Properties

Label 6960.2.a.co.1.1
Level $6960$
Weight $2$
Character 6960.1
Self dual yes
Analytic conductor $55.576$
Analytic rank $1$
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] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 6960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -5.07830 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -5.07830 q^{7} +1.00000 q^{9} +5.07830 q^{11} -3.67096 q^{13} -1.00000 q^{15} -2.60617 q^{17} +6.74926 q^{19} -5.07830 q^{21} +3.21234 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.787665 q^{31} +5.07830 q^{33} +5.07830 q^{35} -5.13021 q^{37} -3.67096 q^{39} -8.81469 q^{41} +2.61968 q^{43} -1.00000 q^{45} +1.11670 q^{47} +18.7892 q^{49} -2.60617 q^{51} +0.619678 q^{53} -5.07830 q^{55} +6.74926 q^{57} -12.7763 q^{59} +8.90587 q^{61} -5.07830 q^{63} +3.67096 q^{65} +0.524047 q^{67} +3.21234 q^{69} -0.195007 q^{71} +1.13021 q^{73} +1.00000 q^{75} -25.7892 q^{77} -15.3035 q^{79} +1.00000 q^{81} +18.1182 q^{83} +2.60617 q^{85} -1.00000 q^{87} -15.5942 q^{89} +18.6423 q^{91} +0.787665 q^{93} -6.74926 q^{95} -12.6671 q^{97} +5.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) 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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −5.07830 −1.91942 −0.959709 0.280995i \(-0.909336\pi\)
−0.959709 + 0.280995i \(0.909336\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.07830 1.53117 0.765583 0.643337i \(-0.222451\pi\)
0.765583 + 0.643337i \(0.222451\pi\)
\(12\) 0 0
\(13\) −3.67096 −1.01814 −0.509071 0.860725i \(-0.670011\pi\)
−0.509071 + 0.860725i \(0.670011\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.60617 −0.632089 −0.316044 0.948744i \(-0.602355\pi\)
−0.316044 + 0.948744i \(0.602355\pi\)
\(18\) 0 0
\(19\) 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 0
\(23\) 3.21234 0.669818 0.334909 0.942250i \(-0.391294\pi\)
0.334909 + 0.942250i \(0.391294\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 0.787665 0.141469 0.0707344 0.997495i \(-0.477466\pi\)
0.0707344 + 0.997495i \(0.477466\pi\)
\(32\) 0 0
\(33\) 5.07830 0.884019
\(34\) 0 0
\(35\) 5.07830 0.858390
\(36\) 0 0
\(37\) −5.13021 −0.843402 −0.421701 0.906735i \(-0.638567\pi\)
−0.421701 + 0.906735i \(0.638567\pi\)
\(38\) 0 0
\(39\) −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 0
\(43\) 2.61968 0.399497 0.199749 0.979847i \(-0.435987\pi\)
0.199749 + 0.979847i \(0.435987\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.11670 0.162888 0.0814440 0.996678i \(-0.474047\pi\)
0.0814440 + 0.996678i \(0.474047\pi\)
\(48\) 0 0
\(49\) 18.7892 2.68417
\(50\) 0 0
\(51\) −2.60617 −0.364936
\(52\) 0 0
\(53\) 0.619678 0.0851194 0.0425597 0.999094i \(-0.486449\pi\)
0.0425597 + 0.999094i \(0.486449\pi\)
\(54\) 0 0
\(55\) −5.07830 −0.684758
\(56\) 0 0
\(57\) 6.74926 0.893962
\(58\) 0 0
\(59\) −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 0
\(63\) −5.07830 −0.639806
\(64\) 0 0
\(65\) 3.67096 0.455327
\(66\) 0 0
\(67\) 0.524047 0.0640225 0.0320112 0.999488i \(-0.489809\pi\)
0.0320112 + 0.999488i \(0.489809\pi\)
\(68\) 0 0
\(69\) 3.21234 0.386720
\(70\) 0 0
\(71\) −0.195007 −0.0231431 −0.0115716 0.999933i \(-0.503683\pi\)
−0.0115716 + 0.999933i \(0.503683\pi\)
\(72\) 0 0
\(73\) 1.13021 0.132282 0.0661408 0.997810i \(-0.478931\pi\)
0.0661408 + 0.997810i \(0.478931\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −25.7892 −2.93895
\(78\) 0 0
\(79\) −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
\(82\) 0 0
