Properties

Label 945.2.a.c.1.1
Level $945$
Weight $2$
Character 945.1
Self dual yes
Analytic conductor $7.546$
Analytic rank $1$
Dimension $2$
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] = [945,2,Mod(1,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, 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("945.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.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: \(7.54586299101\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 945.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} +1.23607 q^{11} -6.47214 q^{13} +1.61803 q^{14} -4.85410 q^{16} -0.236068 q^{17} +1.47214 q^{19} +0.618034 q^{20} -2.00000 q^{22} -1.76393 q^{23} +1.00000 q^{25} +10.4721 q^{26} -0.618034 q^{28} +5.70820 q^{29} -9.00000 q^{31} +3.38197 q^{32} +0.381966 q^{34} -1.00000 q^{35} +3.70820 q^{37} -2.38197 q^{38} +2.23607 q^{40} -0.472136 q^{41} +0.472136 q^{43} +0.763932 q^{44} +2.85410 q^{46} -4.00000 q^{47} +1.00000 q^{49} -1.61803 q^{50} -4.00000 q^{52} -9.94427 q^{53} +1.23607 q^{55} -2.23607 q^{56} -9.23607 q^{58} -1.70820 q^{59} -12.7082 q^{61} +14.5623 q^{62} +4.23607 q^{64} -6.47214 q^{65} -6.94427 q^{67} -0.145898 q^{68} +1.61803 q^{70} +2.00000 q^{71} -8.47214 q^{73} -6.00000 q^{74} +0.909830 q^{76} -1.23607 q^{77} +5.18034 q^{79} -4.85410 q^{80} +0.763932 q^{82} -11.0000 q^{83} -0.236068 q^{85} -0.763932 q^{86} +2.76393 q^{88} +14.1803 q^{89} +6.47214 q^{91} -1.09017 q^{92} +6.47214 q^{94} +1.47214 q^{95} -0.763932 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} - 2 q^{11} - 4 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} - 6 q^{19} - q^{20} - 4 q^{22} - 8 q^{23} + 2 q^{25} + 12 q^{26} + q^{28} - 2 q^{29} - 18 q^{31} + 9 q^{32} + 3 q^{34} - 2 q^{35} - 6 q^{37} - 7 q^{38} + 8 q^{41} - 8 q^{43} + 6 q^{44} - q^{46} - 8 q^{47} + 2 q^{49} - q^{50} - 8 q^{52} - 2 q^{53} - 2 q^{55} - 14 q^{58} + 10 q^{59} - 12 q^{61} + 9 q^{62} + 4 q^{64} - 4 q^{65} + 4 q^{67} - 7 q^{68} + q^{70} + 4 q^{71} - 8 q^{73} - 12 q^{74} + 13 q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 6 q^{82} - 22 q^{83} + 4 q^{85} - 6 q^{86} + 10 q^{88} + 6 q^{89} + 4 q^{91} + 9 q^{92} + 4 q^{94} - 6 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) −1.61803 −0.511667
\(11\) 1.23607 0.372689 0.186344 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 0 0
\(13\) −6.47214 −1.79505 −0.897524 0.440966i \(-0.854636\pi\)
−0.897524 + 0.440966i \(0.854636\pi\)
\(14\) 1.61803 0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −0.236068 −0.0572549 −0.0286274 0.999590i \(-0.509114\pi\)
−0.0286274 + 0.999590i \(0.509114\pi\)
\(18\) 0 0
\(19\) 1.47214 0.337731 0.168866 0.985639i \(-0.445990\pi\)
0.168866 + 0.985639i \(0.445990\pi\)
\(20\) 0.618034 0.138197
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −1.76393 −0.367805 −0.183903 0.982944i \(-0.558873\pi\)
−0.183903 + 0.982944i \(0.558873\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 10.4721 2.05375
\(27\) 0 0
\(28\) −0.618034 −0.116797
\(29\) 5.70820 1.05999 0.529993 0.848002i \(-0.322194\pi\)
0.529993 + 0.848002i \(0.322194\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) 0.381966 0.0655066
