Properties

Label 3450.2.d.x.2899.2
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3450,2,Mod(2899,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.x.2899.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214i q^{7} +1.00000i q^{8} -1.00000 q^{9} -5.23607 q^{11} -1.00000i q^{12} +4.47214i q^{13} +4.47214 q^{14} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} -5.70820 q^{19} -4.47214 q^{21} +5.23607i q^{22} +1.00000i q^{23} -1.00000 q^{24} +4.47214 q^{26} -1.00000i q^{27} -4.47214i q^{28} +4.47214 q^{29} -2.47214 q^{31} -1.00000i q^{32} -5.23607i q^{33} +4.00000 q^{34} +1.00000 q^{36} -11.2361i q^{37} +5.70820i q^{38} -4.47214 q^{39} -2.00000 q^{41} +4.47214i q^{42} -4.76393i q^{43} +5.23607 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -13.0000 q^{49} -4.00000 q^{51} -4.47214i q^{52} +5.23607i q^{53} -1.00000 q^{54} -4.47214 q^{56} -5.70820i q^{57} -4.47214i q^{58} +8.94427 q^{59} +0.763932 q^{61} +2.47214i q^{62} -4.47214i q^{63} -1.00000 q^{64} -5.23607 q^{66} -9.70820i q^{67} -4.00000i q^{68} -1.00000 q^{69} +8.94427 q^{71} -1.00000i q^{72} -4.47214i q^{73} -11.2361 q^{74} +5.70820 q^{76} -23.4164i q^{77} +4.47214i q^{78} -4.47214 q^{79} +1.00000 q^{81} +2.00000i q^{82} +13.2361i q^{83} +4.47214 q^{84} -4.76393 q^{86} +4.47214i q^{87} -5.23607i q^{88} +10.4721 q^{89} -20.0000 q^{91} -1.00000i q^{92} -2.47214i q^{93} -4.00000 q^{94} +1.00000 q^{96} -0.472136i q^{97} +13.0000i q^{98} +5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{16} + 4 q^{19} - 4 q^{24} + 8 q^{31} + 16 q^{34} + 4 q^{36} - 8 q^{41} + 12 q^{44} + 4 q^{46} - 52 q^{49} - 16 q^{51} - 4 q^{54} + 12 q^{61} - 4 q^{64} - 12 q^{66} - 4 q^{69} - 36 q^{74} - 4 q^{76} + 4 q^{81} - 28 q^{86} + 24 q^{89} - 80 q^{91} - 16 q^{94} + 4 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.47214i 1.69031i 0.534522 + 0.845154i \(0.320491\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 4.47214i 1.24035i 0.784465 + 0.620174i \(0.212938\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 4.47214 1.19523
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −5.70820 −1.30955 −0.654776 0.755823i \(-0.727237\pi\)
−0.654776 + 0.755823i \(0.727237\pi\)
\(20\) 0 0
\(21\) −4.47214 −0.975900
\(22\) 5.23607i 1.11633i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.47214 0.877058
\(27\) − 1.00000i − 0.192450i
\(28\) − 4.47214i − 0.845154i
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 5.23607i − 0.911482i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 11.2361i − 1.84720i −0.383360 0.923599i \(-0.625233\pi\)
0.383360 0.923599i \(-0.374767\pi\)
\(38\) 5.70820i 0.925993i
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 4.47214i 0.690066i
\(43\) − 4.76393i − 0.726493i −0.931693 0.363246i \(-0.881668\pi\)
0.931693 0.363246i \(-0.118332\pi\)
\(44\) 5.23607 0.789367
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −13.0000 −1.85714
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) − 4.47214i − 0.620174i
\(53\) 5.23607i 0.719229i 0.933101 + 0.359615i \(0.117092\pi\)
−0.933101 + 0.359615i \(0.882908\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) − 5.70820i − 0.756070i
\(58\) − 4.47214i − 0.587220i
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) 2.47214i 0.313962i
\(63\) − 4.47214i − 0.563436i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.23607 −0.644515
\(67\) − 9.70820i − 1.18605i −0.805186 0.593023i \(-0.797934\pi\)
0.805186 0.593023i \(-0.202066\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 4.47214i − 0.523424i −0.965146 0.261712i \(-0.915713\pi\)
0.965146 0.261712i \(-0.0842870\pi\)
\(74\) −11.2361 −1.30617
\(75\) 0 0
\(76\) 5.70820 0.654776
\(77\) − 23.4164i − 2.66855i
\(78\) 4.47214i 0.506370i
\(79\) −4.47214 −0.503155 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 13.2361i 1.45285i 0.687247 + 0.726424i \(0.258819\pi\)
−0.687247 + 0.726424i \(0.741181\pi\)
\(84\) 4.47214 0.487950
\(85\) 0 0
\(86\) −4.76393 −0.513708
\(87\) 4.47214i 0.479463i
\(88\) − 5.23607i − 0.558167i
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) − 1.00000i − 0.104257i
\(93\) − 2.47214i − 0.256349i
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 0.472136i − 0.0479381i −0.999713 0.0239691i \(-0.992370\pi\)
0.999713 0.0239691i \(-0.00763032\pi\)
\(98\) 13.0000i 1.31320i
\(99\) 5.23607 0.526245
\(100\) 0 0
\(101\) 4.47214 0.444994 0.222497 0.974933i \(-0.428579\pi\)
0.222497 + 0.974933i \(0.428579\pi\)
\(102\) 4.00000i 0.396059i
\(103\) − 6.00000i − 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) 5.23607 0.508572
