Properties

Label 845.2.c.e.506.4
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4$
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] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.e.506.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} +2.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.00000i q^{7} +1.73205i q^{8} +4.46410 q^{9} +O(q^{10})\) \(q+1.73205i q^{2} +2.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.00000i q^{7} +1.73205i q^{8} +4.46410 q^{9} -1.73205 q^{10} -4.73205i q^{11} -2.73205 q^{12} -3.46410 q^{14} +2.73205i q^{15} -5.00000 q^{16} +3.46410 q^{17} +7.73205i q^{18} +6.19615i q^{19} -1.00000i q^{20} +5.46410i q^{21} +8.19615 q^{22} -1.26795 q^{23} +4.73205i q^{24} -1.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -2.53590 q^{29} -4.73205 q^{30} -10.1962i q^{31} -5.19615i q^{32} -12.9282i q^{33} +6.00000i q^{34} -2.00000 q^{35} -4.46410 q^{36} -4.00000i q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410i q^{41} -9.46410 q^{42} +0.196152 q^{43} +4.73205i q^{44} +4.46410i q^{45} -2.19615i q^{46} +6.00000i q^{47} -13.6603 q^{48} +3.00000 q^{49} -1.73205i q^{50} +9.46410 q^{51} +10.3923 q^{53} +6.92820i q^{54} +4.73205 q^{55} -3.46410 q^{56} +16.9282i q^{57} -4.39230i q^{58} +9.12436i q^{59} -2.73205i q^{60} -8.39230 q^{61} +17.6603 q^{62} +8.92820i q^{63} -1.00000 q^{64} +22.3923 q^{66} -6.39230i q^{67} -3.46410 q^{68} -3.46410 q^{69} -3.46410i q^{70} -4.73205i q^{71} +7.73205i q^{72} -4.00000i q^{73} +6.92820 q^{74} -2.73205 q^{75} -6.19615i q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000i q^{80} -2.46410 q^{81} +6.00000 q^{82} +6.00000i q^{83} -5.46410i q^{84} +3.46410i q^{85} +0.339746i q^{86} -6.92820 q^{87} +8.19615 q^{88} -12.9282i q^{89} -7.73205 q^{90} +1.26795 q^{92} -27.8564i q^{93} -10.3923 q^{94} -6.19615 q^{95} -14.1962i q^{96} -2.00000i q^{97} +5.19615i q^{98} -21.1244i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49} + 24 q^{51} + 12 q^{55} + 8 q^{61} + 36 q^{62} - 4 q^{64} + 48 q^{66} - 4 q^{75} + 24 q^{77} + 8 q^{79} + 4 q^{81} + 24 q^{82} + 12 q^{88} - 24 q^{90} + 12 q^{92} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 4.73205i 1.93185i
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.73205i 0.612372i
\(9\) 4.46410 1.48803
\(10\) −1.73205 −0.547723
\(11\) − 4.73205i − 1.42677i −0.700774 0.713384i \(-0.747162\pi\)
0.700774 0.713384i \(-0.252838\pi\)
\(12\) −2.73205 −0.788675
\(13\) 0 0
\(14\) −3.46410 −0.925820
\(15\) 2.73205i 0.705412i
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 7.73205i 1.82246i
\(19\) 6.19615i 1.42149i 0.703447 + 0.710747i \(0.251643\pi\)
−0.703447 + 0.710747i \(0.748357\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 5.46410i 1.19236i
\(22\) 8.19615 1.74743
\(23\) −1.26795 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 4.73205i 0.965926i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) − 2.00000i − 0.377964i
\(29\) −2.53590 −0.470905 −0.235452 0.971886i \(-0.575657\pi\)
−0.235452 + 0.971886i \(0.575657\pi\)
\(30\) −4.73205 −0.863950
\(31\) − 10.1962i − 1.83128i −0.401996 0.915642i \(-0.631683\pi\)
0.401996 0.915642i \(-0.368317\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) − 12.9282i − 2.25051i
\(34\) 6.00000i 1.02899i
\(35\) −2.00000 −0.338062
\(36\) −4.46410 −0.744017
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) −10.7321 −1.74097
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) −9.46410 −1.46034
\(43\) 0.196152 0.0299130 0.0149565 0.999888i \(-0.495239\pi\)
0.0149565 + 0.999888i \(0.495239\pi\)
\(44\) 4.73205i 0.713384i
\(45\) 4.46410i 0.665469i
\(46\) − 2.19615i − 0.323805i
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) −13.6603 −1.97169
\(49\) 3.00000 0.428571
\(50\) − 1.73205i − 0.244949i
\(51\) 9.46410 1.32524
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 6.92820i 0.942809i
\(55\) 4.73205 0.638070
\(56\) −3.46410 −0.462910
\(57\) 16.9282i 2.24220i
\(58\) − 4.39230i − 0.576738i
\(59\) 9.12436i 1.18789i 0.804506 + 0.593945i \(0.202430\pi\)
−0.804506 + 0.593945i \(0.797570\pi\)
\(60\) − 2.73205i − 0.352706i
\(61\) −8.39230 −1.07452 −0.537262 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 17.6603 2.24285
\(63\) 8.92820i 1.12485i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 22.3923 2.75630
\(67\) − 6.39230i − 0.780944i −0.920615 0.390472i \(-0.872312\pi\)
0.920615 0.390472i \(-0.127688\pi\)
\(68\) −3.46410 −0.420084
\(69\) −3.46410 −0.417029
\(70\) − 3.46410i − 0.414039i
\(71\) − 4.73205i − 0.561591i −0.959768 0.280796i \(-0.909402\pi\)
0.959768 0.280796i \(-0.0905983\pi\)
\(72\) 7.73205i 0.911231i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 6.92820 0.805387
\(75\) −2.73205 −0.315470
\(76\) − 6.19615i − 0.710747i
\(77\) 9.46410 1.07853
\(78\) 0 0
\(79\) −8.39230 −0.944208 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(80\) − 5.00000i − 0.559017i
\(81\) −2.46410 −0.273789
\(82\) 6.00000 0.662589
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) − 5.46410i − 0.596182i
