Properties

Label 2175.2.c.n.349.5
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 349.5
Root \(1.13856i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.n.349.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.138564i q^{2} +1.00000i q^{3} +1.98080 q^{4} -0.138564 q^{6} +5.07830i q^{7} +0.551597i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+0.138564i q^{2} +1.00000i q^{3} +1.98080 q^{4} -0.138564 q^{6} +5.07830i q^{7} +0.551597i q^{8} -1.00000 q^{9} -5.07830 q^{11} +1.98080i q^{12} +3.67096i q^{13} -0.703671 q^{14} +3.88517 q^{16} -2.60617i q^{17} -0.138564i q^{18} +6.74926 q^{19} -5.07830 q^{21} -0.703671i q^{22} +3.21234i q^{23} -0.551597 q^{24} -0.508664 q^{26} -1.00000i q^{27} +10.0591i q^{28} +1.00000 q^{29} -0.787665 q^{31} +1.64154i q^{32} -5.07830i q^{33} +0.361122 q^{34} -1.98080 q^{36} -5.13021i q^{37} +0.935207i q^{38} -3.67096 q^{39} -8.81469 q^{41} -0.703671i q^{42} +2.61968i q^{43} -10.0591 q^{44} -0.445115 q^{46} -1.11670i q^{47} +3.88517i q^{48} -18.7892 q^{49} +2.60617 q^{51} +7.27144i q^{52} -0.619678i q^{53} +0.138564 q^{54} -2.80118 q^{56} +6.74926i q^{57} +0.138564i q^{58} -12.7763 q^{59} +8.90587 q^{61} -0.109142i q^{62} -5.07830i q^{63} +7.54288 q^{64} +0.703671 q^{66} -0.524047i q^{67} -5.16230i q^{68} -3.21234 q^{69} +0.195007 q^{71} -0.551597i q^{72} -1.13021i q^{73} +0.710864 q^{74} +13.3689 q^{76} -25.7892i q^{77} -0.508664i q^{78} -15.3035 q^{79} +1.00000 q^{81} -1.22140i q^{82} +18.1182i q^{83} -10.0591 q^{84} -0.362994 q^{86} +1.00000i q^{87} -2.80118i q^{88} +15.5942 q^{89} -18.6423 q^{91} +6.36299i q^{92} -0.787665i q^{93} +0.154735 q^{94} -1.64154 q^{96} -12.6671i q^{97} -2.60351i q^{98} +5.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.138564i 0.0979797i 0.998799 + 0.0489899i \(0.0156002\pi\)
−0.998799 + 0.0489899i \(0.984400\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.98080 0.990400
\(5\) 0 0
\(6\) −0.138564 −0.0565686
\(7\) 5.07830i 1.91942i 0.280995 + 0.959709i \(0.409336\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(8\) 0.551597i 0.195019i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.07830 −1.53117 −0.765583 0.643337i \(-0.777549\pi\)
−0.765583 + 0.643337i \(0.777549\pi\)
\(12\) 1.98080i 0.571808i
\(13\) 3.67096i 1.01814i 0.860725 + 0.509071i \(0.170011\pi\)
−0.860725 + 0.509071i \(0.829989\pi\)
\(14\) −0.703671 −0.188064
\(15\) 0 0
\(16\) 3.88517 0.971292
\(17\) − 2.60617i − 0.632089i −0.948744 0.316044i \(-0.897645\pi\)
0.948744 0.316044i \(-0.102355\pi\)
\(18\) − 0.138564i − 0.0326599i
\(19\) 6.74926 1.54839 0.774194 0.632949i \(-0.218156\pi\)
0.774194 + 0.632949i \(0.218156\pi\)
\(20\) 0 0
\(21\) −5.07830 −1.10818
\(22\) − 0.703671i − 0.150023i
\(23\) 3.21234i 0.669818i 0.942250 + 0.334909i \(0.108706\pi\)
−0.942250 + 0.334909i \(0.891294\pi\)
\(24\) −0.551597 −0.112594
\(25\) 0 0
\(26\) −0.508664 −0.0997572
\(27\) − 1.00000i − 0.192450i
\(28\) 10.0591i 1.90099i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.787665 −0.141469 −0.0707344 0.997495i \(-0.522534\pi\)
−0.0707344 + 0.997495i \(0.522534\pi\)
\(32\) 1.64154i 0.290186i
\(33\) − 5.07830i − 0.884019i
\(34\) 0.361122 0.0619319
\(35\) 0 0
\(36\) −1.98080 −0.330133
\(37\) − 5.13021i − 0.843402i −0.906735 0.421701i \(-0.861433\pi\)
0.906735 0.421701i \(-0.138567\pi\)
\(38\) 0.935207i 0.151711i
\(39\) −3.67096 −0.587824
\(40\) 0 0
\(41\) −8.81469 −1.37662 −0.688311 0.725415i \(-0.741648\pi\)
−0.688311 + 0.725415i \(0.741648\pi\)
\(42\) − 0.703671i − 0.108579i
\(43\) 2.61968i 0.399497i 0.979847 + 0.199749i \(0.0640125\pi\)
−0.979847 + 0.199749i \(0.935987\pi\)
\(44\) −10.0591 −1.51647
\(45\) 0 0
\(46\) −0.445115 −0.0656286
\(47\) − 1.11670i − 0.162888i −0.996678 0.0814440i \(-0.974047\pi\)
0.996678 0.0814440i \(-0.0259532\pi\)
\(48\) 3.88517i 0.560776i
\(49\) −18.7892 −2.68417
\(50\) 0 0
\(51\) 2.60617 0.364936
\(52\) 7.27144i 1.00837i
\(53\) − 0.619678i − 0.0851194i −0.999094 0.0425597i \(-0.986449\pi\)
0.999094 0.0425597i \(-0.0135513\pi\)
\(54\) 0.138564 0.0188562
\(55\) 0 0
\(56\) −2.80118 −0.374323
\(57\) 6.74926i 0.893962i
\(58\) 0.138564i 0.0181944i
\(59\) −12.7763 −1.66333 −0.831665 0.555277i \(-0.812612\pi\)
−0.831665 + 0.555277i \(0.812612\pi\)
\(60\) 0 0
\(61\) 8.90587 1.14028 0.570140 0.821548i \(-0.306889\pi\)
0.570140 + 0.821548i \(0.306889\pi\)
\(62\) − 0.109142i − 0.0138611i
\(63\) − 5.07830i − 0.639806i
\(64\) 7.54288 0.942860
\(65\) 0 0
\(66\) 0.703671 0.0866160
\(67\) − 0.524047i − 0.0640225i −0.999488 0.0320112i \(-0.989809\pi\)
0.999488 0.0320112i \(-0.0101912\pi\)
\(68\) − 5.16230i − 0.626020i
\(69\) −3.21234 −0.386720
\(70\) 0 0
\(71\) 0.195007 0.0231431 0.0115716 0.999933i \(-0.496317\pi\)
0.0115716 + 0.999933i \(0.496317\pi\)
\(72\) − 0.551597i − 0.0650063i
\(73\) − 1.13021i − 0.132282i −0.997810 0.0661408i \(-0.978931\pi\)
0.997810 0.0661408i \(-0.0210686\pi\)
\(74\) 0.710864 0.0826363
\(75\) 0 0
