\(843\) 46.4330i 1.59924i
\(844\) 0 0
\(845\) 10.6397i 0.366017i
\(846\) 0 0
\(847\) −17.4036 −0.597993
\(848\) 0 0
\(849\) −48.1027 −1.65088
\(850\) 0 0
\(851\) − 13.2311i − 0.453556i
\(852\) 0 0
\(853\) 29.8207i 1.02104i 0.859866 + 0.510520i \(0.170547\pi\)
−0.859866 + 0.510520i \(0.829453\pi\)
\(854\) 0 0
\(855\) 3.02760 0.103542
\(856\) 0 0
\(857\) 6.49828 0.221977 0.110989 0.993822i \(-0.464598\pi\)
0.110989 + 0.993822i \(0.464598\pi\)
\(858\) 0 0
\(859\) − 21.0422i − 0.717951i −0.933347 0.358975i \(-0.883126\pi\)
0.933347 0.358975i \(-0.116874\pi\)
\(860\) 0 0
\(861\) − 32.4362i − 1.10542i
\(862\) 0 0
\(863\) −7.80605 −0.265721 −0.132861 0.991135i \(-0.542416\pi\)
−0.132861 + 0.991135i \(0.542416\pi\)
\(864\) 0 0
\(865\) −0.824101 −0.0280203
\(866\) 0 0
\(867\) 31.5829i 1.07261i
\(868\) 0 0
\(869\) 1.75086i 0.0593938i
\(870\) 0 0
\(871\) −26.6448 −0.902823
\(872\) 0 0
\(873\) 14.5311 0.491804
\(874\) 0 0
\(875\) − 16.9509i − 0.573046i
\(876\) 0 0
\(877\) − 4.32582i − 0.146073i −0.997329 0.0730363i \(-0.976731\pi\)
0.997329 0.0730363i \(-0.0232689\pi\)
\(878\) 0 0
\(879\) 55.3484 1.86685
\(880\) 0 0
\(881\) −5.65775 −0.190615 −0.0953073 0.995448i \(-0.530383\pi\)
−0.0953073 + 0.995448i \(0.530383\pi\)
\(882\) 0 0
\(883\) 19.6578i 0.661536i 0.943712 + 0.330768i \(0.107308\pi\)
−0.943712 + 0.330768i \(0.892692\pi\)
\(884\) 0 0
\(885\) 29.9639i 1.00723i
\(886\) 0 0
\(887\) −40.7811 −1.36930 −0.684648 0.728874i \(-0.740044\pi\)
−0.684648 + 0.728874i \(0.740044\pi\)
\(888\) 0 0
\(889\) −1.75086 −0.0587219
\(890\) 0 0
\(891\) − 52.2664i − 1.75099i
\(892\) 0 0
\(893\) − 5.14486i − 0.172166i
\(894\) 0 0
\(895\) 6.42160 0.214650
\(896\) 0 0
\(897\) 40.4691 1.35122
\(898\) 0 0
\(899\) 32.2536i 1.07572i
\(900\) 0 0
\(901\) 22.1250i 0.737091i
\(902\) 0 0
\(903\) −6.07668 −0.202219
\(904\) 0 0
\(905\) −7.43965 −0.247302
\(906\) 0 0
\(907\) 37.1475i 1.23346i 0.787173 + 0.616732i \(0.211544\pi\)
−0.787173 + 0.616732i \(0.788456\pi\)
\(908\) 0 0
\(909\) − 14.2897i − 0.473960i
\(910\) 0 0
\(911\) −20.4070 −0.676114 −0.338057 0.941126i \(-0.609770\pi\)
−0.338057 + 0.941126i \(0.609770\pi\)
\(912\) 0 0
\(913\) 86.2760 2.85532
\(914\) 0 0
\(915\) − 29.4734i − 0.974359i
\(916\) 0 0
\(917\) 3.52588i 0.116435i
\(918\) 0 0
\(919\) −52.9536 −1.74678 −0.873389 0.487023i \(-0.838083\pi\)
−0.873389 + 0.487023i \(0.838083\pi\)
\(920\) 0 0
\(921\) −23.4396 −0.772363
\(922\) 0 0
\(923\) − 27.2380i − 0.896549i
\(924\) 0 0
\(925\) − 9.38445i − 0.308559i
\(926\) 0 0
\(927\) −1.69652 −0.0557210
\(928\) 0 0
\(929\) 56.8984 1.86678 0.933388 0.358869i \(-0.116837\pi\)
0.933388 + 0.358869i \(0.116837\pi\)
\(930\) 0 0
\(931\) 4.83709i 0.158529i
\(932\) 0 0
