Properties

Label 2240.2.b.f.1121.6
Level $2240$
Weight $2$
Character 2240.1121
Analytic conductor $17.886$
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] = [2240,2,Mod(1121,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.b (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3534400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1121.6
Root \(2.19082 + 1.44755i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1121
Dual form 2240.2.b.f.1121.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.89511i q^{3} -1.00000i q^{5} +1.00000 q^{7} -5.38164 q^{9} +O(q^{10})\) \(q+2.89511i q^{3} -1.00000i q^{5} +1.00000 q^{7} -5.38164 q^{9} +2.38164i q^{11} -3.40857i q^{13} +2.89511 q^{15} -1.92204 q^{17} +5.79021i q^{19} +2.89511i q^{21} -8.76328 q^{23} -1.00000 q^{25} -6.89511i q^{27} -5.86818i q^{29} -1.02693 q^{31} -6.89511 q^{33} -1.00000i q^{35} +2.81714i q^{37} +9.86818 q^{39} -11.7902 q^{41} -2.00000i q^{43} +5.38164i q^{45} +5.40857 q^{47} +1.00000 q^{49} -5.56450i q^{51} +6.97307i q^{53} +2.38164 q^{55} -16.7633 q^{57} -6.97307i q^{59} +14.7633i q^{61} -5.38164 q^{63} -3.40857 q^{65} -0.209787i q^{67} -25.3706i q^{69} -13.7902 q^{71} -14.7633 q^{73} -2.89511i q^{75} +2.38164i q^{77} -13.4486 q^{79} +3.81714 q^{81} +5.79021i q^{83} +1.92204i q^{85} +16.9890 q^{87} -1.23672 q^{89} -3.40857i q^{91} -2.97307i q^{93} +5.79021 q^{95} +5.92204 q^{97} -12.8171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} - 10 q^{9} - 4 q^{17} - 8 q^{23} - 6 q^{25} - 16 q^{31} - 24 q^{33} + 32 q^{39} - 36 q^{41} + 20 q^{47} + 6 q^{49} - 8 q^{55} - 56 q^{57} - 10 q^{63} - 8 q^{65} - 48 q^{71} - 44 q^{73} + 16 q^{79} - 2 q^{81} + 20 q^{87} - 52 q^{89} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.89511i 1.67149i 0.549117 + 0.835745i \(0.314964\pi\)
−0.549117 + 0.835745i \(0.685036\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −5.38164 −1.79388
\(10\) 0 0
\(11\) 2.38164i 0.718092i 0.933320 + 0.359046i \(0.116898\pi\)
−0.933320 + 0.359046i \(0.883102\pi\)
\(12\) 0 0
\(13\) − 3.40857i − 0.945368i −0.881232 0.472684i \(-0.843285\pi\)
0.881232 0.472684i \(-0.156715\pi\)
\(14\) 0 0
\(15\) 2.89511 0.747513
\(16\) 0 0
\(17\) −1.92204 −0.466162 −0.233081 0.972457i \(-0.574881\pi\)
−0.233081 + 0.972457i \(0.574881\pi\)
\(18\) 0 0
\(19\) 5.79021i 1.32837i 0.747570 + 0.664183i \(0.231220\pi\)
−0.747570 + 0.664183i \(0.768780\pi\)
\(20\) 0 0
\(21\) 2.89511i 0.631764i
\(22\) 0 0
\(23\) −8.76328 −1.82727 −0.913635 0.406534i \(-0.866737\pi\)
−0.913635 + 0.406534i \(0.866737\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) − 6.89511i − 1.32696i
\(28\) 0 0
\(29\) − 5.86818i − 1.08969i −0.838536 0.544847i \(-0.816588\pi\)
0.838536 0.544847i \(-0.183412\pi\)
\(30\) 0 0
\(31\) −1.02693 −0.184442 −0.0922210 0.995739i \(-0.529397\pi\)
−0.0922210 + 0.995739i \(0.529397\pi\)
\(32\) 0 0
\(33\) −6.89511 −1.20028
\(34\) 0 0
\(35\) − 1.00000i − 0.169031i
\(36\) 0 0
\(37\) 2.81714i 0.463135i 0.972819 + 0.231568i \(0.0743855\pi\)
−0.972819 + 0.231568i \(0.925615\pi\)
\(38\) 0 0
\(39\) 9.86818 1.58017
\(40\) 0 0
\(41\) −11.7902 −1.84132 −0.920661 0.390363i \(-0.872349\pi\)
−0.920661 + 0.390363i \(0.872349\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 5.38164i 0.802248i
\(46\) 0 0
\(47\) 5.40857 0.788921 0.394461 0.918913i \(-0.370931\pi\)
0.394461 + 0.918913i \(0.370931\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 5.56450i − 0.779186i
\(52\) 0 0
\(53\) 6.97307i 0.957825i 0.877863 + 0.478912i \(0.158969\pi\)
−0.877863 + 0.478912i \(0.841031\pi\)
\(54\) 0 0
\(55\) 2.38164 0.321141
\(56\) 0 0
\(57\) −16.7633 −2.22035
\(58\) 0 0
\(59\) − 6.97307i − 0.907816i −0.891048 0.453908i \(-0.850029\pi\)
0.891048 0.453908i \(-0.149971\pi\)
\(60\) 0 0
\(61\) 14.7633i 1.89024i 0.326717 + 0.945122i \(0.394058\pi\)
−0.326717 + 0.945122i \(0.605942\pi\)
\(62\) 0 0
\(63\) −5.38164 −0.678023
\(64\) 0 0
\(65\) −3.40857 −0.422781
\(66\) 0 0
\(67\) − 0.209787i − 0.0256296i −0.999918 0.0128148i \(-0.995921\pi\)
0.999918 0.0128148i \(-0.00407918\pi\)
\(68\) 0 0
\(69\) − 25.3706i − 3.05427i
\(70\) 0 0
\(71\) −13.7902 −1.63660 −0.818299 0.574793i \(-0.805082\pi\)
−0.818299 + 0.574793i \(0.805082\pi\)
\(72\) 0 0
\(73\) −14.7633 −1.72791 −0.863956 0.503568i \(-0.832020\pi\)