\(85\) 3.46410i 0.375735i
\(86\) 0.339746i 0.0366357i
\(87\) −6.92820 −0.742781
\(88\) 8.19615 0.873713
\(89\) − 12.9282i − 1.37039i −0.728361 0.685193i \(-0.759718\pi\)
0.728361 0.685193i \(-0.240282\pi\)
\(90\) −7.73205 −0.815030
\(91\) 0 0
\(92\) 1.26795 0.132193
\(93\) − 27.8564i − 2.88857i
\(94\) −10.3923 −1.07188
\(95\) −6.19615 −0.635712
\(96\) − 14.1962i − 1.44889i
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 5.19615i 0.524891i
\(99\) − 21.1244i − 2.12308i
\(100\) 1.00000 0.100000
\(101\) 0.928203 0.0923597 0.0461798 0.998933i \(-0.485295\pi\)
0.0461798 + 0.998933i \(0.485295\pi\)
\(102\) 16.3923i 1.62308i
\(103\) 0.196152 0.0193275 0.00966374 0.999953i \(-0.496924\pi\)
0.00966374 + 0.999953i \(0.496924\pi\)
\(104\) 0 0
\(105\) −5.46410 −0.533242
\(106\) 18.0000i 1.74831i
\(107\) 17.6603 1.70728 0.853641 0.520862i \(-0.174390\pi\)
0.853641 + 0.520862i \(0.174390\pi\)
\(108\) −4.00000 −0.384900
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 8.19615i 0.781472i
\(111\) − 10.9282i − 1.03726i
\(112\) − 10.0000i − 0.944911i
\(113\) 8.53590 0.802990 0.401495 0.915861i \(-0.368491\pi\)
0.401495 + 0.915861i \(0.368491\pi\)
\(114\) −29.3205 −2.74612
\(115\) − 1.26795i − 0.118237i
\(116\) 2.53590 0.235452
\(117\) 0 0
\(118\) −15.8038 −1.45486
\(119\) 6.92820i 0.635107i
\(120\) −4.73205 −0.431975
\(121\) −11.3923 −1.03566
\(122\) − 14.5359i − 1.31602i
\(123\) − 9.46410i − 0.853349i
\(124\) 10.1962i 0.915642i
\(125\) − 1.00000i − 0.0894427i
\(126\) −15.4641 −1.37765
\(127\) −16.1962 −1.43718 −0.718588 0.695436i \(-0.755211\pi\)
−0.718588 + 0.695436i \(0.755211\pi\)
\(128\) − 12.1244i − 1.07165i
\(129\) 0.535898 0.0471832
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 12.9282i 1.12526i
\(133\) −12.3923 −1.07455
\(134\) 11.0718 0.956458
\(135\) 4.00000i 0.344265i
\(136\) 6.00000i 0.514496i
\(137\) 0.928203i 0.0793018i 0.999214 + 0.0396509i \(0.0126246\pi\)
−0.999214 + 0.0396509i \(0.987375\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 2.00000 0.169031
\(141\) 16.3923i 1.38048i
\(142\) 8.19615 0.687806
\(143\) 0 0
\(144\) −22.3205 −1.86004
\(145\) − 2.53590i − 0.210595i
\(146\) 6.92820 0.573382
\(147\) 8.19615 0.676007
\(148\) 4.00000i 0.328798i
\(149\) 7.85641i 0.643622i 0.946804 + 0.321811i \(0.104292\pi\)
−0.946804 + 0.321811i \(0.895708\pi\)
\(150\) − 4.73205i − 0.386370i
\(151\) − 1.80385i − 0.146795i −0.997303 0.0733975i \(-0.976616\pi\)
0.997303 0.0733975i \(-0.0233842\pi\)
\(152\) −10.7321 −0.870484
\(153\) 15.4641 1.25020
\(154\) 16.3923i 1.32093i
\(155\) 10.1962 0.818975
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) − 14.5359i − 1.15641i
\(159\) 28.3923 2.25166
\(160\) 5.19615 0.410792
\(161\) − 2.53590i − 0.199857i
\(162\) − 4.26795i − 0.335322i
\(163\) − 14.3923i − 1.12729i −0.826016 0.563646i \(-0.809398\pi\)
0.826016 0.563646i \(-0.190602\pi\)
\(164\) 3.46410i 0.270501i
\(165\) 12.9282 1.00646
\(166\) −10.3923 −0.806599
\(167\) − 0.928203i − 0.0718265i −0.999355 0.0359133i \(-0.988566\pi\)
0.999355 0.0359133i \(-0.0114340\pi\)
\(168\) −9.46410 −0.730171
\(169\) 0 0
\(170\) −6.00000 −0.460179
\(171\) 27.6603i 2.11523i
\(172\) −0.196152 −0.0149565
\(173\) −8.53590 −0.648972 −0.324486 0.945890i \(-0.605191\pi\)
−0.324486 + 0.945890i \(0.605191\pi\)
\(174\) − 12.0000i − 0.909718i
\(175\) − 2.00000i − 0.151186i
\(176\) 23.6603i 1.78346i
\(177\) 24.9282i 1.87372i
\(178\) 22.3923 1.67837
\(179\) 18.9282 1.41476 0.707380 0.706833i \(-0.249877\pi\)
0.707380 + 0.706833i \(0.249877\pi\)
\(180\) − 4.46410i − 0.332734i
\(181\) −0.392305 −0.0291598 −0.0145799 0.999894i \(-0.504641\pi\)
−0.0145799 + 0.999894i \(0.504641\pi\)
\(182\) 0 0
\(183\) −22.9282 −1.69490
\(184\) − 2.19615i − 0.161903i
\(185\) 4.00000 0.294086
\(186\) 48.2487 3.53777
\(187\) − 16.3923i − 1.19872i
\(188\) − 6.00000i − 0.437595i
\(189\) 8.00000i 0.581914i
\(190\) − 10.7321i − 0.778585i
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) −2.73205 −0.197169
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 3.46410 0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.9282i 0.921096i 0.887635 + 0.460548i \(0.152347\pi\)
−0.887635 + 0.460548i \(0.847653\pi\)
\(198\) 36.5885 2.60023
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) − 1.73205i − 0.122474i
\(201\) − 17.4641i − 1.23182i
\(202\) 1.60770i 0.113117i
\(203\) − 5.07180i − 0.355970i
\(204\) −9.46410 −0.662620
\(205\) 3.46410 0.241943
\(206\) 0.339746i 0.0236712i
\(207\) −5.66025 −0.393415
\(208\) 0 0
\(209\) 29.3205 2.02814
\(210\) − 9.46410i − 0.653085i
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.3923 −0.713746
\(213\) − 12.9282i − 0.885826i
\(214\) 30.5885i 2.09098i
\(215\) 0.196152i 0.0133775i
\(216\) 6.92820i 0.471405i
\(217\) 20.3923 1.38432
\(218\) 3.46410 0.234619
\(219\) − 10.9282i − 0.738460i
\(220\) −4.73205 −0.319035
\(221\) 0 0
\(222\) 18.9282 1.27038
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 10.3923 0.694365
\(225\) −4.46410 −0.297607
\(226\) 14.7846i 0.983458i
\(227\) 3.46410i 0.229920i 0.993370 + 0.114960i \(0.0366741\pi\)