\(107\) 12.6525i 1.22316i 0.791182 + 0.611581i \(0.209466\pi\)
−0.791182 + 0.611581i \(0.790534\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −4.76393 −0.456302 −0.228151 0.973626i \(-0.573268\pi\)
−0.228151 + 0.973626i \(0.573268\pi\)
\(110\) 0 0
\(111\) 11.2361 1.06648
\(112\) 4.47214i 0.422577i
\(113\) − 5.52786i − 0.520018i −0.965606 0.260009i \(-0.916275\pi\)
0.965606 0.260009i \(-0.0837255\pi\)
\(114\) −5.70820 −0.534622
\(115\) 0 0
\(116\) −4.47214 −0.415227
\(117\) − 4.47214i − 0.413449i
\(118\) − 8.94427i − 0.823387i
\(119\) −17.8885 −1.63984
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) − 0.763932i − 0.0691632i
\(123\) − 2.00000i − 0.180334i
\(124\) 2.47214 0.222004
\(125\) 0 0
\(126\) −4.47214 −0.398410
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.76393 0.419441
\(130\) 0 0
\(131\) −9.52786 −0.832453 −0.416227 0.909261i \(-0.636648\pi\)
−0.416227 + 0.909261i \(0.636648\pi\)
\(132\) 5.23607i 0.455741i
\(133\) − 25.5279i − 2.21355i
\(134\) −9.70820 −0.838661
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) − 3.05573i − 0.261068i −0.991444 0.130534i \(-0.958331\pi\)
0.991444 0.130534i \(-0.0416692\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 16.9443 1.43719 0.718597 0.695427i \(-0.244785\pi\)
0.718597 + 0.695427i \(0.244785\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) − 8.94427i − 0.750587i
\(143\) − 23.4164i − 1.95818i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.47214 −0.370117
\(147\) − 13.0000i − 1.07222i
\(148\) 11.2361i 0.923599i
\(149\) 11.7082 0.959173 0.479587 0.877494i \(-0.340787\pi\)
0.479587 + 0.877494i \(0.340787\pi\)
\(150\) 0 0
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) − 5.70820i − 0.462996i
\(153\) − 4.00000i − 0.323381i
\(154\) −23.4164 −1.88695
\(155\) 0 0
\(156\) 4.47214 0.358057
\(157\) 6.65248i 0.530925i 0.964121 + 0.265463i \(0.0855247\pi\)
−0.964121 + 0.265463i \(0.914475\pi\)
\(158\) 4.47214i 0.355784i
\(159\) −5.23607 −0.415247
\(160\) 0 0
\(161\) −4.47214 −0.352454
\(162\) − 1.00000i − 0.0785674i
\(163\) − 2.47214i − 0.193633i −0.995302 0.0968163i \(-0.969134\pi\)
0.995302 0.0968163i \(-0.0308659\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 13.2361 1.02732
\(167\) − 16.9443i − 1.31119i −0.755114 0.655594i \(-0.772418\pi\)
0.755114 0.655594i \(-0.227582\pi\)
\(168\) − 4.47214i − 0.345033i
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 5.70820 0.436517
\(172\) 4.76393i 0.363246i
\(173\) − 17.4164i − 1.32414i −0.749440 0.662072i \(-0.769677\pi\)
0.749440 0.662072i \(-0.230323\pi\)
\(174\) 4.47214 0.339032
\(175\) 0 0
\(176\) −5.23607 −0.394683
\(177\) 8.94427i 0.672293i
\(178\) − 10.4721i − 0.784920i
\(179\) −19.4164 −1.45125 −0.725625 0.688090i \(-0.758449\pi\)
−0.725625 + 0.688090i \(0.758449\pi\)
\(180\) 0 0
\(181\) −11.2361 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(182\) 20.0000i 1.48250i
\(183\) 0.763932i 0.0564715i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −2.47214 −0.181266
\(187\) − 20.9443i − 1.53160i
\(188\) 4.00000i 0.291730i
\(189\) 4.47214 0.325300
\(190\) 0 0
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 23.8885i 1.71954i 0.510686 + 0.859768i \(0.329392\pi\)
−0.510686 + 0.859768i \(0.670608\pi\)
\(194\) −0.472136 −0.0338974
\(195\) 0 0
\(196\) 13.0000 0.928571
\(197\) − 2.94427i − 0.209771i −0.994484 0.104885i \(-0.966552\pi\)
0.994484 0.104885i \(-0.0334476\pi\)
\(198\) − 5.23607i − 0.372111i
\(199\) −17.4164 −1.23462 −0.617308 0.786721i \(-0.711777\pi\)
−0.617308 + 0.786721i \(0.711777\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) − 4.47214i − 0.314658i
\(203\) 20.0000i 1.40372i
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) − 1.00000i − 0.0695048i
\(208\) 4.47214i 0.310087i
\(209\) 29.8885 2.06743
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) − 5.23607i − 0.359615i
\(213\) 8.94427i 0.612851i
\(214\) 12.6525 0.864905
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 11.0557i − 0.750512i
\(218\) 4.76393i 0.322654i
\(219\) 4.47214 0.302199
\(220\) 0 0
\(221\) −17.8885 −1.20331
\(222\) − 11.2361i − 0.754116i
\(223\) 19.4164i 1.30022i 0.759841 + 0.650109i \(0.225277\pi\)
−0.759841 + 0.650109i \(0.774723\pi\)
\(224\) 4.47214 0.298807
\(225\) 0 0
\(226\) −5.52786 −0.367708
\(227\) 9.23607i 0.613019i 0.951868 + 0.306510i \(0.0991612\pi\)
−0.951868 + 0.306510i \(0.900839\pi\)
\(228\) 5.70820i 0.378035i
\(229\) −17.7082 −1.17019 −0.585096 0.810964i \(-0.698943\pi\)
−0.585096 + 0.810964i \(0.698943\pi\)
\(230\) 0 0
\(231\) 23.4164 1.54069