\(83\) 18.1182 1.98873 0.994366 0.106003i \(-0.0338053\pi\)
0.994366 + 0.106003i \(0.0338053\pi\)
\(84\) 0 0
\(85\) 2.60617 0.282679
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) 0.787665 0.0816770
\(94\) 0 0
\(95\) −6.74926 −0.692460
\(96\) 0 0
\(97\) −12.6671 −1.28615 −0.643077 0.765802i \(-0.722342\pi\)
−0.643077 + 0.765802i \(0.722342\pi\)
\(98\) 0 0
\(99\) 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 0
\(103\) −2.65808 −0.261908 −0.130954 0.991388i \(-0.541804\pi\)
−0.130954 + 0.991388i \(0.541804\pi\)
\(104\) 0 0
\(105\) 5.07830 0.495592
\(106\) 0 0
\(107\) −4.81469 −0.465453 −0.232727 0.972542i \(-0.574765\pi\)
−0.232727 + 0.972542i \(0.574765\pi\)
\(108\) 0 0
\(109\) 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
\(112\) 0 0
\(113\) −8.73575 −0.821791 −0.410895 0.911683i \(-0.634784\pi\)
−0.410895 + 0.911683i \(0.634784\pi\)
\(114\) 0 0
\(115\) −3.21234 −0.299552
\(116\) 0 0
\(117\) −3.67096 −0.339380
\(118\) 0 0
\(119\) 13.2349 1.21324
\(120\) 0 0
\(121\) 14.7892 1.34447
\(122\) 0 0
\(123\) −8.81469 −0.794793
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.7877 −0.957250 −0.478625 0.878019i \(-0.658865\pi\)
−0.478625 + 0.878019i \(0.658865\pi\)
\(128\) 0 0
\(129\) 2.61968 0.230650
\(130\) 0 0
\(131\) −12.9745 −1.13359 −0.566793 0.823860i \(-0.691816\pi\)
−0.566793 + 0.823860i \(0.691816\pi\)
\(132\) 0 0
\(133\) −34.2748 −2.97200
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −4.08212 −0.348759 −0.174380 0.984679i \(-0.555792\pi\)
−0.174380 + 0.984679i \(0.555792\pi\)
\(138\) 0 0
\(139\) 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 0
\(143\) −18.6423 −1.55894
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 18.7892 1.54970
\(148\) 0 0
\(149\) 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 \(-0.428176\pi\)
0.223732 + 0.974651i \(0.428176\pi\)
\(152\) 0 0
\(153\) −2.60617 −0.210696
\(154\) 0 0
\(155\) −0.787665 −0.0632667
\(156\) 0 0
\(157\) 5.77566 0.460948 0.230474 0.973079i \(-0.425972\pi\)
0.230474 + 0.973079i \(0.425972\pi\)
\(158\) 0 0
\(159\) 0.619678 0.0491437
\(160\) 0 0
\(161\) −16.3132 −1.28566
\(162\) 0 0
\(163\) −7.14691 −0.559790 −0.279895 0.960031i \(-0.590300\pi\)
−0.279895 + 0.960031i \(0.590300\pi\)
\(164\) 0 0
\(165\) −5.07830 −0.395345
\(166\) 0 0
\(167\) −17.1009 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(168\) 0 0
\(169\) 0.475953 0.0366118
\(170\) 0 0
\(171\) 6.74926 0.516129
\(172\) 0 0
\(173\) 10.2862 0.782045 0.391022 0.920381i \(-0.372121\pi\)
0.391022 + 0.920381i \(0.372121\pi\)
\(174\) 0 0
\(175\) −5.07830 −0.383884
\(176\) 0 0
\(177\) −12.7763 −0.960324
\(178\) 0 0
\(179\) 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 \(0.784532\pi\)
\(182\) 0 0
\(183\) 8.90587 0.658341
\(184\) 0 0
\(185\) 5.13021 0.377181
\(186\) 0 0
\(187\) −13.2349 −0.967833
\(188\) 0 0
\(189\) −5.07830 −0.369392
\(190\) 0 0
\(191\) −26.2094 −1.89645 −0.948223 0.317607i \(-0.897121\pi\)
−0.948223 + 0.317607i \(0.897121\pi\)
\(192\) 0 0
\(193\) −23.7643 −1.71059 −0.855295 0.518141i \(-0.826624\pi\)
−0.855295 + 0.518141i \(0.826624\pi\)
\(194\) 0 0
\(195\) 3.67096 0.262883
\(196\) 0 0
\(197\) −25.6281 −1.82593 −0.912964 0.408041i \(-0.866212\pi\)
−0.912964 + 0.408041i \(0.866212\pi\)
\(198\) 0 0
\(199\) −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 0
\(203\) 5.07830 0.356427
\(204\) 0 0
\(205\) 8.81469 0.615644
\(206\) 0 0
\(207\) 3.21234 0.223273
\(208\) 0 0
\(209\) 34.2748 2.37084
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) −0.195007 −0.0133617
