Properties

Label 2175.2.a.v.1.2
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(1,2175)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

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: yes
Analytic conductor: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.138564 q^{2} +1.00000 q^{3} -1.98080 q^{4} -0.138564 q^{6} -5.07830 q^{7} +0.551597 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.138564 q^{2} +1.00000 q^{3} -1.98080 q^{4} -0.138564 q^{6} -5.07830 q^{7} +0.551597 q^{8} +1.00000 q^{9} -5.07830 q^{11} -1.98080 q^{12} +3.67096 q^{13} +0.703671 q^{14} +3.88517 q^{16} +2.60617 q^{17} -0.138564 q^{18} -6.74926 q^{19} -5.07830 q^{21} +0.703671 q^{22} +3.21234 q^{23} +0.551597 q^{24} -0.508664 q^{26} +1.00000 q^{27} +10.0591 q^{28} -1.00000 q^{29} -0.787665 q^{31} -1.64154 q^{32} -5.07830 q^{33} -0.361122 q^{34} -1.98080 q^{36} +5.13021 q^{37} +0.935207 q^{38} +3.67096 q^{39} -8.81469 q^{41} +0.703671 q^{42} +2.61968 q^{43} +10.0591 q^{44} -0.445115 q^{46} +1.11670 q^{47} +3.88517 q^{48} +18.7892 q^{49} +2.60617 q^{51} -7.27144 q^{52} -0.619678 q^{53} -0.138564 q^{54} -2.80118 q^{56} -6.74926 q^{57} +0.138564 q^{58} +12.7763 q^{59} +8.90587 q^{61} +0.109142 q^{62} -5.07830 q^{63} -7.54288 q^{64} +0.703671 q^{66} +0.524047 q^{67} -5.16230 q^{68} +3.21234 q^{69} +0.195007 q^{71} +0.551597 q^{72} -1.13021 q^{73} -0.710864 q^{74} +13.3689 q^{76} +25.7892 q^{77} -0.508664 q^{78} +15.3035 q^{79} +1.00000 q^{81} +1.22140 q^{82} +18.1182 q^{83} +10.0591 q^{84} -0.362994 q^{86} -1.00000 q^{87} -2.80118 q^{88} -15.5942 q^{89} -18.6423 q^{91} -6.36299 q^{92} -0.787665 q^{93} -0.154735 q^{94} -1.64154 q^{96} +12.6671 q^{97} -2.60351 q^{98} -5.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} + 8 q^{13} - 3 q^{14} + 11 q^{16} + 10 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{21} - 3 q^{22} + 12 q^{23} + 12 q^{24} - 7 q^{26} + 4 q^{27} + 9 q^{28} - 4 q^{29} - 4 q^{31} + 17 q^{32} - 2 q^{33} - q^{34} + 5 q^{36} + 16 q^{37} + 10 q^{38} + 8 q^{39} - 12 q^{41} - 3 q^{42} - 2 q^{43} + 9 q^{44} - 8 q^{46} + 12 q^{47} + 11 q^{48} + 6 q^{49} + 10 q^{51} + 3 q^{52} + 10 q^{53} + 3 q^{54} - 2 q^{57} - 3 q^{58} + 2 q^{59} - 26 q^{61} - 20 q^{62} - 2 q^{63} + 34 q^{64} - 3 q^{66} - 2 q^{67} - 9 q^{68} + 12 q^{69} - 10 q^{71} + 12 q^{72} + 48 q^{74} + 16 q^{76} + 34 q^{77} - 7 q^{78} + 22 q^{79} + 4 q^{81} - 38 q^{82} + 10 q^{83} + 9 q^{84} - 4 q^{86} - 4 q^{87} - 4 q^{89} - 8 q^{91} - 28 q^{92} - 4 q^{93} + 39 q^{94} + 17 q^{96} + 22 q^{97} - 34 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.138564 −0.0979797 −0.0489899 0.998799i \(-0.515600\pi\)
−0.0489899 + 0.998799i \(0.515600\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.98080 −0.990400
\(5\) 0 0
\(6\) −0.138564 −0.0565686
\(7\) −5.07830 −1.91942 −0.959709 0.280995i \(-0.909336\pi\)
−0.959709 + 0.280995i \(0.909336\pi\)
\(8\) 0.551597 0.195019
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.07830 −1.53117 −0.765583 0.643337i \(-0.777549\pi\)
−0.765583 + 0.643337i \(0.777549\pi\)
\(12\) −1.98080 −0.571808
\(13\) 3.67096 1.01814 0.509071 0.860725i \(-0.329989\pi\)
0.509071 + 0.860725i \(0.329989\pi\)
\(14\) 0.703671 0.188064
\(15\) 0 0
\(16\) 3.88517 0.971292
\(17\) 2.60617 0.632089 0.316044 0.948744i \(-0.397645\pi\)
0.316044 + 0.948744i \(0.397645\pi\)
\(18\) −0.138564 −0.0326599
\(19\) −6.74926 −1.54839 −0.774194 0.632949i \(-0.781844\pi\)
−0.774194 + 0.632949i \(0.781844\pi\)
\(20\) 0 0
\(21\) −5.07830 −1.10818
\(22\) 0.703671 0.150023
\(23\) 3.21234 0.669818 0.334909 0.942250i \(-0.391294\pi\)
0.334909 + 0.942250i \(0.391294\pi\)
\(24\) 0.551597 0.112594
\(25\) 0 0
\(26\) −0.508664 −0.0997572
\(27\) 1.00000 0.192450
\(28\) 10.0591 1.90099
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −0.787665 −0.141469 −0.0707344 0.997495i \(-0.522534\pi\)
−0.0707344 + 0.997495i \(0.522534\pi\)
\(32\) −1.64154 −0.290186
\(33\) −5.07830 −0.884019
\(34\) −0.361122 −0.0619319
\(35\) 0 0
\(36\) −1.98080 −0.330133
\(37\) 5.13021 0.843402 0.421701 0.906735i \(-0.361433\pi\)
0.421701 + 0.906735i \(0.361433\pi\)
\(38\) 0.935207 0.151711
\(39\) 3.67096 0.587824
\(40\) 0 0
\(41\) −8.81469 −1.37662 −0.688311 0.725415i \(-0.741648\pi\)
−0.688311 + 0.725415i \(0.741648\pi\)
\(42\) 0.703671 0.108579
\(43\) 2.61968 0.399497 0.199749 0.979847i \(-0.435987\pi\)
0.199749 + 0.979847i \(0.435987\pi\)
\(44\) 10.0591 1.51647
\(45\) 0 0
\(46\) −0.445115 −0.0656286
\(47\) 1.11670 0.162888 0.0814440 0.996678i \(-0.474047\pi\)
0.0814440 + 0.996678i \(0.474047\pi\)
\(48\) 3.88517 0.560776
\(49\) 18.7892 2.68417
\(50\) 0 0
\(51\) 2.60617 0.364936
\(52\) −7.27144 −1.00837
\(53\) −0.619678 −0.0851194 −0.0425597 0.999094i \(-0.513551\pi\)
−0.0425597 + 0.999094i \(0.513551\pi\)
\(54\) −0.138564 −0.0188562
\(55\) 0 0
\(56\) −2.80118 −0.374323
\(57\) −6.74926 −0.893962