−0.993370 + 0.114960i \(0.963326\pi\)
\(228\) − 16.9282i − 1.12110i
\(229\) 6.39230i 0.422415i 0.977441 + 0.211208i \(0.0677396\pi\)
−0.977441 + 0.211208i \(0.932260\pi\)
\(230\) 2.19615 0.144810
\(231\) 25.8564 1.70123
\(232\) − 4.39230i − 0.288369i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) − 9.12436i − 0.593945i
\(237\) −22.9282 −1.48935
\(238\) −12.0000 −0.777844
\(239\) 14.1962i 0.918273i 0.888366 + 0.459136i \(0.151841\pi\)
−0.888366 + 0.459136i \(0.848159\pi\)
\(240\) − 13.6603i − 0.881766i
\(241\) − 2.39230i − 0.154102i −0.997027 0.0770510i \(-0.975450\pi\)
0.997027 0.0770510i \(-0.0245504\pi\)
\(242\) − 19.7321i − 1.26842i
\(243\) −18.7321 −1.20166
\(244\) 8.39230 0.537262
\(245\) 3.00000i 0.191663i
\(246\) 16.3923 1.04514
\(247\) 0 0
\(248\) 17.6603 1.12143
\(249\) 16.3923i 1.03882i
\(250\) 1.73205 0.109545
\(251\) 21.4641 1.35480 0.677401 0.735614i \(-0.263106\pi\)
0.677401 + 0.735614i \(0.263106\pi\)
\(252\) − 8.92820i − 0.562424i
\(253\) 6.00000i 0.377217i
\(254\) − 28.0526i − 1.76017i
\(255\) 9.46410i 0.592665i
\(256\) 19.0000 1.18750
\(257\) −19.8564 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(258\) 0.928203i 0.0577874i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −11.3205 −0.700722
\(262\) 0 0
\(263\) 1.26795 0.0781851 0.0390925 0.999236i \(-0.487553\pi\)
0.0390925 + 0.999236i \(0.487553\pi\)
\(264\) 22.3923 1.37815
\(265\) 10.3923i 0.638394i
\(266\) − 21.4641i − 1.31605i
\(267\) − 35.3205i − 2.16158i
\(268\) 6.39230i 0.390472i
\(269\) −19.8564 −1.21067 −0.605333 0.795972i \(-0.706960\pi\)
−0.605333 + 0.795972i \(0.706960\pi\)
\(270\) −6.92820 −0.421637
\(271\) 30.9808i 1.88195i 0.338480 + 0.940974i \(0.390087\pi\)
−0.338480 + 0.940974i \(0.609913\pi\)
\(272\) −17.3205 −1.05021
\(273\) 0 0
\(274\) −1.60770 −0.0971244
\(275\) 4.73205i 0.285353i
\(276\) 3.46410 0.208514
\(277\) 26.3923 1.58576 0.792880 0.609378i \(-0.208581\pi\)
0.792880 + 0.609378i \(0.208581\pi\)
\(278\) 21.4641i 1.28733i
\(279\) − 45.5167i − 2.72501i
\(280\) − 3.46410i − 0.207020i
\(281\) 22.3923i 1.33581i 0.744245 + 0.667906i \(0.232809\pi\)
−0.744245 + 0.667906i \(0.767191\pi\)
\(282\) −28.3923 −1.69074
\(283\) −32.5885 −1.93718 −0.968591 0.248658i \(-0.920010\pi\)
−0.968591 + 0.248658i \(0.920010\pi\)
\(284\) 4.73205i 0.280796i
\(285\) −16.9282 −1.00274
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) − 23.1962i − 1.36685i
\(289\) −5.00000 −0.294118
\(290\) 4.39230 0.257925
\(291\) − 5.46410i − 0.320311i
\(292\) 4.00000i 0.234082i
\(293\) − 5.07180i − 0.296298i −0.988965 0.148149i \(-0.952669\pi\)
0.988965 0.148149i \(-0.0473314\pi\)
\(294\) 14.1962i 0.827936i
\(295\) −9.12436 −0.531241
\(296\) 6.92820 0.402694
\(297\) − 18.9282i − 1.09833i
\(298\) −13.6077 −0.788273
\(299\) 0 0
\(300\) 2.73205 0.157735
\(301\) 0.392305i 0.0226121i
\(302\) 3.12436 0.179786
\(303\) 2.53590 0.145684
\(304\) − 30.9808i − 1.77687i
\(305\) − 8.39230i − 0.480542i
\(306\) 26.7846i 1.53117i
\(307\) − 18.7846i − 1.07209i −0.844188 0.536047i \(-0.819917\pi\)
0.844188 0.536047i \(-0.180083\pi\)
\(308\) −9.46410 −0.539267
\(309\) 0.535898 0.0304862
\(310\) 17.6603i 1.00304i
\(311\) 16.3923 0.929522 0.464761 0.885436i \(-0.346140\pi\)
0.464761 + 0.885436i \(0.346140\pi\)
\(312\) 0 0
\(313\) −14.3923 −0.813501 −0.406751 0.913539i \(-0.633338\pi\)
−0.406751 + 0.913539i \(0.633338\pi\)
\(314\) − 17.3205i − 0.977453i
\(315\) −8.92820 −0.503047
\(316\) 8.39230 0.472104
\(317\) − 24.0000i − 1.34797i −0.738743 0.673987i \(-0.764580\pi\)
0.738743 0.673987i \(-0.235420\pi\)
\(318\) 49.1769i 2.75770i
\(319\) 12.0000i 0.671871i
\(320\) − 1.00000i − 0.0559017i
\(321\) 48.2487 2.69298
\(322\) 4.39230 0.244774
\(323\) 21.4641i 1.19429i
\(324\) 2.46410 0.136895
\(325\) 0 0
\(326\) 24.9282 1.38065
\(327\) − 5.46410i − 0.302166i
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) 22.3923i 1.23266i
\(331\) − 2.58846i − 0.142274i −0.997467 0.0711372i \(-0.977337\pi\)
0.997467 0.0711372i \(-0.0226628\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) − 17.8564i − 0.978525i
\(334\) 1.60770 0.0879692
\(335\) 6.39230 0.349249
\(336\) − 27.3205i − 1.49046i
\(337\) 26.3923 1.43768 0.718840 0.695175i \(-0.244673\pi\)
0.718840 + 0.695175i \(0.244673\pi\)
\(338\) 0 0
\(339\) 23.3205 1.26660
\(340\) − 3.46410i − 0.187867i
\(341\) −48.2487 −2.61281
\(342\) −47.9090 −2.59062
\(343\) 20.0000i 1.07990i
\(344\) 0.339746i 0.0183179i
\(345\) − 3.46410i − 0.186501i
\(346\) − 14.7846i − 0.794826i
\(347\) −5.66025 −0.303858 −0.151929 0.988391i \(-0.548549\pi\)
−0.151929 + 0.988391i \(0.548549\pi\)
\(348\) 6.92820 0.371391
\(349\) − 14.3923i − 0.770402i −0.922833 0.385201i \(-0.874132\pi\)
0.922833 0.385201i \(-0.125868\pi\)
\(350\) 3.46410 0.185164
\(351\) 0 0
\(352\) −24.5885 −1.31057
\(353\) 27.7128i 1.47500i 0.675345 + 0.737502i \(0.263995\pi\)
−0.675345 + 0.737502i \(0.736005\pi\)
\(354\) −43.1769 −2.29483
\(355\) 4.73205 0.251151
\(356\) 12.9282i 0.685193i
\(357\) 18.9282i 1.00179i
\(358\) 32.7846i 1.73272i
