Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43828i q^{2} +1.00000i q^{3} -0.0686587 q^{4} -1.43828 q^{6} -2.74301i q^{7} +2.77782i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.43828i q^{2} +1.00000i q^{3} -0.0686587 q^{4} -1.43828 q^{6} -2.74301i q^{7} +2.77782i q^{8} -1.00000 q^{9} +2.74301 q^{11} -0.0686587i q^{12} +5.14744i q^{13} +3.94523 q^{14} -4.13260 q^{16} -3.72913i q^{17} -1.43828i q^{18} +0.404431 q^{19} +2.74301 q^{21} +3.94523i q^{22} +5.45825i q^{23} -2.77782 q^{24} -7.40348 q^{26} -1.00000i q^{27} +0.188331i q^{28} +1.00000 q^{29} +1.45825 q^{31} -0.388222i q^{32} +2.74301i q^{33} +5.36354 q^{34} +0.0686587 q^{36} +6.76702i q^{37} +0.581686i q^{38} -5.14744 q^{39} +9.78090 q^{41} +3.94523i q^{42} -4.43220i q^{43} -0.188331 q^{44} -7.85051 q^{46} +2.60569i q^{47} -4.13260i q^{48} -0.524103 q^{49} +3.72913 q^{51} -0.353416i q^{52} +6.43220i q^{53} +1.43828 q^{54} +7.61958 q^{56} +0.404431i q^{57} +1.43828i q^{58} +9.91822 q^{59} -13.0816 q^{61} +2.09738i q^{62} +2.74301i q^{63} -7.70683 q^{64} -3.94523 q^{66} +12.4961i q^{67} +0.256037i q^{68} -5.45825 q^{69} -11.3487 q^{71} -2.77782i q^{72} +10.7670i q^{73} -9.73289 q^{74} -0.0277677 q^{76} -7.52410i q^{77} -7.40348i q^{78} -14.1576 q^{79} +1.00000 q^{81} +14.0677i q^{82} -1.62334i q^{83} -0.188331 q^{84} +6.37476 q^{86} +1.00000i q^{87} +7.61958i q^{88} +8.87281 q^{89} +14.1195 q^{91} -0.374756i q^{92} +1.45825i q^{93} -3.74772 q^{94} +0.388222 q^{96} +7.82084i q^{97} -0.753809i q^{98} -2.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) 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 \(0.169803\pi\)
−0.861056 + 0.508510i \(0.830197\pi\)
\(3\) 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 \(-0.826535\pi\)
0.855150 0.518380i \(-0.173465\pi\)
\(8\) 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 \(0.253037\pi\)
−0.700328 + 0.713822i \(0.746963\pi\)
\(14\) 3.94523 1.05441
\(15\) 0 0
\(16\) −4.13260 −1.03315
\(17\) − 3.72913i − 0.904446i −0.891905 0.452223i \(-0.850631\pi\)
0.891905 0.452223i \(-0.149369\pi\)
\(18\) − 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 \(0.192694\pi\)
−0.822295 + 0.569062i \(0.807306\pi\)
\(24\) −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 \(0.187759\pi\)
−0.831018 + 0.556245i \(0.812241\pi\)
\(38\) 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 \(-0.890266\pi\)
0.941163 0.337952i \(-0.109734\pi\)
\(44\) −0.188331 −0.0283920
\(45\) 0 0
\(46\) −7.85051 −1.15749
\(47\) 2.60569i 0.380079i 0.981776 + 0.190040i \(0.0608617\pi\)
−0.981776 + 0.190040i \(0.939138\pi\)
\(48\) − 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 \(0.145648\pi\)
−0.897131 + 0.441765i \(0.854352\pi\)
\(54\) 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 \(0.276434\pi\)
−0.646017 + 0.763323i \(0.723566\pi\)
\(68\) 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 \(0.216983\pi\)
−0.776520 + 0.630092i \(0.783017\pi\)
\(74\) −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 \(-0.971603\pi\)
0.996023 0.0890922i \(-0.0283966\pi\)
\(84\) −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 \(0.129964\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(98\) − 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 \(0.00462448\pi\)
−0.999894 + 0.0145277i \(0.995376\pi\)
\(104\) −14.2986 −1.40210
\(105\) 0 0
\(106\) −9.25132 −0.898568
\(107\) − 13.7809i − 1.33225i −0.745840 0.666125i \(-0.767952\pi\)
0.745840 0.666125i \(-0.232048\pi\)
\(108\) 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 \(0.165555\pi\)
−0.867767 + 0.496971i \(0.834445\pi\)
\(114\) −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 \(0.123725\pi\)
−0.925405 + 0.378979i \(0.876275\pi\)
\(128\) − 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 \(-0.715958\pi\)
0.627589 0.778545i \(-0.284042\pi\)
\(138\) − 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 \(-0.944922\pi\)
0.985067 0.172170i \(-0.0550779\pi\)
\(158\) − 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 \(-0.678030\pi\)
0.530591 0.847628i \(-0.321970\pi\)
\(164\) −0.671544 −0.0524388
\(165\) 0 0
\(166\) 2.33482 0.181217
\(167\) − 16.4303i − 1.27141i −0.771930 0.635707i \(-0.780709\pi\)
0.771930 0.635707i \(-0.219291\pi\)
\(168\) 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 \(0.0565563\pi\)
−0.984257 + 0.176743i \(0.943444\pi\)