\(76\) 13.3689 1.53352
\(77\) − 25.7892i − 2.93895i
\(78\) − 0.508664i − 0.0575949i
\(79\) −15.3035 −1.72178 −0.860890 0.508791i \(-0.830093\pi\)
−0.860890 + 0.508791i \(0.830093\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.22140i − 0.134881i
\(83\) 18.1182i 1.98873i 0.106003 + 0.994366i \(0.466195\pi\)
−0.106003 + 0.994366i \(0.533805\pi\)
\(84\) −10.0591 −1.09754
\(85\) 0 0
\(86\) −0.362994 −0.0391426
\(87\) 1.00000i 0.107211i
\(88\) − 2.80118i − 0.298606i
\(89\) 15.5942 1.65298 0.826489 0.562953i \(-0.190335\pi\)
0.826489 + 0.562953i \(0.190335\pi\)
\(90\) 0 0
\(91\) −18.6423 −1.95424
\(92\) 6.36299i 0.663388i
\(93\) − 0.787665i − 0.0816770i
\(94\) 0.154735 0.0159597
\(95\) 0 0
\(96\) −1.64154 −0.167539
\(97\) − 12.6671i − 1.28615i −0.765802 0.643077i \(-0.777658\pi\)
0.765802 0.643077i \(-0.222342\pi\)
\(98\) − 2.60351i − 0.262994i
\(99\) 5.07830 0.510389
\(100\) 0 0
\(101\) 7.03990 0.700497 0.350248 0.936657i \(-0.386097\pi\)
0.350248 + 0.936657i \(0.386097\pi\)
\(102\) 0.361122i 0.0357564i
\(103\) − 2.65808i − 0.261908i −0.991388 0.130954i \(-0.958196\pi\)
0.991388 0.130954i \(-0.0418041\pi\)
\(104\) −2.02489 −0.198557
\(105\) 0 0
\(106\) 0.0858653 0.00833997
\(107\) 4.81469i 0.465453i 0.972542 + 0.232727i \(0.0747647\pi\)
−0.972542 + 0.232727i \(0.925235\pi\)
\(108\) − 1.98080i − 0.190603i
\(109\) −3.86597 −0.370293 −0.185146 0.982711i \(-0.559276\pi\)
−0.185146 + 0.982711i \(0.559276\pi\)
\(110\) 0 0
\(111\) 5.13021 0.486938
\(112\) 19.7301i 1.86432i
\(113\) 8.73575i 0.821791i 0.911683 + 0.410895i \(0.134784\pi\)
−0.911683 + 0.410895i \(0.865216\pi\)
\(114\) −0.935207 −0.0875901
\(115\) 0 0
\(116\) 1.98080 0.183913
\(117\) − 3.67096i − 0.339380i
\(118\) − 1.77034i − 0.162973i
\(119\) 13.2349 1.21324
\(120\) 0 0
\(121\) 14.7892 1.34447
\(122\) 1.23404i 0.111724i
\(123\) − 8.81469i − 0.794793i
\(124\) −1.56021 −0.140111
\(125\) 0 0
\(126\) 0.703671 0.0626880
\(127\) 10.7877i 0.957250i 0.878019 + 0.478625i \(0.158865\pi\)
−0.878019 + 0.478625i \(0.841135\pi\)
\(128\) 4.32825i 0.382567i
\(129\) −2.61968 −0.230650
\(130\) 0 0
\(131\) 12.9745 1.13359 0.566793 0.823860i \(-0.308184\pi\)
0.566793 + 0.823860i \(0.308184\pi\)
\(132\) − 10.0591i − 0.875533i
\(133\) 34.2748i 2.97200i
\(134\) 0.0726141 0.00627291
\(135\) 0 0
\(136\) 1.43755 0.123269
\(137\) − 4.08212i − 0.348759i −0.984679 0.174380i \(-0.944208\pi\)
0.984679 0.174380i \(-0.0557920\pi\)
\(138\) − 0.445115i − 0.0378907i
\(139\) 19.8276 1.68175 0.840876 0.541228i \(-0.182040\pi\)
0.840876 + 0.541228i \(0.182040\pi\)
\(140\) 0 0
\(141\) 1.11670 0.0940434
\(142\) 0.0270211i 0.00226756i
\(143\) − 18.6423i − 1.55894i
\(144\) −3.88517 −0.323764
\(145\) 0 0
\(146\) 0.156607 0.0129609
\(147\) − 18.7892i − 1.54970i
\(148\) − 10.1619i − 0.835305i
\(149\) −10.7763 −0.882828 −0.441414 0.897304i \(-0.645523\pi\)
−0.441414 + 0.897304i \(0.645523\pi\)
\(150\) 0 0
\(151\) −5.49853 −0.447464 −0.223732 0.974651i \(-0.571824\pi\)
−0.223732 + 0.974651i \(0.571824\pi\)
\(152\) 3.72287i 0.301965i
\(153\) 2.60617i 0.210696i
\(154\) 3.57346 0.287957
\(155\) 0 0
\(156\) −7.27144 −0.582181
\(157\) 5.77566i 0.460948i 0.973079 + 0.230474i \(0.0740276\pi\)
−0.973079 + 0.230474i \(0.925972\pi\)
\(158\) − 2.12052i − 0.168700i
\(159\) 0.619678 0.0491437
\(160\) 0 0
\(161\) −16.3132 −1.28566
\(162\) 0.138564i 0.0108866i
\(163\) − 7.14691i − 0.559790i −0.960031 0.279895i \(-0.909700\pi\)
0.960031 0.279895i \(-0.0902996\pi\)
\(164\) −17.4601 −1.36341
\(165\) 0 0
\(166\) −2.51054 −0.194855
\(167\) 17.1009i 1.32331i 0.749810 + 0.661653i \(0.230145\pi\)
−0.749810 + 0.661653i \(0.769855\pi\)
\(168\) − 2.80118i − 0.216115i
\(169\) −0.475953 −0.0366118
\(170\) 0 0
\(171\) −6.74926 −0.516129
\(172\) 5.18906i 0.395662i
\(173\) − 10.2862i − 0.782045i −0.920381 0.391022i \(-0.872121\pi\)
0.920381 0.391022i \(-0.127879\pi\)
\(174\) −0.138564 −0.0105045
\(175\) 0 0
\(176\) −19.7301 −1.48721
\(177\) − 12.7763i − 0.960324i
\(178\) 2.16079i 0.161958i
\(179\) 12.1182 0.905757 0.452879 0.891572i \(-0.350397\pi\)
0.452879 + 0.891572i \(0.350397\pi\)
\(180\) 0 0
\(181\) −20.9745 −1.55902 −0.779510 0.626389i \(-0.784532\pi\)
−0.779510 + 0.626389i \(0.784532\pi\)
\(182\) − 2.58315i − 0.191476i
\(183\) 8.90587i 0.658341i
\(184\) −1.77191 −0.130627
\(185\) 0 0
\(186\) 0.109142 0.00800269
\(187\) 13.2349i 0.967833i
\(188\) − 2.21197i − 0.161324i
\(189\) 5.07830 0.369392
\(190\) 0 0
\(191\) 26.2094 1.89645 0.948223 0.317607i \(-0.102879\pi\)
0.948223 + 0.317607i \(0.102879\pi\)
\(192\) 7.54288i 0.544360i
\(193\) 23.7643i 1.71059i 0.518141 + 0.855295i \(0.326624\pi\)
−0.518141 + 0.855295i \(0.673376\pi\)
\(194\) 1.75521 0.126017
\(195\) 0 0
\(196\) −37.2176 −2.65840
\(197\) − 25.6281i − 1.82593i −0.408041 0.912964i \(-0.633788\pi\)
0.408041 0.912964i \(-0.366212\pi\)
\(198\) 0.703671i 0.0500077i
\(199\) −7.82757 −0.554882 −0.277441 0.960743i \(-0.589486\pi\)