\(359\) − 2.19615i − 0.115908i −0.998319 0.0579542i \(-0.981542\pi\)
0.998319 0.0579542i \(-0.0184578\pi\)
\(360\) −7.73205 −0.407515
\(361\) −19.3923 −1.02065
\(362\) − 0.679492i − 0.0357133i
\(363\) −31.1244 −1.63361
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) − 39.7128i − 2.07582i
\(367\) 11.8038 0.616156 0.308078 0.951361i \(-0.400314\pi\)
0.308078 + 0.951361i \(0.400314\pi\)
\(368\) 6.33975 0.330482
\(369\) − 15.4641i − 0.805029i
\(370\) 6.92820i 0.360180i
\(371\) 20.7846i 1.07908i
\(372\) 27.8564i 1.44429i
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 28.3923 1.46813
\(375\) − 2.73205i − 0.141082i
\(376\) −10.3923 −0.535942
\(377\) 0 0
\(378\) −13.8564 −0.712697
\(379\) − 18.9808i − 0.974976i −0.873130 0.487488i \(-0.837913\pi\)
0.873130 0.487488i \(-0.162087\pi\)
\(380\) 6.19615 0.317856
\(381\) −44.2487 −2.26693
\(382\) − 8.78461i − 0.449460i
\(383\) − 12.9282i − 0.660600i −0.943876 0.330300i \(-0.892850\pi\)
0.943876 0.330300i \(-0.107150\pi\)
\(384\) − 33.1244i − 1.69037i
\(385\) 9.46410i 0.482335i
\(386\) 17.3205 0.881591
\(387\) 0.875644 0.0445115
\(388\) 2.00000i 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −4.39230 −0.222128
\(392\) 5.19615i 0.262445i
\(393\) 0 0
\(394\) −22.3923 −1.12811
\(395\) − 8.39230i − 0.422263i
\(396\) 21.1244i 1.06154i
\(397\) 28.7846i 1.44466i 0.691549 + 0.722329i \(0.256928\pi\)
−0.691549 + 0.722329i \(0.743072\pi\)
\(398\) − 34.6410i − 1.73640i
\(399\) −33.8564 −1.69494
\(400\) 5.00000 0.250000
\(401\) − 36.9282i − 1.84411i −0.387063 0.922053i \(-0.626510\pi\)
0.387063 0.922053i \(-0.373490\pi\)
\(402\) 30.2487 1.50867
\(403\) 0 0
\(404\) −0.928203 −0.0461798
\(405\) − 2.46410i − 0.122442i
\(406\) 8.78461 0.435973
\(407\) −18.9282 −0.938236
\(408\) 16.3923i 0.811540i
\(409\) 17.6077i 0.870644i 0.900275 + 0.435322i \(0.143366\pi\)
−0.900275 + 0.435322i \(0.856634\pi\)
\(410\) 6.00000i 0.296319i
\(411\) 2.53590i 0.125087i
\(412\) −0.196152 −0.00966374
\(413\) −18.2487 −0.897960
\(414\) − 9.80385i − 0.481833i
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 33.8564 1.65796
\(418\) 50.7846i 2.48396i
\(419\) −2.53590 −0.123887 −0.0619434 0.998080i \(-0.519730\pi\)
−0.0619434 + 0.998080i \(0.519730\pi\)
\(420\) 5.46410 0.266621
\(421\) 30.7846i 1.50035i 0.661239 + 0.750175i \(0.270031\pi\)
−0.661239 + 0.750175i \(0.729969\pi\)
\(422\) 13.8564i 0.674519i
\(423\) 26.7846i 1.30231i
\(424\) 18.0000i 0.874157i
\(425\) −3.46410 −0.168034
\(426\) 22.3923 1.08491
\(427\) − 16.7846i − 0.812264i
\(428\) −17.6603 −0.853641
\(429\) 0 0
\(430\) −0.339746 −0.0163840
\(431\) 25.5167i 1.22909i 0.788880 + 0.614547i \(0.210661\pi\)
−0.788880 + 0.614547i \(0.789339\pi\)
\(432\) −20.0000 −0.962250
\(433\) −34.7846 −1.67164 −0.835821 0.549002i \(-0.815008\pi\)
−0.835821 + 0.549002i \(0.815008\pi\)
\(434\) 35.3205i 1.69544i
\(435\) − 6.92820i − 0.332182i
\(436\) 2.00000i 0.0957826i
\(437\) − 7.85641i − 0.375823i
\(438\) 18.9282 0.904425
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 8.19615i 0.390736i
\(441\) 13.3923 0.637729
\(442\) 0 0
\(443\) −16.9808 −0.806780 −0.403390 0.915028i \(-0.632168\pi\)
−0.403390 + 0.915028i \(0.632168\pi\)
\(444\) 10.9282i 0.518630i
\(445\) 12.9282 0.612856
\(446\) 3.46410 0.164030
\(447\) 21.4641i 1.01522i
\(448\) − 2.00000i − 0.0944911i
\(449\) 20.5359i 0.969149i 0.874750 + 0.484574i \(0.161026\pi\)
−0.874750 + 0.484574i \(0.838974\pi\)
\(450\) − 7.73205i − 0.364492i
\(451\) −16.3923 −0.771883
\(452\) −8.53590 −0.401495
\(453\) − 4.92820i − 0.231547i
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) −29.3205 −1.37306
\(457\) − 10.7846i − 0.504483i −0.967664 0.252241i \(-0.918832\pi\)
0.967664 0.252241i \(-0.0811677\pi\)
\(458\) −11.0718 −0.517351
\(459\) 13.8564 0.646762
\(460\) 1.26795i 0.0591184i
\(461\) 3.46410i 0.161339i 0.996741 + 0.0806696i \(0.0257059\pi\)
−0.996741 + 0.0806696i \(0.974294\pi\)
\(462\) 44.7846i 2.08357i
\(463\) − 2.39230i − 0.111180i −0.998454 0.0555899i \(-0.982296\pi\)
0.998454 0.0555899i \(-0.0177039\pi\)
\(464\) 12.6795 0.588631
\(465\) 27.8564 1.29181
\(466\) 10.3923i 0.481414i
\(467\) −27.8038 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(468\) 0 0
\(469\) 12.7846 0.590338
\(470\) − 10.3923i − 0.479361i
\(471\) −27.3205 −1.25886
\(472\) −15.8038 −0.727431
\(473\) − 0.928203i − 0.0426788i
\(474\) − 39.7128i − 1.82407i
\(475\) − 6.19615i − 0.284299i
\(476\) − 6.92820i − 0.317554i
\(477\) 46.3923 2.12416
\(478\) −24.5885 −1.12465
\(479\) 35.6603i 1.62936i 0.579912 + 0.814679i \(0.303087\pi\)
−0.579912 + 0.814679i \(0.696913\pi\)
\(480\) 14.1962 0.647963
\(481\) 0 0
\(482\) 4.14359 0.188736
\(483\) − 6.92820i − 0.315244i
\(484\) 11.3923 0.517832
\(485\) 2.00000 0.0908153
\(486\) − 32.4449i − 1.47173i
\(487\) 26.3923i 1.19595i 0.801515 + 0.597975i \(0.204028\pi\)
−0.801515 + 0.597975i \(0.795972\pi\)
\(488\) − 14.5359i − 0.658009i
\(489\) − 39.3205i − 1.77813i
\(490\) −5.19615 −0.234738
\(491\) 2.53590 0.114443 0.0572217 0.998361i \(-0.481776\pi\)
0.0572217 + 0.998361i \(0.481776\pi\)