−0.863956 + 0.503568i \(0.832020\pi\)
\(74\) 0 0
\(75\) − 2.89511i − 0.334298i
\(76\) 0 0
\(77\) 2.38164i 0.271413i
\(78\) 0 0
\(79\) −13.4486 −1.51309 −0.756543 0.653944i \(-0.773113\pi\)
−0.756543 + 0.653944i \(0.773113\pi\)
\(80\) 0 0
\(81\) 3.81714 0.424127
\(82\) 0 0
\(83\) 5.79021i 0.635558i 0.948165 + 0.317779i \(0.102937\pi\)
−0.948165 + 0.317779i \(0.897063\pi\)
\(84\) 0 0
\(85\) 1.92204i 0.208474i
\(86\) 0 0
\(87\) 16.9890 1.82141
\(88\) 0 0
\(89\) −1.23672 −0.131092 −0.0655458 0.997850i \(-0.520879\pi\)
−0.0655458 + 0.997850i \(0.520879\pi\)
\(90\) 0 0
\(91\) − 3.40857i − 0.357315i
\(92\) 0 0
\(93\) − 2.97307i − 0.308293i
\(94\) 0 0
\(95\) 5.79021 0.594063
\(96\) 0 0
\(97\) 5.92204 0.601292 0.300646 0.953736i \(-0.402798\pi\)
0.300646 + 0.953736i \(0.402798\pi\)
\(98\) 0 0
\(99\) − 12.8171i − 1.28817i
\(100\) 0 0
\(101\) − 9.58043i − 0.953288i −0.879096 0.476644i \(-0.841853\pi\)
0.879096 0.476644i \(-0.158147\pi\)
\(102\) 0 0
\(103\) 6.59143 0.649473 0.324736 0.945805i \(-0.394724\pi\)
0.324736 + 0.945805i \(0.394724\pi\)
\(104\) 0 0
\(105\) 2.89511 0.282533
\(106\) 0 0
\(107\) − 0.0538591i − 0.00520676i −0.999997 0.00260338i \(-0.999171\pi\)
0.999997 0.00260338i \(-0.000828682\pi\)
\(108\) 0 0
\(109\) − 9.44860i − 0.905012i −0.891761 0.452506i \(-0.850530\pi\)
0.891761 0.452506i \(-0.149470\pi\)
\(110\) 0 0
\(111\) −8.15593 −0.774126
\(112\) 0 0
\(113\) 12.5535 1.18093 0.590467 0.807062i \(-0.298944\pi\)
0.590467 + 0.807062i \(0.298944\pi\)
\(114\) 0 0
\(115\) 8.76328i 0.817180i
\(116\) 0 0
\(117\) 18.3437i 1.69588i
\(118\) 0 0
\(119\) −1.92204 −0.176193
\(120\) 0 0
\(121\) 5.32778 0.484344
\(122\) 0 0
\(123\) − 34.1339i − 3.07775i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 2.97307 0.263817 0.131909 0.991262i \(-0.457889\pi\)
0.131909 + 0.991262i \(0.457889\pi\)
\(128\) 0 0
\(129\) 5.79021 0.509800
\(130\) 0 0
\(131\) − 10.8171i − 0.945098i −0.881304 0.472549i \(-0.843334\pi\)
0.881304 0.472549i \(-0.156666\pi\)
\(132\) 0 0
\(133\) 5.79021i 0.502075i
\(134\) 0 0
\(135\) −6.89511 −0.593436
\(136\) 0 0
\(137\) 2.76328 0.236083 0.118042 0.993009i \(-0.462338\pi\)
0.118042 + 0.993009i \(0.462338\pi\)
\(138\) 0 0
\(139\) − 2.81714i − 0.238947i −0.992837 0.119473i \(-0.961879\pi\)
0.992837 0.119473i \(-0.0381206\pi\)
\(140\) 0 0
\(141\) 15.6584i 1.31867i
\(142\) 0 0
\(143\) 8.11800 0.678861
\(144\) 0 0
\(145\) −5.86818 −0.487326
\(146\) 0 0
\(147\) 2.89511i 0.238784i
\(148\) 0 0
\(149\) − 2.05386i − 0.168259i −0.996455 0.0841293i \(-0.973189\pi\)
0.996455 0.0841293i \(-0.0268109\pi\)
\(150\) 0 0
\(151\) 9.71225 0.790372 0.395186 0.918601i \(-0.370680\pi\)
0.395186 + 0.918601i \(0.370680\pi\)
\(152\) 0 0
\(153\) 10.3437 0.836239
\(154\) 0 0
\(155\) 1.02693i 0.0824850i
\(156\) 0 0
\(157\) − 2.41957i − 0.193103i −0.995328 0.0965515i \(-0.969219\pi\)
0.995328 0.0965515i \(-0.0307813\pi\)
\(158\) 0 0
\(159\) −20.1878 −1.60100
\(160\) 0 0
\(161\) −8.76328 −0.690643
\(162\) 0 0
\(163\) 12.5535i 0.983266i 0.870803 + 0.491633i \(0.163600\pi\)
−0.870803 + 0.491633i \(0.836400\pi\)
\(164\) 0 0
\(165\) 6.89511i 0.536783i
\(166\) 0 0
\(167\) 24.5694 1.90124 0.950620 0.310359i \(-0.100449\pi\)
0.950620 + 0.310359i \(0.100449\pi\)
\(168\) 0 0
\(169\) 1.38164 0.106280
\(170\) 0 0
\(171\) − 31.1609i − 2.38293i
\(172\) 0 0
\(173\) 1.46243i 0.111187i 0.998453 + 0.0555933i \(0.0177050\pi\)
−0.998453 + 0.0555933i \(0.982295\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 20.1878 1.51741
\(178\) 0 0
\(179\) 8.81714i 0.659024i 0.944151 + 0.329512i \(0.106884\pi\)
−0.944151 + 0.329512i \(0.893116\pi\)
\(180\) 0 0
\(181\) − 17.3168i − 1.28715i −0.765385 0.643573i \(-0.777451\pi\)
0.765385 0.643573i \(-0.222549\pi\)
\(182\) 0 0
\(183\) −42.7413 −3.15953
\(184\) 0 0
\(185\) 2.81714 0.207120
\(186\) 0 0
\(187\) − 4.57760i − 0.334747i
\(188\) 0 0
\(189\) − 6.89511i − 0.501545i
\(190\) 0 0
\(191\) −15.5025 −1.12172 −0.560859 0.827911i \(-0.689529\pi\)
−0.560859 + 0.827911i \(0.689529\pi\)
\(192\) 0 0
\(193\) −8.39757 −0.604470 −0.302235 0.953233i \(-0.597733\pi\)
−0.302235 + 0.953233i \(0.597733\pi\)
\(194\) 0 0
\(195\) − 9.86818i − 0.706675i
\(196\) 0 0
\(197\) 14.1339i 1.00700i 0.863995 + 0.503500i \(0.167955\pi\)
−0.863995 + 0.503500i \(0.832045\pi\)
\(198\) 0 0