\(58\) 0.138564 0.0181944
\(59\) 12.7763 1.66333 0.831665 0.555277i \(-0.187388\pi\)
0.831665 + 0.555277i \(0.187388\pi\)
\(60\) 0 0
\(61\) 8.90587 1.14028 0.570140 0.821548i \(-0.306889\pi\)
0.570140 + 0.821548i \(0.306889\pi\)
\(62\) 0.109142 0.0138611
\(63\) −5.07830 −0.639806
\(64\) −7.54288 −0.942860
\(65\) 0 0
\(66\) 0.703671 0.0866160
\(67\) 0.524047 0.0640225 0.0320112 0.999488i \(-0.489809\pi\)
0.0320112 + 0.999488i \(0.489809\pi\)
\(68\) −5.16230 −0.626020
\(69\) 3.21234 0.386720
\(70\) 0 0
\(71\) 0.195007 0.0231431 0.0115716 0.999933i \(-0.496317\pi\)
0.0115716 + 0.999933i \(0.496317\pi\)
\(72\) 0.551597 0.0650063
\(73\) −1.13021 −0.132282 −0.0661408 0.997810i \(-0.521069\pi\)
−0.0661408 + 0.997810i \(0.521069\pi\)
\(74\) −0.710864 −0.0826363
\(75\) 0 0
\(76\) 13.3689 1.53352
\(77\) 25.7892 2.93895
\(78\) −0.508664 −0.0575949
\(79\) 15.3035 1.72178 0.860890 0.508791i \(-0.169907\pi\)
0.860890 + 0.508791i \(0.169907\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.22140 0.134881
\(83\) 18.1182 1.98873 0.994366 0.106003i \(-0.0338053\pi\)
0.994366 + 0.106003i \(0.0338053\pi\)
\(84\) 10.0591 1.09754
\(85\) 0 0
\(86\) −0.362994 −0.0391426
\(87\) −1.00000 −0.107211
\(88\) −2.80118 −0.298606
\(89\) −15.5942 −1.65298 −0.826489 0.562953i \(-0.809665\pi\)
−0.826489 + 0.562953i \(0.809665\pi\)
\(90\) 0 0
\(91\) −18.6423 −1.95424
\(92\) −6.36299 −0.663388
\(93\) −0.787665 −0.0816770
\(94\) −0.154735 −0.0159597
\(95\) 0 0
\(96\) −1.64154 −0.167539
\(97\) 12.6671 1.28615 0.643077 0.765802i \(-0.277658\pi\)
0.643077 + 0.765802i \(0.277658\pi\)
\(98\) −2.60351 −0.262994
\(99\) −5.07830 −0.510389
\(100\) 0 0
\(101\) 7.03990 0.700497 0.350248 0.936657i \(-0.386097\pi\)
0.350248 + 0.936657i \(0.386097\pi\)
\(102\) −0.361122 −0.0357564
\(103\) −2.65808 −0.261908 −0.130954 0.991388i \(-0.541804\pi\)
−0.130954 + 0.991388i \(0.541804\pi\)
\(104\) 2.02489 0.198557
\(105\) 0 0
\(106\) 0.0858653 0.00833997
\(107\) −4.81469 −0.465453 −0.232727 0.972542i \(-0.574765\pi\)
−0.232727 + 0.972542i \(0.574765\pi\)
\(108\) −1.98080 −0.190603
\(109\) 3.86597 0.370293 0.185146 0.982711i \(-0.440724\pi\)
0.185146 + 0.982711i \(0.440724\pi\)
\(110\) 0 0
\(111\) 5.13021 0.486938
\(112\) −19.7301 −1.86432
\(113\) 8.73575 0.821791 0.410895 0.911683i \(-0.365216\pi\)
0.410895 + 0.911683i \(0.365216\pi\)
\(114\) 0.935207 0.0875901
\(115\) 0 0
\(116\) 1.98080 0.183913
\(117\) 3.67096 0.339380
\(118\) −1.77034 −0.162973
\(119\) −13.2349 −1.21324
\(120\) 0 0
\(121\) 14.7892 1.34447
\(122\) −1.23404 −0.111724
\(123\) −8.81469 −0.794793
\(124\) 1.56021 0.140111
\(125\) 0 0
\(126\) 0.703671 0.0626880
\(127\) −10.7877 −0.957250 −0.478625 0.878019i \(-0.658865\pi\)
−0.478625 + 0.878019i \(0.658865\pi\)
\(128\) 4.32825 0.382567
\(129\) 2.61968 0.230650
\(130\) 0 0
\(131\) 12.9745 1.13359 0.566793 0.823860i \(-0.308184\pi\)
0.566793 + 0.823860i \(0.308184\pi\)
\(132\) 10.0591 0.875533
\(133\) 34.2748 2.97200
\(134\) −0.0726141 −0.00627291
\(135\) 0 0
\(136\) 1.43755 0.123269
\(137\) 4.08212 0.348759 0.174380 0.984679i \(-0.444208\pi\)
0.174380 + 0.984679i \(0.444208\pi\)
\(138\) −0.445115 −0.0378907
\(139\) −19.8276 −1.68175 −0.840876 0.541228i \(-0.817960\pi\)
−0.840876 + 0.541228i \(0.817960\pi\)
\(140\) 0 0
\(141\) 1.11670 0.0940434
\(142\) −0.0270211 −0.00226756
\(143\) −18.6423 −1.55894
\(144\) 3.88517 0.323764
\(145\) 0 0
\(146\) 0.156607 0.0129609
\(147\) 18.7892 1.54970
\(148\) −10.1619 −0.835305
\(149\) 10.7763 0.882828 0.441414 0.897304i \(-0.354477\pi\)
0.441414 + 0.897304i \(0.354477\pi\)
\(150\) 0 0
\(151\) −5.49853 −0.447464 −0.223732 0.974651i \(-0.571824\pi\)
−0.223732 + 0.974651i \(0.571824\pi\)
\(152\) −3.72287 −0.301965
\(153\) 2.60617 0.210696
\(154\) −3.57346 −0.287957
\(155\) 0 0
\(156\) −7.27144 −0.582181
\(157\) −5.77566 −0.460948 −0.230474 0.973079i \(-0.574028\pi\)
−0.230474 + 0.973079i \(0.574028\pi\)
\(158\) −2.12052 −0.168700
\(159\) −0.619678 −0.0491437
\(160\) 0 0
\(161\) −16.3132 −1.28566
\(162\) −0.138564 −0.0108866
\(163\) −7.14691 −0.559790 −0.279895 0.960031i \(-0.590300\pi\)
−0.279895 + 0.960031i \(0.590300\pi\)
\(164\) 17.4601 1.36341
\(165\) 0 0
\(166\) −2.51054 −0.194855
\(167\) −17.1009 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(168\) −2.80118 −0.216115
\(169\) 0.475953 0.0366118
\(170\) 0 0
\(171\) −6.74926 −0.516129
\(172\) −5.18906 −0.395662
\(173\) −10.2862 −0.782045 −0.391022 0.920381i \(-0.627879\pi\)
−0.391022 + 0.920381i \(0.627879\pi\)
\(174\) 0.138564 0.0105045
\(175\) 0 0
\(176\) −19.7301 −1.48721
\(177\) 12.7763 0.960324
\(178\) 2.16079 0.161958
\(179\) −12.1182 −0.905757 −0.452879 0.891572i \(-0.649603\pi\)
−0.452879 + 0.891572i \(0.649603\pi\)
\(180\) 0 0
\(181\) −20.9745 −1.55902 −0.779510 0.626389i \(-0.784532\pi\)
−0.779510 + 0.626389i \(0.784532\pi\)
\(182\) 2.58315 0.191476
\(183\) 8.90587 0.658341