\(492\) 9.46410i 0.426675i
\(493\) −8.78461 −0.395639
\(494\) 0 0
\(495\) 21.1244 0.949469
\(496\) 50.9808i 2.28910i
\(497\) 9.46410 0.424523
\(498\) −28.3923 −1.27229
\(499\) 38.9808i 1.74502i 0.488598 + 0.872509i \(0.337509\pi\)
−0.488598 + 0.872509i \(0.662491\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 2.53590i − 0.113296i
\(502\) 37.1769i 1.65929i
\(503\) 19.5167 0.870205 0.435102 0.900381i \(-0.356712\pi\)
0.435102 + 0.900381i \(0.356712\pi\)
\(504\) −15.4641 −0.688826
\(505\) 0.928203i 0.0413045i
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) 16.1962 0.718588
\(509\) − 39.4641i − 1.74922i −0.484831 0.874608i \(-0.661119\pi\)
0.484831 0.874608i \(-0.338881\pi\)
\(510\) −16.3923 −0.725863
\(511\) 8.00000 0.353899
\(512\) 8.66025i 0.382733i
\(513\) 24.7846i 1.09427i
\(514\) − 34.3923i − 1.51698i
\(515\) 0.196152i 0.00864351i
\(516\) −0.535898 −0.0235916
\(517\) 28.3923 1.24869
\(518\) 13.8564i 0.608816i
\(519\) −23.3205 −1.02366
\(520\) 0 0
\(521\) −28.3923 −1.24389 −0.621945 0.783061i \(-0.713657\pi\)
−0.621945 + 0.783061i \(0.713657\pi\)
\(522\) − 19.6077i − 0.858206i
\(523\) −24.1962 −1.05802 −0.529012 0.848614i \(-0.677437\pi\)
−0.529012 + 0.848614i \(0.677437\pi\)
\(524\) 0 0
\(525\) − 5.46410i − 0.238473i
\(526\) 2.19615i 0.0957568i
\(527\) − 35.3205i − 1.53859i
\(528\) 64.6410i 2.81314i
\(529\) −21.3923 −0.930100
\(530\) −18.0000 −0.781870
\(531\) 40.7321i 1.76762i
\(532\) 12.3923 0.537275
\(533\) 0 0
\(534\) 61.1769 2.64738
\(535\) 17.6603i 0.763519i
\(536\) 11.0718 0.478229
\(537\) 51.7128 2.23157
\(538\) − 34.3923i − 1.48276i
\(539\) − 14.1962i − 0.611472i
\(540\) − 4.00000i − 0.172133i
\(541\) − 26.3923i − 1.13469i −0.823479 0.567347i \(-0.807970\pi\)
0.823479 0.567347i \(-0.192030\pi\)
\(542\) −53.6603 −2.30491
\(543\) −1.07180 −0.0459952
\(544\) − 18.0000i − 0.771744i
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −12.1962 −0.521470 −0.260735 0.965410i \(-0.583965\pi\)
−0.260735 + 0.965410i \(0.583965\pi\)
\(548\) − 0.928203i − 0.0396509i
\(549\) −37.4641 −1.59893
\(550\) −8.19615 −0.349485
\(551\) − 15.7128i − 0.669388i
\(552\) − 6.00000i − 0.255377i
\(553\) − 16.7846i − 0.713754i
\(554\) 45.7128i 1.94215i
\(555\) 10.9282 0.463876
\(556\) −12.3923 −0.525551
\(557\) 1.85641i 0.0786585i 0.999226 + 0.0393292i \(0.0125221\pi\)
−0.999226 + 0.0393292i \(0.987478\pi\)
\(558\) 78.8372 3.33744
\(559\) 0 0
\(560\) 10.0000 0.422577
\(561\) − 44.7846i − 1.89081i
\(562\) −38.7846 −1.63603
\(563\) 22.0526 0.929405 0.464702 0.885467i \(-0.346161\pi\)
0.464702 + 0.885467i \(0.346161\pi\)
\(564\) − 16.3923i − 0.690241i
\(565\) 8.53590i 0.359108i
\(566\) − 56.4449i − 2.37255i
\(567\) − 4.92820i − 0.206965i
\(568\) 8.19615 0.343903
\(569\) 2.53590 0.106310 0.0531552 0.998586i \(-0.483072\pi\)
0.0531552 + 0.998586i \(0.483072\pi\)
\(570\) − 29.3205i − 1.22810i
\(571\) −36.3923 −1.52297 −0.761485 0.648182i \(-0.775530\pi\)
−0.761485 + 0.648182i \(0.775530\pi\)
\(572\) 0 0
\(573\) −13.8564 −0.578860
\(574\) 12.0000i 0.500870i
\(575\) 1.26795 0.0528771
\(576\) −4.46410 −0.186004
\(577\) 4.00000i 0.166522i 0.996528 + 0.0832611i \(0.0265335\pi\)
−0.996528 + 0.0832611i \(0.973466\pi\)
\(578\) − 8.66025i − 0.360219i
\(579\) − 27.3205i − 1.13540i
\(580\) 2.53590i 0.105297i
\(581\) −12.0000 −0.497844
\(582\) 9.46410 0.392300
\(583\) − 49.1769i − 2.03670i
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) 8.78461 0.362889
\(587\) 8.53590i 0.352314i 0.984362 + 0.176157i \(0.0563667\pi\)
−0.984362 + 0.176157i \(0.943633\pi\)
\(588\) −8.19615 −0.338004
\(589\) 63.1769 2.60316
\(590\) − 15.8038i − 0.650634i
\(591\) 35.3205i 1.45289i
\(592\) 20.0000i 0.821995i
\(593\) 26.7846i 1.09991i 0.835194 + 0.549956i \(0.185356\pi\)
−0.835194 + 0.549956i \(0.814644\pi\)
\(594\) 32.7846 1.34517
\(595\) −6.92820 −0.284029
\(596\) − 7.85641i − 0.321811i
\(597\) −54.6410 −2.23631
\(598\) 0 0
\(599\) −7.60770 −0.310842 −0.155421 0.987848i \(-0.549673\pi\)
−0.155421 + 0.987848i \(0.549673\pi\)
\(600\) − 4.73205i − 0.193185i
\(601\) 43.5692 1.77723 0.888613 0.458658i \(-0.151670\pi\)
0.888613 + 0.458658i \(0.151670\pi\)
\(602\) −0.679492 −0.0276940
\(603\) − 28.5359i − 1.16207i
\(604\) 1.80385i 0.0733975i
\(605\) − 11.3923i − 0.463163i
\(606\) 4.39230i 0.178425i
\(607\) 24.9808 1.01394 0.506969 0.861964i \(-0.330766\pi\)
0.506969 + 0.861964i \(0.330766\pi\)
\(608\) 32.1962 1.30573
\(609\) − 13.8564i − 0.561490i
\(610\) 14.5359 0.588541
\(611\) 0 0
\(612\) −15.4641 −0.625099
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 32.5359 1.31304
\(615\) 9.46410 0.381629
\(616\) 16.3923i 0.660465i
\(617\) − 33.7128i − 1.35723i −0.734496 0.678613i \(-0.762581\pi\)
0.734496 0.678613i \(-0.237419\pi\)
\(618\) 0.928203i 0.0373378i
\(619\) 6.98076i 0.280581i 0.990110 + 0.140290i \(0.0448036\pi\)
−0.990110 + 0.140290i \(0.955196\pi\)
\(620\) −10.1962 −0.409487
\(621\) −5.07180 −0.203524
\(622\) 28.3923i 1.13843i
\(623\) 25.8564 1.03592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 24.9282i − 0.996331i
\(627\) 80.1051 3.19909