\(232\) 4.47214i 0.293610i
\(233\) 19.8885i 1.30294i 0.758674 + 0.651471i \(0.225848\pi\)
−0.758674 + 0.651471i \(0.774152\pi\)
\(234\) −4.47214 −0.292353
\(235\) 0 0
\(236\) −8.94427 −0.582223
\(237\) − 4.47214i − 0.290496i
\(238\) 17.8885i 1.15954i
\(239\) 4.94427 0.319818 0.159909 0.987132i \(-0.448880\pi\)
0.159909 + 0.987132i \(0.448880\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) − 16.4164i − 1.05529i
\(243\) 1.00000i 0.0641500i
\(244\) −0.763932 −0.0489057
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 25.5279i − 1.62430i
\(248\) − 2.47214i − 0.156981i
\(249\) −13.2361 −0.838802
\(250\) 0 0
\(251\) −19.7082 −1.24397 −0.621985 0.783029i \(-0.713674\pi\)
−0.621985 + 0.783029i \(0.713674\pi\)
\(252\) 4.47214i 0.281718i
\(253\) − 5.23607i − 0.329189i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.8885i 0.741587i 0.928715 + 0.370793i \(0.120914\pi\)
−0.928715 + 0.370793i \(0.879086\pi\)
\(258\) − 4.76393i − 0.296589i
\(259\) 50.2492 3.12233
\(260\) 0 0
\(261\) −4.47214 −0.276818
\(262\) 9.52786i 0.588633i
\(263\) 24.9443i 1.53813i 0.639171 + 0.769065i \(0.279278\pi\)
−0.639171 + 0.769065i \(0.720722\pi\)
\(264\) 5.23607 0.322258
\(265\) 0 0
\(266\) −25.5279 −1.56521
\(267\) 10.4721i 0.640884i
\(268\) 9.70820i 0.593023i
\(269\) 13.0557 0.796022 0.398011 0.917381i \(-0.369701\pi\)
0.398011 + 0.917381i \(0.369701\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 4.00000i 0.242536i
\(273\) − 20.0000i − 1.21046i
\(274\) −3.05573 −0.184603
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 20.4721i 1.23005i 0.788507 + 0.615026i \(0.210854\pi\)
−0.788507 + 0.615026i \(0.789146\pi\)
\(278\) − 16.9443i − 1.01625i
\(279\) 2.47214 0.148003
\(280\) 0 0
\(281\) −13.5279 −0.807005 −0.403502 0.914979i \(-0.632207\pi\)
−0.403502 + 0.914979i \(0.632207\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) − 3.81966i − 0.227055i −0.993535 0.113528i \(-0.963785\pi\)
0.993535 0.113528i \(-0.0362150\pi\)
\(284\) −8.94427 −0.530745
\(285\) 0 0
\(286\) −23.4164 −1.38464
\(287\) − 8.94427i − 0.527964i
\(288\) 1.00000i 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0.472136 0.0276771
\(292\) 4.47214i 0.261712i
\(293\) 0.291796i 0.0170469i 0.999964 + 0.00852345i \(0.00271313\pi\)
−0.999964 + 0.00852345i \(0.997287\pi\)
\(294\) −13.0000 −0.758175
\(295\) 0 0
\(296\) 11.2361 0.653083
\(297\) 5.23607i 0.303827i
\(298\) − 11.7082i − 0.678238i
\(299\) −4.47214 −0.258630
\(300\) 0 0
\(301\) 21.3050 1.22800
\(302\) 14.4721i 0.832778i
\(303\) 4.47214i 0.256917i
\(304\) −5.70820 −0.327388
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) − 15.4164i − 0.879861i −0.898032 0.439930i \(-0.855003\pi\)
0.898032 0.439930i \(-0.144997\pi\)
\(308\) 23.4164i 1.33427i
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) −20.9443 −1.18764 −0.593820 0.804598i \(-0.702381\pi\)
−0.593820 + 0.804598i \(0.702381\pi\)
\(312\) − 4.47214i − 0.253185i
\(313\) 15.5279i 0.877687i 0.898564 + 0.438843i \(0.144612\pi\)
−0.898564 + 0.438843i \(0.855388\pi\)
\(314\) 6.65248 0.375421
\(315\) 0 0
\(316\) 4.47214 0.251577
\(317\) − 19.5279i − 1.09679i −0.836218 0.548397i \(-0.815238\pi\)
0.836218 0.548397i \(-0.184762\pi\)
\(318\) 5.23607i 0.293624i
\(319\) −23.4164 −1.31107
\(320\) 0 0
\(321\) −12.6525 −0.706192
\(322\) 4.47214i 0.249222i
\(323\) − 22.8328i − 1.27045i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −2.47214 −0.136919
\(327\) − 4.76393i − 0.263446i
\(328\) − 2.00000i − 0.110432i
\(329\) 17.8885 0.986227
\(330\) 0 0
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) − 13.2361i − 0.726424i
\(333\) 11.2361i 0.615733i
\(334\) −16.9443 −0.927149
\(335\) 0 0
\(336\) −4.47214 −0.243975
\(337\) 19.8885i 1.08340i 0.840573 + 0.541699i \(0.182219\pi\)
−0.840573 + 0.541699i \(0.817781\pi\)
\(338\) 7.00000i 0.380750i
\(339\) 5.52786 0.300232
\(340\) 0 0
\(341\) 12.9443 0.700972
\(342\) − 5.70820i − 0.308664i
\(343\) − 26.8328i − 1.44884i
\(344\) 4.76393 0.256854
\(345\) 0 0
\(346\) −17.4164 −0.936312
\(347\) − 30.4721i − 1.63583i −0.575339 0.817915i \(-0.695130\pi\)
0.575339 0.817915i \(-0.304870\pi\)
\(348\) − 4.47214i − 0.239732i
\(349\) −3.88854 −0.208149 −0.104074 0.994570i \(-0.533188\pi\)
−0.104074 + 0.994570i \(0.533188\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 5.23607i 0.279083i
\(353\) − 3.88854i − 0.206966i −0.994631 0.103483i \(-0.967001\pi\)
0.994631 0.103483i \(-0.0329988\pi\)
\(354\) 8.94427 0.475383
\(355\) 0 0
\(356\) −10.4721 −0.555022
\(357\) − 17.8885i − 0.946762i
\(358\) 19.4164i 1.02619i
\(359\) −29.3050 −1.54666 −0.773328 0.634006i \(-0.781409\pi\)