\(184\) 1.77191 0.130627
\(185\) 0 0
\(186\) 0.109142 0.00800269
\(187\) −13.2349 −0.967833
\(188\) −2.21197 −0.161324
\(189\) −5.07830 −0.369392
\(190\) 0 0
\(191\) 26.2094 1.89645 0.948223 0.317607i \(-0.102879\pi\)
0.948223 + 0.317607i \(0.102879\pi\)
\(192\) −7.54288 −0.544360
\(193\) 23.7643 1.71059 0.855295 0.518141i \(-0.173376\pi\)
0.855295 + 0.518141i \(0.173376\pi\)
\(194\) −1.75521 −0.126017
\(195\) 0 0
\(196\) −37.2176 −2.65840
\(197\) 25.6281 1.82593 0.912964 0.408041i \(-0.133788\pi\)
0.912964 + 0.408041i \(0.133788\pi\)
\(198\) 0.703671 0.0500077
\(199\) 7.82757 0.554882 0.277441 0.960743i \(-0.410514\pi\)
0.277441 + 0.960743i \(0.410514\pi\)
\(200\) 0 0
\(201\) 0.524047 0.0369634
\(202\) −0.975479 −0.0686345
\(203\) 5.07830 0.356427
\(204\) −5.16230 −0.361433
\(205\) 0 0
\(206\) 0.368315 0.0256617
\(207\) 3.21234 0.223273
\(208\) 14.2623 0.988913
\(209\) 34.2748 2.37084
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 1.22746 0.0843022
\(213\) 0.195007 0.0133617
\(214\) 0.667143 0.0456050
\(215\) 0 0
\(216\) 0.551597 0.0375314
\(217\) 4.00000 0.271538
\(218\) −0.535685 −0.0362812
\(219\) −1.13021 −0.0763728
\(220\) 0 0
\(221\) 9.56714 0.643555
\(222\) −0.710864 −0.0477101
\(223\) −7.86597 −0.526744 −0.263372 0.964694i \(-0.584835\pi\)
−0.263372 + 0.964694i \(0.584835\pi\)
\(224\) 8.33623 0.556988
\(225\) 0 0
\(226\) −1.21046 −0.0805188
\(227\) 0.0654212 0.00434216 0.00217108 0.999998i \(-0.499309\pi\)
0.00217108 + 0.999998i \(0.499309\pi\)
\(228\) 13.3689 0.885380
\(229\) −3.87041 −0.255764 −0.127882 0.991789i \(-0.540818\pi\)
−0.127882 + 0.991789i \(0.540818\pi\)
\(230\) 0 0
\(231\) 25.7892 1.69680
\(232\) −0.551597 −0.0362141
\(233\) 8.18363 0.536127 0.268064 0.963401i \(-0.413616\pi\)
0.268064 + 0.963401i \(0.413616\pi\)
\(234\) −0.508664 −0.0332524
\(235\) 0 0
\(236\) −25.3073 −1.64736
\(237\) 15.3035 0.994071
\(238\) 1.83389 0.118873
\(239\) −11.2651 −0.728680 −0.364340 0.931266i \(-0.618705\pi\)
−0.364340 + 0.931266i \(0.618705\pi\)
\(240\) 0 0
\(241\) −15.2619 −0.983107 −0.491554 0.870847i \(-0.663571\pi\)
−0.491554 + 0.870847i \(0.663571\pi\)
\(242\) −2.04925 −0.131731
\(243\) 1.00000 0.0641500
\(244\) −17.6408 −1.12933
\(245\) 0 0
\(246\) 1.22140 0.0778736
\(247\) −24.7763 −1.57648
\(248\) −0.434473 −0.0275891
\(249\) 18.1182 1.14819
\(250\) 0 0
\(251\) 24.9411 1.57427 0.787134 0.616783i \(-0.211564\pi\)
0.787134 + 0.616783i \(0.211564\pi\)
\(252\) 10.0591 0.633664
\(253\) −16.3132 −1.02560
\(254\) 1.49478 0.0937911
\(255\) 0 0
\(256\) 14.4860 0.905376
\(257\) −1.14691 −0.0715425 −0.0357713 0.999360i \(-0.511389\pi\)
−0.0357713 + 0.999360i \(0.511389\pi\)
\(258\) −0.362994 −0.0225990
\(259\) −26.0528 −1.61884
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −1.79780 −0.111068
\(263\) −0.685099 −0.0422450 −0.0211225 0.999777i \(-0.506724\pi\)
−0.0211225 + 0.999777i \(0.506724\pi\)
\(264\) −2.80118 −0.172400
\(265\) 0 0
\(266\) −4.74926 −0.291196
\(267\) −15.5942 −0.954347
\(268\) −1.03803 −0.0634079
\(269\) 4.22522 0.257616 0.128808 0.991670i \(-0.458885\pi\)
0.128808 + 0.991670i \(0.458885\pi\)
\(270\) 0 0
\(271\) −0.814686 −0.0494886 −0.0247443 0.999694i \(-0.507877\pi\)
−0.0247443 + 0.999694i \(0.507877\pi\)
\(272\) 10.1254 0.613943
\(273\) −18.6423 −1.12828
\(274\) −0.565636 −0.0341713
\(275\) 0 0
\(276\) −6.36299 −0.383007
\(277\) −9.85459 −0.592105 −0.296052 0.955172i \(-0.595670\pi\)
−0.296052 + 0.955172i \(0.595670\pi\)
\(278\) 2.74739 0.164778
\(279\) −0.787665 −0.0471562
\(280\) 0 0
\(281\) 12.2297 0.729561 0.364780 0.931094i \(-0.381144\pi\)
0.364780 + 0.931094i \(0.381144\pi\)
\(282\) −0.154735 −0.00921435
\(283\) −4.23341 −0.251650 −0.125825 0.992052i \(-0.540158\pi\)
−0.125825 + 0.992052i \(0.540158\pi\)
\(284\) −0.386271 −0.0229209
\(285\) 0 0
\(286\) 2.58315 0.152745
\(287\) 44.7637 2.64231
\(288\) −1.64154 −0.0967286
\(289\) −10.2079 −0.600464
\(290\) 0 0
\(291\) 12.6671 0.742561
\(292\) 2.23873 0.131012
\(293\) 13.3170 0.777989 0.388995 0.921240i \(-0.372822\pi\)
0.388995 + 0.921240i \(0.372822\pi\)
\(294\) −2.60351 −0.151840
\(295\) 0 0
\(296\) 2.82981 0.164479
\(297\) −5.07830 −0.294673
\(298\) −1.49321 −0.0864992
\(299\) 11.7924 0.681970
\(300\) 0 0
\(301\) −13.3035 −0.766802
\(302\) 0.761900 0.0438424
\(303\) 7.03990 0.404432
\(304\) −26.2220 −1.50394
\(305\) 0 0
\(306\) −0.361122 −0.0206440
\(307\) −14.7379 −0.841136 −0.420568 0.907261i \(-0.638169\pi\)
−0.420568 + 0.907261i \(0.638169\pi\)
\(308\) −51.0832 −2.91073
\(309\) −2.65808 −0.151213
\(310\) 0 0
\(311\) −10.0226 −0.568328 −0.284164 0.958776i \(-0.591716\pi\)
−0.284164 + 0.958776i \(0.591716\pi\)
\(312\) 2.02489 0.114637
\(313\) −4.08800 −0.231067 −0.115534 0.993304i \(-0.536858\pi\)
−0.115534 + 0.993304i \(0.536858\pi\)
\(314\) 0.800300 0.0451635
\(315\) 0 0
\(316\) −30.3132 −1.70525