\(199\) −20.4996 −1.45318 −0.726590 0.687071i \(-0.758896\pi\)
−0.726590 + 0.687071i \(0.758896\pi\)
\(200\) 0 0
\(201\) 0.607356 0.0428396
\(202\) 0 0
\(203\) − 5.86818i − 0.411865i
\(204\) 0 0
\(205\) 11.7902i 0.823464i
\(206\) 0 0
\(207\) 47.1609 3.27791
\(208\) 0 0
\(209\) −13.7902 −0.953889
\(210\) 0 0
\(211\) 25.9621i 1.78730i 0.448762 + 0.893651i \(0.351865\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(212\) 0 0
\(213\) − 39.9241i − 2.73556i
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) −1.02693 −0.0697125
\(218\) 0 0
\(219\) − 42.7413i − 2.88819i
\(220\) 0 0
\(221\) 6.55140i 0.440695i
\(222\) 0 0
\(223\) 24.5694 1.64529 0.822645 0.568555i \(-0.192497\pi\)
0.822645 + 0.568555i \(0.192497\pi\)
\(224\) 0 0
\(225\) 5.38164 0.358776
\(226\) 0 0
\(227\) − 20.8412i − 1.38328i −0.722241 0.691641i \(-0.756888\pi\)
0.722241 0.691641i \(-0.243112\pi\)
\(228\) 0 0
\(229\) − 1.84407i − 0.121860i −0.998142 0.0609299i \(-0.980593\pi\)
0.998142 0.0609299i \(-0.0194066\pi\)
\(230\) 0 0
\(231\) −6.89511 −0.453665
\(232\) 0 0
\(233\) −1.12900 −0.0739631 −0.0369816 0.999316i \(-0.511774\pi\)
−0.0369816 + 0.999316i \(0.511774\pi\)
\(234\) 0 0
\(235\) − 5.40857i − 0.352816i
\(236\) 0 0
\(237\) − 38.9351i − 2.52911i
\(238\) 0 0
\(239\) 6.13182 0.396635 0.198317 0.980138i \(-0.436452\pi\)
0.198317 + 0.980138i \(0.436452\pi\)
\(240\) 0 0
\(241\) 25.5804 1.64778 0.823890 0.566750i \(-0.191799\pi\)
0.823890 + 0.566750i \(0.191799\pi\)
\(242\) 0 0
\(243\) − 9.63429i − 0.618040i
\(244\) 0 0
\(245\) − 1.00000i − 0.0638877i
\(246\) 0 0
\(247\) 19.7364 1.25579
\(248\) 0 0
\(249\) −16.7633 −1.06233
\(250\) 0 0
\(251\) 2.81714i 0.177816i 0.996040 + 0.0889082i \(0.0283378\pi\)
−0.996040 + 0.0889082i \(0.971662\pi\)
\(252\) 0 0
\(253\) − 20.8710i − 1.31215i
\(254\) 0 0
\(255\) −5.56450 −0.348462
\(256\) 0 0
\(257\) −14.3437 −0.894736 −0.447368 0.894350i \(-0.647639\pi\)
−0.447368 + 0.894350i \(0.647639\pi\)
\(258\) 0 0
\(259\) 2.81714i 0.175049i
\(260\) 0 0
\(261\) 31.5804i 1.95478i
\(262\) 0 0
\(263\) 10.2098 0.629562 0.314781 0.949164i \(-0.398069\pi\)
0.314781 + 0.949164i \(0.398069\pi\)
\(264\) 0 0
\(265\) 6.97307 0.428352
\(266\) 0 0
\(267\) − 3.58043i − 0.219119i
\(268\) 0 0
\(269\) 8.55350i 0.521516i 0.965404 + 0.260758i \(0.0839724\pi\)
−0.965404 + 0.260758i \(0.916028\pi\)
\(270\) 0 0
\(271\) −19.5804 −1.18943 −0.594713 0.803938i \(-0.702734\pi\)
−0.594713 + 0.803938i \(0.702734\pi\)
\(272\) 0 0
\(273\) 9.86818 0.597249
\(274\) 0 0
\(275\) − 2.38164i − 0.143618i
\(276\) 0 0
\(277\) 12.7633i 0.766871i 0.923568 + 0.383436i \(0.125259\pi\)
−0.923568 + 0.383436i \(0.874741\pi\)
\(278\) 0 0
\(279\) 5.52657 0.330867
\(280\) 0 0
\(281\) −25.9621 −1.54877 −0.774384 0.632716i \(-0.781940\pi\)
−0.774384 + 0.632716i \(0.781940\pi\)
\(282\) 0 0
\(283\) 28.5776i 1.69876i 0.527780 + 0.849381i \(0.323025\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(284\) 0 0
\(285\) 16.7633i 0.992971i
\(286\) 0 0
\(287\) −11.7902 −0.695954
\(288\) 0 0
\(289\) −13.3058 −0.782693
\(290\) 0 0
\(291\) 17.1449i 1.00505i
\(292\) 0 0
\(293\) − 12.1719i − 0.711087i −0.934660 0.355544i \(-0.884296\pi\)
0.934660 0.355544i \(-0.115704\pi\)
\(294\) 0 0
\(295\) −6.97307 −0.405988
\(296\) 0 0
\(297\) 16.4217 0.952882
\(298\) 0 0
\(299\) 29.8703i 1.72744i
\(300\) 0 0
\(301\) − 2.00000i − 0.115278i
\(302\) 0 0
\(303\) 27.7364 1.59341
\(304\) 0 0
\(305\) 14.7633 0.845343
\(306\) 0 0
\(307\) 12.8412i 0.732889i 0.930440 + 0.366444i \(0.119425\pi\)
−0.930440 + 0.366444i \(0.880575\pi\)
\(308\) 0 0
\(309\) 19.0829i 1.08559i
\(310\) 0 0
\(311\) −17.1829 −0.974350 −0.487175 0.873304i \(-0.661973\pi\)
−0.487175 + 0.873304i \(0.661973\pi\)
\(312\) 0 0
\(313\) −15.2927 −0.864393 −0.432197 0.901779i \(-0.642261\pi\)
−0.432197 + 0.901779i \(0.642261\pi\)
\(314\) 0 0
\(315\) 5.38164i 0.303221i
\(316\) 0 0
\(317\) 30.8972i 1.73536i 0.497123 + 0.867680i \(0.334390\pi\)
−0.497123 + 0.867680i \(0.665610\pi\)
\(318\) 0 0
\(319\) 13.9759 0.782500
\(320\) 0 0
\(321\) 0.155928 0.00870304
\(322\) 0 0
\(323\) − 11.1290i − 0.619234i
\(324\) 0 0
\(325\) 3.40857i 0.189074i
\(326\) 0 0
\(327\) 27.3547 1.51272
\(328\) 0 0
\(329\) 5.40857 0.298184
\(330\) 0 0
\(331\) 26.4514i 1.45390i 0.686689 + 0.726951i \(0.259063\pi\)
−0.686689 + 0.726951i \(0.740937\pi\)
\(332\) 0 0