\(214\) 0 0
\(215\) −2.61968 −0.178661
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 1.13021 0.0763728
\(220\) 0 0
\(221\) 9.56714 0.643555
\(222\) 0 0
\(223\) −7.86597 −0.526744 −0.263372 0.964694i \(-0.584835\pi\)
−0.263372 + 0.964694i \(0.584835\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0.0654212 0.00434216 0.00217108 0.999998i \(-0.499309\pi\)
0.00217108 + 0.999998i \(0.499309\pi\)
\(228\) 0 0
\(229\) −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 0
\(233\) −8.18363 −0.536127 −0.268064 0.963401i \(-0.586384\pi\)
−0.268064 + 0.963401i \(0.586384\pi\)
\(234\) 0 0
\(235\) −1.11670 −0.0728457
\(236\) 0 0
\(237\) −15.3035 −0.994071
\(238\) 0 0
\(239\) 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 \(0.663571\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −18.7892 −1.20040
\(246\) 0 0
\(247\) −24.7763 −1.57648
\(248\) 0 0
\(249\) 18.1182 1.14819
\(250\) 0 0
\(251\) −24.9411 −1.57427 −0.787134 0.616783i \(-0.788436\pi\)
−0.787134 + 0.616783i \(0.788436\pi\)
\(252\) 0 0
\(253\) 16.3132 1.02560
\(254\) 0 0
\(255\) 2.60617 0.163205
\(256\) 0 0
\(257\) 1.14691 0.0715425 0.0357713 0.999360i \(-0.488611\pi\)
0.0357713 + 0.999360i \(0.488611\pi\)
\(258\) 0 0
\(259\) 26.0528 1.61884
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −0.685099 −0.0422450 −0.0211225 0.999777i \(-0.506724\pi\)
−0.0211225 + 0.999777i \(0.506724\pi\)
\(264\) 0 0
\(265\) −0.619678 −0.0380665
\(266\) 0 0
\(267\) −15.5942 −0.954347
\(268\) 0 0
\(269\) 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 \(-0.492123\pi\)
0.0247443 + 0.999694i \(0.492123\pi\)
\(272\) 0 0
\(273\) 18.6423 1.12828
\(274\) 0 0
\(275\) 5.07830 0.306233
\(276\) 0 0
\(277\) 9.85459 0.592105 0.296052 0.955172i \(-0.404330\pi\)
0.296052 + 0.955172i \(0.404330\pi\)
\(278\) 0 0
\(279\) 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 0
\(283\) −4.23341 −0.251650 −0.125825 0.992052i \(-0.540158\pi\)
−0.125825 + 0.992052i \(0.540158\pi\)
\(284\) 0 0
\(285\) −6.74926 −0.399792
\(286\) 0 0
\(287\) 44.7637 2.64231
\(288\) 0 0
\(289\) −10.2079 −0.600464
\(290\) 0 0
\(291\) −12.6671 −0.742561
\(292\) 0 0
\(293\) −13.3170 −0.777989 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(294\) 0 0
\(295\) 12.7763 0.743864
\(296\) 0 0
\(297\) 5.07830 0.294673
\(298\) 0 0
\(299\) −11.7924 −0.681970
\(300\) 0 0
\(301\) −13.3035 −0.766802
\(302\) 0 0
\(303\) 7.03990 0.404432
\(304\) 0 0
\(305\) −8.90587 −0.509949
\(306\) 0 0
\(307\) −14.7379 −0.841136 −0.420568 0.907261i \(-0.638169\pi\)
−0.420568 + 0.907261i \(0.638169\pi\)
\(308\) 0 0
\(309\) −2.65808 −0.151213
\(310\) 0 0
\(311\) 10.0226 0.568328 0.284164 0.958776i \(-0.408284\pi\)
0.284164 + 0.958776i \(0.408284\pi\)
\(312\) 0 0
\(313\) 4.08800 0.231067 0.115534 0.993304i \(-0.463142\pi\)
0.115534 + 0.993304i \(0.463142\pi\)
\(314\) 0 0
\(315\) 5.07830 0.286130
\(316\) 0 0
\(317\) −12.7628 −0.716829 −0.358414 0.933563i \(-0.616683\pi\)
−0.358414 + 0.933563i \(0.616683\pi\)
\(318\) 0 0
\(319\) −5.07830 −0.284330
\(320\) 0 0
\(321\) −4.81469 −0.268730
\(322\) 0 0
\(323\) −17.5897 −0.978718
\(324\) 0 0
\(325\) −3.67096 −0.203628
\(326\) 0 0
\(327\) 3.86597 0.213789
\(328\) 0 0
\(329\) −5.67096 −0.312650
\(330\) 0 0
\(331\) −5.83576 −0.320762 −0.160381 0.987055i \(-0.551272\pi\)
−0.160381 + 0.987055i \(0.551272\pi\)
\(332\) 0 0
\(333\) −5.13021 −0.281134
\(334\) 0 0
\(335\) −0.524047 −0.0286317
\(336\) 0 0
\(337\) 1.95253 0.106361 0.0531807 0.998585i \(-0.483064\pi\)
