Properties

Label 2175.2.c.n.349.3
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1267360000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.3
Root \(-2.43828i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.n.349.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43828i q^{2} -1.00000i q^{3} -0.0686587 q^{4} -1.43828 q^{6} +2.74301i q^{7} -2.77782i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.43828i q^{2} -1.00000i q^{3} -0.0686587 q^{4} -1.43828 q^{6} +2.74301i q^{7} -2.77782i q^{8} -1.00000 q^{9} +2.74301 q^{11} +0.0686587i q^{12} -5.14744i q^{13} +3.94523 q^{14} -4.13260 q^{16} +3.72913i q^{17} +1.43828i q^{18} +0.404431 q^{19} +2.74301 q^{21} -3.94523i q^{22} -5.45825i q^{23} -2.77782 q^{24} -7.40348 q^{26} +1.00000i q^{27} -0.188331i q^{28} +1.00000 q^{29} +1.45825 q^{31} +0.388222i q^{32} -2.74301i q^{33} +5.36354 q^{34} +0.0686587 q^{36} -6.76702i q^{37} -0.581686i q^{38} -5.14744 q^{39} +9.78090 q^{41} -3.94523i q^{42} +4.43220i q^{43} -0.188331 q^{44} -7.85051 q^{46} -2.60569i q^{47} +4.13260i q^{48} -0.524103 q^{49} +3.72913 q^{51} +0.353416i q^{52} -6.43220i q^{53} +1.43828 q^{54} +7.61958 q^{56} -0.404431i q^{57} -1.43828i q^{58} +9.91822 q^{59} -13.0816 q^{61} -2.09738i q^{62} -2.74301i q^{63} -7.70683 q^{64} -3.94523 q^{66} -12.4961i q^{67} -0.256037i q^{68} -5.45825 q^{69} -11.3487 q^{71} +2.77782i q^{72} -10.7670i q^{73} -9.73289 q^{74} -0.0277677 q^{76} +7.52410i q^{77} +7.40348i q^{78} -14.1576 q^{79} +1.00000 q^{81} -14.0677i q^{82} +1.62334i q^{83} -0.188331 q^{84} +6.37476 q^{86} -1.00000i q^{87} -7.61958i q^{88} +8.87281 q^{89} +14.1195 q^{91} +0.374756i q^{92} -1.45825i q^{93} -3.74772 q^{94} +0.388222 q^{96} -7.82084i q^{97} +0.753809i q^{98} -2.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46} - 12 q^{49} + 20 q^{51} - 6 q^{54} - 4 q^{59} - 52 q^{61} - 68 q^{64} - 6 q^{66} - 24 q^{69} - 20 q^{71} - 96 q^{74} + 32 q^{76} - 44 q^{79} + 8 q^{81} - 18 q^{84} - 8 q^{86} + 8 q^{89} - 16 q^{91} - 78 q^{94} + 34 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.43828i − 1.01702i −0.861056 0.508510i \(-0.830197\pi\)
0.861056 0.508510i \(-0.169803\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −0.0686587 −0.0343293
\(5\) 0 0
\(6\) −1.43828 −0.587177
\(7\) 2.74301i 1.03676i 0.855150 + 0.518380i \(0.173465\pi\)
−0.855150 + 0.518380i \(0.826535\pi\)
\(8\) − 2.77782i − 0.982106i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.74301 0.827049 0.413524 0.910493i \(-0.364298\pi\)
0.413524 + 0.910493i \(0.364298\pi\)
\(12\) 0.0686587i 0.0198201i
\(13\) − 5.14744i − 1.42764i −0.700328 0.713822i \(-0.746963\pi\)
0.700328 0.713822i \(-0.253037\pi\)
\(14\) 3.94523 1.05441
\(15\) 0 0
\(16\) −4.13260 −1.03315
\(17\) 3.72913i 0.904446i 0.891905 + 0.452223i \(0.149369\pi\)
−0.891905 + 0.452223i \(0.850631\pi\)
\(18\) 1.43828i 0.339007i
\(19\) 0.404431 0.0927827 0.0463914 0.998923i \(-0.485228\pi\)
0.0463914 + 0.998923i \(0.485228\pi\)
\(20\) 0 0
\(21\) 2.74301 0.598574
\(22\) − 3.94523i − 0.841125i
\(23\) − 5.45825i − 1.13812i −0.822295 0.569062i \(-0.807306\pi\)
0.822295 0.569062i \(-0.192694\pi\)
\(24\) −2.77782 −0.567019
\(25\) 0 0
\(26\) −7.40348 −1.45194
\(27\) 1.00000i 0.192450i
\(28\) − 0.188331i − 0.0355913i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.45825 0.261910 0.130955 0.991388i \(-0.458196\pi\)
0.130955 + 0.991388i \(0.458196\pi\)
\(32\) 0.388222i 0.0686287i
\(33\) − 2.74301i − 0.477497i
\(34\) 5.36354 0.919839
\(35\) 0 0
\(36\) 0.0686587 0.0114431
\(37\) − 6.76702i − 1.11249i −0.831018 0.556245i \(-0.812241\pi\)
0.831018 0.556245i \(-0.187759\pi\)
\(38\) − 0.581686i − 0.0943619i
\(39\) −5.14744 −0.824250
\(40\) 0 0
\(41\) 9.78090 1.52752 0.763760 0.645500i \(-0.223351\pi\)
0.763760 + 0.645500i \(0.223351\pi\)
\(42\) − 3.94523i − 0.608761i
\(43\) 4.43220i 0.675904i 0.941163 + 0.337952i \(0.109734\pi\)
−0.941163 + 0.337952i \(0.890266\pi\)
\(44\) −0.188331 −0.0283920
\(45\) 0 0
\(46\) −7.85051 −1.15749
\(47\) − 2.60569i − 0.380079i −0.981776 0.190040i \(-0.939138\pi\)
0.981776 0.190040i \(-0.0608617\pi\)
\(48\) 4.13260i 0.596490i
\(49\) −0.524103 −0.0748719
\(50\) 0 0
\(51\) 3.72913 0.522182
\(52\) 0.353416i 0.0490100i
\(53\) − 6.43220i − 0.883530i −0.897131 0.441765i \(-0.854352\pi\)
0.897131 0.441765i \(-0.145648\pi\)
\(54\) 1.43828 0.195726
\(55\) 0 0
\(56\) 7.61958 1.01821
\(57\) − 0.404431i − 0.0535681i
\(58\) − 1.43828i − 0.188856i
\(59\) 9.91822 1.29124 0.645621 0.763658i \(-0.276599\pi\)
0.645621 + 0.763658i \(0.276599\pi\)
\(60\) 0 0
\(61\) −13.0816 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(62\) − 2.09738i − 0.266367i
\(63\) − 2.74301i − 0.345587i
\(64\) −7.70683 −0.963354
\(65\) 0 0
\(66\) −3.94523 −0.485624
\(67\) − 12.4961i − 1.52665i −0.646017 0.763323i \(-0.723566\pi\)
0.646017 0.763323i \(-0.276434\pi\)
\(68\) − 0.256037i − 0.0310490i
\(69\) −5.45825 −0.657096
\(70\) 0 0