−0.773328 + 0.634006i \(0.781409\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) 11.2361i 0.590555i
\(363\) 16.4164i 0.861638i
\(364\) 20.0000 1.04828
\(365\) 0 0
\(366\) 0.763932 0.0399314
\(367\) − 9.41641i − 0.491532i −0.969329 0.245766i \(-0.920960\pi\)
0.969329 0.245766i \(-0.0790396\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −23.4164 −1.21572
\(372\) 2.47214i 0.128174i
\(373\) − 35.5967i − 1.84313i −0.388224 0.921565i \(-0.626911\pi\)
0.388224 0.921565i \(-0.373089\pi\)
\(374\) −20.9443 −1.08300
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 20.0000i 1.03005i
\(378\) − 4.47214i − 0.230022i
\(379\) −13.7082 −0.704143 −0.352072 0.935973i \(-0.614523\pi\)
−0.352072 + 0.935973i \(0.614523\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) − 6.47214i − 0.331143i
\(383\) 7.05573i 0.360531i 0.983618 + 0.180265i \(0.0576957\pi\)
−0.983618 + 0.180265i \(0.942304\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 23.8885 1.21589
\(387\) 4.76393i 0.242164i
\(388\) 0.472136i 0.0239691i
\(389\) −23.7082 −1.20205 −0.601027 0.799229i \(-0.705242\pi\)
−0.601027 + 0.799229i \(0.705242\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) − 13.0000i − 0.656599i
\(393\) − 9.52786i − 0.480617i
\(394\) −2.94427 −0.148330
\(395\) 0 0
\(396\) −5.23607 −0.263122
\(397\) − 26.9443i − 1.35229i −0.736767 0.676147i \(-0.763648\pi\)
0.736767 0.676147i \(-0.236352\pi\)
\(398\) 17.4164i 0.873006i
\(399\) 25.5279 1.27799
\(400\) 0 0
\(401\) 8.94427 0.446656 0.223328 0.974743i \(-0.428308\pi\)
0.223328 + 0.974743i \(0.428308\pi\)
\(402\) − 9.70820i − 0.484201i
\(403\) − 11.0557i − 0.550725i
\(404\) −4.47214 −0.222497
\(405\) 0 0
\(406\) 20.0000 0.992583
\(407\) 58.8328i 2.91623i
\(408\) − 4.00000i − 0.198030i
\(409\) 35.8885 1.77457 0.887287 0.461217i \(-0.152587\pi\)
0.887287 + 0.461217i \(0.152587\pi\)
\(410\) 0 0
\(411\) 3.05573 0.150728
\(412\) 6.00000i 0.295599i
\(413\) 40.0000i 1.96827i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 4.47214 0.219265
\(417\) 16.9443i 0.829765i
\(418\) − 29.8885i − 1.46190i
\(419\) −28.0689 −1.37125 −0.685627 0.727953i \(-0.740472\pi\)
−0.685627 + 0.727953i \(0.740472\pi\)
\(420\) 0 0
\(421\) −0.763932 −0.0372318 −0.0186159 0.999827i \(-0.505926\pi\)
−0.0186159 + 0.999827i \(0.505926\pi\)
\(422\) 23.4164i 1.13989i
\(423\) 4.00000i 0.194487i
\(424\) −5.23607 −0.254286
\(425\) 0 0
\(426\) 8.94427 0.433351
\(427\) 3.41641i 0.165332i
\(428\) − 12.6525i − 0.611581i
\(429\) 23.4164 1.13055
\(430\) 0 0
\(431\) −23.4164 −1.12793 −0.563964 0.825799i \(-0.690724\pi\)
−0.563964 + 0.825799i \(0.690724\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 21.4164i − 1.02921i −0.857428 0.514603i \(-0.827939\pi\)
0.857428 0.514603i \(-0.172061\pi\)
\(434\) −11.0557 −0.530692
\(435\) 0 0
\(436\) 4.76393 0.228151
\(437\) − 5.70820i − 0.273060i
\(438\) − 4.47214i − 0.213687i
\(439\) −19.0557 −0.909480 −0.454740 0.890624i \(-0.650268\pi\)
−0.454740 + 0.890624i \(0.650268\pi\)
\(440\) 0 0
\(441\) 13.0000 0.619048
\(442\) 17.8885i 0.850871i
\(443\) − 15.0557i − 0.715319i −0.933852 0.357660i \(-0.883575\pi\)
0.933852 0.357660i \(-0.116425\pi\)
\(444\) −11.2361 −0.533240
\(445\) 0 0
\(446\) 19.4164 0.919394
\(447\) 11.7082i 0.553779i
\(448\) − 4.47214i − 0.211289i
\(449\) −15.8885 −0.749827 −0.374913 0.927060i \(-0.622328\pi\)
−0.374913 + 0.927060i \(0.622328\pi\)
\(450\) 0 0
\(451\) 10.4721 0.493114
\(452\) 5.52786i 0.260009i
\(453\) − 14.4721i − 0.679960i
\(454\) 9.23607 0.433470
\(455\) 0 0
\(456\) 5.70820 0.267311
\(457\) − 27.5279i − 1.28770i −0.765152 0.643850i \(-0.777336\pi\)
0.765152 0.643850i \(-0.222664\pi\)
\(458\) 17.7082i 0.827450i
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) − 23.4164i − 1.08943i
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) 19.8885 0.921319
\(467\) 17.8197i 0.824596i 0.911049 + 0.412298i \(0.135274\pi\)
−0.911049 + 0.412298i \(0.864726\pi\)
\(468\) 4.47214i 0.206725i
\(469\) 43.4164 2.00478
\(470\) 0 0
\(471\) −6.65248 −0.306530
\(472\) 8.94427i 0.411693i
\(473\) 24.9443i 1.14694i
\(474\) −4.47214 −0.205412
\(475\) 0 0
\(476\) 17.8885 0.819920
\(477\) − 5.23607i − 0.239743i
\(478\) − 4.94427i − 0.226146i
\(479\) 28.9443 1.32250 0.661249 0.750167i \(-0.270027\pi\)
0.661249 + 0.750167i \(0.270027\pi\)
\(480\) 0 0
\(481\) 50.2492 2.29117
\(482\) 12.4721i 0.568090i
\(483\) − 4.47214i − 0.203489i
\(484\) −16.4164 −0.746200
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 9.88854i − 0.448093i −0.974578 0.224046i \(-0.928073\pi\)