\(333\) − 15.1609i − 0.830810i
\(334\) 0 0
\(335\) −0.209787 −0.0114619
\(336\) 0 0
\(337\) −0.553497 −0.0301509 −0.0150754 0.999886i \(-0.504799\pi\)
−0.0150754 + 0.999886i \(0.504799\pi\)
\(338\) 0 0
\(339\) 36.3437i 1.97392i
\(340\) 0 0
\(341\) − 2.44578i − 0.132446i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −25.3706 −1.36591
\(346\) 0 0
\(347\) 30.0801i 1.61478i 0.590016 + 0.807391i \(0.299121\pi\)
−0.590016 + 0.807391i \(0.700879\pi\)
\(348\) 0 0
\(349\) − 23.2147i − 1.24266i −0.783551 0.621328i \(-0.786594\pi\)
0.783551 0.621328i \(-0.213406\pi\)
\(350\) 0 0
\(351\) −23.5025 −1.25447
\(352\) 0 0
\(353\) −30.7654 −1.63748 −0.818738 0.574167i \(-0.805326\pi\)
−0.818738 + 0.574167i \(0.805326\pi\)
\(354\) 0 0
\(355\) 13.7902i 0.731909i
\(356\) 0 0
\(357\) − 5.56450i − 0.294505i
\(358\) 0 0
\(359\) −11.4245 −0.602962 −0.301481 0.953472i \(-0.597481\pi\)
−0.301481 + 0.953472i \(0.597481\pi\)
\(360\) 0 0
\(361\) −14.5266 −0.764556
\(362\) 0 0
\(363\) 15.4245i 0.809576i
\(364\) 0 0
\(365\) 14.7633i 0.772746i
\(366\) 0 0
\(367\) −20.9890 −1.09562 −0.547808 0.836604i \(-0.684538\pi\)
−0.547808 + 0.836604i \(0.684538\pi\)
\(368\) 0 0
\(369\) 63.4507 3.30311
\(370\) 0 0
\(371\) 6.97307i 0.362024i
\(372\) 0 0
\(373\) 23.5804i 1.22095i 0.792036 + 0.610474i \(0.209021\pi\)
−0.792036 + 0.610474i \(0.790979\pi\)
\(374\) 0 0
\(375\) −2.89511 −0.149503
\(376\) 0 0
\(377\) −20.0021 −1.03016
\(378\) 0 0
\(379\) 24.3976i 1.25322i 0.779333 + 0.626609i \(0.215558\pi\)
−0.779333 + 0.626609i \(0.784442\pi\)
\(380\) 0 0
\(381\) 8.60736i 0.440968i
\(382\) 0 0
\(383\) −0.311856 −0.0159351 −0.00796754 0.999968i \(-0.502536\pi\)
−0.00796754 + 0.999968i \(0.502536\pi\)
\(384\) 0 0
\(385\) 2.38164 0.121380
\(386\) 0 0
\(387\) 10.7633i 0.547128i
\(388\) 0 0
\(389\) − 0.0779639i − 0.00395293i −0.999998 0.00197646i \(-0.999371\pi\)
0.999998 0.00197646i \(-0.000629128\pi\)
\(390\) 0 0
\(391\) 16.8433 0.851805
\(392\) 0 0
\(393\) 31.3168 1.57972
\(394\) 0 0
\(395\) 13.4486i 0.676673i
\(396\) 0 0
\(397\) − 26.8813i − 1.34913i −0.738214 0.674566i \(-0.764331\pi\)
0.738214 0.674566i \(-0.235669\pi\)
\(398\) 0 0
\(399\) −16.7633 −0.839214
\(400\) 0 0
\(401\) 23.1449 1.15580 0.577901 0.816107i \(-0.303872\pi\)
0.577901 + 0.816107i \(0.303872\pi\)
\(402\) 0 0
\(403\) 3.50036i 0.174365i
\(404\) 0 0
\(405\) − 3.81714i − 0.189675i
\(406\) 0 0
\(407\) −6.70942 −0.332574
\(408\) 0 0
\(409\) 16.5535 0.818518 0.409259 0.912418i \(-0.365787\pi\)
0.409259 + 0.912418i \(0.365787\pi\)
\(410\) 0 0
\(411\) 8.00000i 0.394611i
\(412\) 0 0
\(413\) − 6.97307i − 0.343122i
\(414\) 0 0
\(415\) 5.79021 0.284230
\(416\) 0 0
\(417\) 8.15593 0.399398
\(418\) 0 0
\(419\) 5.44650i 0.266079i 0.991111 + 0.133040i \(0.0424737\pi\)
−0.991111 + 0.133040i \(0.957526\pi\)
\(420\) 0 0
\(421\) − 3.92204i − 0.191148i −0.995422 0.0955742i \(-0.969531\pi\)
0.995422 0.0955742i \(-0.0304687\pi\)
\(422\) 0 0
\(423\) −29.1070 −1.41523
\(424\) 0 0
\(425\) 1.92204 0.0932324
\(426\) 0 0
\(427\) 14.7633i 0.714445i
\(428\) 0 0
\(429\) 23.5025i 1.13471i
\(430\) 0 0
\(431\) −29.1367 −1.40347 −0.701734 0.712439i \(-0.747590\pi\)
−0.701734 + 0.712439i \(0.747590\pi\)
\(432\) 0 0
\(433\) 40.3976 1.94138 0.970692 0.240328i \(-0.0772551\pi\)
0.970692 + 0.240328i \(0.0772551\pi\)
\(434\) 0 0
\(435\) − 16.9890i − 0.814560i
\(436\) 0 0
\(437\) − 50.7413i − 2.42728i
\(438\) 0 0
\(439\) 25.7902 1.23090 0.615450 0.788176i \(-0.288974\pi\)
0.615450 + 0.788176i \(0.288974\pi\)
\(440\) 0 0
\(441\) −5.38164 −0.256269
\(442\) 0 0
\(443\) 36.7413i 1.74563i 0.488050 + 0.872815i \(0.337708\pi\)
−0.488050 + 0.872815i \(0.662292\pi\)
\(444\) 0 0
\(445\) 1.23672i 0.0586260i
\(446\) 0 0
\(447\) 5.94614 0.281243
\(448\) 0 0
\(449\) −21.5425 −1.01665 −0.508327 0.861164i \(-0.669736\pi\)
−0.508327 + 0.861164i \(0.669736\pi\)
\(450\) 0 0
\(451\) − 28.0801i − 1.32224i
\(452\) 0 0
\(453\) 28.1180i 1.32110i
\(454\) 0 0
\(455\) −3.40857 −0.159796
\(456\) 0 0
\(457\) −17.8441 −0.834710 −0.417355 0.908743i \(-0.637043\pi\)
−0.417355 + 0.908743i \(0.637043\pi\)
\(458\) 0 0
\(459\) 13.2526i 0.618580i
\(460\) 0 0
\(461\) 3.07514i 0.143224i 0.997433 + 0.0716118i \(0.0228143\pi\)
−0.997433 + 0.0716118i \(0.977186\pi\)
\(462\) 0 0
\(463\) 21.1070 0.980925 0.490463 0.871462i \(-0.336828\pi\)