\(317\) 12.7628 0.716829 0.358414 0.933563i \(-0.383317\pi\)
0.358414 + 0.933563i \(0.383317\pi\)
\(318\) 0.0858653 0.00481508
\(319\) 5.07830 0.284330
\(320\) 0 0
\(321\) −4.81469 −0.268730
\(322\) 2.26043 0.125969
\(323\) −17.5897 −0.978718
\(324\) −1.98080 −0.110044
\(325\) 0 0
\(326\) 0.990307 0.0548480
\(327\) 3.86597 0.213789
\(328\) −4.86215 −0.268467
\(329\) −5.67096 −0.312650
\(330\) 0 0
\(331\) 5.83576 0.320762 0.160381 0.987055i \(-0.448728\pi\)
0.160381 + 0.987055i \(0.448728\pi\)
\(332\) −35.8885 −1.96964
\(333\) 5.13021 0.281134
\(334\) 2.36957 0.129657
\(335\) 0 0
\(336\) −19.7301 −1.07636
\(337\) −1.95253 −0.106361 −0.0531807 0.998585i \(-0.516936\pi\)
−0.0531807 + 0.998585i \(0.516936\pi\)
\(338\) −0.0659501 −0.00358721
\(339\) 8.73575 0.474461
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0.935207 0.0505702
\(343\) −59.8690 −3.23262
\(344\) 1.44501 0.0779095
\(345\) 0 0
\(346\) 1.42530 0.0766245
\(347\) −17.3960 −0.933864 −0.466932 0.884293i \(-0.654641\pi\)
−0.466932 + 0.884293i \(0.654641\pi\)
\(348\) 1.98080 0.106182
\(349\) −28.7379 −1.53830 −0.769152 0.639066i \(-0.779321\pi\)
−0.769152 + 0.639066i \(0.779321\pi\)
\(350\) 0 0
\(351\) 3.67096 0.195941
\(352\) 8.33623 0.444323
\(353\) 29.7590 1.58391 0.791955 0.610580i \(-0.209064\pi\)
0.791955 + 0.610580i \(0.209064\pi\)
\(354\) −1.77034 −0.0940923
\(355\) 0 0
\(356\) 30.8889 1.63711
\(357\) −13.2349 −0.700466
\(358\) 1.67915 0.0887459
\(359\) 7.76659 0.409905 0.204953 0.978772i \(-0.434296\pi\)
0.204953 + 0.978772i \(0.434296\pi\)
\(360\) 0 0
\(361\) 26.5526 1.39750
\(362\) 2.90631 0.152752
\(363\) 14.7892 0.776230
\(364\) 36.9266 1.93548
\(365\) 0 0
\(366\) −1.23404 −0.0645041
\(367\) 12.8675 0.671677 0.335838 0.941920i \(-0.390980\pi\)
0.335838 + 0.941920i \(0.390980\pi\)
\(368\) 12.4805 0.650589
\(369\) −8.81469 −0.458874
\(370\) 0 0
\(371\) 3.14691 0.163380
\(372\) 1.56021 0.0808929
\(373\) 13.4639 0.697133 0.348566 0.937284i \(-0.386669\pi\)
0.348566 + 0.937284i \(0.386669\pi\)
\(374\) 1.83389 0.0948280
\(375\) 0 0
\(376\) 0.615970 0.0317662
\(377\) −3.67096 −0.189064
\(378\) 0.703671 0.0361930
\(379\) −9.53318 −0.489687 −0.244843 0.969563i \(-0.578737\pi\)
−0.244843 + 0.969563i \(0.578737\pi\)
\(380\) 0 0
\(381\) −10.7877 −0.552669
\(382\) −3.63169 −0.185813
\(383\) 20.0836 1.02622 0.513111 0.858322i \(-0.328493\pi\)
0.513111 + 0.858322i \(0.328493\pi\)
\(384\) 4.32825 0.220875
\(385\) 0 0
\(386\) −3.29288 −0.167603
\(387\) 2.61968 0.133166
\(388\) −25.0911 −1.27381
\(389\) 24.3472 1.23445 0.617225 0.786787i \(-0.288257\pi\)
0.617225 + 0.786787i \(0.288257\pi\)
\(390\) 0 0
\(391\) 8.37188 0.423384
\(392\) 10.3640 0.523463
\(393\) 12.9745 0.654476
\(394\) −3.55114 −0.178904
\(395\) 0 0
\(396\) 10.0591 0.505489
\(397\) 25.9232 1.30105 0.650524 0.759486i \(-0.274549\pi\)
0.650524 + 0.759486i \(0.274549\pi\)
\(398\) −1.08462 −0.0543671
\(399\) 34.2748 1.71589
\(400\) 0 0
\(401\) 31.3576 1.56592 0.782961 0.622071i \(-0.213708\pi\)
0.782961 + 0.622071i \(0.213708\pi\)
\(402\) −0.0726141 −0.00362166
\(403\) −2.89149 −0.144035
\(404\) −13.9446 −0.693772
\(405\) 0 0
\(406\) −0.703671 −0.0349226
\(407\) −26.0528 −1.29139
\(408\) 1.43755 0.0711695
\(409\) 1.67415 0.0827814 0.0413907 0.999143i \(-0.486821\pi\)
0.0413907 + 0.999143i \(0.486821\pi\)
\(410\) 0 0
\(411\) 4.08212 0.201356
\(412\) 5.26512 0.259394
\(413\) −64.8819 −3.19263
\(414\) −0.445115 −0.0218762
\(415\) 0 0
\(416\) −6.02602 −0.295450
\(417\) −19.8276 −0.970960
\(418\) −4.74926 −0.232294
\(419\) 0.658078 0.0321492 0.0160746 0.999871i \(-0.494883\pi\)
0.0160746 + 0.999871i \(0.494883\pi\)
\(420\) 0 0
\(421\) 5.11989 0.249528 0.124764 0.992186i \(-0.460183\pi\)
0.124764 + 0.992186i \(0.460183\pi\)
\(422\) −0.277129 −0.0134904
\(423\) 1.11670 0.0542960
\(424\) −0.341812 −0.0165999
\(425\) 0 0
\(426\) −0.0270211 −0.00130917
\(427\) −45.2267 −2.18867
\(428\) 9.53693 0.460985
\(429\) −18.6423 −0.900056
\(430\) 0 0
\(431\) −0.390015 −0.0187864 −0.00939318 0.999956i \(-0.502990\pi\)
−0.00939318 + 0.999956i \(0.502990\pi\)
\(432\) 3.88517 0.186925
\(433\) 29.6989 1.42724 0.713618 0.700535i \(-0.247055\pi\)
0.713618 + 0.700535i \(0.247055\pi\)
\(434\) −0.554257 −0.0266052
\(435\) 0 0
\(436\) −7.65771 −0.366738
\(437\) −21.6809 −1.03714
\(438\) 0.156607 0.00748299
\(439\) −24.4856 −1.16864 −0.584318 0.811525i \(-0.698638\pi\)
−0.584318 + 0.811525i \(0.698638\pi\)
\(440\) 0 0
\(441\) 18.7892 0.894722
\(442\) −1.32566 −0.0630554
\(443\) 11.1091 0.527808 0.263904 0.964549i \(-0.414990\pi\)
0.263904 + 0.964549i \(0.414990\pi\)
\(444\) −10.1619 −0.482264
\(445\) 0 0
\(446\) 1.08994 0.0516103
\(447\) 10.7763 0.509701
\(448\) 38.3050 1.80974
\(449\) −15.1003 −0.712628 −0.356314 0.934366i \(-0.615967\pi\)
−0.356314 + 0.934366i \(0.615967\pi\)
\(450\) 0 0
\(451\) 44.7637 2.10784
\(452\) −17.3038 −0.813901