0.0531807 + 0.998585i \(0.483064\pi\)
\(338\) 0 0
\(339\) −8.73575 −0.474461
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −59.8690 −3.23262
\(344\) 0 0
\(345\) −3.21234 −0.172946
\(346\) 0 0
\(347\) −17.3960 −0.933864 −0.466932 0.884293i \(-0.654641\pi\)
−0.466932 + 0.884293i \(0.654641\pi\)
\(348\) 0 0
\(349\) −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
\(352\) 0 0
\(353\) −29.7590 −1.58391 −0.791955 0.610580i \(-0.790936\pi\)
−0.791955 + 0.610580i \(0.790936\pi\)
\(354\) 0 0
\(355\) 0.195007 0.0103499
\(356\) 0 0
\(357\) 13.2349 0.700466
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 14.7892 0.776230
\(364\) 0 0
\(365\) −1.13021 −0.0591581
\(366\) 0 0
\(367\) 12.8675 0.671677 0.335838 0.941920i \(-0.390980\pi\)
0.335838 + 0.941920i \(0.390980\pi\)
\(368\) 0 0
\(369\) −8.81469 −0.458874
\(370\) 0 0
\(371\) −3.14691 −0.163380
\(372\) 0 0
\(373\) −13.4639 −0.697133 −0.348566 0.937284i \(-0.613331\pi\)
−0.348566 + 0.937284i \(0.613331\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 3.67096 0.189064
\(378\) 0 0
\(379\) 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
\(382\) 0 0
\(383\) 20.0836 1.02622 0.513111 0.858322i \(-0.328493\pi\)
0.513111 + 0.858322i \(0.328493\pi\)
\(384\) 0 0
\(385\) 25.7892 1.31434
\(386\) 0 0
\(387\) 2.61968 0.133166
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) −12.9745 −0.654476
\(394\) 0 0
\(395\) 15.3035 0.770004
\(396\) 0 0
\(397\) −25.9232 −1.30105 −0.650524 0.759486i \(-0.725451\pi\)
−0.650524 + 0.759486i \(0.725451\pi\)
\(398\) 0 0
\(399\) −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 0
\(403\) −2.89149 −0.144035
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −26.0528 −1.29139
\(408\) 0 0
\(409\) 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
\(412\) 0 0
\(413\) 64.8819 3.19263
\(414\) 0 0
\(415\) −18.1182 −0.889388
\(416\) 0 0
\(417\) 19.8276 0.970960
\(418\) 0 0
\(419\) −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 0
\(423\) 1.11670 0.0542960
\(424\) 0 0
\(425\) −2.60617 −0.126418
\(426\) 0 0
\(427\) −45.2267 −2.18867
\(428\) 0 0
\(429\) −18.6423 −0.900056
\(430\) 0 0
\(431\) 0.390015 0.0187864 0.00939318 0.999956i \(-0.497010\pi\)
0.00939318 + 0.999956i \(0.497010\pi\)
\(432\) 0 0
\(433\) −29.6989 −1.42724 −0.713618 0.700535i \(-0.752945\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) 21.6809 1.03714
\(438\) 0 0
\(439\) 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
\(442\) 0 0
\(443\) 11.1091 0.527808 0.263904 0.964549i \(-0.414990\pi\)
0.263904 + 0.964549i \(0.414990\pi\)
\(444\) 0 0
\(445\) 15.5942 0.739234
\(446\) 0 0
\(447\) 10.7763 0.509701
\(448\) 0 0
\(449\) −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
\(452\) 0 0
\(453\) 5.49853 0.258343
\(454\) 0 0
\(455\) −18.6423 −0.873962
\(456\) 0 0
\(457\) −24.3742 −1.14018 −0.570088 0.821583i \(-0.693091\pi\)
−0.570088 + 0.821583i \(0.693091\pi\)
\(458\) 0 0
\(459\) −2.60617 −0.121645
\(460\) 0 0
\(461\) −12.9789 −0.604489 −0.302244 0.953230i \(-0.597736\pi\)
−0.302244 + 0.953230i \(0.597736\pi\)
\(462\) 0 0
\(463\) 11.2619 0.523386 0.261693 0.965151i \(-0.415719\pi\)
0.261693 + 0.965151i \(0.415719\pi\)
\(464\) 0 0
\(465\) −0.787665 −0.0365271
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −2.66127 −0.122886
\(470\) 0 0
\(471\) 5.77566 0.266128
\(472\) 0 0
\(473\) 13.3035 0.611697
\(474\) 0 0
\(475\) 6.74926 0.309677
\(476\) 0 0
\(477\) 0.619678 0.0283731
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) −16.3132 −0.742277