\(628\) 10.0000 0.399043
\(629\) − 13.8564i − 0.552491i
\(630\) − 15.4641i − 0.616105i
\(631\) 5.80385i 0.231048i 0.993305 + 0.115524i \(0.0368546\pi\)
−0.993305 + 0.115524i \(0.963145\pi\)
\(632\) − 14.5359i − 0.578207i
\(633\) 21.8564 0.868714
\(634\) 41.5692 1.65092
\(635\) − 16.1962i − 0.642725i
\(636\) −28.3923 −1.12583
\(637\) 0 0
\(638\) −20.7846 −0.822871
\(639\) − 21.1244i − 0.835667i
\(640\) 12.1244 0.479257
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 83.5692i 3.29821i
\(643\) 6.78461i 0.267559i 0.991011 + 0.133779i \(0.0427114\pi\)
−0.991011 + 0.133779i \(0.957289\pi\)
\(644\) 2.53590i 0.0999284i
\(645\) 0.535898i 0.0211010i
\(646\) −37.1769 −1.46271
\(647\) −22.0526 −0.866976 −0.433488 0.901159i \(-0.642717\pi\)
−0.433488 + 0.901159i \(0.642717\pi\)
\(648\) − 4.26795i − 0.167661i
\(649\) 43.1769 1.69484
\(650\) 0 0
\(651\) 55.7128 2.18356
\(652\) 14.3923i 0.563646i
\(653\) 7.85641 0.307445 0.153722 0.988114i \(-0.450874\pi\)
0.153722 + 0.988114i \(0.450874\pi\)
\(654\) 9.46410 0.370076
\(655\) 0 0
\(656\) 17.3205i 0.676252i
\(657\) − 17.8564i − 0.696645i
\(658\) − 20.7846i − 0.810268i
\(659\) −21.4641 −0.836123 −0.418061 0.908419i \(-0.637290\pi\)
−0.418061 + 0.908419i \(0.637290\pi\)
\(660\) −12.9282 −0.503230
\(661\) 10.7846i 0.419473i 0.977758 + 0.209736i \(0.0672606\pi\)
−0.977758 + 0.209736i \(0.932739\pi\)
\(662\) 4.48334 0.174250
\(663\) 0 0
\(664\) −10.3923 −0.403300
\(665\) − 12.3923i − 0.480553i
\(666\) 30.9282 1.19844
\(667\) 3.21539 0.124500
\(668\) 0.928203i 0.0359133i
\(669\) − 5.46410i − 0.211254i
\(670\) 11.0718i 0.427741i
\(671\) 39.7128i 1.53310i
\(672\) 28.3923 1.09526
\(673\) 14.3923 0.554783 0.277391 0.960757i \(-0.410530\pi\)
0.277391 + 0.960757i \(0.410530\pi\)
\(674\) 45.7128i 1.76079i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 40.3923i 1.55126i
\(679\) 4.00000 0.153506
\(680\) −6.00000 −0.230089
\(681\) 9.46410i 0.362665i
\(682\) − 83.5692i − 3.20003i
\(683\) 32.5359i 1.24495i 0.782639 + 0.622476i \(0.213873\pi\)
−0.782639 + 0.622476i \(0.786127\pi\)
\(684\) − 27.6603i − 1.05762i
\(685\) −0.928203 −0.0354648
\(686\) −34.6410 −1.32260
\(687\) 17.4641i 0.666297i
\(688\) −0.980762 −0.0373912
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) 47.7654i 1.81708i 0.417797 + 0.908540i \(0.362802\pi\)
−0.417797 + 0.908540i \(0.637198\pi\)
\(692\) 8.53590 0.324486
\(693\) 42.2487 1.60490
\(694\) − 9.80385i − 0.372149i
\(695\) 12.3923i 0.470067i
\(696\) − 12.0000i − 0.454859i
\(697\) − 12.0000i − 0.454532i
\(698\) 24.9282 0.943546
\(699\) 16.3923 0.620014
\(700\) 2.00000i 0.0755929i
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 24.7846 0.934769
\(704\) 4.73205i 0.178346i
\(705\) −16.3923 −0.617370
\(706\) −48.0000 −1.80650
\(707\) 1.85641i 0.0698174i
\(708\) − 24.9282i − 0.936859i
\(709\) 30.3923i 1.14141i 0.821156 + 0.570703i \(0.193329\pi\)
−0.821156 + 0.570703i \(0.806671\pi\)
\(710\) 8.19615i 0.307596i
\(711\) −37.4641 −1.40501
\(712\) 22.3923 0.839187
\(713\) 12.9282i 0.484165i
\(714\) −32.7846 −1.22693
\(715\) 0 0
\(716\) −18.9282 −0.707380
\(717\) 38.7846i 1.44844i
\(718\) 3.80385 0.141958
\(719\) 25.8564 0.964281 0.482141 0.876094i \(-0.339859\pi\)
0.482141 + 0.876094i \(0.339859\pi\)
\(720\) − 22.3205i − 0.831836i
\(721\) 0.392305i 0.0146102i
\(722\) − 33.5885i − 1.25003i
\(723\) − 6.53590i − 0.243073i
\(724\) 0.392305 0.0145799
\(725\) 2.53590 0.0941809
\(726\) − 53.9090i − 2.00075i
\(727\) −44.5885 −1.65369 −0.826847 0.562427i \(-0.809868\pi\)
−0.826847 + 0.562427i \(0.809868\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 6.92820i 0.256424i
\(731\) 0.679492 0.0251319
\(732\) 22.9282 0.847451
\(733\) − 38.0000i − 1.40356i −0.712393 0.701781i \(-0.752388\pi\)
0.712393 0.701781i \(-0.247612\pi\)
\(734\) 20.4449i 0.754634i
\(735\) 8.19615i 0.302320i
\(736\) 6.58846i 0.242854i
\(737\) −30.2487 −1.11423
\(738\) 26.7846 0.985955
\(739\) − 18.1962i − 0.669356i −0.942332 0.334678i \(-0.891372\pi\)
0.942332 0.334678i \(-0.108628\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) 16.1436i 0.592251i 0.955149 + 0.296126i \(0.0956947\pi\)
−0.955149 + 0.296126i \(0.904305\pi\)
\(744\) 48.2487 1.76888
\(745\) −7.85641 −0.287836
\(746\) − 17.3205i − 0.634149i
\(747\) 26.7846i 0.979998i
\(748\) 16.3923i 0.599362i
\(749\) 35.3205i 1.29058i
\(750\) 4.73205 0.172790
\(751\) −36.3923 −1.32797 −0.663987 0.747744i \(-0.731137\pi\)
−0.663987 + 0.747744i \(0.731137\pi\)
\(752\) − 30.0000i − 1.09399i
\(753\) 58.6410 2.13700
\(754\) 0 0
\(755\) 1.80385 0.0656487
\(756\) − 8.00000i − 0.290957i
\(757\) −2.39230 −0.0869498 −0.0434749 0.999055i \(-0.513843\pi\)
−0.0434749 + 0.999055i \(0.513843\pi\)
\(758\) 32.8756 1.19410
\(759\) 16.3923i 0.595003i
\(760\) − 10.7321i − 0.389292i
\(761\) − 19.8564i − 0.719794i −0.932992 0.359897i \(-0.882812\pi\)
0.932992 0.359897i \(-0.117188\pi\)
\(762\) − 76.6410i − 2.77641i
\(763\) 4.00000 0.144810
\(764\) 5.07180 0.183491
\(765\) 15.4641i 0.559106i
\(766\) 22.3923 0.809067
\(767\) 0 0
\(768\) 51.9090 1.87310