\(71\) −11.3487 −1.34684 −0.673422 0.739259i \(-0.735176\pi\)
−0.673422 + 0.739259i \(0.735176\pi\)
\(72\) 2.77782i 0.327369i
\(73\) − 10.7670i − 1.26018i −0.776520 0.630092i \(-0.783017\pi\)
0.776520 0.630092i \(-0.216983\pi\)
\(74\) −9.73289 −1.13143
\(75\) 0 0
\(76\) −0.0277677 −0.00318517
\(77\) 7.52410i 0.857451i
\(78\) 7.40348i 0.838279i
\(79\) −14.1576 −1.59285 −0.796425 0.604737i \(-0.793278\pi\)
−0.796425 + 0.604737i \(0.793278\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 14.0677i − 1.55352i
\(83\) 1.62334i 0.178184i 0.996023 + 0.0890922i \(0.0283966\pi\)
−0.996023 + 0.0890922i \(0.971603\pi\)
\(84\) −0.188331 −0.0205486
\(85\) 0 0
\(86\) 6.37476 0.687408
\(87\) − 1.00000i − 0.107211i
\(88\) − 7.61958i − 0.812250i
\(89\) 8.87281 0.940516 0.470258 0.882529i \(-0.344161\pi\)
0.470258 + 0.882529i \(0.344161\pi\)
\(90\) 0 0
\(91\) 14.1195 1.48012
\(92\) 0.374756i 0.0390711i
\(93\) − 1.45825i − 0.151214i
\(94\) −3.74772 −0.386548
\(95\) 0 0
\(96\) 0.388222 0.0396228
\(97\) − 7.82084i − 0.794086i −0.917800 0.397043i \(-0.870036\pi\)
0.917800 0.397043i \(-0.129964\pi\)
\(98\) 0.753809i 0.0761462i
\(99\) −2.74301 −0.275683
\(100\) 0 0
\(101\) −4.88033 −0.485611 −0.242805 0.970075i \(-0.578068\pi\)
−0.242805 + 0.970075i \(0.578068\pi\)
\(102\) − 5.36354i − 0.531070i
\(103\) − 0.294881i − 0.0290555i −0.999894 0.0145277i \(-0.995376\pi\)
0.999894 0.0145277i \(-0.00462448\pi\)
\(104\) −14.2986 −1.40210
\(105\) 0 0
\(106\) −9.25132 −0.898568
\(107\) 13.7809i 1.33225i 0.745840 + 0.666125i \(0.232048\pi\)
−0.745840 + 0.666125i \(0.767952\pi\)
\(108\) − 0.0686587i − 0.00660668i
\(109\) 6.20126 0.593973 0.296987 0.954882i \(-0.404018\pi\)
0.296987 + 0.954882i \(0.404018\pi\)
\(110\) 0 0
\(111\) −6.76702 −0.642297
\(112\) − 11.3358i − 1.07113i
\(113\) − 10.5658i − 0.993943i −0.867767 0.496971i \(-0.834445\pi\)
0.867767 0.496971i \(-0.165555\pi\)
\(114\) −0.581686 −0.0544799
\(115\) 0 0
\(116\) −0.0686587 −0.00637480
\(117\) 5.14744i 0.475881i
\(118\) − 14.2652i − 1.31322i
\(119\) −10.2290 −0.937694
\(120\) 0 0
\(121\) −3.47590 −0.315991
\(122\) 18.8150i 1.70343i
\(123\) − 9.78090i − 0.881914i
\(124\) −0.100122 −0.00899119
\(125\) 0 0
\(126\) −3.94523 −0.351469
\(127\) − 8.54175i − 0.757958i −0.925405 0.378979i \(-0.876275\pi\)
0.925405 0.378979i \(-0.123725\pi\)
\(128\) 11.8611i 1.04838i
\(129\) 4.43220 0.390233
\(130\) 0 0
\(131\) 13.3050 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(132\) 0.188331i 0.0163921i
\(133\) 1.10936i 0.0961934i
\(134\) −17.9730 −1.55263
\(135\) 0 0
\(136\) 10.3588 0.888262
\(137\) 18.2253i 1.55709i 0.627589 + 0.778545i \(0.284042\pi\)
−0.627589 + 0.778545i \(0.715958\pi\)
\(138\) 7.85051i 0.668280i
\(139\) 5.66142 0.480195 0.240098 0.970749i \(-0.422821\pi\)
0.240098 + 0.970749i \(0.422821\pi\)
\(140\) 0 0
\(141\) −2.60569 −0.219439
\(142\) 16.3226i 1.36977i
\(143\) − 14.1195i − 1.18073i
\(144\) 4.13260 0.344384
\(145\) 0 0
\(146\) −15.4860 −1.28163
\(147\) 0.524103i 0.0432273i
\(148\) 0.464614i 0.0381911i
\(149\) 11.9182 0.976378 0.488189 0.872738i \(-0.337658\pi\)
0.488189 + 0.872738i \(0.337658\pi\)
\(150\) 0 0
\(151\) 7.19114 0.585207 0.292603 0.956234i \(-0.405478\pi\)
0.292603 + 0.956234i \(0.405478\pi\)
\(152\) − 1.12343i − 0.0911225i
\(153\) − 3.72913i − 0.301482i
\(154\) 10.8218 0.872045
\(155\) 0 0
\(156\) 0.353416 0.0282960
\(157\) 4.31457i 0.344340i 0.985067 + 0.172170i \(0.0550779\pi\)
−0.985067 + 0.172170i \(0.944922\pi\)
\(158\) 20.3626i 1.61996i
\(159\) −6.43220 −0.510106
\(160\) 0 0
\(161\) 14.9720 1.17996
\(162\) − 1.43828i − 0.113002i
\(163\) 21.6436i 1.69526i 0.530591 + 0.847628i \(0.321970\pi\)
−0.530591 + 0.847628i \(0.678030\pi\)
\(164\) −0.671544 −0.0524388
\(165\) 0 0
\(166\) 2.33482 0.181217
\(167\) 16.4303i 1.27141i 0.771930 + 0.635707i \(0.219291\pi\)
−0.771930 + 0.635707i \(0.780709\pi\)
\(168\) − 7.61958i − 0.587863i
\(169\) −13.4961 −1.03816
\(170\) 0 0
\(171\) −0.404431 −0.0309276
\(172\) − 0.304309i − 0.0232033i
\(173\) − 4.64939i − 0.353487i −0.984257 0.176743i \(-0.943444\pi\)
0.984257 0.176743i \(-0.0565563\pi\)
\(174\) −1.43828 −0.109036
\(175\) 0 0
\(176\) −11.3358 −0.854466
\(177\) − 9.91822i − 0.745499i
\(178\) − 12.7616i − 0.956523i
\(179\) −7.62334 −0.569795 −0.284897 0.958558i \(-0.591960\pi\)
−0.284897 + 0.958558i \(0.591960\pi\)
\(180\) 0 0
\(181\) −21.3050 −1.58359 −0.791794 0.610788i \(-0.790853\pi\)
−0.791794 + 0.610788i \(0.790853\pi\)
\(182\) − 20.3078i − 1.50532i
\(183\) 13.0816i 0.967019i
\(184\) −15.1620 −1.11776
\(185\) 0 0
\(186\) −2.09738 −0.153787
\(187\) 10.2290i 0.748021i
\(188\) 0.178903i 0.0130479i
\(189\) −2.74301 −0.199525
\(190\) 0 0
\(191\) 3.07597 0.222570 0.111285 0.993789i \(-0.464503\pi\)
0.111285 + 0.993789i \(0.464503\pi\)
\(192\) 7.70683i 0.556193i
\(193\) 6.77454i 0.487642i 0.969820 + 0.243821i \(0.0784009\pi\)
−0.969820 + 0.243821i \(0.921599\pi\)
\(194\) −11.2486 −0.807601
\(195\) 0 0