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 3.70820 0.609625 0.304812 0.952412i \(-0.401406\pi\)
0.304812 + 0.952412i \(0.401406\pi\)
\(38\) −2.38197 −0.386406
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) −0.472136 −0.0737352 −0.0368676 0.999320i \(-0.511738\pi\)
−0.0368676 + 0.999320i \(0.511738\pi\)
\(42\) 0 0
\(43\) 0.472136 0.0720001 0.0360000 0.999352i \(-0.488538\pi\)
0.0360000 + 0.999352i \(0.488538\pi\)
\(44\) 0.763932 0.115167
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.61803 −0.228825
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −9.94427 −1.36595 −0.682975 0.730441i \(-0.739314\pi\)
−0.682975 + 0.730441i \(0.739314\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) −9.23607 −1.21276
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −12.7082 −1.62712 −0.813559 0.581482i \(-0.802473\pi\)
−0.813559 + 0.581482i \(0.802473\pi\)
\(62\) 14.5623 1.84941
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) −6.94427 −0.848378 −0.424189 0.905574i \(-0.639441\pi\)
−0.424189 + 0.905574i \(0.639441\pi\)
\(68\) −0.145898 −0.0176927
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −8.47214 −0.991589 −0.495794 0.868440i \(-0.665123\pi\)
−0.495794 + 0.868440i \(0.665123\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0.909830 0.104365
\(77\) −1.23607 −0.140863
\(78\) 0 0
\(79\) 5.18034 0.582834 0.291417 0.956596i \(-0.405873\pi\)
0.291417 + 0.956596i \(0.405873\pi\)
\(80\) −4.85410 −0.542705
\(81\) 0 0
\(82\) 0.763932 0.0843622
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) −0.236068 −0.0256052
\(86\) −0.763932 −0.0823769
\(87\) 0 0
\(88\) 2.76393 0.294636
\(89\) 14.1803 1.50311 0.751557 0.659669i \(-0.229303\pi\)
0.751557 + 0.659669i \(0.229303\pi\)
\(90\) 0 0
\(91\) 6.47214 0.678464
\(92\) −1.09017 −0.113658
\(93\) 0 0
\(94\) 6.47214 0.667550
\(95\) 1.47214 0.151038
\(96\) 0 0
\(97\) −0.763932 −0.0775655 −0.0387828 0.999248i \(-0.512348\pi\)
−0.0387828 + 0.999248i \(0.512348\pi\)
\(98\) −1.61803 −0.163446
\(99\) 0 0
\(100\) 0.618034 0.0618034
\(101\) −16.1803 −1.61000 −0.805002 0.593272i \(-0.797836\pi\)
−0.805002 + 0.593272i \(0.797836\pi\)
\(102\) 0 0
\(103\) −15.2361 −1.50125 −0.750627 0.660726i \(-0.770249\pi\)
−0.750627 + 0.660726i \(0.770249\pi\)
\(104\) −14.4721 −1.41911
\(105\) 0 0
\(106\) 16.0902 1.56282
\(107\) −15.4164 −1.49036 −0.745180 0.666863i \(-0.767637\pi\)
−0.745180 + 0.666863i \(0.767637\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 4.85410 0.458670
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −1.76393 −0.164488
\(116\) 3.52786 0.327554
\(117\) 0 0
\(118\) 2.76393 0.254441
\(119\) 0.236068 0.0216403
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 20.5623 1.86162
\(123\) 0 0
\(124\) −5.56231 −0.499510
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.18034 −0.548416 −0.274208 0.961670i \(-0.588416\pi\)
−0.274208 + 0.961670i \(0.588416\pi\)
\(128\) −13.6180 −1.20368
\(129\) 0 0
\(130\) 10.4721 0.918467
\(131\) 21.2361 1.85540 0.927702 0.373322i \(-0.121781\pi\)
0.927702 + 0.373322i \(0.121781\pi\)
\(132\) 0 0
\(133\) −1.47214 −0.127650
\(134\) 11.2361 0.970648
\(135\) 0 0
\(136\) −0.527864 −0.0452640
\(137\) −15.9443 −1.36221 −0.681106 0.732185i \(-0.738501\pi\)