\(453\) −5.49853 −0.258343
\(454\) −0.00906504 −0.000425443 0
\(455\) 0 0
\(456\) −3.72287 −0.174339
\(457\) 24.3742 1.14018 0.570088 0.821583i \(-0.306909\pi\)
0.570088 + 0.821583i \(0.306909\pi\)
\(458\) 0.536301 0.0250597
\(459\) 2.60617 0.121645
\(460\) 0 0
\(461\) −12.9789 −0.604489 −0.302244 0.953230i \(-0.597736\pi\)
−0.302244 + 0.953230i \(0.597736\pi\)
\(462\) −3.57346 −0.166252
\(463\) 11.2619 0.523386 0.261693 0.965151i \(-0.415719\pi\)
0.261693 + 0.965151i \(0.415719\pi\)
\(464\) −3.88517 −0.180364
\(465\) 0 0
\(466\) −1.13396 −0.0525296
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −7.27144 −0.336122
\(469\) −2.66127 −0.122886
\(470\) 0 0
\(471\) −5.77566 −0.266128
\(472\) 7.04736 0.324381
\(473\) −13.3035 −0.611697
\(474\) −2.12052 −0.0973988
\(475\) 0 0
\(476\) 26.2157 1.20160
\(477\) −0.619678 −0.0283731
\(478\) 1.56094 0.0713959
\(479\) −23.8615 −1.09026 −0.545130 0.838351i \(-0.683520\pi\)
−0.545130 + 0.838351i \(0.683520\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) 2.11476 0.0963246
\(483\) −16.3132 −0.742277
\(484\) −29.2944 −1.33156
\(485\) 0 0
\(486\) −0.138564 −0.00628540
\(487\) 20.8417 0.944428 0.472214 0.881484i \(-0.343455\pi\)
0.472214 + 0.881484i \(0.343455\pi\)
\(488\) 4.91245 0.222376
\(489\) −7.14691 −0.323195
\(490\) 0 0
\(491\) −19.1021 −0.862067 −0.431034 0.902336i \(-0.641851\pi\)
−0.431034 + 0.902336i \(0.641851\pi\)
\(492\) 17.4601 0.787163
\(493\) −2.60617 −0.117376
\(494\) 3.43311 0.154463
\(495\) 0 0
\(496\) −3.06021 −0.137407
\(497\) −0.990307 −0.0444213
\(498\) −2.51054 −0.112500
\(499\) −7.82757 −0.350410 −0.175205 0.984532i \(-0.556059\pi\)
−0.175205 + 0.984532i \(0.556059\pi\)
\(500\) 0 0
\(501\) −17.1009 −0.764011
\(502\) −3.45594 −0.154246
\(503\) −31.5865 −1.40837 −0.704187 0.710015i \(-0.748688\pi\)
−0.704187 + 0.710015i \(0.748688\pi\)
\(504\) −2.80118 −0.124774
\(505\) 0 0
\(506\) 2.26043 0.100488
\(507\) 0.475953 0.0211378
\(508\) 21.3682 0.948061
\(509\) 19.6167 0.869497 0.434748 0.900552i \(-0.356837\pi\)
0.434748 + 0.900552i \(0.356837\pi\)
\(510\) 0 0
\(511\) 5.73957 0.253904
\(512\) −10.6637 −0.471275
\(513\) −6.74926 −0.297987
\(514\) 0.158921 0.00700972
\(515\) 0 0
\(516\) −5.18906 −0.228436
\(517\) −5.67096 −0.249409
\(518\) 3.60999 0.158614
\(519\) −10.2862 −0.451514
\(520\) 0 0
\(521\) 24.4057 1.06923 0.534616 0.845095i \(-0.320456\pi\)
0.534616 + 0.845095i \(0.320456\pi\)
\(522\) 0.138564 0.00606479
\(523\) 39.9715 1.74783 0.873917 0.486076i \(-0.161572\pi\)
0.873917 + 0.486076i \(0.161572\pi\)
\(524\) −25.6999 −1.12270
\(525\) 0 0
\(526\) 0.0949303 0.00413916
\(527\) −2.05279 −0.0894208
\(528\) −19.7301 −0.858641
\(529\) −12.6809 −0.551344
\(530\) 0 0
\(531\) 12.7763 0.554444
\(532\) −67.8916 −2.94347
\(533\) −32.3584 −1.40160
\(534\) 2.16079 0.0935067
\(535\) 0 0
\(536\) 0.289062 0.0124856
\(537\) −12.1182 −0.522939
\(538\) −0.585464 −0.0252412
\(539\) −95.4171 −4.10991
\(540\) 0 0
\(541\) −8.76064 −0.376649 −0.188325 0.982107i \(-0.560306\pi\)
−0.188325 + 0.982107i \(0.560306\pi\)
\(542\) 0.112886 0.00484888
\(543\) −20.9745 −0.900101
\(544\) −4.27813 −0.183423
\(545\) 0 0
\(546\) 2.58315 0.110549
\(547\) −31.3569 −1.34072 −0.670361 0.742035i \(-0.733861\pi\)
−0.670361 + 0.742035i \(0.733861\pi\)
\(548\) −8.08587 −0.345411
\(549\) 8.90587 0.380093
\(550\) 0 0
\(551\) 6.74926 0.287528
\(552\) 1.77191 0.0754176
\(553\) −77.7159 −3.30482
\(554\) 1.36549 0.0580143
\(555\) 0 0
\(556\) 39.2744 1.66561
\(557\) −37.6514 −1.59534 −0.797670 0.603094i \(-0.793934\pi\)
−0.797670 + 0.603094i \(0.793934\pi\)
\(558\) 0.109142 0.00462036
\(559\) 9.61674 0.406745
\(560\) 0 0
\(561\) −13.2349 −0.558778
\(562\) −1.69459 −0.0714821
\(563\) 39.0323 1.64501 0.822507 0.568755i \(-0.192575\pi\)
0.822507 + 0.568755i \(0.192575\pi\)
\(564\) −2.21197 −0.0931406
\(565\) 0 0
\(566\) 0.586599 0.0246566
\(567\) −5.07830 −0.213269
\(568\) 0.107565 0.00451335
\(569\) −3.03990 −0.127439 −0.0637197 0.997968i \(-0.520296\pi\)
−0.0637197 + 0.997968i \(0.520296\pi\)
\(570\) 0 0
\(571\) 40.1056 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(572\) 36.9266 1.54398
\(573\) 26.2094 1.09491
\(574\) −6.20264 −0.258893
\(575\) 0 0
\(576\) −7.54288 −0.314287
\(577\) 16.1091 0.670632 0.335316 0.942106i \(-0.391157\pi\)
0.335316 + 0.942106i \(0.391157\pi\)
\(578\) 1.41445 0.0588333
\(579\) 23.7643 0.987610
\(580\) 0 0
\(581\) −92.0098 −3.81721
\(582\) −1.75521 −0.0727559
\(583\) 3.14691 0.130332
\(584\) −0.623422 −0.0257974
\(585\) 0 0
\(586\) −1.84526 −0.0762272
\(587\) 21.4331 0.884639 0.442320 0.896858i \(-0.354156\pi\)
0.442320 + 0.896858i \(0.354156\pi\)
\(588\) −37.2176 −1.53483
\(589\) 5.31616 0.219048
\(590\) 0 0
\(591\) 25.6281 1.05420
\(592\) 19.9317 0.819190
\(593\) −23.9886 −0.985095 −0.492547 0.870286i \(-0.663934\pi\)
−0.492547 + 0.870286i \(0.663934\pi\)
\(594\) 0.703671 0.0288720