\(484\) 0 0
\(485\) 12.6671 0.575185
\(486\) 0 0
\(487\) 20.8417 0.944428 0.472214 0.881484i \(-0.343455\pi\)
0.472214 + 0.881484i \(0.343455\pi\)
\(488\) 0 0
\(489\) −7.14691 −0.323195
\(490\) 0 0
\(491\) 19.1021 0.862067 0.431034 0.902336i \(-0.358149\pi\)
0.431034 + 0.902336i \(0.358149\pi\)
\(492\) 0 0
\(493\) 2.60617 0.117376
\(494\) 0 0
\(495\) −5.07830 −0.228253
\(496\) 0 0
\(497\) 0.990307 0.0444213
\(498\) 0 0
\(499\) 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
\(502\) 0 0
\(503\) −31.5865 −1.40837 −0.704187 0.710015i \(-0.748688\pi\)
−0.704187 + 0.710015i \(0.748688\pi\)
\(504\) 0 0
\(505\) −7.03990 −0.313272
\(506\) 0 0
\(507\) 0.475953 0.0211378
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) 6.74926 0.297987
\(514\) 0 0
\(515\) 2.65808 0.117129
\(516\) 0 0
\(517\) 5.67096 0.249409
\(518\) 0 0
\(519\) 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 0
\(523\) 39.9715 1.74783 0.873917 0.486076i \(-0.161572\pi\)
0.873917 + 0.486076i \(0.161572\pi\)
\(524\) 0 0
\(525\) −5.07830 −0.221635
\(526\) 0 0
\(527\) −2.05279 −0.0894208
\(528\) 0 0
\(529\) −12.6809 −0.551344
\(530\) 0 0
\(531\) −12.7763 −0.554444
\(532\) 0 0
\(533\) 32.3584 1.40160
\(534\) 0 0
\(535\) 4.81469 0.208157
\(536\) 0 0
\(537\) 12.1182 0.522939
\(538\) 0 0
\(539\) 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 0
\(543\) −20.9745 −0.900101
\(544\) 0 0
\(545\) −3.86597 −0.165600
\(546\) 0 0
\(547\) −31.3569 −1.34072 −0.670361 0.742035i \(-0.733861\pi\)
−0.670361 + 0.742035i \(0.733861\pi\)
\(548\) 0 0
\(549\) 8.90587 0.380093
\(550\) 0 0
\(551\) −6.74926 −0.287528
\(552\) 0 0
\(553\) 77.7159 3.30482
\(554\) 0 0
\(555\) 5.13021 0.217765
\(556\) 0 0
\(557\) 37.6514 1.59534 0.797670 0.603094i \(-0.206066\pi\)
0.797670 + 0.603094i \(0.206066\pi\)
\(558\) 0 0
\(559\) −9.61674 −0.406745
\(560\) 0 0
\(561\) −13.2349 −0.558778
\(562\) 0 0
\(563\) 39.0323 1.64501 0.822507 0.568755i \(-0.192575\pi\)
0.822507 + 0.568755i \(0.192575\pi\)
\(564\) 0 0
\(565\) 8.73575 0.367516
\(566\) 0 0
\(567\) −5.07830 −0.213269
\(568\) 0 0
\(569\) −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 \(-0.816966\pi\)
−0.839183 + 0.543849i \(0.816966\pi\)
\(572\) 0 0
\(573\) −26.2094 −1.09491
\(574\) 0 0
\(575\) 3.21234 0.133964
\(576\) 0 0
\(577\) −16.1091 −0.670632 −0.335316 0.942106i \(-0.608843\pi\)
−0.335316 + 0.942106i \(0.608843\pi\)
\(578\) 0 0
\(579\) −23.7643 −0.987610
\(580\) 0 0
\(581\) −92.0098 −3.81721
\(582\) 0 0
\(583\) 3.14691 0.130332
\(584\) 0 0
\(585\) 3.67096 0.151776
\(586\) 0 0
\(587\) 21.4331 0.884639 0.442320 0.896858i \(-0.354156\pi\)
0.442320 + 0.896858i \(0.354156\pi\)
\(588\) 0 0
\(589\) 5.31616 0.219048
\(590\) 0 0
\(591\) −25.6281 −1.05420
\(592\) 0 0
\(593\) 23.9886 0.985095 0.492547 0.870286i \(-0.336066\pi\)
0.492547 + 0.870286i \(0.336066\pi\)
\(594\) 0 0
\(595\) −13.2349 −0.542578
\(596\) 0 0
\(597\) −7.82757 −0.320361
\(598\) 0 0
\(599\) 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 \(0.634117\pi\)
\(602\) 0 0
\(603\) 0.524047 0.0213408
\(604\) 0 0
\(605\) −14.7892 −0.601265
\(606\) 0 0
\(607\) 38.7523 1.57291 0.786453 0.617650i \(-0.211915\pi\)
0.786453 + 0.617650i \(0.211915\pi\)
\(608\) 0 0
\(609\) 5.07830 0.205783
\(610\) 0 0
\(611\) −4.09938 −0.165843
\(612\) 0 0
\(613\) 24.6757 0.996640 0.498320 0.866993i \(-0.333950\pi\)
0.498320 + 0.866993i \(0.333950\pi\)
\(614\) 0 0
\(615\) 8.81469 0.355442
\(616\) 0 0
\(617\) 10.5249 0.423717 0.211859 0.977300i \(-0.432048\pi\)