−0.681106 + 0.732185i \(0.738501\pi\)
\(138\) 0 0
\(139\) 14.4721 1.22751 0.613755 0.789496i \(-0.289658\pi\)
0.613755 + 0.789496i \(0.289658\pi\)
\(140\) −0.618034 −0.0522334
\(141\) 0 0
\(142\) −3.23607 −0.271565
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 5.70820 0.474041
\(146\) 13.7082 1.13450
\(147\) 0 0
\(148\) 2.29180 0.188384
\(149\) −10.9443 −0.896590 −0.448295 0.893886i \(-0.647969\pi\)
−0.448295 + 0.893886i \(0.647969\pi\)
\(150\) 0 0
\(151\) −17.8885 −1.45575 −0.727875 0.685710i \(-0.759492\pi\)
−0.727875 + 0.685710i \(0.759492\pi\)
\(152\) 3.29180 0.267000
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) −5.23607 −0.417884 −0.208942 0.977928i \(-0.567002\pi\)
−0.208942 + 0.977928i \(0.567002\pi\)
\(158\) −8.38197 −0.666833
\(159\) 0 0
\(160\) 3.38197 0.267368
\(161\) 1.76393 0.139017
\(162\) 0 0
\(163\) 6.76393 0.529792 0.264896 0.964277i \(-0.414662\pi\)
0.264896 + 0.964277i \(0.414662\pi\)
\(164\) −0.291796 −0.0227854
\(165\) 0 0
\(166\) 17.7984 1.38142
\(167\) 11.9443 0.924276 0.462138 0.886808i \(-0.347083\pi\)
0.462138 + 0.886808i \(0.347083\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 0.381966 0.0292955
\(171\) 0 0
\(172\) 0.291796 0.0222492
\(173\) 9.18034 0.697968 0.348984 0.937129i \(-0.386527\pi\)
0.348984 + 0.937129i \(0.386527\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −22.9443 −1.71975
\(179\) −11.4164 −0.853302 −0.426651 0.904416i \(-0.640307\pi\)
−0.426651 + 0.904416i \(0.640307\pi\)
\(180\) 0 0
\(181\) 8.23607 0.612182 0.306091 0.952002i \(-0.400979\pi\)
0.306091 + 0.952002i \(0.400979\pi\)
\(182\) −10.4721 −0.776246
\(183\) 0 0
\(184\) −3.94427 −0.290776
\(185\) 3.70820 0.272633
\(186\) 0 0
\(187\) −0.291796 −0.0213382
\(188\) −2.47214 −0.180299
\(189\) 0 0
\(190\) −2.38197 −0.172806
\(191\) 25.2361 1.82602 0.913009 0.407940i \(-0.133753\pi\)
0.913009 + 0.407940i \(0.133753\pi\)
\(192\) 0 0
\(193\) 23.4164 1.68555 0.842775 0.538266i \(-0.180920\pi\)
0.842775 + 0.538266i \(0.180920\pi\)
\(194\) 1.23607 0.0887445
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) 17.9443 1.27848 0.639238 0.769009i \(-0.279250\pi\)
0.639238 + 0.769009i \(0.279250\pi\)
\(198\) 0 0
\(199\) −17.5279 −1.24252 −0.621259 0.783606i \(-0.713378\pi\)
−0.621259 + 0.783606i \(0.713378\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 26.1803 1.84204
\(203\) −5.70820 −0.400637
\(204\) 0 0
\(205\) −0.472136 −0.0329754
\(206\) 24.6525 1.71762
\(207\) 0 0
\(208\) 31.4164 2.17834
\(209\) 1.81966 0.125869
\(210\) 0 0
\(211\) −12.7082 −0.874869 −0.437434 0.899250i \(-0.644113\pi\)
−0.437434 + 0.899250i \(0.644113\pi\)
\(212\) −6.14590 −0.422102
\(213\) 0 0
\(214\) 24.9443 1.70516
\(215\) 0.472136 0.0321994
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) −1.61803 −0.109587
\(219\) 0 0
\(220\) 0.763932 0.0515043
\(221\) 1.52786 0.102775
\(222\) 0 0
\(223\) −12.6525 −0.847272 −0.423636 0.905832i \(-0.639247\pi\)
−0.423636 + 0.905832i \(0.639247\pi\)
\(224\) −3.38197 −0.225967
\(225\) 0 0
\(226\) 3.23607 0.215260
\(227\) −11.0000 −0.730096 −0.365048 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(228\) 0 0
\(229\) 3.76393 0.248728 0.124364 0.992237i \(-0.460311\pi\)
0.124364 + 0.992237i \(0.460311\pi\)