\(196\) 0.0359842 0.00257030
\(197\) 13.6455i 0.972201i 0.873903 + 0.486100i \(0.161581\pi\)
−0.873903 + 0.486100i \(0.838419\pi\)
\(198\) 3.94523i 0.280375i
\(199\) 6.33858 0.449330 0.224665 0.974436i \(-0.427871\pi\)
0.224665 + 0.974436i \(0.427871\pi\)
\(200\) 0 0
\(201\) −12.4961 −0.881410
\(202\) 7.01929i 0.493876i
\(203\) 2.74301i 0.192522i
\(204\) −0.256037 −0.0179262
\(205\) 0 0
\(206\) −0.424122 −0.0295500
\(207\) 5.45825i 0.379375i
\(208\) 21.2723i 1.47497i
\(209\) 1.10936 0.0767358
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0.441626i 0.0303310i
\(213\) 11.3487i 0.777600i
\(214\) 19.8208 1.35492
\(215\) 0 0
\(216\) 2.77782 0.189006
\(217\) 4.00000i 0.271538i
\(218\) − 8.91917i − 0.604082i
\(219\) −10.7670 −0.727568
\(220\) 0 0
\(221\) 19.1955 1.29123
\(222\) 9.73289i 0.653229i
\(223\) − 2.20126i − 0.147407i −0.997280 0.0737037i \(-0.976518\pi\)
0.997280 0.0737037i \(-0.0234819\pi\)
\(224\) −1.06490 −0.0711515
\(225\) 0 0
\(226\) −15.1965 −1.01086
\(227\) − 12.1853i − 0.808769i −0.914589 0.404384i \(-0.867486\pi\)
0.914589 0.404384i \(-0.132514\pi\)
\(228\) 0.0277677i 0.00183896i
\(229\) 3.16337 0.209041 0.104521 0.994523i \(-0.466669\pi\)
0.104521 + 0.994523i \(0.466669\pi\)
\(230\) 0 0
\(231\) 7.52410 0.495050
\(232\) − 2.77782i − 0.182373i
\(233\) 23.8087i 1.55976i 0.625930 + 0.779879i \(0.284719\pi\)
−0.625930 + 0.779879i \(0.715281\pi\)
\(234\) 7.40348 0.483980
\(235\) 0 0
\(236\) −0.680972 −0.0443275
\(237\) 14.1576i 0.919633i
\(238\) 14.7122i 0.953653i
\(239\) 6.02025 0.389417 0.194709 0.980861i \(-0.437624\pi\)
0.194709 + 0.980861i \(0.437624\pi\)
\(240\) 0 0
\(241\) 24.5517 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(242\) 4.99932i 0.321369i
\(243\) − 1.00000i − 0.0641500i
\(244\) 0.898165 0.0574991
\(245\) 0 0
\(246\) −14.0677 −0.896924
\(247\) − 2.08178i − 0.132461i
\(248\) − 4.05076i − 0.257223i
\(249\) 1.62334 0.102875
\(250\) 0 0
\(251\) −27.5162 −1.73681 −0.868403 0.495858i \(-0.834854\pi\)
−0.868403 + 0.495858i \(0.834854\pi\)
\(252\) 0.188331i 0.0118638i
\(253\) − 14.9720i − 0.941284i
\(254\) −12.2855 −0.770858
\(255\) 0 0
\(256\) 1.64589 0.102868
\(257\) − 15.6436i − 0.975820i −0.872894 0.487910i \(-0.837759\pi\)
0.872894 0.487910i \(-0.162241\pi\)
\(258\) − 6.37476i − 0.396875i
\(259\) 18.5620 1.15339
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) − 19.1364i − 1.18225i
\(263\) − 18.6175i − 1.14801i −0.818853 0.574003i \(-0.805390\pi\)
0.818853 0.574003i \(-0.194610\pi\)
\(264\) −7.61958 −0.468953
\(265\) 0 0
\(266\) 1.59557 0.0978306
\(267\) − 8.87281i − 0.543007i
\(268\) 0.857969i 0.0524088i
\(269\) −10.9006 −0.664620 −0.332310 0.943170i \(-0.607828\pi\)
−0.332310 + 0.943170i \(0.607828\pi\)
\(270\) 0 0
\(271\) 17.7809 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(272\) − 15.4110i − 0.934429i
\(273\) − 14.1195i − 0.854550i
\(274\) 26.2131 1.58359
\(275\) 0 0
\(276\) 0.374756 0.0225577
\(277\) 20.6612i 1.24141i 0.784043 + 0.620706i \(0.213154\pi\)
−0.784043 + 0.620706i \(0.786846\pi\)
\(278\) − 8.14273i − 0.488368i
\(279\) −1.45825 −0.0873033
\(280\) 0 0
\(281\) 28.2652 1.68616 0.843080 0.537787i \(-0.180740\pi\)
0.843080 + 0.537787i \(0.180740\pi\)
\(282\) 3.74772i 0.223174i
\(283\) − 3.21139i − 0.190897i −0.995434 0.0954485i \(-0.969571\pi\)
0.995434 0.0954485i \(-0.0304285\pi\)
\(284\) 0.779187 0.0462362
\(285\) 0 0
\(286\) −20.3078 −1.20083
\(287\) 26.8291i 1.58367i
\(288\) − 0.388222i − 0.0228762i
\(289\) 3.09362 0.181978
\(290\) 0 0
\(291\) −7.82084 −0.458466
\(292\) 0.739249i 0.0432613i
\(293\) − 3.99624i − 0.233463i −0.993164 0.116731i \(-0.962758\pi\)
0.993164 0.116731i \(-0.0372417\pi\)
\(294\) 0.753809 0.0439630
\(295\) 0 0
\(296\) −18.7975 −1.09258
\(297\) 2.74301i 0.159166i
\(298\) − 17.1418i − 0.992996i
\(299\) −28.0960 −1.62484
\(300\) 0 0
\(301\) −12.1576 −0.700750
\(302\) − 10.3429i − 0.595167i
\(303\) 4.88033i 0.280367i
\(304\) −1.67135 −0.0958586
\(305\) 0 0
\(306\) −5.36354 −0.306613
\(307\) 12.0555i 0.688046i 0.938961 + 0.344023i \(0.111790\pi\)
−0.938961 + 0.344023i \(0.888210\pi\)
\(308\) − 0.516595i − 0.0294357i
\(309\) −0.294881 −0.0167752
\(310\) 0 0
\(311\) 15.6873 0.889544 0.444772 0.895644i \(-0.353285\pi\)
0.444772 + 0.895644i \(0.353285\pi\)
\(312\) 14.2986i 0.809501i
\(313\) − 33.8726i − 1.91459i −0.289107 0.957297i \(-0.593359\pi\)
0.289107 0.957297i \(-0.406641\pi\)
\(314\) 6.20558 0.350201
\(315\) 0 0
\(316\) 0.972040 0.0546815
\(317\) − 1.75689i − 0.0986770i −0.998782 0.0493385i \(-0.984289\pi\)
0.998782 0.0493385i \(-0.0157113\pi\)
\(318\) 9.25132i 0.518788i
\(319\) 2.74301 0.153579
\(320\) 0 0
\(321\) 13.7809 0.769175
\(322\) − 21.5340i − 1.20004i
\(323\) 1.50817i 0.0839170i
\(324\) −0.0686587 −0.00381437
\(325\) 0 0
\(326\) 31.1296 1.72411
\(327\) − 6.20126i − 0.342931i
\(328\) − 27.1695i − 1.50019i
\(329\) 7.14744 0.394051
\(330\) 0 0
\(331\) −22.4505 −1.23399 −0.616997 0.786966i \(-0.711651\pi\)