−0.277441 + 0.960743i \(0.589486\pi\)
\(200\) 0 0
\(201\) 0.524047 0.0369634
\(202\) 0.975479i 0.0686345i
\(203\) 5.07830i 0.356427i
\(204\) 5.16230 0.361433
\(205\) 0 0
\(206\) 0.368315 0.0256617
\(207\) − 3.21234i − 0.223273i
\(208\) 14.2623i 0.988913i
\(209\) −34.2748 −2.37084
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) − 1.22746i − 0.0843022i
\(213\) 0.195007i 0.0133617i
\(214\) −0.667143 −0.0456050
\(215\) 0 0
\(216\) 0.551597 0.0375314
\(217\) − 4.00000i − 0.271538i
\(218\) − 0.535685i − 0.0362812i
\(219\) 1.13021 0.0763728
\(220\) 0 0
\(221\) 9.56714 0.643555
\(222\) 0.710864i 0.0477101i
\(223\) − 7.86597i − 0.526744i −0.964694 0.263372i \(-0.915165\pi\)
0.964694 0.263372i \(-0.0848347\pi\)
\(224\) −8.33623 −0.556988
\(225\) 0 0
\(226\) −1.21046 −0.0805188
\(227\) − 0.0654212i − 0.00434216i −0.999998 0.00217108i \(-0.999309\pi\)
0.999998 0.00217108i \(-0.000691076\pi\)
\(228\) 13.3689i 0.885380i
\(229\) 3.87041 0.255764 0.127882 0.991789i \(-0.459182\pi\)
0.127882 + 0.991789i \(0.459182\pi\)
\(230\) 0 0
\(231\) 25.7892 1.69680
\(232\) 0.551597i 0.0362141i
\(233\) 8.18363i 0.536127i 0.963401 + 0.268064i \(0.0863838\pi\)
−0.963401 + 0.268064i \(0.913616\pi\)
\(234\) 0.508664 0.0332524
\(235\) 0 0
\(236\) −25.3073 −1.64736
\(237\) − 15.3035i − 0.994071i
\(238\) 1.83389i 0.118873i
\(239\) 11.2651 0.728680 0.364340 0.931266i \(-0.381295\pi\)
0.364340 + 0.931266i \(0.381295\pi\)
\(240\) 0 0
\(241\) −15.2619 −0.983107 −0.491554 0.870847i \(-0.663571\pi\)
−0.491554 + 0.870847i \(0.663571\pi\)
\(242\) 2.04925i 0.131731i
\(243\) 1.00000i 0.0641500i
\(244\) 17.6408 1.12933
\(245\) 0 0
\(246\) 1.22140 0.0778736
\(247\) 24.7763i 1.57648i
\(248\) − 0.434473i − 0.0275891i
\(249\) −18.1182 −1.14819
\(250\) 0 0
\(251\) 24.9411 1.57427 0.787134 0.616783i \(-0.211564\pi\)
0.787134 + 0.616783i \(0.211564\pi\)
\(252\) − 10.0591i − 0.633664i
\(253\) − 16.3132i − 1.02560i
\(254\) −1.49478 −0.0937911
\(255\) 0 0
\(256\) 14.4860 0.905376
\(257\) 1.14691i 0.0715425i 0.999360 + 0.0357713i \(0.0113888\pi\)
−0.999360 + 0.0357713i \(0.988611\pi\)
\(258\) − 0.362994i − 0.0225990i
\(259\) 26.0528 1.61884
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 1.79780i 0.111068i
\(263\) − 0.685099i − 0.0422450i −0.999777 0.0211225i \(-0.993276\pi\)
0.999777 0.0211225i \(-0.00672401\pi\)
\(264\) 2.80118 0.172400
\(265\) 0 0
\(266\) −4.74926 −0.291196
\(267\) 15.5942i 0.954347i
\(268\) − 1.03803i − 0.0634079i
\(269\) −4.22522 −0.257616 −0.128808 0.991670i \(-0.541115\pi\)
−0.128808 + 0.991670i \(0.541115\pi\)
\(270\) 0 0
\(271\) −0.814686 −0.0494886 −0.0247443 0.999694i \(-0.507877\pi\)
−0.0247443 + 0.999694i \(0.507877\pi\)
\(272\) − 10.1254i − 0.613943i
\(273\) − 18.6423i − 1.12828i
\(274\) 0.565636 0.0341713
\(275\) 0 0
\(276\) −6.36299 −0.383007
\(277\) 9.85459i 0.592105i 0.955172 + 0.296052i \(0.0956703\pi\)
−0.955172 + 0.296052i \(0.904330\pi\)
\(278\) 2.74739i 0.164778i
\(279\) 0.787665 0.0471562
\(280\) 0 0
\(281\) 12.2297 0.729561 0.364780 0.931094i \(-0.381144\pi\)
0.364780 + 0.931094i \(0.381144\pi\)
\(282\) 0.154735i 0.00921435i
\(283\) − 4.23341i − 0.251650i −0.992052 0.125825i \(-0.959842\pi\)
0.992052 0.125825i \(-0.0401578\pi\)
\(284\) 0.386271 0.0229209
\(285\) 0 0
\(286\) 2.58315 0.152745
\(287\) − 44.7637i − 2.64231i
\(288\) − 1.64154i − 0.0967286i
\(289\) 10.2079 0.600464
\(290\) 0 0
\(291\) 12.6671 0.742561
\(292\) − 2.23873i − 0.131012i
\(293\) 13.3170i 0.777989i 0.921240 + 0.388995i \(0.127178\pi\)
−0.921240 + 0.388995i \(0.872822\pi\)
\(294\) 2.60351 0.151840
\(295\) 0 0
\(296\) 2.82981 0.164479
\(297\) 5.07830i 0.294673i
\(298\) − 1.49321i − 0.0864992i
\(299\) −11.7924 −0.681970
\(300\) 0 0
\(301\) −13.3035 −0.766802
\(302\) − 0.761900i − 0.0438424i
\(303\) 7.03990i 0.404432i
\(304\) 26.2220 1.50394
\(305\) 0 0
\(306\) −0.361122 −0.0206440
\(307\) 14.7379i 0.841136i 0.907261 + 0.420568i \(0.138169\pi\)
−0.907261 + 0.420568i \(0.861831\pi\)
\(308\) − 51.0832i − 2.91073i
\(309\) 2.65808 0.151213
\(310\) 0 0
\(311\) −10.0226 −0.568328 −0.284164 0.958776i \(-0.591716\pi\)
−0.284164 + 0.958776i \(0.591716\pi\)
\(312\) − 2.02489i − 0.114637i
\(313\) − 4.08800i − 0.231067i −0.993304 0.115534i \(-0.963142\pi\)
0.993304 0.115534i \(-0.0368578\pi\)
\(314\) −0.800300 −0.0451635
\(315\) 0 0
\(316\) −30.3132 −1.70525
\(317\) − 12.7628i − 0.716829i −0.933563 0.358414i \(-0.883317\pi\)
0.933563 0.358414i \(-0.116683\pi\)
\(318\) 0.0858653i 0.00481508i
\(319\) −5.07830 −0.284330
\(320\) 0 0
\(321\) −4.81469 −0.268730
\(322\) − 2.26043i − 0.125969i
\(323\) − 17.5897i − 0.978718i
\(324\) 1.98080 0.110044
\(325\) 0 0
\(326\) 0.990307 0.0548480
\(327\) − 3.86597i − 0.213789i
\(328\) − 4.86215i − 0.268467i
\(329\) 5.67096 0.312650
\(330\) 0 0
\(331\) 5.83576 0.320762 0.160381 0.987055i \(-0.448728\pi\)
0.160381 + 0.987055i \(0.448728\pi\)
\(332\) 35.8885i 1.96964i
\(333\) 5.13021i 0.281134i