0.211859 + 0.977300i \(0.432048\pi\)
\(618\) 0 0
\(619\) −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
\(622\) 0 0
\(623\) 79.1919 3.17276
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 34.2748 1.36880
\(628\) 0 0
\(629\) 13.3702 0.533105
\(630\) 0 0
\(631\) 13.2041 0.525649 0.262824 0.964844i \(-0.415346\pi\)
0.262824 + 0.964844i \(0.415346\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 10.7877 0.428095
\(636\) 0 0
\(637\) −68.9743 −2.73286
\(638\) 0 0
\(639\) −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 0
\(643\) −30.1370 −1.18849 −0.594244 0.804285i \(-0.702549\pi\)
−0.594244 + 0.804285i \(0.702549\pi\)
\(644\) 0 0
\(645\) −2.61968 −0.103150
\(646\) 0 0
\(647\) 42.7535 1.68081 0.840407 0.541955i \(-0.182316\pi\)
0.840407 + 0.541955i \(0.182316\pi\)
\(648\) 0 0
\(649\) −64.8819 −2.54684
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 33.8983 1.32654 0.663272 0.748379i \(-0.269167\pi\)
0.663272 + 0.748379i \(0.269167\pi\)
\(654\) 0 0
\(655\) 12.9745 0.506955
\(656\) 0 0
\(657\) 1.13021 0.0440939
\(658\) 0 0
\(659\) −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 0
\(663\) 9.56714 0.371557
\(664\) 0 0
\(665\) 34.2748 1.32912
\(666\) 0 0
\(667\) −3.21234 −0.124382
\(668\) 0 0
\(669\) −7.86597 −0.304116
\(670\) 0 0
\(671\) 45.2267 1.74596
\(672\) 0 0
\(673\) 30.6410 1.18112 0.590562 0.806992i \(-0.298906\pi\)
0.590562 + 0.806992i \(0.298906\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −40.8954 −1.57174 −0.785868 0.618394i \(-0.787784\pi\)
−0.785868 + 0.618394i \(0.787784\pi\)
\(678\) 0 0
\(679\) 64.3276 2.46867
\(680\) 0 0
\(681\) 0.0654212 0.00250694
\(682\) 0 0
\(683\) −21.2317 −0.812409 −0.406205 0.913782i \(-0.633148\pi\)
−0.406205 + 0.913782i \(0.633148\pi\)
\(684\) 0 0
\(685\) 4.08212 0.155970
\(686\) 0 0
\(687\) −3.87041 −0.147665
\(688\) 0 0
\(689\) −2.27481 −0.0866635
\(690\) 0 0
\(691\) 13.9842 0.531983 0.265992 0.963975i \(-0.414301\pi\)
0.265992 + 0.963975i \(0.414301\pi\)
\(692\) 0 0
\(693\) −25.7892 −0.979649
\(694\) 0 0
\(695\) −19.8276 −0.752103
\(696\) 0 0
\(697\) 22.9725 0.870147
\(698\) 0 0
\(699\) −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 0
\(703\) −34.6252 −1.30591
\(704\) 0 0
\(705\) −1.11670 −0.0420575
\(706\) 0 0
\(707\) −35.7508 −1.34455
\(708\) 0 0
\(709\) 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
\(712\) 0 0
\(713\) 2.53024 0.0947583
\(714\) 0 0
\(715\) 18.6423 0.697181
\(716\) 0 0
\(717\) 11.2651 0.420704
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) −15.2619 −0.567597
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 17.7243 0.657358 0.328679 0.944442i \(-0.393397\pi\)
0.328679 + 0.944442i \(0.393397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.82732 −0.252518
\(732\) 0 0
\(733\) −5.67184 −0.209494 −0.104747 0.994499i \(-0.533403\pi\)
−0.104747 + 0.994499i \(0.533403\pi\)
\(734\) 0 0
\(735\) −18.7892 −0.693049
\(736\) 0 0
\(737\) 2.66127 0.0980291
\(738\) 0 0
\(739\) −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 0
\(743\) 0.413474 0.0151689 0.00758445 0.999971i \(-0.497586\pi\)
0.00758445 + 0.999971i \(0.497586\pi\)
\(744\) 0 0
\(745\) −10.7763 −0.394813
\(746\) 0 0
\(747\) 18.1182 0.662911
\(748\) 0 0
\(749\) 24.4504 0.893399
\(750\) 0 0
\(751\) −9.71381 −0.354462 −0.177231 0.984169i \(-0.556714\pi\)
−0.177231 + 0.984169i \(0.556714\pi\)
\(752\) 0 0
\(753\) −24.9411 −0.908904
\(754\) 0 0
\(755\) −5.49853 −0.200112
\(756\) 0 0
\(757\) 22.3877 0.813695 0.406847 0.913496i \(-0.366628\pi\)