\(769\) − 34.7846i − 1.25437i −0.778872 0.627183i \(-0.784208\pi\)
0.778872 0.627183i \(-0.215792\pi\)
\(770\) −16.3923 −0.590738
\(771\) −54.2487 −1.95372
\(772\) 10.0000i 0.359908i
\(773\) − 6.92820i − 0.249190i −0.992208 0.124595i \(-0.960237\pi\)
0.992208 0.124595i \(-0.0397632\pi\)
\(774\) 1.51666i 0.0545152i
\(775\) 10.1962i 0.366257i
\(776\) 3.46410 0.124354
\(777\) 21.8564 0.784094
\(778\) − 10.3923i − 0.372582i
\(779\) 21.4641 0.769031
\(780\) 0 0
\(781\) −22.3923 −0.801260
\(782\) − 7.60770i − 0.272051i
\(783\) −10.1436 −0.362502
\(784\) −15.0000 −0.535714
\(785\) − 10.0000i − 0.356915i
\(786\) 0 0
\(787\) 31.5692i 1.12532i 0.826688 + 0.562661i \(0.190222\pi\)
−0.826688 + 0.562661i \(0.809778\pi\)
\(788\) − 12.9282i − 0.460548i
\(789\) 3.46410 0.123325
\(790\) 14.5359 0.517164
\(791\) 17.0718i 0.607003i
\(792\) 36.5885 1.30011
\(793\) 0 0
\(794\) −49.8564 −1.76934
\(795\) 28.3923i 1.00697i
\(796\) 20.0000 0.708881
\(797\) 40.6410 1.43958 0.719789 0.694193i \(-0.244238\pi\)
0.719789 + 0.694193i \(0.244238\pi\)
\(798\) − 58.6410i − 2.07587i
\(799\) 20.7846i 0.735307i
\(800\) 5.19615i 0.183712i
\(801\) − 57.7128i − 2.03918i
\(802\) 63.9615 2.25856
\(803\) −18.9282 −0.667962
\(804\) 17.4641i 0.615911i
\(805\) 2.53590 0.0893787
\(806\) 0 0
\(807\) −54.2487 −1.90965
\(808\) 1.60770i 0.0565585i
\(809\) −2.53590 −0.0891574 −0.0445787 0.999006i \(-0.514195\pi\)
−0.0445787 + 0.999006i \(0.514195\pi\)
\(810\) 4.26795 0.149960
\(811\) − 17.8038i − 0.625178i −0.949889 0.312589i \(-0.898804\pi\)
0.949889 0.312589i \(-0.101196\pi\)
\(812\) 5.07180i 0.177985i
\(813\) 84.6410i 2.96849i
\(814\) − 32.7846i − 1.14910i
\(815\) 14.3923 0.504140
\(816\) −47.3205 −1.65655
\(817\) 1.21539i 0.0425211i
\(818\) −30.4974 −1.06632
\(819\) 0 0
\(820\) −3.46410 −0.120972
\(821\) − 28.6410i − 0.999578i −0.866147 0.499789i \(-0.833411\pi\)
0.866147 0.499789i \(-0.166589\pi\)
\(822\) −4.39230 −0.153199
\(823\) 15.4115 0.537213 0.268606 0.963250i \(-0.413437\pi\)
0.268606 + 0.963250i \(0.413437\pi\)
\(824\) 0.339746i 0.0118356i
\(825\) 12.9282i 0.450102i
\(826\) − 31.6077i − 1.09977i
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) 5.66025 0.196707
\(829\) −0.392305 −0.0136253 −0.00681266 0.999977i \(-0.502169\pi\)
−0.00681266 + 0.999977i \(0.502169\pi\)
\(830\) − 10.3923i − 0.360722i
\(831\) 72.1051 2.50130
\(832\) 0 0
\(833\) 10.3923 0.360072
\(834\) 58.6410i 2.03057i
\(835\) 0.928203 0.0321218
\(836\) −29.3205 −1.01407
\(837\) − 40.7846i − 1.40972i
\(838\) − 4.39230i − 0.151730i
\(839\) 0.339746i 0.0117293i 0.999983 + 0.00586467i \(0.00186679\pi\)
−0.999983 + 0.00586467i \(0.998133\pi\)
\(840\) − 9.46410i − 0.326543i
\(841\) −22.5692 −0.778249
\(842\) −53.3205 −1.83755
\(843\) 61.1769i 2.10704i
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −46.3923 −1.59500
\(847\) − 22.7846i − 0.782888i
\(848\) −51.9615 −1.78437
\(849\) −89.0333 −3.05562
\(850\) − 6.00000i − 0.205798i
\(851\) 5.07180i 0.173859i
\(852\) 12.9282i 0.442913i
\(853\) 8.00000i 0.273915i 0.990577 + 0.136957i \(0.0437323\pi\)
−0.990577 + 0.136957i \(0.956268\pi\)
\(854\) 29.0718 0.994816
\(855\) −27.6603 −0.945961
\(856\) 30.5885i 1.04549i
\(857\) 35.5692 1.21502 0.607511 0.794311i \(-0.292168\pi\)
0.607511 + 0.794311i \(0.292168\pi\)
\(858\) 0 0
\(859\) −17.1769 −0.586069 −0.293034 0.956102i \(-0.594665\pi\)
−0.293034 + 0.956102i \(0.594665\pi\)
\(860\) − 0.196152i − 0.00668874i
\(861\) 18.9282 0.645071
\(862\) −44.1962 −1.50533
\(863\) 38.7846i 1.32024i 0.751159 + 0.660122i \(0.229495\pi\)
−0.751159 + 0.660122i \(0.770505\pi\)
\(864\) − 20.7846i − 0.707107i
\(865\) − 8.53590i − 0.290229i
\(866\) − 60.2487i − 2.04733i
\(867\) −13.6603 −0.463927
\(868\) −20.3923 −0.692160
\(869\) 39.7128i 1.34716i
\(870\) 12.0000 0.406838
\(871\) 0 0
\(872\) 3.46410 0.117309
\(873\) − 8.92820i − 0.302174i
\(874\) 13.6077 0.460287
\(875\) 2.00000 0.0676123
\(876\) 10.9282i 0.369230i
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) − 55.4256i − 1.87052i
\(879\) − 13.8564i − 0.467365i
\(880\) −23.6603 −0.797587
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 23.1962i 0.781055i
\(883\) −23.8038 −0.801063 −0.400532 0.916283i \(-0.631175\pi\)
−0.400532 + 0.916283i \(0.631175\pi\)
\(884\) 0 0
\(885\) −24.9282 −0.837952
\(886\) − 29.4115i − 0.988100i
\(887\) −47.9090 −1.60863 −0.804313 0.594206i \(-0.797466\pi\)
−0.804313 + 0.594206i \(0.797466\pi\)
\(888\) 18.9282 0.635189
\(889\) − 32.3923i − 1.08640i
\(890\) 22.3923i 0.750592i
\(891\) 11.6603i 0.390633i
\(892\) 2.00000i 0.0669650i
\(893\) −37.1769 −1.24408
\(894\) −37.1769 −1.24338
\(895\) 18.9282i 0.632700i
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) −35.5692 −1.18696
\(899\) 25.8564i 0.862359i
\(900\) 4.46410 0.148803
\(901\) 36.0000 1.19933
\(902\) − 28.3923i − 0.945360i
\(903\) 1.07180i 0.0356672i
\(904\) 14.7846i 0.491729i
\(905\) − 0.392305i − 0.0130407i
\(906\) 8.53590 0.283586
\(907\) 53.7654 1.78525 0.892625 0.450800i \(-0.148861\pi\)
0.892625 + 0.450800i \(0.148861\pi\)
\(908\) − 3.46410i − 0.114960i