\(334\) −2.36957 −0.129657
\(335\) 0 0
\(336\) −19.7301 −1.07636
\(337\) 1.95253i 0.106361i 0.998585 + 0.0531807i \(0.0169359\pi\)
−0.998585 + 0.0531807i \(0.983064\pi\)
\(338\) − 0.0659501i − 0.00358721i
\(339\) −8.73575 −0.474461
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) − 0.935207i − 0.0505702i
\(343\) − 59.8690i − 3.23262i
\(344\) −1.44501 −0.0779095
\(345\) 0 0
\(346\) 1.42530 0.0766245
\(347\) 17.3960i 0.933864i 0.884293 + 0.466932i \(0.154641\pi\)
−0.884293 + 0.466932i \(0.845359\pi\)
\(348\) 1.98080i 0.106182i
\(349\) 28.7379 1.53830 0.769152 0.639066i \(-0.220679\pi\)
0.769152 + 0.639066i \(0.220679\pi\)
\(350\) 0 0
\(351\) 3.67096 0.195941
\(352\) − 8.33623i − 0.444323i
\(353\) 29.7590i 1.58391i 0.610580 + 0.791955i \(0.290936\pi\)
−0.610580 + 0.791955i \(0.709064\pi\)
\(354\) 1.77034 0.0940923
\(355\) 0 0
\(356\) 30.8889 1.63711
\(357\) 13.2349i 0.700466i
\(358\) 1.67915i 0.0887459i
\(359\) −7.76659 −0.409905 −0.204953 0.978772i \(-0.565704\pi\)
−0.204953 + 0.978772i \(0.565704\pi\)
\(360\) 0 0
\(361\) 26.5526 1.39750
\(362\) − 2.90631i − 0.152752i
\(363\) 14.7892i 0.776230i
\(364\) −36.9266 −1.93548
\(365\) 0 0
\(366\) −1.23404 −0.0645041
\(367\) − 12.8675i − 0.671677i −0.941920 0.335838i \(-0.890980\pi\)
0.941920 0.335838i \(-0.109020\pi\)
\(368\) 12.4805i 0.650589i
\(369\) 8.81469 0.458874
\(370\) 0 0
\(371\) 3.14691 0.163380
\(372\) − 1.56021i − 0.0808929i
\(373\) 13.4639i 0.697133i 0.937284 + 0.348566i \(0.113331\pi\)
−0.937284 + 0.348566i \(0.886669\pi\)
\(374\) −1.83389 −0.0948280
\(375\) 0 0
\(376\) 0.615970 0.0317662
\(377\) 3.67096i 0.189064i
\(378\) 0.703671i 0.0361930i
\(379\) 9.53318 0.489687 0.244843 0.969563i \(-0.421263\pi\)
0.244843 + 0.969563i \(0.421263\pi\)
\(380\) 0 0
\(381\) −10.7877 −0.552669
\(382\) 3.63169i 0.185813i
\(383\) 20.0836i 1.02622i 0.858322 + 0.513111i \(0.171507\pi\)
−0.858322 + 0.513111i \(0.828493\pi\)
\(384\) −4.32825 −0.220875
\(385\) 0 0
\(386\) −3.29288 −0.167603
\(387\) − 2.61968i − 0.133166i
\(388\) − 25.0911i − 1.27381i
\(389\) −24.3472 −1.23445 −0.617225 0.786787i \(-0.711743\pi\)
−0.617225 + 0.786787i \(0.711743\pi\)
\(390\) 0 0
\(391\) 8.37188 0.423384
\(392\) − 10.3640i − 0.523463i
\(393\) 12.9745i 0.654476i
\(394\) 3.55114 0.178904
\(395\) 0 0
\(396\) 10.0591 0.505489
\(397\) − 25.9232i − 1.30105i −0.759486 0.650524i \(-0.774549\pi\)
0.759486 0.650524i \(-0.225451\pi\)
\(398\) − 1.08462i − 0.0543671i
\(399\) −34.2748 −1.71589
\(400\) 0 0
\(401\) 31.3576 1.56592 0.782961 0.622071i \(-0.213708\pi\)
0.782961 + 0.622071i \(0.213708\pi\)
\(402\) 0.0726141i 0.00362166i
\(403\) − 2.89149i − 0.144035i
\(404\) 13.9446 0.693772
\(405\) 0 0
\(406\) −0.703671 −0.0349226
\(407\) 26.0528i 1.29139i
\(408\) 1.43755i 0.0711695i
\(409\) −1.67415 −0.0827814 −0.0413907 0.999143i \(-0.513179\pi\)
−0.0413907 + 0.999143i \(0.513179\pi\)
\(410\) 0 0
\(411\) 4.08212 0.201356
\(412\) − 5.26512i − 0.259394i
\(413\) − 64.8819i − 3.19263i
\(414\) 0.445115 0.0218762
\(415\) 0 0
\(416\) −6.02602 −0.295450
\(417\) 19.8276i 0.970960i
\(418\) − 4.74926i − 0.232294i
\(419\) −0.658078 −0.0321492 −0.0160746 0.999871i \(-0.505117\pi\)
−0.0160746 + 0.999871i \(0.505117\pi\)
\(420\) 0 0
\(421\) 5.11989 0.249528 0.124764 0.992186i \(-0.460183\pi\)
0.124764 + 0.992186i \(0.460183\pi\)
\(422\) 0.277129i 0.0134904i
\(423\) 1.11670i 0.0542960i
\(424\) 0.341812 0.0165999
\(425\) 0 0
\(426\) −0.0270211 −0.00130917
\(427\) 45.2267i 2.18867i
\(428\) 9.53693i 0.460985i
\(429\) 18.6423 0.900056
\(430\) 0 0
\(431\) −0.390015 −0.0187864 −0.00939318 0.999956i \(-0.502990\pi\)
−0.00939318 + 0.999956i \(0.502990\pi\)
\(432\) − 3.88517i − 0.186925i
\(433\) 29.6989i 1.42724i 0.700535 + 0.713618i \(0.252945\pi\)
−0.700535 + 0.713618i \(0.747055\pi\)
\(434\) 0.554257 0.0266052
\(435\) 0 0
\(436\) −7.65771 −0.366738
\(437\) 21.6809i 1.03714i
\(438\) 0.156607i 0.00748299i
\(439\) 24.4856 1.16864 0.584318 0.811525i \(-0.301362\pi\)
0.584318 + 0.811525i \(0.301362\pi\)
\(440\) 0 0
\(441\) 18.7892 0.894722
\(442\) 1.32566i 0.0630554i
\(443\) 11.1091i 0.527808i 0.964549 + 0.263904i \(0.0850102\pi\)
−0.964549 + 0.263904i \(0.914990\pi\)
\(444\) 10.1619 0.482264
\(445\) 0 0
\(446\) 1.08994 0.0516103
\(447\) − 10.7763i − 0.509701i
\(448\) 38.3050i 1.80974i
\(449\) 15.1003 0.712628 0.356314 0.934366i \(-0.384033\pi\)
0.356314 + 0.934366i \(0.384033\pi\)
\(450\) 0 0
\(451\) 44.7637 2.10784
\(452\) 17.3038i 0.813901i
\(453\) − 5.49853i − 0.258343i
\(454\) 0.00906504 0.000425443 0
\(455\) 0 0
\(456\) −3.72287 −0.174339
\(457\) − 24.3742i − 1.14018i −0.821583 0.570088i \(-0.806909\pi\)
0.821583 0.570088i \(-0.193091\pi\)
\(458\) 0.536301i 0.0250597i
\(459\) −2.60617 −0.121645
\(460\) 0 0
\(461\) −12.9789 −0.604489 −0.302244 0.953230i \(-0.597736\pi\)
−0.302244 + 0.953230i \(0.597736\pi\)
\(462\) 3.57346i 0.166252i
\(463\) 11.2619i 0.523386i 0.965151 + 0.261693i \(0.0842809\pi\)