−0.616997 + 0.786966i \(0.711651\pi\)
\(332\) − 0.111456i − 0.00611695i
\(333\) 6.76702i 0.370830i
\(334\) 23.6314 1.29305
\(335\) 0 0
\(336\) −11.3358 −0.618417
\(337\) − 15.3886i − 0.838273i −0.907923 0.419136i \(-0.862333\pi\)
0.907923 0.419136i \(-0.137667\pi\)
\(338\) 19.4113i 1.05583i
\(339\) −10.5658 −0.573853
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0.581686i 0.0314540i
\(343\) 17.7634i 0.959136i
\(344\) 12.3118 0.663809
\(345\) 0 0
\(346\) −6.68714 −0.359503
\(347\) 12.3504i 0.663005i 0.943454 + 0.331503i \(0.107556\pi\)
−0.943454 + 0.331503i \(0.892444\pi\)
\(348\) 0.0686587i 0.00368049i
\(349\) 1.94446 0.104085 0.0520424 0.998645i \(-0.483427\pi\)
0.0520424 + 0.998645i \(0.483427\pi\)
\(350\) 0 0
\(351\) 5.14744 0.274750
\(352\) 1.06490i 0.0567592i
\(353\) 6.72517i 0.357945i 0.983854 + 0.178972i \(0.0572773\pi\)
−0.983854 + 0.178972i \(0.942723\pi\)
\(354\) −14.2652 −0.758187
\(355\) 0 0
\(356\) −0.609195 −0.0322873
\(357\) 10.2290i 0.541378i
\(358\) 10.9645i 0.579493i
\(359\) −15.2114 −0.802826 −0.401413 0.915897i \(-0.631481\pi\)
−0.401413 + 0.915897i \(0.631481\pi\)
\(360\) 0 0
\(361\) −18.8364 −0.991391
\(362\) 30.6426i 1.61054i
\(363\) 3.47590i 0.182437i
\(364\) −0.969425 −0.0508117
\(365\) 0 0
\(366\) 18.8150 0.983477
\(367\) − 13.2189i − 0.690021i −0.938599 0.345011i \(-0.887875\pi\)
0.938599 0.345011i \(-0.112125\pi\)
\(368\) 22.5568i 1.17585i
\(369\) −9.78090 −0.509173
\(370\) 0 0
\(371\) 17.6436 0.916009
\(372\) 0.100122i 0.00519107i
\(373\) 26.8050i 1.38791i 0.720017 + 0.693956i \(0.244134\pi\)
−0.720017 + 0.693956i \(0.755866\pi\)
\(374\) 14.7122 0.760752
\(375\) 0 0
\(376\) −7.23813 −0.373278
\(377\) − 5.14744i − 0.265107i
\(378\) 3.94523i 0.202920i
\(379\) 24.4228 1.25451 0.627257 0.778813i \(-0.284178\pi\)
0.627257 + 0.778813i \(0.284178\pi\)
\(380\) 0 0
\(381\) −8.54175 −0.437607
\(382\) − 4.42412i − 0.226358i
\(383\) 27.2372i 1.39176i 0.718159 + 0.695879i \(0.244985\pi\)
−0.718159 + 0.695879i \(0.755015\pi\)
\(384\) 11.8611 0.605282
\(385\) 0 0
\(386\) 9.74370 0.495942
\(387\) − 4.43220i − 0.225301i
\(388\) 0.536968i 0.0272604i
\(389\) 12.1994 0.618532 0.309266 0.950976i \(-0.399917\pi\)
0.309266 + 0.950976i \(0.399917\pi\)
\(390\) 0 0
\(391\) 20.3545 1.02937
\(392\) 1.45586i 0.0735322i
\(393\) − 13.3050i − 0.671149i
\(394\) 19.6261 0.988748
\(395\) 0 0
\(396\) 0.188331 0.00946401
\(397\) 17.7254i 0.889611i 0.895627 + 0.444805i \(0.146727\pi\)
−0.895627 + 0.444805i \(0.853273\pi\)
\(398\) − 9.11667i − 0.456977i
\(399\) 1.10936 0.0555373
\(400\) 0 0
\(401\) −2.48773 −0.124231 −0.0621157 0.998069i \(-0.519785\pi\)
−0.0621157 + 0.998069i \(0.519785\pi\)
\(402\) 17.9730i 0.896411i
\(403\) − 7.50627i − 0.373914i
\(404\) 0.335077 0.0166707
\(405\) 0 0
\(406\) 3.94523 0.195798
\(407\) − 18.5620i − 0.920084i
\(408\) − 10.3588i − 0.512838i
\(409\) −37.7194 −1.86510 −0.932551 0.361038i \(-0.882423\pi\)
−0.932551 + 0.361038i \(0.882423\pi\)
\(410\) 0 0
\(411\) 18.2253 0.898986
\(412\) 0.0202461i 0 0.000997455i
\(413\) 27.2058i 1.33871i
\(414\) 7.85051 0.385832
\(415\) 0 0
\(416\) 1.99835 0.0979773
\(417\) − 5.66142i − 0.277241i
\(418\) − 1.59557i − 0.0780419i
\(419\) 2.29488 0.112112 0.0560561 0.998428i \(-0.482147\pi\)
0.0560561 + 0.998428i \(0.482147\pi\)
\(420\) 0 0
\(421\) 35.9662 1.75289 0.876443 0.481505i \(-0.159910\pi\)
0.876443 + 0.481505i \(0.159910\pi\)
\(422\) − 2.87657i − 0.140029i
\(423\) 2.60569i 0.126693i
\(424\) −17.8675 −0.867721
\(425\) 0 0
\(426\) 16.3226 0.790835
\(427\) − 35.8829i − 1.73650i
\(428\) − 0.946178i − 0.0457353i
\(429\) −14.1195 −0.681695
\(430\) 0 0
\(431\) 22.6974 1.09330 0.546648 0.837363i \(-0.315904\pi\)
0.546648 + 0.837363i \(0.315904\pi\)
\(432\) − 4.13260i − 0.198830i
\(433\) − 11.4108i − 0.548368i −0.961677 0.274184i \(-0.911592\pi\)
0.961677 0.274184i \(-0.0884077\pi\)
\(434\) 5.75313 0.276159
\(435\) 0 0
\(436\) −0.425770 −0.0203907
\(437\) − 2.20748i − 0.105598i
\(438\) 15.4860i 0.739951i
\(439\) 7.36654 0.351586 0.175793 0.984427i \(-0.443751\pi\)
0.175793 + 0.984427i \(0.443751\pi\)
\(440\) 0 0
\(441\) 0.524103 0.0249573
\(442\) − 27.6085i − 1.31320i
\(443\) 36.5423i 1.73617i 0.496411 + 0.868087i \(0.334651\pi\)
−0.496411 + 0.868087i \(0.665349\pi\)
\(444\) 0.464614 0.0220496
\(445\) 0 0
\(446\) −3.16604 −0.149916
\(447\) − 11.9182i − 0.563712i
\(448\) − 21.1399i − 0.998767i
\(449\) 39.6182 1.86970 0.934850 0.355043i \(-0.115534\pi\)
0.934850 + 0.355043i \(0.115534\pi\)
\(450\) 0 0
\(451\) 26.8291 1.26333
\(452\) 0.725431i 0.0341214i
\(453\) − 7.19114i − 0.337869i
\(454\) −17.5260 −0.822534
\(455\) 0 0
\(456\) −1.12343 −0.0526096
\(457\) − 28.5220i − 1.33420i −0.744967 0.667102i \(-0.767535\pi\)
0.744967 0.667102i \(-0.232465\pi\)
\(458\) − 4.54982i − 0.212599i
\(459\) −3.72913 −0.174061
\(460\) 0 0
\(461\) −22.6696 −1.05583 −0.527915 0.849297i \(-0.677026\pi\)
−0.527915 + 0.849297i \(0.677026\pi\)