0.490463 + 0.871462i \(0.336828\pi\)
\(464\) 0 0
\(465\) −2.97307 −0.137873
\(466\) 0 0
\(467\) 3.31468i 0.153385i 0.997055 + 0.0766926i \(0.0244360\pi\)
−0.997055 + 0.0766926i \(0.975564\pi\)
\(468\) 0 0
\(469\) − 0.209787i − 0.00968706i
\(470\) 0 0
\(471\) 7.00492 0.322770
\(472\) 0 0
\(473\) 4.76328 0.219016
\(474\) 0 0
\(475\) − 5.79021i − 0.265673i
\(476\) 0 0
\(477\) − 37.5266i − 1.71822i
\(478\) 0 0
\(479\) 19.3168 0.882606 0.441303 0.897358i \(-0.354516\pi\)
0.441303 + 0.897358i \(0.354516\pi\)
\(480\) 0 0
\(481\) 9.60243 0.437833
\(482\) 0 0
\(483\) − 25.3706i − 1.15440i
\(484\) 0 0
\(485\) − 5.92204i − 0.268906i
\(486\) 0 0
\(487\) 3.65629 0.165682 0.0828412 0.996563i \(-0.473601\pi\)
0.0828412 + 0.996563i \(0.473601\pi\)
\(488\) 0 0
\(489\) −36.3437 −1.64352
\(490\) 0 0
\(491\) 4.32778i 0.195310i 0.995220 + 0.0976550i \(0.0311342\pi\)
−0.995220 + 0.0976550i \(0.968866\pi\)
\(492\) 0 0
\(493\) 11.2788i 0.507974i
\(494\) 0 0
\(495\) −12.8171 −0.576088
\(496\) 0 0
\(497\) −13.7902 −0.618576
\(498\) 0 0
\(499\) − 29.1229i − 1.30372i −0.758339 0.651860i \(-0.773989\pi\)
0.758339 0.651860i \(-0.226011\pi\)
\(500\) 0 0
\(501\) 71.1311i 3.17790i
\(502\) 0 0
\(503\) −30.1719 −1.34530 −0.672648 0.739962i \(-0.734843\pi\)
−0.672648 + 0.739962i \(0.734843\pi\)
\(504\) 0 0
\(505\) −9.58043 −0.426323
\(506\) 0 0
\(507\) 4.00000i 0.177646i
\(508\) 0 0
\(509\) − 12.5535i − 0.556424i −0.960520 0.278212i \(-0.910258\pi\)
0.960520 0.278212i \(-0.0897418\pi\)
\(510\) 0 0
\(511\) −14.7633 −0.653089
\(512\) 0 0
\(513\) 39.9241 1.76269
\(514\) 0 0
\(515\) − 6.59143i − 0.290453i
\(516\) 0 0
\(517\) 12.8813i 0.566518i
\(518\) 0 0
\(519\) −4.23389 −0.185847
\(520\) 0 0
\(521\) −8.66121 −0.379455 −0.189727 0.981837i \(-0.560760\pi\)
−0.189727 + 0.981837i \(0.560760\pi\)
\(522\) 0 0
\(523\) 9.79021i 0.428096i 0.976823 + 0.214048i \(0.0686649\pi\)
−0.976823 + 0.214048i \(0.931335\pi\)
\(524\) 0 0
\(525\) − 2.89511i − 0.126353i
\(526\) 0 0
\(527\) 1.97380 0.0859799
\(528\) 0 0
\(529\) 53.7951 2.33892
\(530\) 0 0
\(531\) 37.5266i 1.62851i
\(532\) 0 0
\(533\) 40.1878i 1.74073i
\(534\) 0 0
\(535\) −0.0538591 −0.00232853
\(536\) 0 0
\(537\) −25.5266 −1.10155
\(538\) 0 0
\(539\) 2.38164i 0.102585i
\(540\) 0 0
\(541\) 31.5025i 1.35440i 0.735801 + 0.677198i \(0.236806\pi\)
−0.735801 + 0.677198i \(0.763194\pi\)
\(542\) 0 0
\(543\) 50.1339 2.15145
\(544\) 0 0
\(545\) −9.44860 −0.404734
\(546\) 0 0
\(547\) 30.7633i 1.31534i 0.753305 + 0.657672i \(0.228458\pi\)
−0.753305 + 0.657672i \(0.771542\pi\)
\(548\) 0 0
\(549\) − 79.4507i − 3.39087i
\(550\) 0 0
\(551\) 33.9780 1.44751
\(552\) 0 0
\(553\) −13.4486 −0.571893
\(554\) 0 0
\(555\) 8.15593i 0.346200i
\(556\) 0 0
\(557\) − 17.1070i − 0.724847i −0.932014 0.362423i \(-0.881949\pi\)
0.932014 0.362423i \(-0.118051\pi\)
\(558\) 0 0
\(559\) −6.81714 −0.288334
\(560\) 0 0
\(561\) 13.2526 0.559527
\(562\) 0 0
\(563\) 2.62936i 0.110814i 0.998464 + 0.0554072i \(0.0176457\pi\)
−0.998464 + 0.0554072i \(0.982354\pi\)
\(564\) 0 0
\(565\) − 12.5535i − 0.528130i
\(566\) 0 0
\(567\) 3.81714 0.160305
\(568\) 0 0
\(569\) −28.7413 −1.20490 −0.602449 0.798158i \(-0.705808\pi\)
−0.602449 + 0.798158i \(0.705808\pi\)
\(570\) 0 0
\(571\) 7.18286i 0.300593i 0.988641 + 0.150297i \(0.0480229\pi\)
−0.988641 + 0.150297i \(0.951977\pi\)
\(572\) 0 0
\(573\) − 44.8813i − 1.87494i
\(574\) 0 0
\(575\) 8.76328 0.365454
\(576\) 0 0
\(577\) 20.2878 0.844590 0.422295 0.906458i \(-0.361225\pi\)
0.422295 + 0.906458i \(0.361225\pi\)
\(578\) 0 0
\(579\) − 24.3119i − 1.01037i
\(580\) 0 0
\(581\) 5.79021i 0.240219i
\(582\) 0 0
\(583\) −16.6074 −0.687806
\(584\) 0 0
\(585\) 18.3437 0.758419
\(586\) 0 0
\(587\) − 43.7364i − 1.80519i −0.430488 0.902596i \(-0.641659\pi\)
0.430488 0.902596i \(-0.358341\pi\)
\(588\) 0 0
\(589\) − 5.94614i − 0.245006i
\(590\) 0 0
\(591\) −40.9192 −1.68319
\(592\) 0 0
\(593\) −3.86818 −0.158847 −0.0794235 0.996841i \(-0.525308\pi\)
−0.0794235 + 0.996841i \(0.525308\pi\)
\(594\) 0 0
\(595\) 1.92204i 0.0787958i
\(596\) 0 0
\(597\) − 59.3486i − 2.42898i
\(598\) 0 0
\(599\) 41.8682 1.71069 0.855344 0.518061i \(-0.173346\pi\)
0.855344 + 0.518061i \(0.173346\pi\)
\(600\) 0 0
\(601\) 22.2955 0.909452 0.454726 0.890631i \(-0.349737\pi\)
0.454726 + 0.890631i \(0.349737\pi\)