0.406847 + 0.913496i \(0.366628\pi\)
\(758\) 0 0
\(759\) 16.3132 0.592132
\(760\) 0 0
\(761\) 44.0836 1.59803 0.799014 0.601313i \(-0.205355\pi\)
0.799014 + 0.601313i \(0.205355\pi\)
\(762\) 0 0
\(763\) −19.6326 −0.710746
\(764\) 0 0
\(765\) 2.60617 0.0942262
\(766\) 0 0
\(767\) 46.9012 1.69351
\(768\) 0 0
\(769\) −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
\(772\) 0 0
\(773\) −25.5807 −0.920072 −0.460036 0.887900i \(-0.652164\pi\)
−0.460036 + 0.887900i \(0.652164\pi\)
\(774\) 0 0
\(775\) 0.787665 0.0282937
\(776\) 0 0
\(777\) 26.0528 0.934638
\(778\) 0 0
\(779\) −59.4926 −2.13155
\(780\) 0 0
\(781\) −0.990307 −0.0354360
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −5.77566 −0.206142
\(786\) 0 0
\(787\) 8.73362 0.311320 0.155660 0.987811i \(-0.450250\pi\)
0.155660 + 0.987811i \(0.450250\pi\)
\(788\) 0 0
\(789\) −0.685099 −0.0243902
\(790\) 0 0
\(791\) 44.3628 1.57736
\(792\) 0 0
\(793\) −32.6931 −1.16097
\(794\) 0 0
\(795\) −0.619678 −0.0219777
\(796\) 0 0
\(797\) 8.43874 0.298915 0.149458 0.988768i \(-0.452247\pi\)
0.149458 + 0.988768i \(0.452247\pi\)
\(798\) 0 0
\(799\) −2.91032 −0.102960
\(800\) 0 0
\(801\) −15.5942 −0.550993
\(802\) 0 0
\(803\) 5.73957 0.202545
\(804\) 0 0
\(805\) 16.3132 0.574965
\(806\) 0 0
\(807\) 4.22522 0.148735
\(808\) 0 0
\(809\) −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 \(-0.481848\pi\)
0.0569963 + 0.998374i \(0.481848\pi\)
\(812\) 0 0
\(813\) 0.814686 0.0285723
\(814\) 0 0
\(815\) 7.14691 0.250345
\(816\) 0 0
\(817\) 17.6809 0.618576
\(818\) 0 0
\(819\) 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 0
\(823\) 36.4648 1.27108 0.635542 0.772066i \(-0.280777\pi\)
0.635542 + 0.772066i \(0.280777\pi\)
\(824\) 0 0
\(825\) 5.07830 0.176804
\(826\) 0 0
\(827\) 15.9772 0.555583 0.277792 0.960641i \(-0.410398\pi\)
0.277792 + 0.960641i \(0.410398\pi\)
\(828\) 0 0
\(829\) 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
\(832\) 0 0
\(833\) −48.9677 −1.69663
\(834\) 0 0
\(835\) 17.1009 0.591800
\(836\) 0 0
\(837\) 0.787665 0.0272257
\(838\) 0 0
\(839\) −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 0
\(843\) 12.2297 0.421212
\(844\) 0 0
\(845\) −0.475953 −0.0163733
\(846\) 0 0
\(847\) −75.1039 −2.58060
\(848\) 0 0
\(849\) −4.23341 −0.145290
\(850\) 0 0
\(851\) −16.4800 −0.564926
\(852\) 0 0
\(853\) −36.0197 −1.23329 −0.616646 0.787241i \(-0.711509\pi\)
−0.616646 + 0.787241i \(0.711509\pi\)
\(854\) 0 0
\(855\) −6.74926 −0.230820
\(856\) 0 0
\(857\) −27.4217 −0.936708 −0.468354 0.883541i \(-0.655153\pi\)
−0.468354 + 0.883541i \(0.655153\pi\)
\(858\) 0 0
\(859\) −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 0
\(863\) −19.7540 −0.672433 −0.336216 0.941785i \(-0.609147\pi\)
−0.336216 + 0.941785i \(0.609147\pi\)
\(864\) 0 0
\(865\) −10.2862 −0.349741
\(866\) 0 0
\(867\) −10.2079 −0.346678
\(868\) 0 0
\(869\) −77.7159 −2.63633
\(870\) 0 0
\(871\) −1.92375 −0.0651839
\(872\) 0 0
\(873\) −12.6671 −0.428718
\(874\) 0 0
\(875\) 5.07830 0.171678
\(876\) 0 0
\(877\) 42.2030 1.42509 0.712547 0.701624i \(-0.247541\pi\)
0.712547 + 0.701624i \(0.247541\pi\)
\(878\) 0 0
\(879\) −13.3170 −0.449172
\(880\) 0 0
\(881\) −33.0927 −1.11492 −0.557461 0.830203i \(-0.688224\pi\)
−0.557461 + 0.830203i \(0.688224\pi\)
\(882\) 0 0
\(883\) 26.2634 0.883835 0.441917 0.897056i \(-0.354298\pi\)
0.441917 + 0.897056i \(0.354298\pi\)
\(884\) 0 0
\(885\) 12.7763 0.429470
\(886\) 0 0
\(887\) −14.0716 −0.472479 −0.236239 0.971695i \(-0.575915\pi\)