0.974578 0.224046i \(-0.0719267\pi\)
\(488\) 0.763932i 0.0345816i
\(489\) 2.47214 0.111794
\(490\) 0 0
\(491\) 29.3050 1.32251 0.661257 0.750159i \(-0.270023\pi\)
0.661257 + 0.750159i \(0.270023\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 17.8885i 0.805659i
\(494\) −25.5279 −1.14855
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) 40.0000i 1.79425i
\(498\) 13.2361i 0.593122i
\(499\) −0.583592 −0.0261252 −0.0130626 0.999915i \(-0.504158\pi\)
−0.0130626 + 0.999915i \(0.504158\pi\)
\(500\) 0 0
\(501\) 16.9443 0.757014
\(502\) 19.7082i 0.879620i
\(503\) 10.4721i 0.466929i 0.972365 + 0.233465i \(0.0750063\pi\)
−0.972365 + 0.233465i \(0.924994\pi\)
\(504\) 4.47214 0.199205
\(505\) 0 0
\(506\) −5.23607 −0.232772
\(507\) − 7.00000i − 0.310881i
\(508\) − 4.00000i − 0.177471i
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) − 1.00000i − 0.0441942i
\(513\) 5.70820i 0.252023i
\(514\) 11.8885 0.524381
\(515\) 0 0
\(516\) −4.76393 −0.209720
\(517\) 20.9443i 0.921128i
\(518\) − 50.2492i − 2.20782i
\(519\) 17.4164 0.764495
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 4.47214i 0.195740i
\(523\) 25.7082i 1.12414i 0.827089 + 0.562071i \(0.189995\pi\)
−0.827089 + 0.562071i \(0.810005\pi\)
\(524\) 9.52786 0.416227
\(525\) 0 0
\(526\) 24.9443 1.08762
\(527\) − 9.88854i − 0.430752i
\(528\) − 5.23607i − 0.227871i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −8.94427 −0.388148
\(532\) 25.5279i 1.10677i
\(533\) − 8.94427i − 0.387419i
\(534\) 10.4721 0.453174
\(535\) 0 0
\(536\) 9.70820 0.419331
\(537\) − 19.4164i − 0.837880i
\(538\) − 13.0557i − 0.562872i
\(539\) 68.0689 2.93193
\(540\) 0 0
\(541\) 8.11146 0.348739 0.174369 0.984680i \(-0.444211\pi\)
0.174369 + 0.984680i \(0.444211\pi\)
\(542\) 16.9443i 0.727819i
\(543\) − 11.2361i − 0.482186i
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −20.0000 −0.855921
\(547\) − 41.3050i − 1.76607i −0.469305 0.883036i \(-0.655496\pi\)
0.469305 0.883036i \(-0.344504\pi\)
\(548\) 3.05573i 0.130534i
\(549\) −0.763932 −0.0326038
\(550\) 0 0
\(551\) −25.5279 −1.08752
\(552\) − 1.00000i − 0.0425628i
\(553\) − 20.0000i − 0.850487i
\(554\) 20.4721 0.869778
\(555\) 0 0
\(556\) −16.9443 −0.718597
\(557\) 15.1246i 0.640850i 0.947274 + 0.320425i \(0.103826\pi\)
−0.947274 + 0.320425i \(0.896174\pi\)
\(558\) − 2.47214i − 0.104654i
\(559\) 21.3050 0.901103
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 13.5279i 0.570639i
\(563\) 24.6525i 1.03898i 0.854477 + 0.519489i \(0.173878\pi\)
−0.854477 + 0.519489i \(0.826122\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −3.81966 −0.160552
\(567\) 4.47214i 0.187812i
\(568\) 8.94427i 0.375293i
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) 23.4164i 0.979089i
\(573\) 6.47214i 0.270377i
\(574\) −8.94427 −0.373327
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.3607i 1.43045i 0.698892 + 0.715227i \(0.253677\pi\)
−0.698892 + 0.715227i \(0.746323\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) −23.8885 −0.992774
\(580\) 0 0
\(581\) −59.1935 −2.45576
\(582\) − 0.472136i − 0.0195707i
\(583\) − 27.4164i − 1.13547i
\(584\) 4.47214 0.185058
\(585\) 0 0
\(586\) 0.291796 0.0120540
\(587\) − 6.47214i − 0.267134i −0.991040 0.133567i \(-0.957357\pi\)
0.991040 0.133567i \(-0.0426431\pi\)
\(588\) 13.0000i 0.536111i
\(589\) 14.1115 0.581452
\(590\) 0 0
\(591\) 2.94427 0.121111
\(592\) − 11.2361i − 0.461800i
\(593\) 33.7771i 1.38706i 0.720428 + 0.693529i \(0.243945\pi\)
−0.720428 + 0.693529i \(0.756055\pi\)
\(594\) 5.23607 0.214838
\(595\) 0 0
\(596\) −11.7082 −0.479587
\(597\) − 17.4164i − 0.712806i
\(598\) 4.47214i 0.182879i
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) − 21.3050i − 0.868325i
\(603\) 9.70820i 0.395349i
\(604\) 14.4721 0.588863
\(605\) 0 0
\(606\) 4.47214 0.181668
\(607\) 17.5279i 0.711434i 0.934594 + 0.355717i \(0.115763\pi\)
−0.934594 + 0.355717i \(0.884237\pi\)
\(608\) 5.70820i 0.231498i
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 17.8885 0.723693
\(612\) 4.00000i 0.161690i
\(613\) − 25.1246i − 1.01477i −0.861718 0.507387i \(-0.830612\pi\)
0.861718 0.507387i \(-0.169388\pi\)
\(614\) −15.4164 −0.622156
\(615\) 0 0
\(616\) 23.4164 0.943474
\(617\) − 20.3607i − 0.819690i −0.912155 0.409845i \(-0.865583\pi\)
0.912155 0.409845i \(-0.134417\pi\)
\(618\) − 6.00000i − 0.241355i
\(619\) −18.2918 −0.735209 −0.367605 0.929982i \(-0.619822\pi\)
−0.367605 + 0.929982i \(0.619822\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 20.9443i 0.839789i
\(623\) 46.8328i 1.87632i
\(624\) −4.47214 −0.179029
\(625\) 0 0
\(626\) 15.5279 0.620618