\(462\) − 10.8218i − 0.503475i
\(463\) 28.5517i 1.32691i 0.748217 + 0.663455i \(0.230910\pi\)
−0.748217 + 0.663455i \(0.769090\pi\)
\(464\) −4.13260 −0.191851
\(465\) 0 0
\(466\) 34.2436 1.58630
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) − 0.353416i − 0.0163367i
\(469\) 34.2770 1.58277
\(470\) 0 0
\(471\) 4.31457 0.198805
\(472\) − 27.5510i − 1.26814i
\(473\) 12.1576i 0.559005i
\(474\) 20.3626 0.935285
\(475\) 0 0
\(476\) 0.702312 0.0321904
\(477\) 6.43220i 0.294510i
\(478\) − 8.65882i − 0.396045i
\(479\) 4.43410 0.202599 0.101300 0.994856i \(-0.467700\pi\)
0.101300 + 0.994856i \(0.467700\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) − 35.3123i − 1.60843i
\(483\) − 14.9720i − 0.681251i
\(484\) 0.238650 0.0108477
\(485\) 0 0
\(486\) −1.43828 −0.0652419
\(487\) − 14.1035i − 0.639093i −0.947571 0.319546i \(-0.896469\pi\)
0.947571 0.319546i \(-0.103531\pi\)
\(488\) 36.3382i 1.64496i
\(489\) 21.6436 0.978757
\(490\) 0 0
\(491\) 39.6376 1.78882 0.894410 0.447249i \(-0.147596\pi\)
0.894410 + 0.447249i \(0.147596\pi\)
\(492\) 0.671544i 0.0302755i
\(493\) 3.72913i 0.167951i
\(494\) −2.99419 −0.134715
\(495\) 0 0
\(496\) −6.02638 −0.270592
\(497\) − 31.1296i − 1.39635i
\(498\) − 2.33482i − 0.104626i
\(499\) −6.33858 −0.283754 −0.141877 0.989884i \(-0.545314\pi\)
−0.141877 + 0.989884i \(0.545314\pi\)
\(500\) 0 0
\(501\) 16.4303 0.734051
\(502\) 39.5761i 1.76637i
\(503\) − 19.0638i − 0.850011i −0.905191 0.425005i \(-0.860272\pi\)
0.905191 0.425005i \(-0.139728\pi\)
\(504\) −7.61958 −0.339403
\(505\) 0 0
\(506\) −21.5340 −0.957305
\(507\) 13.4961i 0.599385i
\(508\) 0.586465i 0.0260202i
\(509\) 12.8145 0.567992 0.283996 0.958826i \(-0.408340\pi\)
0.283996 + 0.958826i \(0.408340\pi\)
\(510\) 0 0
\(511\) 29.5340 1.30651
\(512\) 21.3549i 0.943760i
\(513\) 0.404431i 0.0178560i
\(514\) −22.4999 −0.992428
\(515\) 0 0
\(516\) −0.304309 −0.0133965
\(517\) − 7.14744i − 0.314344i
\(518\) − 26.6974i − 1.17302i
\(519\) −4.64939 −0.204086
\(520\) 0 0
\(521\) −35.4800 −1.55441 −0.777204 0.629249i \(-0.783363\pi\)
−0.777204 + 0.629249i \(0.783363\pi\)
\(522\) 1.43828i 0.0629519i
\(523\) − 14.9227i − 0.652525i −0.945279 0.326263i \(-0.894211\pi\)
0.945279 0.326263i \(-0.105789\pi\)
\(524\) −0.913504 −0.0399066
\(525\) 0 0
\(526\) −26.7773 −1.16754
\(527\) 5.43801i 0.236883i
\(528\) 11.3358i 0.493326i
\(529\) −6.79252 −0.295327
\(530\) 0 0
\(531\) −9.91822 −0.430414
\(532\) − 0.0761670i − 0.00330226i
\(533\) − 50.3466i − 2.18075i
\(534\) −12.7616 −0.552249
\(535\) 0 0
\(536\) −34.7120 −1.49933
\(537\) 7.62334i 0.328971i
\(538\) 15.6781i 0.675931i
\(539\) −1.43762 −0.0619227
\(540\) 0 0
\(541\) −22.8644 −0.983017 −0.491509 0.870873i \(-0.663554\pi\)
−0.491509 + 0.870873i \(0.663554\pi\)
\(542\) − 25.5740i − 1.09850i
\(543\) 21.3050i 0.914285i
\(544\) −1.44773 −0.0620709
\(545\) 0 0
\(546\) −20.3078 −0.869094
\(547\) 35.3290i 1.51056i 0.655404 + 0.755279i \(0.272498\pi\)
−0.655404 + 0.755279i \(0.727502\pi\)
\(548\) − 1.25132i − 0.0534539i
\(549\) 13.0816 0.558309
\(550\) 0 0
\(551\) 0.404431 0.0172293
\(552\) 15.1620i 0.645338i
\(553\) − 38.8343i − 1.65140i
\(554\) 29.7167 1.26254
\(555\) 0 0
\(556\) −0.388706 −0.0164848
\(557\) − 32.7994i − 1.38976i −0.719127 0.694878i \(-0.755458\pi\)
0.719127 0.694878i \(-0.244542\pi\)
\(558\) 2.09738i 0.0887892i
\(559\) 22.8145 0.964950
\(560\) 0 0
\(561\) 10.2290 0.431870
\(562\) − 40.6534i − 1.71486i
\(563\) 16.8169i 0.708747i 0.935104 + 0.354374i \(0.115306\pi\)
−0.935104 + 0.354374i \(0.884694\pi\)
\(564\) 0.178903 0.00753319
\(565\) 0 0
\(566\) −4.61888 −0.194146
\(567\) 2.74301i 0.115196i
\(568\) 31.5246i 1.32274i
\(569\) −8.88033 −0.372283 −0.186141 0.982523i \(-0.559598\pi\)
−0.186141 + 0.982523i \(0.559598\pi\)
\(570\) 0 0
\(571\) 25.1240 1.05141 0.525703 0.850668i \(-0.323802\pi\)
0.525703 + 0.850668i \(0.323802\pi\)
\(572\) 0.969425i 0.0405337i
\(573\) − 3.07597i − 0.128501i
\(574\) 38.5879 1.61063
\(575\) 0 0
\(576\) 7.70683 0.321118
\(577\) 13.9026i 0.578774i 0.957212 + 0.289387i \(0.0934514\pi\)
−0.957212 + 0.289387i \(0.906549\pi\)
\(578\) − 4.44950i − 0.185075i
\(579\) 6.77454 0.281540
\(580\) 0 0
\(581\) −4.45283 −0.184735
\(582\) 11.2486i 0.466269i
\(583\) − 17.6436i − 0.730723i
\(584\) −29.9088 −1.23763
\(585\) 0 0
\(586\) −5.74772 −0.237436
\(587\) 20.9942i 0.866523i 0.901268 + 0.433262i \(0.142637\pi\)
−0.901268 + 0.433262i \(0.857363\pi\)
\(588\) − 0.0359842i − 0.00148396i
\(589\) 0.589762 0.0243007
\(590\) 0 0
\(591\) 13.6455 0.561300
\(592\) 27.9654i 1.14937i
\(593\) 3.54003i 0.145372i 0.997355 + 0.0726859i \(0.0231571\pi\)
−0.997355 + 0.0726859i \(0.976843\pi\)
\(594\) 3.94523 0.161875
\(595\) 0 0
\(596\) −0.818289 −0.0335184
\(597\) − 6.33858i − 0.259421i
\(598\) 40.4100i 1.65249i
\(599\) 32.4886 1.32745 0.663725 0.747977i \(-0.268975\pi\)
0.663725 + 0.747977i \(0.268975\pi\)
\(600\) 0 0
\(601\) −12.5620 −0.512414 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(602\) 17.4860i 0.712677i