−0.236239 + 0.971695i \(0.575915\pi\)
\(888\) 0 0
\(889\) 54.7830 1.83736
\(890\) 0 0
\(891\) 5.07830 0.170130
\(892\) 0 0
\(893\) 7.53693 0.252214
\(894\) 0 0
\(895\) −12.1182 −0.405067
\(896\) 0 0
\(897\) −11.7924 −0.393735
\(898\) 0 0
\(899\) −0.787665 −0.0262701
\(900\) 0 0
\(901\) −1.61499 −0.0538030
\(902\) 0 0
\(903\) −13.3035 −0.442713
\(904\) 0 0
\(905\) 20.9745 0.697215
\(906\) 0 0
\(907\) −11.9810 −0.397822 −0.198911 0.980018i \(-0.563740\pi\)
−0.198911 + 0.980018i \(0.563740\pi\)
\(908\) 0 0
\(909\) 7.03990 0.233499
\(910\) 0 0
\(911\) −27.5300 −0.912109 −0.456055 0.889952i \(-0.650738\pi\)
−0.456055 + 0.889952i \(0.650738\pi\)
\(912\) 0 0
\(913\) 92.0098 3.04508
\(914\) 0 0
\(915\) −8.90587 −0.294419
\(916\) 0 0
\(917\) 65.8884 2.17583
\(918\) 0 0
\(919\) 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
\(922\) 0 0
\(923\) 0.715865 0.0235630
\(924\) 0 0
\(925\) −5.13021 −0.168680
\(926\) 0 0
\(927\) −2.65808 −0.0873027
\(928\) 0 0
\(929\) 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
\(932\) 0 0
\(933\) 10.0226 0.328124
\(934\) 0 0
\(935\) 13.2349 0.432828
\(936\) 0 0
\(937\) −36.0446 −1.17753 −0.588763 0.808306i \(-0.700385\pi\)
−0.588763 + 0.808306i \(0.700385\pi\)
\(938\) 0 0
\(939\) 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 0
\(943\) −28.3157 −0.922087
\(944\) 0 0
\(945\) 5.07830 0.165197
\(946\) 0 0
\(947\) −18.9555 −0.615970 −0.307985 0.951391i \(-0.599655\pi\)
−0.307985 + 0.951391i \(0.599655\pi\)
\(948\) 0 0
\(949\) −4.14897 −0.134681
\(950\) 0 0
\(951\) −12.7628 −0.413861
\(952\) 0 0
\(953\) −30.2761 −0.980738 −0.490369 0.871515i \(-0.663138\pi\)
−0.490369 + 0.871515i \(0.663138\pi\)
\(954\) 0 0
\(955\) 26.2094 0.848116
\(956\) 0 0
\(957\) −5.07830 −0.164158
\(958\) 0 0
\(959\) 20.7303 0.669415
\(960\) 0 0
\(961\) −30.3796 −0.979987
\(962\) 0 0
\(963\) −4.81469 −0.155151
\(964\) 0 0
\(965\) 23.7643 0.764999
\(966\) 0 0
\(967\) 20.4812 0.658631 0.329316 0.944220i \(-0.393182\pi\)
0.329316 + 0.944220i \(0.393182\pi\)
\(968\) 0 0
\(969\) −17.5897 −0.565063
\(970\) 0 0
\(971\) −44.1566 −1.41705 −0.708526 0.705684i \(-0.750640\pi\)
−0.708526 + 0.705684i \(0.750640\pi\)
\(972\) 0 0
\(973\) −100.690 −3.22799
\(974\) 0 0
\(975\) −3.67096 −0.117565
\(976\) 0 0
\(977\) 0.492580 0.0157590 0.00787952 0.999969i \(-0.497492\pi\)
0.00787952 + 0.999969i \(0.497492\pi\)
\(978\) 0 0
\(979\) −79.1919 −2.53098
\(980\) 0 0
\(981\) 3.86597 0.123431
\(982\) 0 0
\(983\) −27.5513 −0.878750 −0.439375 0.898304i \(-0.644800\pi\)
−0.439375 + 0.898304i \(0.644800\pi\)
\(984\) 0 0
\(985\) 25.6281 0.816580
\(986\) 0 0
\(987\) −5.67096 −0.180509
\(988\) 0 0
\(989\) 8.41529 0.267590
\(990\) 0 0
\(991\) −23.0927 −0.733563 −0.366782 0.930307i \(-0.619540\pi\)
−0.366782 + 0.930307i \(0.619540\pi\)
\(992\) 0 0
\(993\) −5.83576 −0.185192
\(994\) 0 0
\(995\) 7.82757 0.248151
\(996\) 0 0
\(997\) 7.79593 0.246900 0.123450 0.992351i \(-0.460604\pi\)
0.123450 + 0.992351i \(0.460604\pi\)
\(998\) 0 0
\(999\) −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 6960.2.a.co.1.1 4
4.3 odd 2 435.2.a.j.1.3 4
12.11 even 2 1305.2.a.r.1.2 4
20.3 even 4 2175.2.c.n.349.4 8
20.7 even 4 2175.2.c.n.349.5 8
20.19 odd 2 2175.2.a.v.1.2 4
60.59 even 2 6525.2.a.bi.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 4.3 odd 2
1305.2.a.r.1.2 4 12.11 even 2
2175.2.a.v.1.2 4 20.19 odd 2
2175.2.c.n.349.4 8 20.3 even 4
2175.2.c.n.349.5 8 20.7 even 4
6525.2.a.bi.1.3 4 60.59 even 2
6960.2.a.co.1.1 4 1.1 even 1 trivial