\(230\) 2.85410 0.188194
\(231\) 0 0
\(232\) 12.7639 0.837993
\(233\) 26.3607 1.72695 0.863473 0.504395i \(-0.168285\pi\)
0.863473 + 0.504395i \(0.168285\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) −1.05573 −0.0687220
\(237\) 0 0
\(238\) −0.381966 −0.0247592
\(239\) 3.52786 0.228199 0.114099 0.993469i \(-0.463602\pi\)
0.114099 + 0.993469i \(0.463602\pi\)
\(240\) 0 0
\(241\) 25.6525 1.65242 0.826211 0.563361i \(-0.190492\pi\)
0.826211 + 0.563361i \(0.190492\pi\)
\(242\) 15.3262 0.985208
\(243\) 0 0
\(244\) −7.85410 −0.502807
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −9.52786 −0.606243
\(248\) −20.1246 −1.27791
\(249\) 0 0
\(250\) −1.61803 −0.102333
\(251\) 7.23607 0.456737 0.228368 0.973575i \(-0.426661\pi\)
0.228368 + 0.973575i \(0.426661\pi\)
\(252\) 0 0
\(253\) −2.18034 −0.137077
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 24.5967 1.53430 0.767151 0.641466i \(-0.221673\pi\)
0.767151 + 0.641466i \(0.221673\pi\)
\(258\) 0 0
\(259\) −3.70820 −0.230417
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −34.3607 −2.12281
\(263\) −18.4721 −1.13904 −0.569520 0.821977i \(-0.692871\pi\)
−0.569520 + 0.821977i \(0.692871\pi\)
\(264\) 0 0
\(265\) −9.94427 −0.610872
\(266\) 2.38197 0.146048
\(267\) 0 0
\(268\) −4.29180 −0.262163
\(269\) 7.70820 0.469977 0.234989 0.971998i \(-0.424495\pi\)
0.234989 + 0.971998i \(0.424495\pi\)
\(270\) 0 0
\(271\) −20.4164 −1.24021 −0.620104 0.784519i \(-0.712910\pi\)
−0.620104 + 0.784519i \(0.712910\pi\)
\(272\) 1.14590 0.0694803
\(273\) 0 0
\(274\) 25.7984 1.55854
\(275\) 1.23607 0.0745377
\(276\) 0 0
\(277\) 30.3607 1.82420 0.912098 0.409972i \(-0.134461\pi\)
0.912098 + 0.409972i \(0.134461\pi\)
\(278\) −23.4164 −1.40442
\(279\) 0 0
\(280\) −2.23607 −0.133631
\(281\) 10.6525 0.635473 0.317737 0.948179i \(-0.397077\pi\)
0.317737 + 0.948179i \(0.397077\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 1.23607 0.0733471
\(285\) 0 0
\(286\) 12.9443 0.765411
\(287\) 0.472136 0.0278693
\(288\) 0 0
\(289\) −16.9443 −0.996722
\(290\) −9.23607 −0.542361
\(291\) 0 0
\(292\) −5.23607 −0.306418
\(293\) −9.29180 −0.542833 −0.271416 0.962462i \(-0.587492\pi\)
−0.271416 + 0.962462i \(0.587492\pi\)
\(294\) 0 0
\(295\) −1.70820 −0.0994555
\(296\) 8.29180 0.481951
\(297\) 0 0
\(298\) 17.7082 1.02581
\(299\) 11.4164 0.660228
\(300\) 0 0
\(301\) −0.472136 −0.0272135
\(302\) 28.9443 1.66556
\(303\) 0 0
\(304\) −7.14590 −0.409845
\(305\) −12.7082 −0.727670
\(306\) 0 0
\(307\) 23.1246 1.31979 0.659896 0.751357i \(-0.270600\pi\)
0.659896 + 0.751357i \(0.270600\pi\)
\(308\) −0.763932 −0.0435291
\(309\) 0 0
\(310\) 14.5623 0.827083
\(311\) −0.944272 −0.0535447 −0.0267724 0.999642i \(-0.508523\pi\)
−0.0267724 + 0.999642i \(0.508523\pi\)
\(312\) 0 0
\(313\) −10.4721 −0.591920 −0.295960 0.955200i \(-0.595640\pi\)
−0.295960 + 0.955200i \(0.595640\pi\)
\(314\) 8.47214 0.478110
\(315\) 0 0
\(316\) 3.20163 0.180106
\(317\) 11.9443 0.670857 0.335429 0.942066i \(-0.391119\pi\)
0.335429 + 0.942066i \(0.391119\pi\)
\(318\) 0 0
\(319\) 7.05573 0.395045
\(320\) 4.23607 0.236803
\(321\) 0 0
\(322\) −2.85410 −0.159053
\(323\) −0.347524 −0.0193368
\(324\) 0 0
\(325\) −6.47214 −0.359010