−0.965151 + 0.261693i \(0.915719\pi\)
\(464\) 3.88517 0.180364
\(465\) 0 0
\(466\) −1.13396 −0.0525296
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) − 7.27144i − 0.336122i
\(469\) 2.66127 0.122886
\(470\) 0 0
\(471\) −5.77566 −0.266128
\(472\) − 7.04736i − 0.324381i
\(473\) − 13.3035i − 0.611697i
\(474\) 2.12052 0.0973988
\(475\) 0 0
\(476\) 26.2157 1.20160
\(477\) 0.619678i 0.0283731i
\(478\) 1.56094i 0.0713959i
\(479\) 23.8615 1.09026 0.545130 0.838351i \(-0.316480\pi\)
0.545130 + 0.838351i \(0.316480\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) − 2.11476i − 0.0963246i
\(483\) − 16.3132i − 0.742277i
\(484\) 29.2944 1.33156
\(485\) 0 0
\(486\) −0.138564 −0.00628540
\(487\) − 20.8417i − 0.944428i −0.881484 0.472214i \(-0.843455\pi\)
0.881484 0.472214i \(-0.156545\pi\)
\(488\) 4.91245i 0.222376i
\(489\) 7.14691 0.323195
\(490\) 0 0
\(491\) −19.1021 −0.862067 −0.431034 0.902336i \(-0.641851\pi\)
−0.431034 + 0.902336i \(0.641851\pi\)
\(492\) − 17.4601i − 0.787163i
\(493\) − 2.60617i − 0.117376i
\(494\) −3.43311 −0.154463
\(495\) 0 0
\(496\) −3.06021 −0.137407
\(497\) 0.990307i 0.0444213i
\(498\) − 2.51054i − 0.112500i
\(499\) 7.82757 0.350410 0.175205 0.984532i \(-0.443941\pi\)
0.175205 + 0.984532i \(0.443941\pi\)
\(500\) 0 0
\(501\) −17.1009 −0.764011
\(502\) 3.45594i 0.154246i
\(503\) − 31.5865i − 1.40837i −0.710015 0.704187i \(-0.751312\pi\)
0.710015 0.704187i \(-0.248688\pi\)
\(504\) 2.80118 0.124774
\(505\) 0 0
\(506\) 2.26043 0.100488
\(507\) − 0.475953i − 0.0211378i
\(508\) 21.3682i 0.948061i
\(509\) −19.6167 −0.869497 −0.434748 0.900552i \(-0.643163\pi\)
−0.434748 + 0.900552i \(0.643163\pi\)
\(510\) 0 0
\(511\) 5.73957 0.253904
\(512\) 10.6637i 0.471275i
\(513\) − 6.74926i − 0.297987i
\(514\) −0.158921 −0.00700972
\(515\) 0 0
\(516\) −5.18906 −0.228436
\(517\) 5.67096i 0.249409i
\(518\) 3.60999i 0.158614i
\(519\) 10.2862 0.451514
\(520\) 0 0
\(521\) 24.4057 1.06923 0.534616 0.845095i \(-0.320456\pi\)
0.534616 + 0.845095i \(0.320456\pi\)
\(522\) − 0.138564i − 0.00606479i
\(523\) 39.9715i 1.74783i 0.486076 + 0.873917i \(0.338428\pi\)
−0.486076 + 0.873917i \(0.661572\pi\)
\(524\) 25.6999 1.12270
\(525\) 0 0
\(526\) 0.0949303 0.00413916
\(527\) 2.05279i 0.0894208i
\(528\) − 19.7301i − 0.858641i
\(529\) 12.6809 0.551344
\(530\) 0 0
\(531\) 12.7763 0.554444
\(532\) 67.8916i 2.94347i
\(533\) − 32.3584i − 1.40160i
\(534\) −2.16079 −0.0935067
\(535\) 0 0
\(536\) 0.289062 0.0124856
\(537\) 12.1182i 0.522939i
\(538\) − 0.585464i − 0.0252412i
\(539\) 95.4171 4.10991
\(540\) 0 0
\(541\) −8.76064 −0.376649 −0.188325 0.982107i \(-0.560306\pi\)
−0.188325 + 0.982107i \(0.560306\pi\)
\(542\) − 0.112886i − 0.00484888i
\(543\) − 20.9745i − 0.900101i
\(544\) 4.27813 0.183423
\(545\) 0 0
\(546\) 2.58315 0.110549
\(547\) 31.3569i 1.34072i 0.742035 + 0.670361i \(0.233861\pi\)
−0.742035 + 0.670361i \(0.766139\pi\)
\(548\) − 8.08587i − 0.345411i
\(549\) −8.90587 −0.380093
\(550\) 0 0
\(551\) 6.74926 0.287528
\(552\) − 1.77191i − 0.0754176i
\(553\) − 77.7159i − 3.30482i
\(554\) −1.36549 −0.0580143
\(555\) 0 0
\(556\) 39.2744 1.66561
\(557\) 37.6514i 1.59534i 0.603094 + 0.797670i \(0.293934\pi\)
−0.603094 + 0.797670i \(0.706066\pi\)
\(558\) 0.109142i 0.00462036i
\(559\) −9.61674 −0.406745
\(560\) 0 0
\(561\) −13.2349 −0.558778
\(562\) 1.69459i 0.0714821i
\(563\) 39.0323i 1.64501i 0.568755 + 0.822507i \(0.307425\pi\)
−0.568755 + 0.822507i \(0.692575\pi\)
\(564\) 2.21197 0.0931406
\(565\) 0 0
\(566\) 0.586599 0.0246566
\(567\) 5.07830i 0.213269i
\(568\) 0.107565i 0.00451335i
\(569\) 3.03990 0.127439 0.0637197 0.997968i \(-0.479704\pi\)
0.0637197 + 0.997968i \(0.479704\pi\)
\(570\) 0 0
\(571\) 40.1056 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(572\) − 36.9266i − 1.54398i
\(573\) 26.2094i 1.09491i
\(574\) 6.20264 0.258893
\(575\) 0 0
\(576\) −7.54288 −0.314287
\(577\) − 16.1091i − 0.670632i −0.942106 0.335316i \(-0.891157\pi\)
0.942106 0.335316i \(-0.108843\pi\)
\(578\) 1.41445i 0.0588333i
\(579\) −23.7643 −0.987610
\(580\) 0 0
\(581\) −92.0098 −3.81721
\(582\) 1.75521i 0.0727559i
\(583\) 3.14691i 0.130332i
\(584\) 0.623422 0.0257974
\(585\) 0 0
\(586\) −1.84526 −0.0762272
\(587\) − 21.4331i − 0.884639i −0.896858 0.442320i \(-0.854156\pi\)
0.896858 0.442320i \(-0.145844\pi\)
\(588\) − 37.2176i − 1.53483i
\(589\) −5.31616 −0.219048
\(590\) 0 0
\(591\) 25.6281 1.05420
\(592\) − 19.9317i − 0.819190i
\(593\) − 23.9886i − 0.985095i −0.870286 0.492547i \(-0.836066\pi\)
0.870286 0.492547i \(-0.163934\pi\)
\(594\) −0.703671 −0.0288720
\(595\) 0 0
\(596\) −21.3457 −0.874353
\(597\) − 7.82757i − 0.320361i
\(598\) − 1.63400i − 0.0668192i
\(599\) 38.1100 1.55713 0.778567 0.627562i \(-0.215947\pi\)
0.778567 + 0.627562i \(0.215947\pi\)
\(600\) 0 0
\(601\) −20.0528 −0.817970 −0.408985 0.912541i \(-0.634117\pi\)
−0.408985 + 0.912541i \(0.634117\pi\)