\(602\) 0 0
\(603\) 1.12900i 0.0459764i
\(604\) 0 0
\(605\) − 5.32778i − 0.216605i
\(606\) 0 0
\(607\) −11.6984 −0.474824 −0.237412 0.971409i \(-0.576299\pi\)
−0.237412 + 0.971409i \(0.576299\pi\)
\(608\) 0 0
\(609\) 16.9890 0.688429
\(610\) 0 0
\(611\) − 18.4355i − 0.745821i
\(612\) 0 0
\(613\) 29.4784i 1.19062i 0.803496 + 0.595310i \(0.202971\pi\)
−0.803496 + 0.595310i \(0.797029\pi\)
\(614\) 0 0
\(615\) −34.1339 −1.37641
\(616\) 0 0
\(617\) 29.7682 1.19842 0.599211 0.800591i \(-0.295481\pi\)
0.599211 + 0.800591i \(0.295481\pi\)
\(618\) 0 0
\(619\) 34.0857i 1.37002i 0.728533 + 0.685010i \(0.240202\pi\)
−0.728533 + 0.685010i \(0.759798\pi\)
\(620\) 0 0
\(621\) 60.4238i 2.42472i
\(622\) 0 0
\(623\) −1.23672 −0.0495480
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) − 39.9241i − 1.59442i
\(628\) 0 0
\(629\) − 5.41465i − 0.215896i
\(630\) 0 0
\(631\) −30.1318 −1.19953 −0.599764 0.800177i \(-0.704739\pi\)
−0.599764 + 0.800177i \(0.704739\pi\)
\(632\) 0 0
\(633\) −75.1630 −2.98746
\(634\) 0 0
\(635\) − 2.97307i − 0.117983i
\(636\) 0 0
\(637\) − 3.40857i − 0.135053i
\(638\) 0 0
\(639\) 74.2140 2.93586
\(640\) 0 0
\(641\) −41.1609 −1.62576 −0.812878 0.582434i \(-0.802100\pi\)
−0.812878 + 0.582434i \(0.802100\pi\)
\(642\) 0 0
\(643\) − 20.1098i − 0.793054i −0.918023 0.396527i \(-0.870215\pi\)
0.918023 0.396527i \(-0.129785\pi\)
\(644\) 0 0
\(645\) − 5.79021i − 0.227989i
\(646\) 0 0
\(647\) 4.31186 0.169517 0.0847583 0.996402i \(-0.472988\pi\)
0.0847583 + 0.996402i \(0.472988\pi\)
\(648\) 0 0
\(649\) 16.6074 0.651896
\(650\) 0 0
\(651\) − 2.97307i − 0.116524i
\(652\) 0 0
\(653\) 26.8972i 1.05257i 0.850309 + 0.526285i \(0.176415\pi\)
−0.850309 + 0.526285i \(0.823585\pi\)
\(654\) 0 0
\(655\) −10.8171 −0.422661
\(656\) 0 0
\(657\) 79.4507 3.09967
\(658\) 0 0
\(659\) − 28.4674i − 1.10893i −0.832207 0.554465i \(-0.812923\pi\)
0.832207 0.554465i \(-0.187077\pi\)
\(660\) 0 0
\(661\) 5.12900i 0.199495i 0.995013 + 0.0997475i \(0.0318035\pi\)
−0.995013 + 0.0997475i \(0.968197\pi\)
\(662\) 0 0
\(663\) −18.9670 −0.736617
\(664\) 0 0
\(665\) 5.79021 0.224535
\(666\) 0 0
\(667\) 51.4245i 1.99116i
\(668\) 0 0
\(669\) 71.1311i 2.75009i
\(670\) 0 0
\(671\) −35.1609 −1.35737
\(672\) 0 0
\(673\) 46.3437 1.78642 0.893209 0.449641i \(-0.148448\pi\)
0.893209 + 0.449641i \(0.148448\pi\)
\(674\) 0 0
\(675\) 6.89511i 0.265393i
\(676\) 0 0
\(677\) − 13.8061i − 0.530613i −0.964164 0.265307i \(-0.914527\pi\)
0.964164 0.265307i \(-0.0854732\pi\)
\(678\) 0 0
\(679\) 5.92204 0.227267
\(680\) 0 0
\(681\) 60.3376 2.31214
\(682\) 0 0
\(683\) − 0.817143i − 0.0312671i −0.999878 0.0156335i \(-0.995023\pi\)
0.999878 0.0156335i \(-0.00497651\pi\)
\(684\) 0 0
\(685\) − 2.76328i − 0.105580i
\(686\) 0 0
\(687\) 5.33879 0.203687
\(688\) 0 0
\(689\) 23.7682 0.905497
\(690\) 0 0
\(691\) − 21.1070i − 0.802948i −0.915870 0.401474i \(-0.868498\pi\)
0.915870 0.401474i \(-0.131502\pi\)
\(692\) 0 0
\(693\) − 12.8171i − 0.486883i
\(694\) 0 0
\(695\) −2.81714 −0.106860
\(696\) 0 0
\(697\) 22.6612 0.858355
\(698\) 0 0
\(699\) − 3.26857i − 0.123629i
\(700\) 0 0
\(701\) − 38.5556i − 1.45622i −0.685458 0.728112i \(-0.740398\pi\)
0.685458 0.728112i \(-0.259602\pi\)
\(702\) 0 0
\(703\) −16.3119 −0.615213
\(704\) 0 0
\(705\) 15.6584 0.589729
\(706\) 0 0
\(707\) − 9.58043i − 0.360309i
\(708\) 0 0
\(709\) − 40.9213i − 1.53683i −0.639951 0.768416i \(-0.721045\pi\)
0.639951 0.768416i \(-0.278955\pi\)
\(710\) 0 0
\(711\) 72.3756 2.71430
\(712\) 0 0
\(713\) 8.99927 0.337025
\(714\) 0 0
\(715\) − 8.11800i − 0.303596i
\(716\) 0 0
\(717\) 17.7523i 0.662971i
\(718\) 0 0
\(719\) 40.9511 1.52722 0.763609 0.645680i \(-0.223426\pi\)
0.763609 + 0.645680i \(0.223426\pi\)
\(720\) 0 0
\(721\) 6.59143 0.245478
\(722\) 0 0
\(723\) 74.0581i 2.75425i
\(724\) 0 0
\(725\) 5.86818i 0.217939i
\(726\) 0 0
\(727\) −42.6336 −1.58119 −0.790596 0.612339i \(-0.790229\pi\)
−0.790596 + 0.612339i \(0.790229\pi\)
\(728\) 0 0
\(729\) 39.3437 1.45717
\(730\) 0 0
\(731\) 3.84407i 0.142178i
\(732\) 0 0
\(733\) 26.8813i 0.992883i 0.868070 + 0.496441i \(0.165360\pi\)
−0.868070 + 0.496441i \(0.834640\pi\)
\(734\) 0 0
\(735\) 2.89511 0.106788
\(736\) 0 0
\(737\) 0.499637 0.0184044
\(738\) 0 0
\(739\) 28.2201i 1.03809i 0.854746 + 0.519046i \(0.173713\pi\)