\(326\) −10.9443 −0.606147
\(327\) 0 0
\(328\) −1.05573 −0.0582928
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −19.4164 −1.06722 −0.533611 0.845730i \(-0.679165\pi\)
−0.533611 + 0.845730i \(0.679165\pi\)
\(332\) −6.79837 −0.373109
\(333\) 0 0
\(334\) −19.3262 −1.05748
\(335\) −6.94427 −0.379406
\(336\) 0 0
\(337\) 20.9443 1.14091 0.570454 0.821330i \(-0.306767\pi\)
0.570454 + 0.821330i \(0.306767\pi\)
\(338\) −46.7426 −2.54246
\(339\) 0 0
\(340\) −0.145898 −0.00791243
\(341\) −11.1246 −0.602432
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.05573 0.0569210
\(345\) 0 0
\(346\) −14.8541 −0.798561
\(347\) 20.9443 1.12435 0.562174 0.827019i \(-0.309965\pi\)
0.562174 + 0.827019i \(0.309965\pi\)
\(348\) 0 0
\(349\) −11.1803 −0.598470 −0.299235 0.954179i \(-0.596731\pi\)
−0.299235 + 0.954179i \(0.596731\pi\)
\(350\) 1.61803 0.0864876
\(351\) 0 0
\(352\) 4.18034 0.222813
\(353\) −4.47214 −0.238028 −0.119014 0.992893i \(-0.537973\pi\)
−0.119014 + 0.992893i \(0.537973\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 8.76393 0.464487
\(357\) 0 0
\(358\) 18.4721 0.976283
\(359\) −17.4164 −0.919203 −0.459601 0.888125i \(-0.652008\pi\)
−0.459601 + 0.888125i \(0.652008\pi\)
\(360\) 0 0
\(361\) −16.8328 −0.885938
\(362\) −13.3262 −0.700412
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −8.47214 −0.443452
\(366\) 0 0
\(367\) 3.41641 0.178335 0.0891675 0.996017i \(-0.471579\pi\)
0.0891675 + 0.996017i \(0.471579\pi\)
\(368\) 8.56231 0.446341
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) 9.94427 0.516281
\(372\) 0 0
\(373\) 15.4164 0.798231 0.399116 0.916901i \(-0.369317\pi\)
0.399116 + 0.916901i \(0.369317\pi\)
\(374\) 0.472136 0.0244136
\(375\) 0 0
\(376\) −8.94427 −0.461266
\(377\) −36.9443 −1.90273
\(378\) 0 0
\(379\) −33.6525 −1.72861 −0.864306 0.502967i \(-0.832242\pi\)
−0.864306 + 0.502967i \(0.832242\pi\)
\(380\) 0.909830 0.0466733
\(381\) 0 0
\(382\) −40.8328 −2.08919
\(383\) 10.0557 0.513824 0.256912 0.966435i \(-0.417295\pi\)
0.256912 + 0.966435i \(0.417295\pi\)
\(384\) 0 0
\(385\) −1.23607 −0.0629959
\(386\) −37.8885 −1.92848
\(387\) 0 0
\(388\) −0.472136 −0.0239691
\(389\) 24.6525 1.24993 0.624965 0.780653i \(-0.285113\pi\)
0.624965 + 0.780653i \(0.285113\pi\)
\(390\) 0 0
\(391\) 0.416408 0.0210587
\(392\) 2.23607 0.112938
\(393\) 0 0
\(394\) −29.0344 −1.46273
\(395\) 5.18034 0.260651
\(396\) 0 0
\(397\) −11.2361 −0.563922 −0.281961 0.959426i \(-0.590985\pi\)
−0.281961 + 0.959426i \(0.590985\pi\)
\(398\) 28.3607 1.42159
\(399\) 0 0
\(400\) −4.85410 −0.242705
\(401\) −15.5279 −0.775425 −0.387712 0.921780i \(-0.626735\pi\)
−0.387712 + 0.921780i \(0.626735\pi\)
\(402\) 0 0
\(403\) 58.2492 2.90160
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 9.23607 0.458378
\(407\) 4.58359 0.227200
\(408\) 0 0
\(409\) 4.23607 0.209460 0.104730 0.994501i \(-0.466602\pi\)
0.104730 + 0.994501i \(0.466602\pi\)
\(410\) 0.763932 0.0377279
\(411\) 0 0
\(412\) −9.41641 −0.463913
\(413\) 1.70820 0.0840552
\(414\) 0 0
\(415\) −11.0000 −0.539969
\(416\) −21.8885 −1.07317
\(417\) 0 0
\(418\) −2.94427 −0.144009
\(419\) −12.6525 −0.618114 −0.309057 0.951044i \(-0.600013\pi\)
−0.309057 + 0.951044i \(0.600013\pi\)
\(420\) 0 0