\(595\) 0 0
\(596\) −21.3457 −0.874353
\(597\) 7.82757 0.320361
\(598\) −1.63400 −0.0668192
\(599\) −38.1100 −1.55713 −0.778567 0.627562i \(-0.784053\pi\)
−0.778567 + 0.627562i \(0.784053\pi\)
\(600\) 0 0
\(601\) −20.0528 −0.817970 −0.408985 0.912541i \(-0.634117\pi\)
−0.408985 + 0.912541i \(0.634117\pi\)
\(602\) 1.84339 0.0751311
\(603\) 0.524047 0.0213408
\(604\) 10.8915 0.443168
\(605\) 0 0
\(606\) −0.975479 −0.0396261
\(607\) 38.7523 1.57291 0.786453 0.617650i \(-0.211915\pi\)
0.786453 + 0.617650i \(0.211915\pi\)
\(608\) 11.0792 0.449320
\(609\) 5.07830 0.205783
\(610\) 0 0
\(611\) 4.09938 0.165843
\(612\) −5.16230 −0.208673
\(613\) −24.6757 −0.996640 −0.498320 0.866993i \(-0.666050\pi\)
−0.498320 + 0.866993i \(0.666050\pi\)
\(614\) 2.04214 0.0824142
\(615\) 0 0
\(616\) 14.2252 0.573150
\(617\) −10.5249 −0.423717 −0.211859 0.977300i \(-0.567952\pi\)
−0.211859 + 0.977300i \(0.567952\pi\)
\(618\) 0.368315 0.0148158
\(619\) 26.5496 1.06712 0.533560 0.845762i \(-0.320854\pi\)
0.533560 + 0.845762i \(0.320854\pi\)
\(620\) 0 0
\(621\) 3.21234 0.128907
\(622\) 1.38877 0.0556846
\(623\) 79.1919 3.17276
\(624\) 14.2623 0.570949
\(625\) 0 0
\(626\) 0.566450 0.0226399
\(627\) 34.2748 1.36880
\(628\) 11.4404 0.456523
\(629\) 13.3702 0.533105
\(630\) 0 0
\(631\) −13.2041 −0.525649 −0.262824 0.964844i \(-0.584654\pi\)
−0.262824 + 0.964844i \(0.584654\pi\)
\(632\) 8.44137 0.335780
\(633\) 2.00000 0.0794929
\(634\) −1.76846 −0.0702347
\(635\) 0 0
\(636\) 1.22746 0.0486719
\(637\) 68.9743 2.73286
\(638\) −0.703671 −0.0278586
\(639\) 0.195007 0.00771437
\(640\) 0 0
\(641\) 9.51435 0.375794 0.187897 0.982189i \(-0.439833\pi\)
0.187897 + 0.982189i \(0.439833\pi\)
\(642\) 0.667143 0.0263300
\(643\) −30.1370 −1.18849 −0.594244 0.804285i \(-0.702549\pi\)
−0.594244 + 0.804285i \(0.702549\pi\)
\(644\) 32.3132 1.27332
\(645\) 0 0
\(646\) 2.43731 0.0958945
\(647\) 42.7535 1.68081 0.840407 0.541955i \(-0.182316\pi\)
0.840407 + 0.541955i \(0.182316\pi\)
\(648\) 0.551597 0.0216688
\(649\) −64.8819 −2.54684
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 14.1566 0.554416
\(653\) −33.8983 −1.32654 −0.663272 0.748379i \(-0.730833\pi\)
−0.663272 + 0.748379i \(0.730833\pi\)
\(654\) −0.535685 −0.0209469
\(655\) 0 0
\(656\) −34.2465 −1.33710
\(657\) −1.13021 −0.0440939
\(658\) 0.785793 0.0306334
\(659\) 11.7287 0.456887 0.228444 0.973557i \(-0.426636\pi\)
0.228444 + 0.973557i \(0.426636\pi\)
\(660\) 0 0
\(661\) −38.6807 −1.50450 −0.752252 0.658876i \(-0.771032\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) −0.808627 −0.0314282
\(663\) 9.56714 0.371557
\(664\) 9.99394 0.387840
\(665\) 0 0
\(666\) −0.710864 −0.0275454
\(667\) −3.21234 −0.124382
\(668\) 33.8734 1.31060
\(669\) −7.86597 −0.304116
\(670\) 0 0
\(671\) −45.2267 −1.74596
\(672\) 8.33623 0.321577
\(673\) −30.6410 −1.18112 −0.590562 0.806992i \(-0.701094\pi\)
−0.590562 + 0.806992i \(0.701094\pi\)
\(674\) 0.270552 0.0104213
\(675\) 0 0
\(676\) −0.942768 −0.0362603
\(677\) 40.8954 1.57174 0.785868 0.618394i \(-0.212216\pi\)
0.785868 + 0.618394i \(0.212216\pi\)
\(678\) −1.21046 −0.0464876
\(679\) −64.3276 −2.46867
\(680\) 0 0
\(681\) 0.0654212 0.00250694
\(682\) −0.554257 −0.0212236
\(683\) −21.2317 −0.812409 −0.406205 0.913782i \(-0.633148\pi\)
−0.406205 + 0.913782i \(0.633148\pi\)
\(684\) 13.3689 0.511174
\(685\) 0 0
\(686\) 8.29570 0.316731
\(687\) −3.87041 −0.147665
\(688\) 10.1779 0.388028
\(689\) −2.27481 −0.0866635
\(690\) 0 0
\(691\) −13.9842 −0.531983 −0.265992 0.963975i \(-0.585699\pi\)
−0.265992 + 0.963975i \(0.585699\pi\)
\(692\) 20.3749 0.774537
\(693\) 25.7892 0.979649
\(694\) 2.41046 0.0914998
\(695\) 0 0
\(696\) −0.551597 −0.0209082
\(697\) −22.9725 −0.870147
\(698\) 3.98204 0.150723
\(699\) 8.18363 0.309533
\(700\) 0 0
\(701\) 3.16630 0.119590 0.0597948 0.998211i \(-0.480955\pi\)
0.0597948 + 0.998211i \(0.480955\pi\)
\(702\) −0.508664 −0.0191983
\(703\) −34.6252 −1.30591
\(704\) 38.3050 1.44367
\(705\) 0 0
\(706\) −4.12353 −0.155191
\(707\) −35.7508 −1.34455
\(708\) −25.3073 −0.951105
\(709\) 23.3677 0.877592 0.438796 0.898587i \(-0.355405\pi\)
0.438796 + 0.898587i \(0.355405\pi\)
\(710\) 0 0
\(711\) 15.3035 0.573927
\(712\) −8.60169 −0.322362
\(713\) −2.53024 −0.0947583
\(714\) 1.83389 0.0686314
\(715\) 0 0
\(716\) 24.0037 0.897062
\(717\) −11.2651 −0.420704
\(718\) −1.07617 −0.0401624
\(719\) −0.731134 −0.0272667 −0.0136334 0.999907i \(-0.504340\pi\)
−0.0136334 + 0.999907i \(0.504340\pi\)
\(720\) 0 0
\(721\) 13.4985 0.502711
\(722\) −3.67924 −0.136927
\(723\) −15.2619 −0.567597
\(724\) 41.5463 1.54405
\(725\) 0 0
\(726\) −2.04925 −0.0760548
\(727\) 17.7243 0.657358 0.328679 0.944442i \(-0.393397\pi\)
0.328679 + 0.944442i \(0.393397\pi\)
\(728\) −10.2830 −0.381113
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.82732 0.252518
\(732\) −17.6408 −0.652021