\(603\) 12.4961i 0.508882i
\(604\) −0.493734 −0.0200898
\(605\) 0 0
\(606\) 7.01929 0.285139
\(607\) 0.369141i 0.0149830i 0.999972 + 0.00749149i \(0.00238464\pi\)
−0.999972 + 0.00749149i \(0.997615\pi\)
\(608\) 0.157009i 0.00636756i
\(609\) 2.74301 0.111152
\(610\) 0 0
\(611\) −13.4126 −0.542618
\(612\) 0.256037i 0.0103497i
\(613\) 44.7017i 1.80549i 0.430181 + 0.902743i \(0.358450\pi\)
−0.430181 + 0.902743i \(0.641550\pi\)
\(614\) 17.3393 0.699756
\(615\) 0 0
\(616\) 20.9006 0.842108
\(617\) 5.91014i 0.237933i 0.992898 + 0.118967i \(0.0379582\pi\)
−0.992898 + 0.118967i \(0.962042\pi\)
\(618\) 0.424122i 0.0170607i
\(619\) 44.2187 1.77730 0.888650 0.458586i \(-0.151644\pi\)
0.888650 + 0.458586i \(0.151644\pi\)
\(620\) 0 0
\(621\) 5.45825 0.219032
\(622\) − 22.5628i − 0.904684i
\(623\) 24.3382i 0.975089i
\(624\) 21.2723 0.851575
\(625\) 0 0
\(626\) −48.7184 −1.94718
\(627\) − 1.10936i − 0.0443034i
\(628\) − 0.296233i − 0.0118210i
\(629\) 25.2351 1.00619
\(630\) 0 0
\(631\) −29.5702 −1.17717 −0.588586 0.808435i \(-0.700315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(632\) 39.3271i 1.56435i
\(633\) − 2.00000i − 0.0794929i
\(634\) −2.52691 −0.100356
\(635\) 0 0
\(636\) 0.441626 0.0175116
\(637\) 2.69779i 0.106890i
\(638\) − 3.94523i − 0.156193i
\(639\) 11.3487 0.448948
\(640\) 0 0
\(641\) 26.6335 1.05196 0.525979 0.850497i \(-0.323699\pi\)
0.525979 + 0.850497i \(0.323699\pi\)
\(642\) − 19.8208i − 0.782266i
\(643\) 8.16597i 0.322035i 0.986952 + 0.161017i \(0.0514775\pi\)
−0.986952 + 0.161017i \(0.948523\pi\)
\(644\) −1.02796 −0.0405073
\(645\) 0 0
\(646\) 2.16918 0.0853452
\(647\) − 20.8381i − 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(648\) − 2.77782i − 0.109123i
\(649\) 27.2058 1.06792
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) − 1.48602i − 0.0581970i
\(653\) 13.4267i 0.525428i 0.964874 + 0.262714i \(0.0846176\pi\)
−0.964874 + 0.262714i \(0.915382\pi\)
\(654\) −8.91917 −0.348767
\(655\) 0 0
\(656\) −40.4206 −1.57816
\(657\) 10.7670i 0.420061i
\(658\) − 10.2800i − 0.400758i
\(659\) 42.9744 1.67405 0.837023 0.547167i \(-0.184294\pi\)
0.837023 + 0.547167i \(0.184294\pi\)
\(660\) 0 0
\(661\) −10.0178 −0.389649 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(662\) 32.2902i 1.25500i
\(663\) − 19.1955i − 0.745490i
\(664\) 4.50933 0.174996
\(665\) 0 0
\(666\) 9.73289 0.377142
\(667\) − 5.45825i − 0.211344i
\(668\) − 1.12808i − 0.0436468i
\(669\) −2.20126 −0.0851057
\(670\) 0 0
\(671\) −35.8829 −1.38525
\(672\) 1.06490i 0.0410793i
\(673\) 23.0878i 0.889970i 0.895538 + 0.444985i \(0.146791\pi\)
−0.895538 + 0.444985i \(0.853209\pi\)
\(674\) −22.1332 −0.852540
\(675\) 0 0
\(676\) 0.926627 0.0356395
\(677\) − 4.95555i − 0.190457i −0.995455 0.0952287i \(-0.969642\pi\)
0.995455 0.0952287i \(-0.0303582\pi\)
\(678\) 15.1965i 0.583620i
\(679\) 21.4526 0.823277
\(680\) 0 0
\(681\) −12.1853 −0.466943
\(682\) − 5.75313i − 0.220299i
\(683\) − 36.8010i − 1.40815i −0.710126 0.704075i \(-0.751362\pi\)
0.710126 0.704075i \(-0.248638\pi\)
\(684\) 0.0277677 0.00106172
\(685\) 0 0
\(686\) 25.5489 0.975460
\(687\) − 3.16337i − 0.120690i
\(688\) − 18.3165i − 0.698311i
\(689\) −33.1094 −1.26137
\(690\) 0 0
\(691\) 15.8246 0.601996 0.300998 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(692\) 0.319221i 0.0121350i
\(693\) − 7.52410i − 0.285817i
\(694\) 17.7634 0.674289
\(695\) 0 0
\(696\) −2.77782 −0.105293
\(697\) 36.4742i 1.38156i
\(698\) − 2.79669i − 0.105856i
\(699\) 23.8087 0.900527
\(700\) 0 0
\(701\) −42.6156 −1.60957 −0.804785 0.593567i \(-0.797719\pi\)
−0.804785 + 0.593567i \(0.797719\pi\)
\(702\) − 7.40348i − 0.279426i
\(703\) − 2.73679i − 0.103220i
\(704\) −21.1399 −0.796741
\(705\) 0 0
\(706\) 9.67270 0.364037
\(707\) − 13.3868i − 0.503462i
\(708\) 0.680972i 0.0255925i
\(709\) −35.1795 −1.32119 −0.660597 0.750740i \(-0.729697\pi\)
−0.660597 + 0.750740i \(0.729697\pi\)
\(710\) 0 0
\(711\) 14.1576 0.530950
\(712\) − 24.6470i − 0.923686i
\(713\) − 7.95951i − 0.298086i
\(714\) 14.7122 0.550592
\(715\) 0 0
\(716\) 0.523408 0.0195607
\(717\) − 6.02025i − 0.224830i
\(718\) 21.8783i 0.816490i
\(719\) 29.4563 1.09854 0.549268 0.835646i \(-0.314907\pi\)
0.549268 + 0.835646i \(0.314907\pi\)
\(720\) 0 0
\(721\) 0.808861 0.0301236
\(722\) 27.0921i 1.00826i
\(723\) − 24.5517i − 0.913087i
\(724\) 1.46277 0.0543635
\(725\) 0 0
\(726\) 4.99932 0.185542
\(727\) − 46.3391i − 1.71862i −0.511454 0.859311i \(-0.670893\pi\)
0.511454 0.859311i \(-0.329107\pi\)
\(728\) − 39.2213i − 1.45364i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −16.5282 −0.611319
\(732\) − 0.898165i − 0.0331971i
\(733\) − 3.73344i − 0.137898i −0.997620 0.0689489i \(-0.978035\pi\)
0.997620 0.0689489i \(-0.0219645\pi\)
\(734\) −19.0125 −0.701765
\(735\) 0 0
\(736\) 2.11902 0.0781080
\(737\) − 34.2770i − 1.26261i
\(738\) 14.0677i 0.517839i