\(421\) 25.8328 1.25901 0.629507 0.776995i \(-0.283257\pi\)
0.629507 + 0.776995i \(0.283257\pi\)
\(422\) 20.5623 1.00096
\(423\) 0 0
\(424\) −22.2361 −1.07988
\(425\) −0.236068 −0.0114510
\(426\) 0 0
\(427\) 12.7082 0.614993
\(428\) −9.52786 −0.460547
\(429\) 0 0
\(430\) −0.763932 −0.0368401
\(431\) −9.52786 −0.458941 −0.229471 0.973316i \(-0.573699\pi\)
−0.229471 + 0.973316i \(0.573699\pi\)
\(432\) 0 0
\(433\) 0.763932 0.0367122 0.0183561 0.999832i \(-0.494157\pi\)
0.0183561 + 0.999832i \(0.494157\pi\)
\(434\) −14.5623 −0.699013
\(435\) 0 0
\(436\) 0.618034 0.0295985
\(437\) −2.59675 −0.124219
\(438\) 0 0
\(439\) −21.0000 −1.00228 −0.501138 0.865368i \(-0.667085\pi\)
−0.501138 + 0.865368i \(0.667085\pi\)
\(440\) 2.76393 0.131765
\(441\) 0 0
\(442\) −2.47214 −0.117588
\(443\) 30.7082 1.45899 0.729495 0.683986i \(-0.239755\pi\)
0.729495 + 0.683986i \(0.239755\pi\)
\(444\) 0 0
\(445\) 14.1803 0.672213
\(446\) 20.4721 0.969384
\(447\) 0 0
\(448\) −4.23607 −0.200135
\(449\) 7.05573 0.332980 0.166490 0.986043i \(-0.446757\pi\)
0.166490 + 0.986043i \(0.446757\pi\)
\(450\) 0 0
\(451\) −0.583592 −0.0274803
\(452\) −1.23607 −0.0581397
\(453\) 0 0
\(454\) 17.7984 0.835319
\(455\) 6.47214 0.303418
\(456\) 0 0
\(457\) −18.0689 −0.845227 −0.422613 0.906310i \(-0.638887\pi\)
−0.422613 + 0.906310i \(0.638887\pi\)
\(458\) −6.09017 −0.284575
\(459\) 0 0
\(460\) −1.09017 −0.0508294
\(461\) 28.4721 1.32608 0.663040 0.748584i \(-0.269266\pi\)
0.663040 + 0.748584i \(0.269266\pi\)
\(462\) 0 0
\(463\) −32.4721 −1.50911 −0.754554 0.656238i \(-0.772147\pi\)
−0.754554 + 0.656238i \(0.772147\pi\)
\(464\) −27.7082 −1.28632
\(465\) 0 0
\(466\) −42.6525 −1.97584
\(467\) −41.4721 −1.91910 −0.959551 0.281536i \(-0.909156\pi\)
−0.959551 + 0.281536i \(0.909156\pi\)
\(468\) 0 0
\(469\) 6.94427 0.320657
\(470\) 6.47214 0.298537
\(471\) 0 0
\(472\) −3.81966 −0.175814
\(473\) 0.583592 0.0268336
\(474\) 0 0
\(475\) 1.47214 0.0675462
\(476\) 0.145898 0.00668723
\(477\) 0 0
\(478\) −5.70820 −0.261087
\(479\) 21.5967 0.986781 0.493390 0.869808i \(-0.335757\pi\)
0.493390 + 0.869808i \(0.335757\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) −41.5066 −1.89057
\(483\) 0 0
\(484\) −5.85410 −0.266096
\(485\) −0.763932 −0.0346884
\(486\) 0 0
\(487\) −1.23607 −0.0560116 −0.0280058 0.999608i \(-0.508916\pi\)
−0.0280058 + 0.999608i \(0.508916\pi\)
\(488\) −28.4164 −1.28635
\(489\) 0 0
\(490\) −1.61803 −0.0730953
\(491\) 3.88854 0.175488 0.0877438 0.996143i \(-0.472034\pi\)
0.0877438 + 0.996143i \(0.472034\pi\)
\(492\) 0 0
\(493\) −1.34752 −0.0606894
\(494\) 15.4164 0.693617
\(495\) 0 0
\(496\) 43.6869 1.96160
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) 10.2361 0.458229 0.229115 0.973399i \(-0.426417\pi\)
0.229115 + 0.973399i \(0.426417\pi\)
\(500\) 0.618034 0.0276393
\(501\) 0 0
\(502\) −11.7082 −0.522563
\(503\) −9.94427 −0.443393 −0.221697 0.975116i \(-0.571159\pi\)
−0.221697 + 0.975116i \(0.571159\pi\)
\(504\) 0 0
\(505\) −16.1803 −0.720016
\(506\) 3.52786 0.156833
\(507\) 0 0
\(508\) −3.81966 −0.169470
\(509\) −36.3607 −1.61166 −0.805829 0.592148i \(-0.798280\pi\)
−0.805829 + 0.592148i \(0.798280\pi\)
\(510\) 0 0
\(511\) 8.47214 0.374785
\(512\) 5.29180 0.233867
\(513\) 0 0