\(602\) − 1.84339i − 0.0751311i
\(603\) 0.524047i 0.0213408i
\(604\) −10.8915 −0.443168
\(605\) 0 0
\(606\) −0.975479 −0.0396261
\(607\) − 38.7523i − 1.57291i −0.617650 0.786453i \(-0.711915\pi\)
0.617650 0.786453i \(-0.288085\pi\)
\(608\) 11.0792i 0.449320i
\(609\) −5.07830 −0.205783
\(610\) 0 0
\(611\) 4.09938 0.165843
\(612\) 5.16230i 0.208673i
\(613\) − 24.6757i − 0.996640i −0.866993 0.498320i \(-0.833950\pi\)
0.866993 0.498320i \(-0.166050\pi\)
\(614\) −2.04214 −0.0824142
\(615\) 0 0
\(616\) 14.2252 0.573150
\(617\) 10.5249i 0.423717i 0.977300 + 0.211859i \(0.0679517\pi\)
−0.977300 + 0.211859i \(0.932048\pi\)
\(618\) 0.368315i 0.0148158i
\(619\) −26.5496 −1.06712 −0.533560 0.845762i \(-0.679146\pi\)
−0.533560 + 0.845762i \(0.679146\pi\)
\(620\) 0 0
\(621\) 3.21234 0.128907
\(622\) − 1.38877i − 0.0556846i
\(623\) 79.1919i 3.17276i
\(624\) −14.2623 −0.570949
\(625\) 0 0
\(626\) 0.566450 0.0226399
\(627\) − 34.2748i − 1.36880i
\(628\) 11.4404i 0.456523i
\(629\) −13.3702 −0.533105
\(630\) 0 0
\(631\) −13.2041 −0.525649 −0.262824 0.964844i \(-0.584654\pi\)
−0.262824 + 0.964844i \(0.584654\pi\)
\(632\) − 8.44137i − 0.335780i
\(633\) 2.00000i 0.0794929i
\(634\) 1.76846 0.0702347
\(635\) 0 0
\(636\) 1.22746 0.0486719
\(637\) − 68.9743i − 2.73286i
\(638\) − 0.703671i − 0.0278586i
\(639\) −0.195007 −0.00771437
\(640\) 0 0
\(641\) 9.51435 0.375794 0.187897 0.982189i \(-0.439833\pi\)
0.187897 + 0.982189i \(0.439833\pi\)
\(642\) − 0.667143i − 0.0263300i
\(643\) − 30.1370i − 1.18849i −0.804285 0.594244i \(-0.797451\pi\)
0.804285 0.594244i \(-0.202549\pi\)
\(644\) −32.3132 −1.27332
\(645\) 0 0
\(646\) 2.43731 0.0958945
\(647\) − 42.7535i − 1.68081i −0.541955 0.840407i \(-0.682316\pi\)
0.541955 0.840407i \(-0.317684\pi\)
\(648\) 0.551597i 0.0216688i
\(649\) 64.8819 2.54684
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) − 14.1566i − 0.554416i
\(653\) − 33.8983i − 1.32654i −0.748379 0.663272i \(-0.769167\pi\)
0.748379 0.663272i \(-0.230833\pi\)
\(654\) 0.535685 0.0209469
\(655\) 0 0
\(656\) −34.2465 −1.33710
\(657\) 1.13021i 0.0440939i
\(658\) 0.785793i 0.0306334i
\(659\) −11.7287 −0.456887 −0.228444 0.973557i \(-0.573364\pi\)
−0.228444 + 0.973557i \(0.573364\pi\)
\(660\) 0 0
\(661\) −38.6807 −1.50450 −0.752252 0.658876i \(-0.771032\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) 0.808627i 0.0314282i
\(663\) 9.56714i 0.371557i
\(664\) −9.99394 −0.387840
\(665\) 0 0
\(666\) −0.710864 −0.0275454
\(667\) 3.21234i 0.124382i
\(668\) 33.8734i 1.31060i
\(669\) 7.86597 0.304116
\(670\) 0 0
\(671\) −45.2267 −1.74596
\(672\) − 8.33623i − 0.321577i
\(673\) − 30.6410i − 1.18112i −0.806992 0.590562i \(-0.798906\pi\)
0.806992 0.590562i \(-0.201094\pi\)
\(674\) −0.270552 −0.0104213
\(675\) 0 0
\(676\) −0.942768 −0.0362603
\(677\) − 40.8954i − 1.57174i −0.618394 0.785868i \(-0.712216\pi\)
0.618394 0.785868i \(-0.287784\pi\)
\(678\) − 1.21046i − 0.0464876i
\(679\) 64.3276 2.46867
\(680\) 0 0
\(681\) 0.0654212 0.00250694
\(682\) 0.554257i 0.0212236i
\(683\) − 21.2317i − 0.812409i −0.913782 0.406205i \(-0.866852\pi\)
0.913782 0.406205i \(-0.133148\pi\)
\(684\) −13.3689 −0.511174
\(685\) 0 0
\(686\) 8.29570 0.316731
\(687\) 3.87041i 0.147665i
\(688\) 10.1779i 0.388028i
\(689\) 2.27481 0.0866635
\(690\) 0 0
\(691\) −13.9842 −0.531983 −0.265992 0.963975i \(-0.585699\pi\)
−0.265992 + 0.963975i \(0.585699\pi\)
\(692\) − 20.3749i − 0.774537i
\(693\) 25.7892i 0.979649i
\(694\) −2.41046 −0.0914998
\(695\) 0 0
\(696\) −0.551597 −0.0209082
\(697\) 22.9725i 0.870147i
\(698\) 3.98204i 0.150723i
\(699\) −8.18363 −0.309533
\(700\) 0 0
\(701\) 3.16630 0.119590 0.0597948 0.998211i \(-0.480955\pi\)
0.0597948 + 0.998211i \(0.480955\pi\)
\(702\) 0.508664i 0.0191983i
\(703\) − 34.6252i − 1.30591i
\(704\) −38.3050 −1.44367
\(705\) 0 0
\(706\) −4.12353 −0.155191
\(707\) 35.7508i 1.34455i
\(708\) − 25.3073i − 0.951105i
\(709\) −23.3677 −0.877592 −0.438796 0.898587i \(-0.644595\pi\)
−0.438796 + 0.898587i \(0.644595\pi\)
\(710\) 0 0
\(711\) 15.3035 0.573927
\(712\) 8.60169i 0.322362i
\(713\) − 2.53024i − 0.0947583i
\(714\) −1.83389 −0.0686314
\(715\) 0 0
\(716\) 24.0037 0.897062
\(717\) 11.2651i 0.420704i
\(718\) − 1.07617i − 0.0401624i
\(719\) 0.731134 0.0272667 0.0136334 0.999907i \(-0.495660\pi\)
0.0136334 + 0.999907i \(0.495660\pi\)
\(720\) 0 0
\(721\) 13.4985 0.502711
\(722\) 3.67924i 0.136927i
\(723\) − 15.2619i − 0.567597i
\(724\) −41.5463 −1.54405
\(725\) 0 0
\(726\) −2.04925 −0.0760548
\(727\) − 17.7243i − 0.657358i −0.944442 0.328679i \(-0.893397\pi\)
0.944442 0.328679i \(-0.106603\pi\)
\(728\) − 10.2830i − 0.381113i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.82732 0.252518
\(732\) 17.6408i 0.652021i
\(733\) 5.67184i 0.209494i 0.994499 + 0.104747i \(0.0334033\pi\)
−0.994499 + 0.104747i \(0.966597\pi\)
\(734\) 1.78297 0.0658107
\(735\) 0 0
\(736\) −5.27317 −0.194372
\(737\) 2.66127i 0.0980291i