\(733\) 5.67184 0.209494 0.104747 0.994499i \(-0.466597\pi\)
0.104747 + 0.994499i \(0.466597\pi\)
\(734\) −1.78297 −0.0658107
\(735\) 0 0
\(736\) −5.27317 −0.194372
\(737\) −2.66127 −0.0980291
\(738\) 1.22140 0.0449604
\(739\) 8.72350 0.320899 0.160450 0.987044i \(-0.448706\pi\)
0.160450 + 0.987044i \(0.448706\pi\)
\(740\) 0 0
\(741\) −24.7763 −0.910180
\(742\) −0.436050 −0.0160079
\(743\) 0.413474 0.0151689 0.00758445 0.999971i \(-0.497586\pi\)
0.00758445 + 0.999971i \(0.497586\pi\)
\(744\) −0.434473 −0.0159286
\(745\) 0 0
\(746\) −1.86561 −0.0683049
\(747\) 18.1182 0.662911
\(748\) 26.2157 0.958541
\(749\) 24.4504 0.893399
\(750\) 0 0
\(751\) 9.71381 0.354462 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(752\) 4.33858 0.158212
\(753\) 24.9411 0.908904
\(754\) 0.508664 0.0185244
\(755\) 0 0
\(756\) 10.0591 0.365846
\(757\) −22.3877 −0.813695 −0.406847 0.913496i \(-0.633372\pi\)
−0.406847 + 0.913496i \(0.633372\pi\)
\(758\) 1.32096 0.0479794
\(759\) −16.3132 −0.592132
\(760\) 0 0
\(761\) 44.0836 1.59803 0.799014 0.601313i \(-0.205355\pi\)
0.799014 + 0.601313i \(0.205355\pi\)
\(762\) 1.49478 0.0541503
\(763\) −19.6326 −0.710746
\(764\) −51.9156 −1.87824
\(765\) 0 0
\(766\) −2.78286 −0.100549
\(767\) 46.9012 1.69351
\(768\) 14.4860 0.522719
\(769\) −14.2761 −0.514808 −0.257404 0.966304i \(-0.582867\pi\)
−0.257404 + 0.966304i \(0.582867\pi\)
\(770\) 0 0
\(771\) −1.14691 −0.0413051
\(772\) −47.0723 −1.69417
\(773\) 25.5807 0.920072 0.460036 0.887900i \(-0.347836\pi\)
0.460036 + 0.887900i \(0.347836\pi\)
\(774\) −0.362994 −0.0130475
\(775\) 0 0
\(776\) 6.98715 0.250824
\(777\) −26.0528 −0.934638
\(778\) −3.37365 −0.120951
\(779\) 59.4926 2.13155
\(780\) 0 0
\(781\) −0.990307 −0.0354360
\(782\) −1.16004 −0.0414831
\(783\) −1.00000 −0.0357371
\(784\) 72.9991 2.60711
\(785\) 0 0
\(786\) −1.79780 −0.0641254
\(787\) 8.73362 0.311320 0.155660 0.987811i \(-0.450250\pi\)
0.155660 + 0.987811i \(0.450250\pi\)
\(788\) −50.7642 −1.80840
\(789\) −0.685099 −0.0243902
\(790\) 0 0
\(791\) −44.3628 −1.57736
\(792\) −2.80118 −0.0995354
\(793\) 32.6931 1.16097
\(794\) −3.59203 −0.127476
\(795\) 0 0
\(796\) −15.5048 −0.549555
\(797\) −8.43874 −0.298915 −0.149458 0.988768i \(-0.547753\pi\)
−0.149458 + 0.988768i \(0.547753\pi\)
\(798\) −4.74926 −0.168122
\(799\) 2.91032 0.102960
\(800\) 0 0
\(801\) −15.5942 −0.550993
\(802\) −4.34504 −0.153429
\(803\) 5.73957 0.202545
\(804\) −1.03803 −0.0366085
\(805\) 0 0
\(806\) 0.400657 0.0141125
\(807\) 4.22522 0.148735
\(808\) 3.88319 0.136610
\(809\) −21.7508 −0.764716 −0.382358 0.924014i \(-0.624888\pi\)
−0.382358 + 0.924014i \(0.624888\pi\)
\(810\) 0 0
\(811\) −3.24629 −0.113993 −0.0569963 0.998374i \(-0.518152\pi\)
−0.0569963 + 0.998374i \(0.518152\pi\)
\(812\) −10.0591 −0.353005
\(813\) −0.814686 −0.0285723
\(814\) 3.60999 0.126530
\(815\) 0 0
\(816\) 10.1254 0.354460
\(817\) −17.6809 −0.618576
\(818\) −0.231977 −0.00811090
\(819\) −18.6423 −0.651413
\(820\) 0 0
\(821\) 33.9232 1.18393 0.591964 0.805964i \(-0.298353\pi\)
0.591964 + 0.805964i \(0.298353\pi\)
\(822\) −0.565636 −0.0197288
\(823\) 36.4648 1.27108 0.635542 0.772066i \(-0.280777\pi\)
0.635542 + 0.772066i \(0.280777\pi\)
\(824\) −1.46619 −0.0510770
\(825\) 0 0
\(826\) 8.99031 0.312813
\(827\) 15.9772 0.555583 0.277792 0.960641i \(-0.410398\pi\)
0.277792 + 0.960641i \(0.410398\pi\)
\(828\) −6.36299 −0.221129
\(829\) 5.00595 0.173864 0.0869319 0.996214i \(-0.472294\pi\)
0.0869319 + 0.996214i \(0.472294\pi\)
\(830\) 0 0
\(831\) −9.85459 −0.341852
\(832\) −27.6896 −0.959964
\(833\) 48.9677 1.69663
\(834\) 2.74739 0.0951344
\(835\) 0 0
\(836\) −67.8916 −2.34808
\(837\) −0.787665 −0.0272257
\(838\) −0.0911861 −0.00314997
\(839\) 31.4713 1.08651 0.543255 0.839567i \(-0.317192\pi\)
0.543255 + 0.839567i \(0.317192\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −0.709434 −0.0244487
\(843\) 12.2297 0.421212
\(844\) −3.96160 −0.136364
\(845\) 0 0
\(846\) −0.154735 −0.00531991
\(847\) −75.1039 −2.58060
\(848\) −2.40755 −0.0826758
\(849\) −4.23341 −0.145290
\(850\) 0 0
\(851\) 16.4800 0.564926
\(852\) −0.386271 −0.0132334
\(853\) 36.0197 1.23329 0.616646 0.787241i \(-0.288491\pi\)
0.616646 + 0.787241i \(0.288491\pi\)
\(854\) 6.26681 0.214446
\(855\) 0 0
\(856\) −2.65576 −0.0907722
\(857\) 27.4217 0.936708 0.468354 0.883541i \(-0.344847\pi\)
0.468354 + 0.883541i \(0.344847\pi\)
\(858\) 2.58315 0.0881873
\(859\) 14.0642 0.479863 0.239932 0.970790i \(-0.422875\pi\)
0.239932 + 0.970790i \(0.422875\pi\)
\(860\) 0 0
\(861\) 44.7637 1.52554
\(862\) 0.0540421 0.00184068
\(863\) −19.7540 −0.672433 −0.336216 0.941785i \(-0.609147\pi\)
−0.336216 + 0.941785i \(0.609147\pi\)
\(864\) −1.64154 −0.0558463
\(865\) 0 0
\(866\) −4.11520 −0.139840
\(867\) −10.2079 −0.346678
\(868\) −7.92320 −0.268931
\(869\) −77.7159 −2.63633
\(870\) 0 0
\(871\) 1.92375 0.0651839