−0.854746 + 0.519046i \(0.826287\pi\)
\(740\) 0 0
\(741\) 57.1388i 2.09905i
\(742\) 0 0
\(743\) 35.6605 1.30826 0.654128 0.756384i \(-0.273036\pi\)
0.654128 + 0.756384i \(0.273036\pi\)
\(744\) 0 0
\(745\) −2.05386 −0.0752476
\(746\) 0 0
\(747\) − 31.1609i − 1.14012i
\(748\) 0 0
\(749\) − 0.0538591i − 0.00196797i
\(750\) 0 0
\(751\) −28.7654 −1.04966 −0.524832 0.851206i \(-0.675872\pi\)
−0.524832 + 0.851206i \(0.675872\pi\)
\(752\) 0 0
\(753\) −8.15593 −0.297219
\(754\) 0 0
\(755\) − 9.71225i − 0.353465i
\(756\) 0 0
\(757\) − 50.7413i − 1.84422i −0.386924 0.922112i \(-0.626462\pi\)
0.386924 0.922112i \(-0.373538\pi\)
\(758\) 0 0
\(759\) 60.4238 2.19324
\(760\) 0 0
\(761\) 21.8923 0.793595 0.396797 0.917906i \(-0.370122\pi\)
0.396797 + 0.917906i \(0.370122\pi\)
\(762\) 0 0
\(763\) − 9.44860i − 0.342062i
\(764\) 0 0
\(765\) − 10.3437i − 0.373978i
\(766\) 0 0
\(767\) −23.7682 −0.858220
\(768\) 0 0
\(769\) −43.8221 −1.58026 −0.790132 0.612937i \(-0.789988\pi\)
−0.790132 + 0.612937i \(0.789988\pi\)
\(770\) 0 0
\(771\) − 41.5266i − 1.49554i
\(772\) 0 0
\(773\) 40.9351i 1.47233i 0.676800 + 0.736167i \(0.263366\pi\)
−0.676800 + 0.736167i \(0.736634\pi\)
\(774\) 0 0
\(775\) 1.02693 0.0368884
\(776\) 0 0
\(777\) −8.15593 −0.292592
\(778\) 0 0
\(779\) − 68.2678i − 2.44595i
\(780\) 0 0
\(781\) − 32.8433i − 1.17523i
\(782\) 0 0
\(783\) −40.4617 −1.44598
\(784\) 0 0
\(785\) −2.41957 −0.0863583
\(786\) 0 0
\(787\) − 10.3196i − 0.367854i −0.982940 0.183927i \(-0.941119\pi\)
0.982940 0.183927i \(-0.0588810\pi\)
\(788\) 0 0
\(789\) 29.5584i 1.05231i
\(790\) 0 0
\(791\) 12.5535 0.446351
\(792\) 0 0
\(793\) 50.3217 1.78698
\(794\) 0 0
\(795\) 20.1878i 0.715987i
\(796\) 0 0
\(797\) − 38.9890i − 1.38106i −0.723303 0.690531i \(-0.757377\pi\)
0.723303 0.690531i \(-0.242623\pi\)
\(798\) 0 0
\(799\) −10.3955 −0.367765
\(800\) 0 0
\(801\) 6.65557 0.235163
\(802\) 0 0
\(803\) − 35.1609i − 1.24080i
\(804\) 0 0
\(805\) 8.76328i 0.308865i
\(806\) 0 0
\(807\) −24.7633 −0.871709
\(808\) 0 0
\(809\) −19.6722 −0.691638 −0.345819 0.938301i \(-0.612399\pi\)
−0.345819 + 0.938301i \(0.612399\pi\)
\(810\) 0 0
\(811\) 33.8703i 1.18935i 0.803968 + 0.594673i \(0.202719\pi\)
−0.803968 + 0.594673i \(0.797281\pi\)
\(812\) 0 0
\(813\) − 56.6874i − 1.98811i
\(814\) 0 0
\(815\) 12.5535 0.439730
\(816\) 0 0
\(817\) 11.5804 0.405148
\(818\) 0 0
\(819\) 18.3437i 0.640981i
\(820\) 0 0
\(821\) − 24.8731i − 0.868077i −0.900894 0.434039i \(-0.857088\pi\)
0.900894 0.434039i \(-0.142912\pi\)
\(822\) 0 0
\(823\) 1.44650 0.0504219 0.0252110 0.999682i \(-0.491974\pi\)
0.0252110 + 0.999682i \(0.491974\pi\)
\(824\) 0 0
\(825\) 6.89511 0.240057
\(826\) 0 0
\(827\) 24.2416i 0.842964i 0.906837 + 0.421482i \(0.138490\pi\)
−0.906837 + 0.421482i \(0.861510\pi\)
\(828\) 0 0
\(829\) 40.5854i 1.40959i 0.709412 + 0.704794i \(0.248960\pi\)
−0.709412 + 0.704794i \(0.751040\pi\)
\(830\) 0 0
\(831\) −36.9511 −1.28182
\(832\) 0 0
\(833\) −1.92204 −0.0665946
\(834\) 0 0
\(835\) − 24.5694i − 0.850260i
\(836\) 0 0
\(837\) 7.08079i 0.244748i
\(838\) 0 0
\(839\) −12.0482 −0.415950 −0.207975 0.978134i \(-0.566687\pi\)
−0.207975 + 0.978134i \(0.566687\pi\)
\(840\) 0 0
\(841\) −5.43550 −0.187431
\(842\) 0 0
\(843\) − 75.1630i − 2.58875i
\(844\) 0 0
\(845\) − 1.38164i − 0.0475299i
\(846\) 0 0
\(847\) 5.32778 0.183065
\(848\) 0 0
\(849\) −82.7352 −2.83946
\(850\) 0 0
\(851\) − 24.6874i − 0.846274i
\(852\) 0 0
\(853\) 36.3217i 1.24363i 0.783164 + 0.621816i \(0.213605\pi\)
−0.783164 + 0.621816i \(0.786395\pi\)
\(854\) 0 0
\(855\) −31.1609 −1.06568
\(856\) 0 0
\(857\) 39.9780 1.36562 0.682811 0.730595i \(-0.260757\pi\)
0.682811 + 0.730595i \(0.260757\pi\)
\(858\) 0 0
\(859\) 46.9454i 1.60176i 0.598827 + 0.800878i \(0.295634\pi\)
−0.598827 + 0.800878i \(0.704366\pi\)
\(860\) 0 0
\(861\) − 34.1339i − 1.16328i
\(862\) 0 0
\(863\) 16.2955 0.554705 0.277353 0.960768i \(-0.410543\pi\)
0.277353 + 0.960768i \(0.410543\pi\)
\(864\) 0 0
\(865\) 1.46243 0.0497241
\(866\) 0 0
\(867\) − 38.5216i − 1.30826i
\(868\) 0 0
\(869\) − 32.0298i − 1.08653i
\(870\) 0 0
\(871\) −0.715074 −0.0242294
\(872\) 0 0
\(873\) −31.8703 −1.07865
\(874\) 0 0
\(875\) 1.00000i 0.0338062i
\(876\) 0 0
\(877\) − 36.9511i − 1.24775i −0.781525 0.623874i \(-0.785558\pi\)
0.781525 0.623874i \(-0.214442\pi\)