\(627\) 29.8885i 1.19363i
\(628\) − 6.65248i − 0.265463i
\(629\) 44.9443 1.79205
\(630\) 0 0
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) − 4.47214i − 0.177892i
\(633\) − 23.4164i − 0.930719i
\(634\) −19.5279 −0.775551
\(635\) 0 0
\(636\) 5.23607 0.207624
\(637\) − 58.1378i − 2.30350i
\(638\) 23.4164i 0.927064i
\(639\) −8.94427 −0.353830
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 12.6525i 0.499353i
\(643\) − 32.5410i − 1.28329i −0.767001 0.641646i \(-0.778252\pi\)
0.767001 0.641646i \(-0.221748\pi\)
\(644\) 4.47214 0.176227
\(645\) 0 0
\(646\) −22.8328 −0.898345
\(647\) − 12.9443i − 0.508892i −0.967087 0.254446i \(-0.918107\pi\)
0.967087 0.254446i \(-0.0818931\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −46.8328 −1.83835
\(650\) 0 0
\(651\) 11.0557 0.433308
\(652\) 2.47214i 0.0968163i
\(653\) 9.41641i 0.368493i 0.982880 + 0.184246i \(0.0589844\pi\)
−0.982880 + 0.184246i \(0.941016\pi\)
\(654\) −4.76393 −0.186284
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 4.47214i 0.174475i
\(658\) − 17.8885i − 0.697368i
\(659\) 32.6525 1.27196 0.635980 0.771706i \(-0.280596\pi\)
0.635980 + 0.771706i \(0.280596\pi\)
\(660\) 0 0
\(661\) 3.81966 0.148568 0.0742838 0.997237i \(-0.476333\pi\)
0.0742838 + 0.997237i \(0.476333\pi\)
\(662\) 10.4721i 0.407011i
\(663\) − 17.8885i − 0.694733i
\(664\) −13.2361 −0.513659
\(665\) 0 0
\(666\) 11.2361 0.435389
\(667\) 4.47214i 0.173162i
\(668\) 16.9443i 0.655594i
\(669\) −19.4164 −0.750682
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 4.47214i 0.172516i
\(673\) 4.47214i 0.172388i 0.996278 + 0.0861941i \(0.0274705\pi\)
−0.996278 + 0.0861941i \(0.972529\pi\)
\(674\) 19.8885 0.766078
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) − 19.1246i − 0.735019i −0.930020 0.367509i \(-0.880211\pi\)
0.930020 0.367509i \(-0.119789\pi\)
\(678\) − 5.52786i − 0.212296i
\(679\) 2.11146 0.0810303
\(680\) 0 0
\(681\) −9.23607 −0.353927
\(682\) − 12.9443i − 0.495662i
\(683\) 14.4721i 0.553761i 0.960904 + 0.276880i \(0.0893006\pi\)
−0.960904 + 0.276880i \(0.910699\pi\)
\(684\) −5.70820 −0.218259
\(685\) 0 0
\(686\) −26.8328 −1.02448
\(687\) − 17.7082i − 0.675610i
\(688\) − 4.76393i − 0.181623i
\(689\) −23.4164 −0.892094
\(690\) 0 0
\(691\) 16.5836 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(692\) 17.4164i 0.662072i
\(693\) 23.4164i 0.889516i
\(694\) −30.4721 −1.15671
\(695\) 0 0
\(696\) −4.47214 −0.169516
\(697\) − 8.00000i − 0.303022i
\(698\) 3.88854i 0.147184i
\(699\) −19.8885 −0.752254
\(700\) 0 0
\(701\) −3.12461 −0.118015 −0.0590075 0.998258i \(-0.518794\pi\)
−0.0590075 + 0.998258i \(0.518794\pi\)
\(702\) − 4.47214i − 0.168790i
\(703\) 64.1378i 2.41900i
\(704\) 5.23607 0.197342
\(705\) 0 0
\(706\) −3.88854 −0.146347
\(707\) 20.0000i 0.752177i
\(708\) − 8.94427i − 0.336146i
\(709\) −35.0132 −1.31495 −0.657473 0.753478i \(-0.728375\pi\)
−0.657473 + 0.753478i \(0.728375\pi\)
\(710\) 0 0
\(711\) 4.47214 0.167718
\(712\) 10.4721i 0.392460i
\(713\) − 2.47214i − 0.0925822i
\(714\) −17.8885 −0.669462
\(715\) 0 0
\(716\) 19.4164 0.725625
\(717\) 4.94427i 0.184647i
\(718\) 29.3050i 1.09365i
\(719\) 3.05573 0.113959 0.0569797 0.998375i \(-0.481853\pi\)
0.0569797 + 0.998375i \(0.481853\pi\)
\(720\) 0 0
\(721\) 26.8328 0.999306
\(722\) − 13.5836i − 0.505529i
\(723\) − 12.4721i − 0.463844i
\(724\) 11.2361 0.417585
\(725\) 0 0
\(726\) 16.4164 0.609270
\(727\) 19.3050i 0.715981i 0.933725 + 0.357991i \(0.116538\pi\)
−0.933725 + 0.357991i \(0.883462\pi\)
\(728\) − 20.0000i − 0.741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 19.0557 0.704802
\(732\) − 0.763932i − 0.0282357i
\(733\) − 21.7082i − 0.801811i −0.916119 0.400905i \(-0.868696\pi\)
0.916119 0.400905i \(-0.131304\pi\)
\(734\) −9.41641 −0.347566
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 50.8328i 1.87245i
\(738\) − 2.00000i − 0.0736210i
\(739\) 32.9443 1.21187 0.605937 0.795512i \(-0.292798\pi\)
0.605937 + 0.795512i \(0.292798\pi\)
\(740\) 0 0
\(741\) 25.5279 0.937790
\(742\) 23.4164i 0.859643i
\(743\) − 4.00000i − 0.146746i −0.997305 0.0733729i \(-0.976624\pi\)
0.997305 0.0733729i \(-0.0233763\pi\)
\(744\) 2.47214 0.0906329
\(745\) 0 0
\(746\) −35.5967 −1.30329
\(747\) − 13.2361i − 0.484282i
\(748\) 20.9443i 0.765798i
\(749\) −56.5836 −2.06752
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) − 4.00000i − 0.145865i
\(753\) − 19.7082i − 0.718207i
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) −4.47214 −0.162650
\(757\) 1.34752i 0.0489766i 0.999700 + 0.0244883i \(0.00779565\pi\)