\(739\) 6.48021 0.238378 0.119189 0.992872i \(-0.461970\pi\)
0.119189 + 0.992872i \(0.461970\pi\)
\(740\) 0 0
\(741\) −2.08178 −0.0764762
\(742\) − 25.3765i − 0.931600i
\(743\) − 51.0638i − 1.87335i −0.350203 0.936674i \(-0.613888\pi\)
0.350203 0.936674i \(-0.386112\pi\)
\(744\) −4.05076 −0.148508
\(745\) 0 0
\(746\) 38.5533 1.41153
\(747\) − 1.62334i − 0.0593948i
\(748\) − 0.702312i − 0.0256791i
\(749\) −37.8011 −1.38122
\(750\) 0 0
\(751\) 24.6494 0.899469 0.449735 0.893162i \(-0.351519\pi\)
0.449735 + 0.893162i \(0.351519\pi\)
\(752\) 10.7683i 0.392679i
\(753\) 27.5162i 1.00275i
\(754\) −7.40348 −0.269619
\(755\) 0 0
\(756\) 0.188331 0.00684955
\(757\) 38.6833i 1.40597i 0.711205 + 0.702985i \(0.248150\pi\)
−0.711205 + 0.702985i \(0.751850\pi\)
\(758\) − 35.1269i − 1.27587i
\(759\) −14.9720 −0.543451
\(760\) 0 0
\(761\) −3.23725 −0.117350 −0.0586750 0.998277i \(-0.518688\pi\)
−0.0586750 + 0.998277i \(0.518688\pi\)
\(762\) 12.2855i 0.445055i
\(763\) 17.0101i 0.615808i
\(764\) −0.211192 −0.00764067
\(765\) 0 0
\(766\) 39.1749 1.41545
\(767\) − 51.0534i − 1.84343i
\(768\) − 1.64589i − 0.0593909i
\(769\) −46.3166 −1.67022 −0.835111 0.550082i \(-0.814596\pi\)
−0.835111 + 0.550082i \(0.814596\pi\)
\(770\) 0 0
\(771\) −15.6436 −0.563390
\(772\) − 0.465131i − 0.0167404i
\(773\) − 27.0341i − 0.972350i −0.873861 0.486175i \(-0.838392\pi\)
0.873861 0.486175i \(-0.161608\pi\)
\(774\) −6.37476 −0.229136
\(775\) 0 0
\(776\) −21.7248 −0.779877
\(777\) − 18.5620i − 0.665908i
\(778\) − 17.5461i − 0.629059i
\(779\) 3.95569 0.141727
\(780\) 0 0
\(781\) −31.1296 −1.11390
\(782\) − 29.2756i − 1.04689i
\(783\) 1.00000i 0.0357371i
\(784\) 2.16591 0.0773540
\(785\) 0 0
\(786\) −19.1364 −0.682572
\(787\) 39.1870i 1.39687i 0.715675 + 0.698434i \(0.246119\pi\)
−0.715675 + 0.698434i \(0.753881\pi\)
\(788\) − 0.936881i − 0.0333750i
\(789\) −18.6175 −0.662802
\(790\) 0 0
\(791\) 28.9820 1.03048
\(792\) 7.61958i 0.270750i
\(793\) 67.3367i 2.39120i
\(794\) 25.4941 0.904752
\(795\) 0 0
\(796\) −0.435198 −0.0154252
\(797\) 53.2933i 1.88775i 0.330307 + 0.943873i \(0.392848\pi\)
−0.330307 + 0.943873i \(0.607152\pi\)
\(798\) − 1.59557i − 0.0564825i
\(799\) 9.71696 0.343761
\(800\) 0 0
\(801\) −8.87281 −0.313505
\(802\) 3.57807i 0.126346i
\(803\) − 29.5340i − 1.04223i
\(804\) 0.857969 0.0302582
\(805\) 0 0
\(806\) −10.7961 −0.380278
\(807\) 10.9006i 0.383718i
\(808\) 13.5567i 0.476921i
\(809\) −0.613214 −0.0215595 −0.0107797 0.999942i \(-0.503431\pi\)
−0.0107797 + 0.999942i \(0.503431\pi\)
\(810\) 0 0
\(811\) −0.230936 −0.00810927 −0.00405463 0.999992i \(-0.501291\pi\)
−0.00405463 + 0.999992i \(0.501291\pi\)
\(812\) − 0.188331i − 0.00660914i
\(813\) − 17.7809i − 0.623603i
\(814\) −26.6974 −0.935744
\(815\) 0 0
\(816\) −15.4110 −0.539493
\(817\) 1.79252i 0.0627122i
\(818\) 54.2511i 1.89685i
\(819\) −14.1195 −0.493375
\(820\) 0 0
\(821\) 25.7254 0.897821 0.448911 0.893577i \(-0.351812\pi\)
0.448911 + 0.893577i \(0.351812\pi\)
\(822\) − 26.2131i − 0.914287i
\(823\) − 38.2258i − 1.33247i −0.745743 0.666234i \(-0.767905\pi\)
0.745743 0.666234i \(-0.232095\pi\)
\(824\) −0.819125 −0.0285356
\(825\) 0 0
\(826\) 39.1296 1.36149
\(827\) − 24.9199i − 0.866551i −0.901262 0.433275i \(-0.857358\pi\)
0.901262 0.433275i \(-0.142642\pi\)
\(828\) − 0.374756i − 0.0130237i
\(829\) 1.65301 0.0574115 0.0287057 0.999588i \(-0.490861\pi\)
0.0287057 + 0.999588i \(0.490861\pi\)
\(830\) 0 0
\(831\) 20.6612 0.716730
\(832\) 39.6705i 1.37533i
\(833\) − 1.95445i − 0.0677176i
\(834\) −8.14273 −0.281960
\(835\) 0 0
\(836\) −0.0761670 −0.00263429
\(837\) 1.45825i 0.0504046i
\(838\) − 3.30069i − 0.114020i
\(839\) 31.4757 1.08666 0.543331 0.839519i \(-0.317163\pi\)
0.543331 + 0.839519i \(0.317163\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 51.7296i − 1.78272i
\(843\) − 28.2652i − 0.973505i
\(844\) −0.137317 −0.00472666
\(845\) 0 0
\(846\) 3.74772 0.128849
\(847\) − 9.53442i − 0.327607i
\(848\) 26.5817i 0.912820i
\(849\) −3.21139 −0.110214
\(850\) 0 0
\(851\) −36.9361 −1.26615
\(852\) − 0.779187i − 0.0266945i
\(853\) − 30.3753i − 1.04003i −0.854157 0.520015i \(-0.825926\pi\)
0.854157 0.520015i \(-0.174074\pi\)
\(854\) −51.6098 −1.76605
\(855\) 0 0
\(856\) 38.2808 1.30841
\(857\) 6.53423i 0.223205i 0.993753 + 0.111602i \(0.0355983\pi\)
−0.993753 + 0.111602i \(0.964402\pi\)
\(858\) 20.3078i 0.693297i
\(859\) −27.0220 −0.921977 −0.460989 0.887406i \(-0.652505\pi\)
−0.460989 + 0.887406i \(0.652505\pi\)
\(860\) 0 0
\(861\) 26.8291 0.914334
\(862\) − 32.6453i − 1.11190i
\(863\) 31.9587i 1.08789i 0.839122 + 0.543944i \(0.183069\pi\)
−0.839122 + 0.543944i \(0.816931\pi\)
\(864\) −0.388222 −0.0132076
\(865\) 0 0
\(866\) −16.4120 −0.557701
\(867\) − 3.09362i − 0.105065i
\(868\) − 0.274635i − 0.00932171i
\(869\) −38.8343 −1.31736
\(870\) 0 0
\(871\) −64.3232 −2.17951
\(872\) − 17.2260i − 0.583345i
\(873\) 7.82084i 0.264695i
\(874\) −3.17499 −0.107396
\(875\) 0 0
\(876\) 0.739249 0.0249769