\(878\) 0 0
\(879\) 35.2388 1.18858
\(880\) 0 0
\(881\) 11.9780 0.403549 0.201774 0.979432i \(-0.435329\pi\)
0.201774 + 0.979432i \(0.435329\pi\)
\(882\) 0 0
\(883\) − 26.6556i − 0.897031i −0.893775 0.448516i \(-0.851953\pi\)
0.893775 0.448516i \(-0.148047\pi\)
\(884\) 0 0
\(885\) − 20.1878i − 0.678605i
\(886\) 0 0
\(887\) 4.31186 0.144778 0.0723890 0.997376i \(-0.476938\pi\)
0.0723890 + 0.997376i \(0.476938\pi\)
\(888\) 0 0
\(889\) 2.97307 0.0997136
\(890\) 0 0
\(891\) 9.09107i 0.304562i
\(892\) 0 0
\(893\) 31.3168i 1.04798i
\(894\) 0 0
\(895\) 8.81714 0.294725
\(896\) 0 0
\(897\) −86.4776 −2.88740
\(898\) 0 0
\(899\) 6.02620i 0.200985i
\(900\) 0 0
\(901\) − 13.4025i − 0.446502i
\(902\) 0 0
\(903\) 5.79021 0.192686
\(904\) 0 0
\(905\) −17.3168 −0.575629
\(906\) 0 0
\(907\) − 11.0269i − 0.366143i −0.983100 0.183072i \(-0.941396\pi\)
0.983100 0.183072i \(-0.0586040\pi\)
\(908\) 0 0
\(909\) 51.5584i 1.71008i
\(910\) 0 0
\(911\) 23.0049 0.762187 0.381094 0.924536i \(-0.375548\pi\)
0.381094 + 0.924536i \(0.375548\pi\)
\(912\) 0 0
\(913\) −13.7902 −0.456389
\(914\) 0 0
\(915\) 42.7413i 1.41298i
\(916\) 0 0
\(917\) − 10.8171i − 0.357214i
\(918\) 0 0
\(919\) 5.13675 0.169446 0.0847228 0.996405i \(-0.473000\pi\)
0.0847228 + 0.996405i \(0.473000\pi\)
\(920\) 0 0
\(921\) −37.1768 −1.22502
\(922\) 0 0
\(923\) 47.0049i 1.54719i
\(924\) 0 0
\(925\) − 2.81714i − 0.0926271i
\(926\) 0 0
\(927\) −35.4727 −1.16508
\(928\) 0 0
\(929\) 15.0588 0.494063 0.247031 0.969007i \(-0.420545\pi\)
0.247031 + 0.969007i \(0.420545\pi\)
\(930\) 0 0
\(931\) 5.79021i 0.189767i
\(932\) 0 0
\(933\) − 49.7462i − 1.62862i
\(934\) 0 0
\(935\) −4.57760 −0.149704
\(936\) 0 0
\(937\) 6.76118 0.220878 0.110439 0.993883i \(-0.464774\pi\)
0.110439 + 0.993883i \(0.464774\pi\)
\(938\) 0 0
\(939\) − 44.2739i − 1.44482i
\(940\) 0 0
\(941\) 43.5584i 1.41996i 0.704220 + 0.709982i \(0.251297\pi\)
−0.704220 + 0.709982i \(0.748703\pi\)
\(942\) 0 0
\(943\) 103.321 3.36459
\(944\) 0 0
\(945\) −6.89511 −0.224298
\(946\) 0 0
\(947\) − 29.2091i − 0.949167i −0.880210 0.474583i \(-0.842599\pi\)
0.880210 0.474583i \(-0.157401\pi\)
\(948\) 0 0
\(949\) 50.3217i 1.63351i
\(950\) 0 0
\(951\) −89.4507 −2.90064
\(952\) 0 0
\(953\) −37.3168 −1.20881 −0.604405 0.796678i \(-0.706589\pi\)
−0.604405 + 0.796678i \(0.706589\pi\)
\(954\) 0 0
\(955\) 15.5025i 0.501648i
\(956\) 0 0
\(957\) 40.4617i 1.30794i
\(958\) 0 0
\(959\) 2.76328 0.0892311
\(960\) 0 0
\(961\) −29.9454 −0.965981
\(962\) 0 0
\(963\) 0.289850i 0.00934030i
\(964\) 0 0
\(965\) 8.39757i 0.270327i
\(966\) 0 0
\(967\) −6.62936 −0.213186 −0.106593 0.994303i \(-0.533994\pi\)
−0.106593 + 0.994303i \(0.533994\pi\)
\(968\) 0 0
\(969\) 32.2196 1.03504
\(970\) 0 0
\(971\) − 40.4996i − 1.29970i −0.760065 0.649848i \(-0.774833\pi\)
0.760065 0.649848i \(-0.225167\pi\)
\(972\) 0 0
\(973\) − 2.81714i − 0.0903134i
\(974\) 0 0
\(975\) −9.86818 −0.316035
\(976\) 0 0
\(977\) −33.0531 −1.05746 −0.528732 0.848789i \(-0.677332\pi\)
−0.528732 + 0.848789i \(0.677332\pi\)
\(978\) 0 0
\(979\) − 2.94542i − 0.0941359i
\(980\) 0 0
\(981\) 50.8490i 1.62348i
\(982\) 0 0
\(983\) −9.40857 −0.300087 −0.150043 0.988679i \(-0.547941\pi\)
−0.150043 + 0.988679i \(0.547941\pi\)
\(984\) 0 0
\(985\) 14.1339 0.450344
\(986\) 0 0
\(987\) 15.6584i 0.498412i
\(988\) 0 0
\(989\) 17.5266i 0.557312i
\(990\) 0 0
\(991\) −15.0049 −0.476647 −0.238324 0.971186i \(-0.576598\pi\)
−0.238324 + 0.971186i \(0.576598\pi\)
\(992\) 0 0
\(993\) −76.5797 −2.43018
\(994\) 0 0
\(995\) 20.4996i 0.649882i
\(996\) 0 0
\(997\) − 36.1719i − 1.14557i −0.819704 0.572787i \(-0.805862\pi\)
0.819704 0.572787i \(-0.194138\pi\)
\(998\) 0 0
\(999\) 19.4245 0.614564
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 2240.2.b.f.1121.6 yes 6
4.3 odd 2 2240.2.b.e.1121.1 6
8.3 odd 2 2240.2.b.e.1121.6 yes 6
8.5 even 2 inner 2240.2.b.f.1121.1 yes 6
16.3 odd 4 8960.2.a.bk.1.3 3
16.5 even 4 8960.2.a.bn.1.3 3
16.11 odd 4 8960.2.a.bq.1.1 3
16.13 even 4 8960.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.b.e.1121.1 6 4.3 odd 2
2240.2.b.e.1121.6 yes 6 8.3 odd 2
2240.2.b.f.1121.1 yes 6 8.5 even 2 inner
2240.2.b.f.1121.6 yes 6 1.1 even 1 trivial
8960.2.a.bh.1.1 3 16.13 even 4
8960.2.a.bk.1.3 3 16.3 odd 4
8960.2.a.bn.1.3 3 16.5 even 4
8960.2.a.bq.1.1 3 16.11 odd 4