\(738\) 1.22140i 0.0449604i
\(739\) −8.72350 −0.320899 −0.160450 0.987044i \(-0.551294\pi\)
−0.160450 + 0.987044i \(0.551294\pi\)
\(740\) 0 0
\(741\) −24.7763 −0.910180
\(742\) 0.436050i 0.0160079i
\(743\) 0.413474i 0.0151689i 0.999971 + 0.00758445i \(0.00241423\pi\)
−0.999971 + 0.00758445i \(0.997586\pi\)
\(744\) 0.434473 0.0159286
\(745\) 0 0
\(746\) −1.86561 −0.0683049
\(747\) − 18.1182i − 0.662911i
\(748\) 26.2157i 0.958541i
\(749\) −24.4504 −0.893399
\(750\) 0 0
\(751\) 9.71381 0.354462 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(752\) − 4.33858i − 0.158212i
\(753\) 24.9411i 0.908904i
\(754\) −0.508664 −0.0185244
\(755\) 0 0
\(756\) 10.0591 0.365846
\(757\) 22.3877i 0.813695i 0.913496 + 0.406847i \(0.133372\pi\)
−0.913496 + 0.406847i \(0.866628\pi\)
\(758\) 1.32096i 0.0479794i
\(759\) 16.3132 0.592132
\(760\) 0 0
\(761\) 44.0836 1.59803 0.799014 0.601313i \(-0.205355\pi\)
0.799014 + 0.601313i \(0.205355\pi\)
\(762\) − 1.49478i − 0.0541503i
\(763\) − 19.6326i − 0.710746i
\(764\) 51.9156 1.87824
\(765\) 0 0
\(766\) −2.78286 −0.100549
\(767\) − 46.9012i − 1.69351i
\(768\) 14.4860i 0.522719i
\(769\) 14.2761 0.514808 0.257404 0.966304i \(-0.417133\pi\)
0.257404 + 0.966304i \(0.417133\pi\)
\(770\) 0 0
\(771\) −1.14691 −0.0413051
\(772\) 47.0723i 1.69417i
\(773\) 25.5807i 0.920072i 0.887900 + 0.460036i \(0.152164\pi\)
−0.887900 + 0.460036i \(0.847836\pi\)
\(774\) 0.362994 0.0130475
\(775\) 0 0
\(776\) 6.98715 0.250824
\(777\) 26.0528i 0.934638i
\(778\) − 3.37365i − 0.120951i
\(779\) −59.4926 −2.13155
\(780\) 0 0
\(781\) −0.990307 −0.0354360
\(782\) 1.16004i 0.0414831i
\(783\) − 1.00000i − 0.0357371i
\(784\) −72.9991 −2.60711
\(785\) 0 0
\(786\) −1.79780 −0.0641254
\(787\) − 8.73362i − 0.311320i −0.987811 0.155660i \(-0.950250\pi\)
0.987811 0.155660i \(-0.0497504\pi\)
\(788\) − 50.7642i − 1.80840i
\(789\) 0.685099 0.0243902
\(790\) 0 0
\(791\) −44.3628 −1.57736
\(792\) 2.80118i 0.0995354i
\(793\) 32.6931i 1.16097i
\(794\) 3.59203 0.127476
\(795\) 0 0
\(796\) −15.5048 −0.549555
\(797\) 8.43874i 0.298915i 0.988768 + 0.149458i \(0.0477528\pi\)
−0.988768 + 0.149458i \(0.952247\pi\)
\(798\) − 4.74926i − 0.168122i
\(799\) −2.91032 −0.102960
\(800\) 0 0
\(801\) −15.5942 −0.550993
\(802\) 4.34504i 0.153429i
\(803\) 5.73957i 0.202545i
\(804\) 1.03803 0.0366085
\(805\) 0 0
\(806\) 0.400657 0.0141125
\(807\) − 4.22522i − 0.148735i
\(808\) 3.88319i 0.136610i
\(809\) 21.7508 0.764716 0.382358 0.924014i \(-0.375112\pi\)
0.382358 + 0.924014i \(0.375112\pi\)
\(810\) 0 0
\(811\) −3.24629 −0.113993 −0.0569963 0.998374i \(-0.518152\pi\)
−0.0569963 + 0.998374i \(0.518152\pi\)
\(812\) 10.0591i 0.353005i
\(813\) − 0.814686i − 0.0285723i
\(814\) −3.60999 −0.126530
\(815\) 0 0
\(816\) 10.1254 0.354460
\(817\) 17.6809i 0.618576i
\(818\) − 0.231977i − 0.00811090i
\(819\) 18.6423 0.651413
\(820\) 0 0
\(821\) 33.9232 1.18393 0.591964 0.805964i \(-0.298353\pi\)
0.591964 + 0.805964i \(0.298353\pi\)
\(822\) 0.565636i 0.0197288i
\(823\) 36.4648i 1.27108i 0.772066 + 0.635542i \(0.219223\pi\)
−0.772066 + 0.635542i \(0.780777\pi\)
\(824\) 1.46619 0.0510770
\(825\) 0 0
\(826\) 8.99031 0.312813
\(827\) − 15.9772i − 0.555583i −0.960641 0.277792i \(-0.910398\pi\)
0.960641 0.277792i \(-0.0896025\pi\)
\(828\) − 6.36299i − 0.221129i
\(829\) −5.00595 −0.173864 −0.0869319 0.996214i \(-0.527706\pi\)
−0.0869319 + 0.996214i \(0.527706\pi\)
\(830\) 0 0
\(831\) −9.85459 −0.341852
\(832\) 27.6896i 0.959964i
\(833\) 48.9677i 1.69663i
\(834\) −2.74739 −0.0951344
\(835\) 0 0
\(836\) −67.8916 −2.34808
\(837\) 0.787665i 0.0272257i
\(838\) − 0.0911861i − 0.00314997i
\(839\) −31.4713 −1.08651 −0.543255 0.839567i \(-0.682808\pi\)
−0.543255 + 0.839567i \(0.682808\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0.709434i 0.0244487i
\(843\) 12.2297i 0.421212i
\(844\) 3.96160 0.136364
\(845\) 0 0
\(846\) −0.154735 −0.00531991
\(847\) 75.1039i 2.58060i
\(848\) − 2.40755i − 0.0826758i
\(849\) 4.23341 0.145290
\(850\) 0 0
\(851\) 16.4800 0.564926
\(852\) 0.386271i 0.0132334i
\(853\) 36.0197i 1.23329i 0.787241 + 0.616646i \(0.211509\pi\)
−0.787241 + 0.616646i \(0.788491\pi\)
\(854\) −6.26681 −0.214446
\(855\) 0 0
\(856\) −2.65576 −0.0907722
\(857\) − 27.4217i − 0.936708i −0.883541 0.468354i \(-0.844847\pi\)
0.883541 0.468354i \(-0.155153\pi\)
\(858\) 2.58315i 0.0881873i
\(859\) −14.0642 −0.479863 −0.239932 0.970790i \(-0.577125\pi\)
−0.239932 + 0.970790i \(0.577125\pi\)
\(860\) 0 0
\(861\) 44.7637 1.52554
\(862\) − 0.0540421i − 0.00184068i
\(863\) − 19.7540i − 0.672433i −0.941785 0.336216i \(-0.890853\pi\)
0.941785 0.336216i \(-0.109147\pi\)
\(864\) 1.64154 0.0558463
\(865\) 0 0
\(866\) −4.11520 −0.139840
\(867\) 10.2079i 0.346678i
\(868\) − 7.92320i − 0.268931i
\(869\) 77.7159 2.63633
\(870\) 0 0
\(871\) 1.92375 0.0651839
\(872\) − 2.13246i − 0.0722140i
\(873\) 12.6671i 0.428718i
\(874\) −3.00420 −0.101619
\(875\) 0 0