\(514\) −39.7984 −1.75543
\(515\) −15.2361 −0.671381
\(516\) 0 0
\(517\) −4.94427 −0.217449
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) −14.4721 −0.634645
\(521\) −28.2918 −1.23949 −0.619743 0.784805i \(-0.712763\pi\)
−0.619743 + 0.784805i \(0.712763\pi\)
\(522\) 0 0
\(523\) 38.0689 1.66464 0.832318 0.554298i \(-0.187013\pi\)
0.832318 + 0.554298i \(0.187013\pi\)
\(524\) 13.1246 0.573351
\(525\) 0 0
\(526\) 29.8885 1.30320
\(527\) 2.12461 0.0925495
\(528\) 0 0
\(529\) −19.8885 −0.864719
\(530\) 16.0902 0.698912
\(531\) 0 0
\(532\) −0.909830 −0.0394461
\(533\) 3.05573 0.132358
\(534\) 0 0
\(535\) −15.4164 −0.666509
\(536\) −15.5279 −0.670702
\(537\) 0 0
\(538\) −12.4721 −0.537712
\(539\) 1.23607 0.0532412
\(540\) 0 0
\(541\) 8.47214 0.364246 0.182123 0.983276i \(-0.441703\pi\)
0.182123 + 0.983276i \(0.441703\pi\)
\(542\) 33.0344 1.41895
\(543\) 0 0
\(544\) −0.798374 −0.0342300
\(545\) 1.00000 0.0428353
\(546\) 0 0
\(547\) −6.76393 −0.289205 −0.144602 0.989490i \(-0.546190\pi\)
−0.144602 + 0.989490i \(0.546190\pi\)
\(548\) −9.85410 −0.420946
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 8.40325 0.357991
\(552\) 0 0
\(553\) −5.18034 −0.220290
\(554\) −49.1246 −2.08710
\(555\) 0 0
\(556\) 8.94427 0.379322
\(557\) −14.9443 −0.633209 −0.316605 0.948558i \(-0.602543\pi\)
−0.316605 + 0.948558i \(0.602543\pi\)
\(558\) 0 0
\(559\) −3.05573 −0.129244
\(560\) 4.85410 0.205123
\(561\) 0 0
\(562\) −17.2361 −0.727060
\(563\) 27.4164 1.15546 0.577732 0.816227i \(-0.303938\pi\)
0.577732 + 0.816227i \(0.303938\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 35.5967 1.49624
\(567\) 0 0
\(568\) 4.47214 0.187647
\(569\) 33.5279 1.40556 0.702781 0.711407i \(-0.251942\pi\)
0.702781 + 0.711407i \(0.251942\pi\)
\(570\) 0 0
\(571\) −5.18034 −0.216790 −0.108395 0.994108i \(-0.534571\pi\)
−0.108395 + 0.994108i \(0.534571\pi\)
\(572\) −4.94427 −0.206730
\(573\) 0 0
\(574\) −0.763932 −0.0318859
\(575\) −1.76393 −0.0735611
\(576\) 0 0
\(577\) 38.5410 1.60448 0.802242 0.596999i \(-0.203640\pi\)
0.802242 + 0.596999i \(0.203640\pi\)
\(578\) 27.4164 1.14037
\(579\) 0 0
\(580\) 3.52786 0.146487
\(581\) 11.0000 0.456357
\(582\) 0 0
\(583\) −12.2918 −0.509074
\(584\) −18.9443 −0.783920
\(585\) 0 0
\(586\) 15.0344 0.621067
\(587\) −46.3050 −1.91121 −0.955605 0.294651i \(-0.904797\pi\)
−0.955605 + 0.294651i \(0.904797\pi\)
\(588\) 0 0
\(589\) −13.2492 −0.545925
\(590\) 2.76393 0.113789
\(591\) 0 0
\(592\) −18.0000 −0.739795
\(593\) −23.1803 −0.951902 −0.475951 0.879472i \(-0.657896\pi\)
−0.475951 + 0.879472i \(0.657896\pi\)
\(594\) 0 0
\(595\) 0.236068 0.00967784
\(596\) −6.76393 −0.277061
\(597\) 0 0
\(598\) −18.4721 −0.755382
\(599\) 41.8885 1.71152 0.855760 0.517373i \(-0.173090\pi\)
0.855760 + 0.517373i \(0.173090\pi\)
\(600\) 0 0
\(601\) 4.23607 0.172793 0.0863964 0.996261i \(-0.472465\pi\)
0.0863964 + 0.996261i \(0.472465\pi\)
\(602\) 0.763932 0.0311355
\(603\) 0 0
\(604\) −11.0557 −0.449851
\(605\) −9.47214 −0.385097
\(606\) 0 0
\(607\) −2.58359 −0.104865 −0.0524324 0.998624i \(-0.516697\pi\)
−0.0524324 + 0.998624i \(0.516697\pi\)
\(608\) 4.97871 0.201914
\(609\) 0 0
\(610\) 20.5623 0.832543
\(611\) 25.8885 1.04734
\(612\) 0 0