\(933\) − 17.7992i − 0.582719i
\(934\) 0 0
\(935\) 12.0862 0.395262
\(936\) 0 0
\(937\) −40.9148 −1.33663 −0.668315 0.743879i \(-0.732984\pi\)
−0.668315 + 0.743879i \(0.732984\pi\)
\(938\) 0 0
\(939\) − 40.0000i − 1.30535i
\(940\) 0 0
\(941\) 2.59644i 0.0846416i 0.999104 + 0.0423208i \(0.0134752\pi\)
−0.999104 + 0.0423208i \(0.986525\pi\)
\(942\) 0 0
\(943\) 39.2242 1.27732
\(944\) 0 0
\(945\) −4.57946 −0.148970
\(946\) 0 0
\(947\) 25.7586i 0.837042i 0.908207 + 0.418521i \(0.137451\pi\)
−0.908207 + 0.418521i \(0.862549\pi\)
\(948\) 0 0
\(949\) 1.50859i 0.0489711i
\(950\) 0 0
\(951\) −67.5984 −2.19203
\(952\) 0 0
\(953\) −2.25602 −0.0730795 −0.0365398 0.999332i \(-0.511634\pi\)
−0.0365398 + 0.999332i \(0.511634\pi\)
\(954\) 0 0
\(955\) − 3.62854i − 0.117417i
\(956\) 0 0
\(957\) 83.8950i 2.71194i
\(958\) 0 0
\(959\) 13.8156 0.446129
\(960\) 0 0
\(961\) −13.9276 −0.449278
\(962\) 0 0
\(963\) 23.5500i 0.758889i
\(964\) 0 0
\(965\) − 18.0146i − 0.579911i
\(966\) 0 0
\(967\) 45.0449 1.44855 0.724273 0.689513i \(-0.242176\pi\)
0.724273 + 0.689513i \(0.242176\pi\)
\(968\) 0 0
\(969\) −3.86813 −0.124262
\(970\) 0 0
\(971\) − 17.5760i − 0.564041i −0.959408 0.282021i \(-0.908995\pi\)
0.959408 0.282021i \(-0.0910047\pi\)
\(972\) 0 0
\(973\) 17.9690i 0.576059i
\(974\) 0 0
\(975\) 28.7036 0.919251
\(976\) 0 0
\(977\) −15.9042 −0.508821 −0.254410 0.967096i \(-0.581881\pi\)
−0.254410 + 0.967096i \(0.581881\pi\)
\(978\) 0 0
\(979\) − 11.3078i − 0.361398i
\(980\) 0 0
\(981\) 8.50601i 0.271576i
\(982\) 0 0
\(983\) 41.7294 1.33096 0.665480 0.746416i \(-0.268227\pi\)
0.665480 + 0.746416i \(0.268227\pi\)
\(984\) 0 0
\(985\) 1.81055 0.0576889
\(986\) 0 0
\(987\) − 17.0180i − 0.541690i
\(988\) 0 0
\(989\) − 7.34836i − 0.233664i
\(990\) 0 0
\(991\) 0.193945 0.00616087 0.00308044 0.999995i \(-0.499019\pi\)
0.00308044 + 0.999995i \(0.499019\pi\)
\(992\) 0 0
\(993\) 42.3052 1.34251
\(994\) 0 0
\(995\) − 39.7490i − 1.26013i
\(996\) 0 0
\(997\) − 40.6949i − 1.28882i −0.764680 0.644410i \(-0.777103\pi\)
0.764680 0.644410i \(-0.222897\pi\)
\(998\) 0 0
\(999\) −7.00344 −0.221579
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 2432.2.c.h.1217.2 yes 6
4.3 odd 2 2432.2.c.e.1217.5 yes 6
8.3 odd 2 2432.2.c.e.1217.2 6
8.5 even 2 inner 2432.2.c.h.1217.5 yes 6
16.3 odd 4 4864.2.a.bg.1.1 3
16.5 even 4 4864.2.a.bh.1.1 3
16.11 odd 4 4864.2.a.bb.1.3 3
16.13 even 4 4864.2.a.ba.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.e.1217.2 6 8.3 odd 2
2432.2.c.e.1217.5 yes 6 4.3 odd 2
2432.2.c.h.1217.2 yes 6 1.1 even 1 trivial
2432.2.c.h.1217.5 yes 6 8.5 even 2 inner
4864.2.a.ba.1.3 3 16.13 even 4
4864.2.a.bb.1.3 3 16.11 odd 4
4864.2.a.bg.1.1 3 16.3 odd 4
4864.2.a.bh.1.1 3 16.5 even 4