−0.999700 + 0.0244883i \(0.992204\pi\)
\(758\) 13.7082i 0.497904i
\(759\) 5.23607 0.190057
\(760\) 0 0
\(761\) −5.05573 −0.183270 −0.0916350 0.995793i \(-0.529209\pi\)
−0.0916350 + 0.995793i \(0.529209\pi\)
\(762\) 4.00000i 0.144905i
\(763\) − 21.3050i − 0.771291i
\(764\) −6.47214 −0.234154
\(765\) 0 0
\(766\) 7.05573 0.254934
\(767\) 40.0000i 1.44432i
\(768\) 1.00000i 0.0360844i
\(769\) −12.4721 −0.449757 −0.224878 0.974387i \(-0.572198\pi\)
−0.224878 + 0.974387i \(0.572198\pi\)
\(770\) 0 0
\(771\) −11.8885 −0.428155
\(772\) − 23.8885i − 0.859768i
\(773\) − 38.1803i − 1.37325i −0.727011 0.686626i \(-0.759091\pi\)
0.727011 0.686626i \(-0.240909\pi\)
\(774\) 4.76393 0.171236
\(775\) 0 0
\(776\) 0.472136 0.0169487
\(777\) 50.2492i 1.80268i
\(778\) 23.7082i 0.849980i
\(779\) 11.4164 0.409035
\(780\) 0 0
\(781\) −46.8328 −1.67581
\(782\) 4.00000i 0.143040i
\(783\) − 4.47214i − 0.159821i
\(784\) −13.0000 −0.464286
\(785\) 0 0
\(786\) −9.52786 −0.339848
\(787\) − 35.2361i − 1.25603i −0.778201 0.628015i \(-0.783868\pi\)
0.778201 0.628015i \(-0.216132\pi\)
\(788\) 2.94427i 0.104885i
\(789\) −24.9443 −0.888040
\(790\) 0 0
\(791\) 24.7214 0.878990
\(792\) 5.23607i 0.186056i
\(793\) 3.41641i 0.121320i
\(794\) −26.9443 −0.956216
\(795\) 0 0
\(796\) 17.4164 0.617308
\(797\) − 41.5967i − 1.47343i −0.676202 0.736716i \(-0.736375\pi\)
0.676202 0.736716i \(-0.263625\pi\)
\(798\) − 25.5279i − 0.903677i
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −10.4721 −0.370015
\(802\) − 8.94427i − 0.315833i
\(803\) 23.4164i 0.826347i
\(804\) −9.70820 −0.342382
\(805\) 0 0
\(806\) −11.0557 −0.389421
\(807\) 13.0557i 0.459583i
\(808\) 4.47214i 0.157329i
\(809\) −42.9443 −1.50984 −0.754920 0.655817i \(-0.772324\pi\)
−0.754920 + 0.655817i \(0.772324\pi\)
\(810\) 0 0
\(811\) 41.3050 1.45041 0.725207 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(812\) − 20.0000i − 0.701862i
\(813\) − 16.9443i − 0.594262i
\(814\) 58.8328 2.06209
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 27.1935i 0.951380i
\(818\) − 35.8885i − 1.25481i
\(819\) 20.0000 0.698857
\(820\) 0 0
\(821\) 39.5279 1.37953 0.689766 0.724032i \(-0.257713\pi\)
0.689766 + 0.724032i \(0.257713\pi\)
\(822\) − 3.05573i − 0.106581i
\(823\) − 52.3607i − 1.82518i −0.408877 0.912589i \(-0.634080\pi\)
0.408877 0.912589i \(-0.365920\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 40.0000 1.39178
\(827\) 21.5967i 0.750993i 0.926824 + 0.375496i \(0.122528\pi\)
−0.926824 + 0.375496i \(0.877472\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) −20.4721 −0.710171
\(832\) − 4.47214i − 0.155043i
\(833\) − 52.0000i − 1.80169i
\(834\) 16.9443 0.586732
\(835\) 0 0
\(836\) −29.8885 −1.03372
\(837\) 2.47214i 0.0854495i
\(838\) 28.0689i 0.969623i
\(839\) 45.8885 1.58425 0.792124 0.610360i \(-0.208975\pi\)
0.792124 + 0.610360i \(0.208975\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0.763932i 0.0263268i
\(843\) − 13.5279i − 0.465924i
\(844\) 23.4164 0.806026
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 73.4164i 2.52262i
\(848\) 5.23607i 0.179807i
\(849\) 3.81966 0.131090
\(850\) 0 0
\(851\) 11.2361 0.385167
\(852\) − 8.94427i − 0.306426i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 3.41641 0.116907
\(855\) 0 0
\(856\) −12.6525 −0.432453
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) − 23.4164i − 0.799423i
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 8.94427 0.304820
\(862\) 23.4164i 0.797566i
\(863\) 14.8328i 0.504915i 0.967608 + 0.252457i \(0.0812388\pi\)
−0.967608 + 0.252457i \(0.918761\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −21.4164 −0.727759
\(867\) 1.00000i 0.0339618i
\(868\) 11.0557i 0.375256i
\(869\) 23.4164 0.794347
\(870\) 0 0
\(871\) 43.4164 1.47111
\(872\) − 4.76393i − 0.161327i
\(873\) 0.472136i 0.0159794i
\(874\) −5.70820 −0.193083
\(875\) 0 0
\(876\) −4.47214 −0.151099
\(877\) 48.2492i 1.62926i 0.579981 + 0.814630i \(0.303060\pi\)
−0.579981 + 0.814630i \(0.696940\pi\)
\(878\) 19.0557i 0.643100i
\(879\) −0.291796 −0.00984204
\(880\) 0 0
\(881\) 39.4164 1.32797 0.663986 0.747745i \(-0.268863\pi\)
0.663986 + 0.747745i \(0.268863\pi\)
\(882\) − 13.0000i − 0.437733i
\(883\) 13.5279i 0.455249i 0.973749 + 0.227624i \(0.0730958\pi\)
−0.973749 + 0.227624i \(0.926904\pi\)
\(884\) 17.8885 0.601657
\(885\) 0 0
\(886\) −15.0557 −0.505807
\(887\) 16.9443i 0.568933i 0.958686 + 0.284466i \(0.0918164\pi\)
−0.958686 + 0.284466i \(0.908184\pi\)
\(888\) 11.2361i 0.377058i
\(889\) −17.8885 −0.599963
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) − 19.4164i − 0.650109i