\(909\) 4.14359 0.137434
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) − 84.6410i − 2.80274i
\(913\) 28.3923 0.939648
\(914\) 18.6795 0.617863
\(915\) − 22.9282i − 0.757983i
\(916\) − 6.39230i − 0.211208i
\(917\) 0 0
\(918\) 24.0000i 0.792118i
\(919\) 9.17691 0.302718 0.151359 0.988479i \(-0.451635\pi\)
0.151359 + 0.988479i \(0.451635\pi\)
\(920\) 2.19615 0.0724050
\(921\) − 51.3205i − 1.69107i
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) −25.8564 −0.850613
\(925\) 4.00000i 0.131519i
\(926\) 4.14359 0.136167
\(927\) 0.875644 0.0287599
\(928\) 13.1769i 0.432553i
\(929\) − 44.5359i − 1.46118i −0.682819 0.730588i \(-0.739246\pi\)
0.682819 0.730588i \(-0.260754\pi\)
\(930\) 48.2487i 1.58214i
\(931\) 18.5885i 0.609212i
\(932\) −6.00000 −0.196537
\(933\) 44.7846 1.46618
\(934\) − 48.1577i − 1.57577i
\(935\) 16.3923 0.536086
\(936\) 0 0
\(937\) 34.7846 1.13636 0.568182 0.822903i \(-0.307647\pi\)
0.568182 + 0.822903i \(0.307647\pi\)
\(938\) 22.1436i 0.723014i
\(939\) −39.3205 −1.28318
\(940\) 6.00000 0.195698
\(941\) − 31.1769i − 1.01634i −0.861257 0.508169i \(-0.830322\pi\)
0.861257 0.508169i \(-0.169678\pi\)
\(942\) − 47.3205i − 1.54179i
\(943\) 4.39230i 0.143033i
\(944\) − 45.6218i − 1.48486i
\(945\) −8.00000 −0.260240
\(946\) 1.60770 0.0522707
\(947\) 40.6410i 1.32066i 0.750978 + 0.660328i \(0.229583\pi\)
−0.750978 + 0.660328i \(0.770417\pi\)
\(948\) 22.9282 0.744673
\(949\) 0 0
\(950\) 10.7321 0.348194
\(951\) − 65.5692i − 2.12623i
\(952\) −12.0000 −0.388922
\(953\) −0.928203 −0.0300675 −0.0150337 0.999887i \(-0.504786\pi\)
−0.0150337 + 0.999887i \(0.504786\pi\)
\(954\) 80.3538i 2.60155i
\(955\) − 5.07180i − 0.164119i
\(956\) − 14.1962i − 0.459136i
\(957\) 32.7846i 1.05978i
\(958\) −61.7654 −1.99555
\(959\) −1.85641 −0.0599465
\(960\) − 2.73205i − 0.0881766i
\(961\) −72.9615 −2.35360
\(962\) 0 0
\(963\) 78.8372 2.54049
\(964\) 2.39230i 0.0770510i
\(965\) 10.0000 0.321911
\(966\) 12.0000 0.386094
\(967\) 50.3923i 1.62051i 0.586079 + 0.810254i \(0.300671\pi\)
−0.586079 + 0.810254i \(0.699329\pi\)
\(968\) − 19.7321i − 0.634212i
\(969\) 58.6410i 1.88382i
\(970\) 3.46410i 0.111226i
\(971\) 18.9282 0.607435 0.303717 0.952762i \(-0.401772\pi\)
0.303717 + 0.952762i \(0.401772\pi\)
\(972\) 18.7321 0.600831
\(973\) 24.7846i 0.794558i
\(974\) −45.7128 −1.46473
\(975\) 0 0
\(976\) 41.9615 1.34316
\(977\) 15.7128i 0.502697i 0.967897 + 0.251349i \(0.0808741\pi\)
−0.967897 + 0.251349i \(0.919126\pi\)
\(978\) 68.1051 2.17776
\(979\) −61.1769 −1.95522
\(980\) − 3.00000i − 0.0958315i
\(981\) − 8.92820i − 0.285056i
\(982\) 4.39230i 0.140164i
\(983\) − 34.3923i − 1.09694i −0.836169 0.548472i \(-0.815210\pi\)
0.836169 0.548472i \(-0.184790\pi\)
\(984\) 16.3923 0.522568
\(985\) −12.9282 −0.411927
\(986\) − 15.2154i − 0.484557i
\(987\) −32.7846 −1.04355
\(988\) 0 0
\(989\) −0.248711 −0.00790856
\(990\) 36.5885i 1.16286i
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −52.9808 −1.68214
\(993\) − 7.07180i − 0.224417i
\(994\) 16.3923i 0.519932i
\(995\) − 20.0000i − 0.634043i
\(996\) − 16.3923i − 0.519410i
\(997\) 33.6077 1.06437 0.532183 0.846629i \(-0.321372\pi\)
0.532183 + 0.846629i \(0.321372\pi\)
\(998\) −67.5167 −2.13720
\(999\) − 16.0000i − 0.506218i
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 845.2.c.e.506.4 4
13.2 odd 12 845.2.e.f.191.1 4
13.3 even 3 845.2.m.a.316.1 4
13.4 even 6 845.2.m.a.361.1 4
13.5 odd 4 845.2.a.d.1.2 2
13.6 odd 12 845.2.e.f.146.1 4
13.7 odd 12 845.2.e.e.146.2 4
13.8 odd 4 65.2.a.c.1.1 2
13.9 even 3 845.2.m.c.361.1 4
13.10 even 6 845.2.m.c.316.1 4
13.11 odd 12 845.2.e.e.191.2 4
13.12 even 2 inner 845.2.c.e.506.2 4
39.5 even 4 7605.2.a.be.1.1 2
39.8 even 4 585.2.a.k.1.2 2
52.47 even 4 1040.2.a.h.1.1 2
65.8 even 4 325.2.b.e.274.4 4
65.34 odd 4 325.2.a.g.1.2 2
65.44 odd 4 4225.2.a.w.1.1 2
65.47 even 4 325.2.b.e.274.1 4
91.34 even 4 3185.2.a.k.1.1 2
104.21 odd 4 4160.2.a.y.1.1 2
104.99 even 4 4160.2.a.bj.1.2 2
143.21 even 4 7865.2.a.h.1.2 2
156.47 odd 4 9360.2.a.cm.1.1 2
195.8 odd 4 2925.2.c.v.2224.1 4
195.47 odd 4 2925.2.c.v.2224.4 4
195.164 even 4 2925.2.a.z.1.1 2
260.99 even 4 5200.2.a.ca.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.1 2 13.8 odd 4
325.2.a.g.1.2 2 65.34 odd 4
325.2.b.e.274.1 4 65.47 even 4
325.2.b.e.274.4 4 65.8 even 4
585.2.a.k.1.2 2 39.8 even 4
845.2.a.d.1.2 2 13.5 odd 4
845.2.c.e.506.2 4 13.12 even 2 inner
845.2.c.e.506.4 4 1.1 even 1 trivial
845.2.e.e.146.2 4 13.7 odd 12
845.2.e.e.191.2 4 13.11 odd 12
845.2.e.f.146.1 4 13.6 odd 12
845.2.e.f.191.1 4 13.2 odd 12
845.2.m.a.316.1 4 13.3 even 3
845.2.m.a.361.1 4 13.4 even 6
845.2.m.c.316.1 4 13.10 even 6
845.2.m.c.361.1 4 13.9 even 3
1040.2.a.h.1.1 2 52.47 even 4
2925.2.a.z.1.1 2 195.164 even 4
2925.2.c.v.2224.1 4 195.8 odd 4
2925.2.c.v.2224.4 4 195.47 odd 4
3185.2.a.k.1.1 2 91.34 even 4
4160.2.a.y.1.1 2 104.21 odd 4
4160.2.a.bj.1.2 2 104.99 even 4
4225.2.a.w.1.1 2 65.44 odd 4
5200.2.a.ca.1.2 2 260.99 even 4
7605.2.a.be.1.1 2 39.5 even 4
7865.2.a.h.1.2 2 143.21 even 4
9360.2.a.cm.1.1 2 156.47 odd 4