\(876\) 2.23873 0.0756396
\(877\) 42.2030i 1.42509i 0.701624 + 0.712547i \(0.252459\pi\)
−0.701624 + 0.712547i \(0.747541\pi\)
\(878\) 3.39284i 0.114503i
\(879\) −13.3170 −0.449172
\(880\) 0 0
\(881\) −33.0927 −1.11492 −0.557461 0.830203i \(-0.688224\pi\)
−0.557461 + 0.830203i \(0.688224\pi\)
\(882\) 2.60351i 0.0876646i
\(883\) 26.2634i 0.883835i 0.897056 + 0.441917i \(0.145702\pi\)
−0.897056 + 0.441917i \(0.854298\pi\)
\(884\) 18.9506 0.637377
\(885\) 0 0
\(886\) −1.53932 −0.0517145
\(887\) 14.0716i 0.472479i 0.971695 + 0.236239i \(0.0759149\pi\)
−0.971695 + 0.236239i \(0.924085\pi\)
\(888\) 2.82981i 0.0949622i
\(889\) −54.7830 −1.83736
\(890\) 0 0
\(891\) −5.07830 −0.170130
\(892\) − 15.5809i − 0.521687i
\(893\) − 7.53693i − 0.252214i
\(894\) 1.49321 0.0499403
\(895\) 0 0
\(896\) −21.9802 −0.734306
\(897\) − 11.7924i − 0.393735i
\(898\) 2.09237i 0.0698231i
\(899\) −0.787665 −0.0262701
\(900\) 0 0
\(901\) −1.61499 −0.0538030
\(902\) 6.20264i 0.206525i
\(903\) − 13.3035i − 0.442713i
\(904\) −4.81861 −0.160265
\(905\) 0 0
\(906\) 0.761900 0.0253124
\(907\) 11.9810i 0.397822i 0.980018 + 0.198911i \(0.0637405\pi\)
−0.980018 + 0.198911i \(0.936260\pi\)
\(908\) − 0.129586i − 0.00430047i
\(909\) −7.03990 −0.233499
\(910\) 0 0
\(911\) 27.5300 0.912109 0.456055 0.889952i \(-0.349262\pi\)
0.456055 + 0.889952i \(0.349262\pi\)
\(912\) 26.2220i 0.868298i
\(913\) − 92.0098i − 3.04508i
\(914\) 3.37739 0.111714
\(915\) 0 0
\(916\) 7.66652 0.253309
\(917\) 65.8884i 2.17583i
\(918\) − 0.361122i − 0.0119188i
\(919\) 17.7250 0.584694 0.292347 0.956312i \(-0.405564\pi\)
0.292347 + 0.956312i \(0.405564\pi\)
\(920\) 0 0
\(921\) −14.7379 −0.485630
\(922\) − 1.79842i − 0.0592277i
\(923\) 0.715865i 0.0235630i
\(924\) 51.0832 1.68051
\(925\) 0 0
\(926\) −1.56050 −0.0512813
\(927\) 2.65808i 0.0873027i
\(928\) 1.64154i 0.0538861i
\(929\) −36.9971 −1.21383 −0.606917 0.794765i \(-0.707594\pi\)
−0.606917 + 0.794765i \(0.707594\pi\)
\(930\) 0 0
\(931\) −126.813 −4.15613
\(932\) 16.2101i 0.530980i
\(933\) − 10.0226i − 0.328124i
\(934\) 1.10851 0.0362717
\(935\) 0 0
\(936\) 2.02489 0.0661856
\(937\) − 36.0446i − 1.17753i −0.808306 0.588763i \(-0.799615\pi\)
0.808306 0.588763i \(-0.200385\pi\)
\(938\) 0.368757i 0.0120403i
\(939\) 4.08800 0.133407
\(940\) 0 0
\(941\) −19.6256 −0.639777 −0.319889 0.947455i \(-0.603645\pi\)
−0.319889 + 0.947455i \(0.603645\pi\)
\(942\) − 0.800300i − 0.0260752i
\(943\) − 28.3157i − 0.922087i
\(944\) −49.6380 −1.61558
\(945\) 0 0
\(946\) 1.84339 0.0599339
\(947\) 18.9555i 0.615970i 0.951391 + 0.307985i \(0.0996547\pi\)
−0.951391 + 0.307985i \(0.900345\pi\)
\(948\) − 30.3132i − 0.984527i
\(949\) 4.14897 0.134681
\(950\) 0 0
\(951\) 12.7628 0.413861
\(952\) 7.30033i 0.236605i
\(953\) 30.2761i 0.980738i 0.871515 + 0.490369i \(0.163138\pi\)
−0.871515 + 0.490369i \(0.836862\pi\)
\(954\) −0.0858653 −0.00277999
\(955\) 0 0
\(956\) 22.3140 0.721685
\(957\) − 5.07830i − 0.164158i
\(958\) 3.30635i 0.106823i
\(959\) 20.7303 0.669415
\(960\) 0 0
\(961\) −30.3796 −0.979987
\(962\) 2.60956i 0.0841354i
\(963\) − 4.81469i − 0.155151i
\(964\) −30.2308 −0.973670
\(965\) 0 0
\(966\) 2.26043 0.0727281
\(967\) − 20.4812i − 0.658631i −0.944220 0.329316i \(-0.893182\pi\)
0.944220 0.329316i \(-0.106818\pi\)
\(968\) 8.15766i 0.262197i
\(969\) 17.5897 0.565063
\(970\) 0 0
\(971\) 44.1566 1.41705 0.708526 0.705684i \(-0.249360\pi\)
0.708526 + 0.705684i \(0.249360\pi\)
\(972\) 1.98080i 0.0635342i
\(973\) 100.690i 3.22799i
\(974\) 2.88792 0.0925348
\(975\) 0 0
\(976\) 34.6008 1.10754
\(977\) 0.492580i 0.0157590i 0.999969 + 0.00787952i \(0.00250815\pi\)
−0.999969 + 0.00787952i \(0.997492\pi\)
\(978\) 0.990307i 0.0316665i
\(979\) −79.1919 −2.53098
\(980\) 0 0
\(981\) 3.86597 0.123431
\(982\) − 2.64687i − 0.0844651i
\(983\) − 27.5513i − 0.878750i −0.898304 0.439375i \(-0.855200\pi\)
0.898304 0.439375i \(-0.144800\pi\)
\(984\) 4.86215 0.155000
\(985\) 0 0
\(986\) 0.361122 0.0115005
\(987\) 5.67096i 0.180509i
\(988\) 49.0769i 1.56134i
\(989\) −8.41529 −0.267590
\(990\) 0 0
\(991\) 23.0927 0.733563 0.366782 0.930307i \(-0.380460\pi\)
0.366782 + 0.930307i \(0.380460\pi\)
\(992\) − 1.29298i − 0.0410522i
\(993\) 5.83576i 0.185192i
\(994\) −0.137221 −0.00435239
\(995\) 0 0
\(996\) −35.8885 −1.13717
\(997\) 7.79593i 0.246900i 0.992351 + 0.123450i \(0.0393958\pi\)
−0.992351 + 0.123450i \(0.960604\pi\)
\(998\) 1.08462i 0.0343331i
\(999\) −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.c.n.349.5 8
5.2 odd 4 2175.2.a.v.1.2 4
5.3 odd 4 435.2.a.j.1.3 4
5.4 even 2 inner 2175.2.c.n.349.4 8
15.2 even 4 6525.2.a.bi.1.3 4
15.8 even 4 1305.2.a.r.1.2 4
20.3 even 4 6960.2.a.co.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 5.3 odd 4
1305.2.a.r.1.2 4 15.8 even 4
2175.2.a.v.1.2 4 5.2 odd 4
2175.2.c.n.349.4 8 5.4 even 2 inner
2175.2.c.n.349.5 8 1.1 even 1 trivial
6525.2.a.bi.1.3 4 15.2 even 4
6960.2.a.co.1.1 4 20.3 even 4