\(174\) −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 \(-0.921599\pi\)
0.969820 0.243821i \(-0.0784009\pi\)
\(194\) −11.2486 −0.807601
\(195\) 0 0
\(196\) 0.0359842 0.00257030
\(197\) − 13.6455i − 0.972201i −0.873903 0.486100i \(-0.838419\pi\)
0.873903 0.486100i \(-0.161581\pi\)
\(198\) − 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 \(0.0234819\pi\)
−0.997280 + 0.0737037i \(0.976518\pi\)
\(224\) −1.06490 −0.0711515
\(225\) 0 0
\(226\) −15.1965 −1.01086
\(227\) 12.1853i 0.808769i 0.914589 + 0.404384i \(0.132514\pi\)
−0.914589 + 0.404384i \(0.867486\pi\)
\(228\) − 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 \(-0.715281\pi\)
0.625930 0.779879i \(-0.284719\pi\)
\(234\) 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 \(0.162241\pi\)
−0.872894 + 0.487910i \(0.837759\pi\)
\(258\) 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 \(0.194610\pi\)
−0.818853 + 0.574003i \(0.805390\pi\)
\(264\) −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 \(-0.786846\pi\)
0.784043 0.620706i \(-0.213154\pi\)
\(278\) 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 \(0.0304285\pi\)
−0.995434 + 0.0954485i \(0.969571\pi\)
\(284\) 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 \(0.0372417\pi\)
−0.993164 + 0.116731i \(0.962758\pi\)
\(294\) 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 \(-0.888210\pi\)
0.938961 0.344023i \(-0.111790\pi\)
\(308\) 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 \(0.406641\pi\)
−0.289107 + 0.957297i \(0.593359\pi\)
\(314\) 6.20558 0.350201
\(315\) 0 0
\(316\) 0.972040 0.0546815
\(317\) 1.75689i 0.0986770i 0.998782 + 0.0493385i \(0.0157113\pi\)
−0.998782 + 0.0493385i \(0.984289\pi\)
\(318\) − 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 \(0.137667\pi\)
−0.907923 + 0.419136i \(0.862333\pi\)
\(338\) − 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 \(-0.892444\pi\)
0.943454 0.331503i \(-0.107556\pi\)
\(348\) − 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 \(-0.942723\pi\)
0.983854 0.178972i \(-0.0572773\pi\)
\(354\) −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 \(0.112125\pi\)
−0.938599 + 0.345011i \(0.887875\pi\)
\(368\) − 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 \(-0.755866\pi\)
0.720017 0.693956i \(-0.244134\pi\)
\(374\) 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 \(-0.755015\pi\)
0.718159 0.695879i \(-0.244985\pi\)
\(384\) 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 \(-0.853273\pi\)
0.895627 0.444805i \(-0.146727\pi\)
\(398\) 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 \(0.0884077\pi\)
−0.961677 + 0.274184i \(0.911592\pi\)
\(434\) 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 \(-0.665349\pi\)
0.496411 0.868087i \(-0.334651\pi\)
\(444\) 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 \(0.232465\pi\)
−0.744967 + 0.667102i \(0.767535\pi\)
\(458\) 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 \(-0.769090\pi\)
0.748217 0.663455i \(-0.230910\pi\)
\(464\) −4.13260 −0.191851
\(465\) 0 0
\(466\) 34.2436 1.58630
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 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 \(0.103531\pi\)
−0.947571 + 0.319546i \(0.896469\pi\)
\(488\) − 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 \(0.139728\pi\)
−0.905191 + 0.425005i \(0.860272\pi\)
\(504\) −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 \(0.105789\pi\)
−0.945279 + 0.326263i \(0.894211\pi\)
\(524\) −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 \(-0.727502\pi\)
0.655404 0.755279i \(-0.272498\pi\)
\(548\) 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 \(0.244542\pi\)
−0.719127 + 0.694878i \(0.755458\pi\)
\(558\) − 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 \(-0.884694\pi\)
0.935104 0.354374i \(-0.115306\pi\)
\(564\) 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 \(-0.906549\pi\)
0.957212 0.289387i \(-0.0934514\pi\)
\(578\) 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 \(-0.857363\pi\)
0.901268 0.433262i \(-0.142637\pi\)
\(588\) 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 \(-0.976843\pi\)
0.997355 0.0726859i \(-0.0231571\pi\)
\(594\) 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 \(-0.997615\pi\)
0.999972 0.00749149i \(-0.00238464\pi\)
\(608\) − 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 \(-0.641550\pi\)