\(893\) 22.8328i 0.764071i
\(894\) 11.7082 0.391581
\(895\) 0 0
\(896\) −4.47214 −0.149404
\(897\) − 4.47214i − 0.149320i
\(898\) 15.8885i 0.530208i
\(899\) −11.0557 −0.368729
\(900\) 0 0
\(901\) −20.9443 −0.697755
\(902\) − 10.4721i − 0.348684i
\(903\) 21.3050i 0.708984i
\(904\) 5.52786 0.183854
\(905\) 0 0
\(906\) −14.4721 −0.480805
\(907\) − 4.18034i − 0.138806i −0.997589 0.0694030i \(-0.977891\pi\)
0.997589 0.0694030i \(-0.0221094\pi\)
\(908\) − 9.23607i − 0.306510i
\(909\) −4.47214 −0.148331
\(910\) 0 0
\(911\) 16.5836 0.549439 0.274719 0.961524i \(-0.411415\pi\)
0.274719 + 0.961524i \(0.411415\pi\)
\(912\) − 5.70820i − 0.189018i
\(913\) − 69.3050i − 2.29366i
\(914\) −27.5279 −0.910541
\(915\) 0 0
\(916\) 17.7082 0.585096
\(917\) − 42.6099i − 1.40710i
\(918\) − 4.00000i − 0.132020i
\(919\) 16.4721 0.543366 0.271683 0.962387i \(-0.412420\pi\)
0.271683 + 0.962387i \(0.412420\pi\)
\(920\) 0 0
\(921\) 15.4164 0.507988
\(922\) − 22.0000i − 0.724531i
\(923\) 40.0000i 1.31662i
\(924\) −23.4164 −0.770343
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) 6.00000i 0.197066i
\(928\) − 4.47214i − 0.146805i
\(929\) 24.8328 0.814738 0.407369 0.913264i \(-0.366446\pi\)
0.407369 + 0.913264i \(0.366446\pi\)
\(930\) 0 0
\(931\) 74.2067 2.43202
\(932\) − 19.8885i − 0.651471i
\(933\) − 20.9443i − 0.685685i
\(934\) 17.8197 0.583077
\(935\) 0 0
\(936\) 4.47214 0.146176
\(937\) 30.0000i 0.980057i 0.871706 + 0.490029i \(0.163014\pi\)
−0.871706 + 0.490029i \(0.836986\pi\)
\(938\) − 43.4164i − 1.41760i
\(939\) −15.5279 −0.506733
\(940\) 0 0
\(941\) 38.1803 1.24464 0.622322 0.782762i \(-0.286190\pi\)
0.622322 + 0.782762i \(0.286190\pi\)
\(942\) 6.65248i 0.216749i
\(943\) − 2.00000i − 0.0651290i
\(944\) 8.94427 0.291111
\(945\) 0 0
\(946\) 24.9443 0.811008
\(947\) 47.1935i 1.53358i 0.641897 + 0.766791i \(0.278148\pi\)
−0.641897 + 0.766791i \(0.721852\pi\)
\(948\) 4.47214i 0.145248i
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 19.5279 0.633234
\(952\) − 17.8885i − 0.579771i
\(953\) − 47.7771i − 1.54765i −0.633398 0.773826i \(-0.718341\pi\)
0.633398 0.773826i \(-0.281659\pi\)
\(954\) −5.23607 −0.169524
\(955\) 0 0
\(956\) −4.94427 −0.159909
\(957\) − 23.4164i − 0.756945i
\(958\) − 28.9443i − 0.935147i
\(959\) 13.6656 0.441286
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) − 50.2492i − 1.62010i
\(963\) − 12.6525i − 0.407720i
\(964\) 12.4721 0.401700
\(965\) 0 0
\(966\) −4.47214 −0.143889
\(967\) − 15.4164i − 0.495758i −0.968791 0.247879i \(-0.920266\pi\)
0.968791 0.247879i \(-0.0797336\pi\)
\(968\) 16.4164i 0.527643i
\(969\) 22.8328 0.733496
\(970\) 0 0
\(971\) −19.1246 −0.613738 −0.306869 0.951752i \(-0.599281\pi\)
−0.306869 + 0.951752i \(0.599281\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 75.7771i 2.42930i
\(974\) −9.88854 −0.316849
\(975\) 0 0
\(976\) 0.763932 0.0244529
\(977\) − 1.16718i − 0.0373415i −0.999826 0.0186708i \(-0.994057\pi\)
0.999826 0.0186708i \(-0.00594343\pi\)
\(978\) − 2.47214i − 0.0790502i
\(979\) −54.8328 −1.75246
\(980\) 0 0
\(981\) 4.76393 0.152101
\(982\) − 29.3050i − 0.935159i
\(983\) 0.583592i 0.0186137i 0.999957 + 0.00930685i \(0.00296250\pi\)
−0.999957 + 0.00930685i \(0.997037\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 17.8885 0.569687
\(987\) 17.8885i 0.569399i
\(988\) 25.5279i 0.812150i
\(989\) 4.76393 0.151484
\(990\) 0 0
\(991\) −10.4721 −0.332658 −0.166329 0.986070i \(-0.553191\pi\)
−0.166329 + 0.986070i \(0.553191\pi\)
\(992\) 2.47214i 0.0784904i
\(993\) − 10.4721i − 0.332323i
\(994\) 40.0000 1.26872
\(995\) 0 0
\(996\) 13.2361 0.419401
\(997\) − 5.05573i − 0.160117i −0.996790 0.0800583i \(-0.974489\pi\)
0.996790 0.0800583i \(-0.0255106\pi\)
\(998\) 0.583592i 0.0184733i
\(999\) −11.2361 −0.355493
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 3450.2.d.x.2899.2 4
5.2 odd 4 138.2.a.d.1.2 2
5.3 odd 4 3450.2.a.be.1.2 2
5.4 even 2 inner 3450.2.d.x.2899.3 4
15.2 even 4 414.2.a.f.1.1 2
20.7 even 4 1104.2.a.j.1.2 2
35.27 even 4 6762.2.a.cb.1.1 2
40.27 even 4 4416.2.a.bl.1.1 2
40.37 odd 4 4416.2.a.bh.1.1 2
60.47 odd 4 3312.2.a.bc.1.1 2
115.22 even 4 3174.2.a.s.1.1 2
345.137 odd 4 9522.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 5.2 odd 4
414.2.a.f.1.1 2 15.2 even 4
1104.2.a.j.1.2 2 20.7 even 4
3174.2.a.s.1.1 2 115.22 even 4
3312.2.a.bc.1.1 2 60.47 odd 4
3450.2.a.be.1.2 2 5.3 odd 4
3450.2.d.x.2899.2 4 1.1 even 1 trivial
3450.2.d.x.2899.3 4 5.4 even 2 inner
4416.2.a.bh.1.1 2 40.37 odd 4
4416.2.a.bl.1.1 2 40.27 even 4
6762.2.a.cb.1.1 2 35.27 even 4
9522.2.a.q.1.2 2 345.137 odd 4