\(877\) 50.0679i 1.69067i 0.534235 + 0.845336i \(0.320600\pi\)
−0.534235 + 0.845336i \(0.679400\pi\)
\(878\) − 10.5952i − 0.357570i
\(879\) −3.99624 −0.134790
\(880\) 0 0
\(881\) −13.6817 −0.460947 −0.230474 0.973079i \(-0.574028\pi\)
−0.230474 + 0.973079i \(0.574028\pi\)
\(882\) − 0.753809i − 0.0253821i
\(883\) 29.5693i 0.995087i 0.867439 + 0.497543i \(0.165764\pi\)
−0.867439 + 0.497543i \(0.834236\pi\)
\(884\) −1.31793 −0.0443269
\(885\) 0 0
\(886\) 52.5581 1.76572
\(887\) − 4.35130i − 0.146102i −0.997328 0.0730512i \(-0.976726\pi\)
0.997328 0.0730512i \(-0.0232737\pi\)
\(888\) 18.7975i 0.630804i
\(889\) 23.4301 0.785820
\(890\) 0 0
\(891\) 2.74301 0.0918943
\(892\) 0.151136i 0.00506040i
\(893\) − 1.05382i − 0.0352648i
\(894\) −17.1418 −0.573307
\(895\) 0 0
\(896\) −32.5350 −1.08692
\(897\) 28.0960i 0.938099i
\(898\) − 56.9822i − 1.90152i
\(899\) 1.45825 0.0486354
\(900\) 0 0
\(901\) 23.9865 0.799105
\(902\) − 38.5879i − 1.28484i
\(903\) 12.1576i 0.404578i
\(904\) −29.3497 −0.976157
\(905\) 0 0
\(906\) −10.3429 −0.343620
\(907\) 52.3965i 1.73980i 0.493230 + 0.869899i \(0.335816\pi\)
−0.493230 + 0.869899i \(0.664184\pi\)
\(908\) 0.836629i 0.0277645i
\(909\) 4.88033 0.161870
\(910\) 0 0
\(911\) 7.85085 0.260110 0.130055 0.991507i \(-0.458485\pi\)
0.130055 + 0.991507i \(0.458485\pi\)
\(912\) 1.67135i 0.0553440i
\(913\) 4.45283i 0.147367i
\(914\) −41.0227 −1.35691
\(915\) 0 0
\(916\) −0.217193 −0.00717625
\(917\) 36.4958i 1.20520i
\(918\) 5.36354i 0.177023i
\(919\) −13.4979 −0.445253 −0.222627 0.974904i \(-0.571463\pi\)
−0.222627 + 0.974904i \(0.571463\pi\)
\(920\) 0 0
\(921\) 12.0555 0.397243
\(922\) 32.6054i 1.07380i
\(923\) 58.4168i 1.92281i
\(924\) −0.516595 −0.0169947
\(925\) 0 0
\(926\) 41.0654 1.34949
\(927\) 0.294881i 0.00968516i
\(928\) 0.388222i 0.0127440i
\(929\) −11.6177 −0.381165 −0.190583 0.981671i \(-0.561038\pi\)
−0.190583 + 0.981671i \(0.561038\pi\)
\(930\) 0 0
\(931\) −0.211963 −0.00694682
\(932\) − 1.63467i − 0.0535455i
\(933\) − 15.6873i − 0.513579i
\(934\) 11.5063 0.376497
\(935\) 0 0
\(936\) 14.2986 0.467366
\(937\) 42.6740i 1.39410i 0.717024 + 0.697049i \(0.245504\pi\)
−0.717024 + 0.697049i \(0.754496\pi\)
\(938\) − 49.3001i − 1.60971i
\(939\) −33.8726 −1.10539
\(940\) 0 0
\(941\) −5.91479 −0.192817 −0.0964083 0.995342i \(-0.530735\pi\)
−0.0964083 + 0.995342i \(0.530735\pi\)
\(942\) − 6.20558i − 0.202189i
\(943\) − 53.3866i − 1.73851i
\(944\) −40.9881 −1.33405
\(945\) 0 0
\(946\) 17.4860 0.568520
\(947\) 45.0915i 1.46528i 0.680618 + 0.732639i \(0.261711\pi\)
−0.680618 + 0.732639i \(0.738289\pi\)
\(948\) − 0.972040i − 0.0315704i
\(949\) −55.4226 −1.79909
\(950\) 0 0
\(951\) −1.75689 −0.0569712
\(952\) 28.4144i 0.920915i
\(953\) 30.3166i 0.982053i 0.871145 + 0.491026i \(0.163378\pi\)
−0.871145 + 0.491026i \(0.836622\pi\)
\(954\) 9.25132 0.299523
\(955\) 0 0
\(956\) −0.413342 −0.0133684
\(957\) − 2.74301i − 0.0886689i
\(958\) − 6.37750i − 0.206048i
\(959\) −49.9921 −1.61433
\(960\) 0 0
\(961\) −28.8735 −0.931403
\(962\) 50.0995i 1.61527i
\(963\) − 13.7809i − 0.444083i
\(964\) −1.68569 −0.0542923
\(965\) 0 0
\(966\) −21.5340 −0.692846
\(967\) − 5.99809i − 0.192886i −0.995339 0.0964428i \(-0.969254\pi\)
0.995339 0.0964428i \(-0.0307465\pi\)
\(968\) 9.65540i 0.310336i
\(969\) 1.50817 0.0484495
\(970\) 0 0
\(971\) 28.5140 0.915057 0.457529 0.889195i \(-0.348735\pi\)
0.457529 + 0.889195i \(0.348735\pi\)
\(972\) 0.0686587i 0.00220223i
\(973\) 15.5293i 0.497848i
\(974\) −20.2849 −0.649970
\(975\) 0 0
\(976\) 54.0610 1.73045
\(977\) 5.53813i 0.177180i 0.996068 + 0.0885902i \(0.0282362\pi\)
−0.996068 + 0.0885902i \(0.971764\pi\)
\(978\) − 31.1296i − 0.995415i
\(979\) 24.3382 0.777852
\(980\) 0 0
\(981\) −6.20126 −0.197991
\(982\) − 57.0101i − 1.81926i
\(983\) 7.37086i 0.235094i 0.993067 + 0.117547i \(0.0375030\pi\)
−0.993067 + 0.117547i \(0.962497\pi\)
\(984\) −27.1695 −0.866133
\(985\) 0 0
\(986\) 5.36354 0.170810
\(987\) − 7.14744i − 0.227506i
\(988\) 0.142932i 0.00454729i
\(989\) 24.1921 0.769263
\(990\) 0 0
\(991\) 3.68167 0.116952 0.0584760 0.998289i \(-0.481376\pi\)
0.0584760 + 0.998289i \(0.481376\pi\)
\(992\) 0.566126i 0.0179745i
\(993\) 22.4505i 0.712446i
\(994\) −44.7732 −1.42012
\(995\) 0 0
\(996\) −0.111456 −0.00353162
\(997\) − 36.8747i − 1.16783i −0.811814 0.583916i \(-0.801520\pi\)
0.811814 0.583916i \(-0.198480\pi\)
\(998\) 9.11667i 0.288583i
\(999\) 6.76702 0.214099
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 2175.2.c.n.349.3 8
5.2 odd 4 435.2.a.j.1.4 4
5.3 odd 4 2175.2.a.v.1.1 4
5.4 even 2 inner 2175.2.c.n.349.6 8
15.2 even 4 1305.2.a.r.1.1 4
15.8 even 4 6525.2.a.bi.1.4 4
20.7 even 4 6960.2.a.co.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 5.2 odd 4
1305.2.a.r.1.1 4 15.2 even 4
2175.2.a.v.1.1 4 5.3 odd 4
2175.2.c.n.349.3 8 1.1 even 1 trivial
2175.2.c.n.349.6 8 5.4 even 2 inner
6525.2.a.bi.1.4 4 15.8 even 4
6960.2.a.co.1.4 4 20.7 even 4