\(872\) 2.13246 0.0722140
\(873\) 12.6671 0.428718
\(874\) 3.00420 0.101619
\(875\) 0 0
\(876\) 2.23873 0.0756396
\(877\) −42.2030 −1.42509 −0.712547 0.701624i \(-0.752459\pi\)
−0.712547 + 0.701624i \(0.752459\pi\)
\(878\) 3.39284 0.114503
\(879\) 13.3170 0.449172
\(880\) 0 0
\(881\) −33.0927 −1.11492 −0.557461 0.830203i \(-0.688224\pi\)
−0.557461 + 0.830203i \(0.688224\pi\)
\(882\) −2.60351 −0.0876646
\(883\) 26.2634 0.883835 0.441917 0.897056i \(-0.354298\pi\)
0.441917 + 0.897056i \(0.354298\pi\)
\(884\) −18.9506 −0.637377
\(885\) 0 0
\(886\) −1.53932 −0.0517145
\(887\) −14.0716 −0.472479 −0.236239 0.971695i \(-0.575915\pi\)
−0.236239 + 0.971695i \(0.575915\pi\)
\(888\) 2.82981 0.0949622
\(889\) 54.7830 1.83736
\(890\) 0 0
\(891\) −5.07830 −0.170130
\(892\) 15.5809 0.521687
\(893\) −7.53693 −0.252214
\(894\) −1.49321 −0.0499403
\(895\) 0 0
\(896\) −21.9802 −0.734306
\(897\) 11.7924 0.393735
\(898\) 2.09237 0.0698231
\(899\) 0.787665 0.0262701
\(900\) 0 0
\(901\) −1.61499 −0.0538030
\(902\) −6.20264 −0.206525
\(903\) −13.3035 −0.442713
\(904\) 4.81861 0.160265
\(905\) 0 0
\(906\) 0.761900 0.0253124
\(907\) −11.9810 −0.397822 −0.198911 0.980018i \(-0.563740\pi\)
−0.198911 + 0.980018i \(0.563740\pi\)
\(908\) −0.129586 −0.00430047
\(909\) 7.03990 0.233499
\(910\) 0 0
\(911\) 27.5300 0.912109 0.456055 0.889952i \(-0.349262\pi\)
0.456055 + 0.889952i \(0.349262\pi\)
\(912\) −26.2220 −0.868298
\(913\) −92.0098 −3.04508
\(914\) −3.37739 −0.111714
\(915\) 0 0
\(916\) 7.66652 0.253309
\(917\) −65.8884 −2.17583
\(918\) −0.361122 −0.0119188
\(919\) −17.7250 −0.584694 −0.292347 0.956312i \(-0.594436\pi\)
−0.292347 + 0.956312i \(0.594436\pi\)
\(920\) 0 0
\(921\) −14.7379 −0.485630
\(922\) 1.79842 0.0592277
\(923\) 0.715865 0.0235630
\(924\) −51.0832 −1.68051
\(925\) 0 0
\(926\) −1.56050 −0.0512813
\(927\) −2.65808 −0.0873027
\(928\) 1.64154 0.0538861
\(929\) 36.9971 1.21383 0.606917 0.794765i \(-0.292406\pi\)
0.606917 + 0.794765i \(0.292406\pi\)
\(930\) 0 0
\(931\) −126.813 −4.15613
\(932\) −16.2101 −0.530980
\(933\) −10.0226 −0.328124
\(934\) −1.10851 −0.0362717
\(935\) 0 0
\(936\) 2.02489 0.0661856
\(937\) 36.0446 1.17753 0.588763 0.808306i \(-0.299615\pi\)
0.588763 + 0.808306i \(0.299615\pi\)
\(938\) 0.368757 0.0120403
\(939\) −4.08800 −0.133407
\(940\) 0 0
\(941\) −19.6256 −0.639777 −0.319889 0.947455i \(-0.603645\pi\)
−0.319889 + 0.947455i \(0.603645\pi\)
\(942\) 0.800300 0.0260752
\(943\) −28.3157 −0.922087
\(944\) 49.6380 1.61558
\(945\) 0 0
\(946\) 1.84339 0.0599339
\(947\) −18.9555 −0.615970 −0.307985 0.951391i \(-0.599655\pi\)
−0.307985 + 0.951391i \(0.599655\pi\)
\(948\) −30.3132 −0.984527
\(949\) −4.14897 −0.134681
\(950\) 0 0
\(951\) 12.7628 0.413861
\(952\) −7.30033 −0.236605
\(953\) 30.2761 0.980738 0.490369 0.871515i \(-0.336862\pi\)
0.490369 + 0.871515i \(0.336862\pi\)
\(954\) 0.0858653 0.00277999
\(955\) 0 0
\(956\) 22.3140 0.721685
\(957\) 5.07830 0.164158
\(958\) 3.30635 0.106823
\(959\) −20.7303 −0.669415
\(960\) 0 0
\(961\) −30.3796 −0.979987
\(962\) −2.60956 −0.0841354
\(963\) −4.81469 −0.155151
\(964\) 30.2308 0.973670
\(965\) 0 0
\(966\) 2.26043 0.0727281
\(967\) 20.4812 0.658631 0.329316 0.944220i \(-0.393182\pi\)
0.329316 + 0.944220i \(0.393182\pi\)
\(968\) 8.15766 0.262197
\(969\) −17.5897 −0.565063
\(970\) 0 0
\(971\) 44.1566 1.41705 0.708526 0.705684i \(-0.249360\pi\)
0.708526 + 0.705684i \(0.249360\pi\)
\(972\) −1.98080 −0.0635342
\(973\) 100.690 3.22799
\(974\) −2.88792 −0.0925348
\(975\) 0 0
\(976\) 34.6008 1.10754
\(977\) −0.492580 −0.0157590 −0.00787952 0.999969i \(-0.502508\pi\)
−0.00787952 + 0.999969i \(0.502508\pi\)
\(978\) 0.990307 0.0316665
\(979\) 79.1919 2.53098
\(980\) 0 0
\(981\) 3.86597 0.123431
\(982\) 2.64687 0.0844651
\(983\) −27.5513 −0.878750 −0.439375 0.898304i \(-0.644800\pi\)
−0.439375 + 0.898304i \(0.644800\pi\)
\(984\) −4.86215 −0.155000
\(985\) 0 0
\(986\) 0.361122 0.0115005
\(987\) −5.67096 −0.180509
\(988\) 49.0769 1.56134
\(989\) 8.41529 0.267590
\(990\) 0 0
\(991\) 23.0927 0.733563 0.366782 0.930307i \(-0.380460\pi\)
0.366782 + 0.930307i \(0.380460\pi\)
\(992\) 1.29298 0.0410522
\(993\) 5.83576 0.185192
\(994\) 0.137221 0.00435239
\(995\) 0 0
\(996\) −35.8885 −1.13717
\(997\) −7.79593 −0.246900 −0.123450 0.992351i \(-0.539396\pi\)
−0.123450 + 0.992351i \(0.539396\pi\)
\(998\) 1.08462 0.0343331
\(999\) 5.13021 0.162313
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.a.v.1.2 4
3.2 odd 2 6525.2.a.bi.1.3 4
5.2 odd 4 2175.2.c.n.349.4 8
5.3 odd 4 2175.2.c.n.349.5 8
5.4 even 2 435.2.a.j.1.3 4
15.14 odd 2 1305.2.a.r.1.2 4
20.19 odd 2 6960.2.a.co.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 5.4 even 2
1305.2.a.r.1.2 4 15.14 odd 2
2175.2.a.v.1.2 4 1.1 even 1 trivial
2175.2.c.n.349.4 8 5.2 odd 4
2175.2.c.n.349.5 8 5.3 odd 4
6525.2.a.bi.1.3 4 3.2 odd 2
6960.2.a.co.1.1 4 20.19 odd 2