0.430181 0.902743i \(-0.358450\pi\)
\(614\) 17.3393 0.699756
\(615\) 0 0
\(616\) 20.9006 0.842108
\(617\) − 5.91014i − 0.237933i −0.992898 0.118967i \(-0.962042\pi\)
0.992898 0.118967i \(-0.0379582\pi\)
\(618\) − 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 \(-0.948523\pi\)
0.986952 0.161017i \(-0.0514775\pi\)
\(644\) −1.02796 −0.0405073
\(645\) 0 0
\(646\) 2.16918 0.0853452
\(647\) 20.8381i 0.819232i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(648\) 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 \(-0.915382\pi\)
0.964874 0.262714i \(-0.0846176\pi\)
\(654\) −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 \(-0.853209\pi\)
0.895538 0.444985i \(-0.146791\pi\)
\(674\) −22.1332 −0.852540
\(675\) 0 0
\(676\) 0.926627 0.0356395
\(677\) 4.95555i 0.190457i 0.995455 + 0.0952287i \(0.0303582\pi\)
−0.995455 + 0.0952287i \(0.969642\pi\)
\(678\) − 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 \(0.248638\pi\)
−0.710126 + 0.704075i \(0.751362\pi\)
\(684\) 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 \(0.329107\pi\)
−0.511454 + 0.859311i \(0.670893\pi\)
\(728\) 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 \(0.0219645\pi\)
−0.997620 + 0.0689489i \(0.978035\pi\)
\(734\) −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 \(0.386112\pi\)
−0.350203 + 0.936674i \(0.613888\pi\)
\(744\) −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 \(-0.751850\pi\)
0.711205 0.702985i \(-0.248150\pi\)
\(758\) 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 \(0.161608\pi\)
−0.873861 + 0.486175i \(0.838392\pi\)
\(774\) −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 \(-0.753881\pi\)
0.715675 0.698434i \(-0.246119\pi\)
\(788\) 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 \(-0.607152\pi\)
0.330307 0.943873i \(-0.392848\pi\)
\(798\) 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 \(0.232095\pi\)
−0.745743 + 0.666234i \(0.767905\pi\)
\(824\) −0.819125 −0.0285356
\(825\) 0 0
\(826\) 39.1296 1.36149
\(827\) 24.9199i 0.866551i 0.901262 + 0.433275i \(0.142642\pi\)
−0.901262 + 0.433275i \(0.857358\pi\)
\(828\) 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 \(0.174074\pi\)
−0.854157 + 0.520015i \(0.825926\pi\)
\(854\) −51.6098 −1.76605
\(855\) 0 0
\(856\) 38.2808 1.30841
\(857\) − 6.53423i − 0.223205i −0.993753 0.111602i \(-0.964402\pi\)
0.993753 0.111602i \(-0.0355983\pi\)
\(858\) − 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 \(-0.816931\pi\)
0.839122 0.543944i \(-0.183069\pi\)
\(864\) −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 \(-0.679400\pi\)
0.534235 0.845336i \(-0.320600\pi\)
\(878\) 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 \(-0.834236\pi\)
0.867439 0.497543i \(-0.165764\pi\)
\(884\) −1.31793 −0.0443269
\(885\) 0 0
\(886\) 52.5581 1.76572
\(887\) 4.35130i 0.146102i 0.997328 + 0.0730512i \(0.0232737\pi\)
−0.997328 + 0.0730512i \(0.976726\pi\)
\(888\) − 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 \(-0.664184\pi\)
0.493230 0.869899i \(-0.335816\pi\)
\(908\) − 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 \(-0.754496\pi\)
0.717024 0.697049i \(-0.245504\pi\)
\(938\) 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 \(-0.738289\pi\)
0.680618 0.732639i \(-0.261711\pi\)
\(948\) 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 \(-0.836622\pi\)
0.871145 0.491026i \(-0.163378\pi\)
\(954\) 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 \(0.0307465\pi\)
−0.995339 + 0.0964428i \(0.969254\pi\)
\(968\) − 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 \(-0.971764\pi\)
0.996068 0.0885902i \(-0.0282362\pi\)
\(978\) 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 \(-0.962497\pi\)
0.993067 0.117547i \(-0.0375030\pi\)
\(984\) −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 \(0.198480\pi\)
−0.811814 + 0.583916i \(0.801520\pi\)
\(998\) − 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.349.6 8
5.2 odd 4 2175.2.a.v.1.1 4
5.3 odd 4 435.2.a.j.1.4 4
5.4 even 2 inner 2175.2.c.n.349.3 8
15.2 even 4 6525.2.a.bi.1.4 4
15.8 even 4 1305.2.a.r.1.1 4
20.3 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 5.3 odd 4
1305.2.a.r.1.1 4 15.8 even 4
2175.2.a.v.1.1 4 5.2 odd 4
2175.2.c.n.349.3 8 5.4 even 2 inner
2175.2.c.n.349.6 8 1.1 even 1 trivial
6525.2.a.bi.1.4 4 15.2 even 4
6960.2.a.co.1.4 4 20.3 even 4