\(613\) 25.7082 1.03834 0.519172 0.854670i \(-0.326240\pi\)
0.519172 + 0.854670i \(0.326240\pi\)
\(614\) −37.4164 −1.51000
\(615\) 0 0
\(616\) −2.76393 −0.111362
\(617\) 8.88854 0.357839 0.178920 0.983864i \(-0.442740\pi\)
0.178920 + 0.983864i \(0.442740\pi\)
\(618\) 0 0
\(619\) −10.8328 −0.435408 −0.217704 0.976015i \(-0.569857\pi\)
−0.217704 + 0.976015i \(0.569857\pi\)
\(620\) −5.56231 −0.223388
\(621\) 0 0
\(622\) 1.52786 0.0612618
\(623\) −14.1803 −0.568123
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 16.9443 0.677229
\(627\) 0 0
\(628\) −3.23607 −0.129133
\(629\) −0.875388 −0.0349040
\(630\) 0 0
\(631\) 17.1803 0.683939 0.341969 0.939711i \(-0.388906\pi\)
0.341969 + 0.939711i \(0.388906\pi\)
\(632\) 11.5836 0.460771
\(633\) 0 0
\(634\) −19.3262 −0.767543
\(635\) −6.18034 −0.245259
\(636\) 0 0
\(637\) −6.47214 −0.256435
\(638\) −11.4164 −0.451980
\(639\) 0 0
\(640\) −13.6180 −0.538300
\(641\) 13.1246 0.518391 0.259195 0.965825i \(-0.416543\pi\)
0.259195 + 0.965825i \(0.416543\pi\)
\(642\) 0 0
\(643\) −14.6525 −0.577837 −0.288919 0.957354i \(-0.593296\pi\)
−0.288919 + 0.957354i \(0.593296\pi\)
\(644\) 1.09017 0.0429587
\(645\) 0 0
\(646\) 0.562306 0.0221236
\(647\) −11.4721 −0.451016 −0.225508 0.974241i \(-0.572404\pi\)
−0.225508 + 0.974241i \(0.572404\pi\)
\(648\) 0 0
\(649\) −2.11146 −0.0828819
\(650\) 10.4721 0.410751
\(651\) 0 0
\(652\) 4.18034 0.163715
\(653\) 4.88854 0.191304 0.0956518 0.995415i \(-0.469506\pi\)
0.0956518 + 0.995415i \(0.469506\pi\)
\(654\) 0 0
\(655\) 21.2361 0.829762
\(656\) 2.29180 0.0894796
\(657\) 0 0
\(658\) −6.47214 −0.252310
\(659\) 13.5279 0.526971 0.263485 0.964663i \(-0.415128\pi\)
0.263485 + 0.964663i \(0.415128\pi\)
\(660\) 0 0
\(661\) −26.3607 −1.02531 −0.512656 0.858594i \(-0.671338\pi\)
−0.512656 + 0.858594i \(0.671338\pi\)
\(662\) 31.4164 1.22103
\(663\) 0 0
\(664\) −24.5967 −0.954539
\(665\) −1.47214 −0.0570870
\(666\) 0 0
\(667\) −10.0689 −0.389869
\(668\) 7.38197 0.285617
\(669\) 0 0
\(670\) 11.2361 0.434087
\(671\) −15.7082 −0.606408
\(672\) 0 0
\(673\) 35.2361 1.35825 0.679125 0.734022i \(-0.262359\pi\)
0.679125 + 0.734022i \(0.262359\pi\)
\(674\) −33.8885 −1.30534
\(675\) 0 0
\(676\) 17.8541 0.686696
\(677\) −15.5279 −0.596784 −0.298392 0.954443i \(-0.596450\pi\)
−0.298392 + 0.954443i \(0.596450\pi\)
\(678\) 0 0
\(679\) 0.763932 0.0293170
\(680\) −0.527864 −0.0202427
\(681\) 0 0
\(682\) 18.0000 0.689256
\(683\) 38.1246 1.45880 0.729399 0.684089i \(-0.239800\pi\)
0.729399 + 0.684089i \(0.239800\pi\)
\(684\) 0 0
\(685\) −15.9443 −0.609199
\(686\) 1.61803 0.0617768
\(687\) 0 0
\(688\) −2.29180 −0.0873739
\(689\) 64.3607 2.45195
\(690\) 0 0
\(691\) −27.3607 −1.04085 −0.520425 0.853908i \(-0.674226\pi\)
−0.520425 + 0.853908i \(0.674226\pi\)
\(692\) 5.67376 0.215684
\(693\) 0 0
\(694\) −33.8885 −1.28639
\(695\) 14.4721 0.548959
\(696\) 0 0
\(697\) 0.111456 0.00422170
\(698\) 18.0902 0.684723
\(699\) 0 0
\(700\) −0.618034 −0.0233595
\(701\) −46.9443 −1.77306 −0.886530 0.462670i \(-0.846891\pi\)
−0.886530 + 0.462670i \(0.846891\pi\)
\(702\) 0 0
\(703\) 5.45898 0.205889
\(704\) 5.23607 0.197342
\(705\) 0 0
\(706\) 7.23607 0.272333
\(707\) 16.1803 0.608524
\(708\) 0 0
\(709\) 38.9443 1.46258 0.731291 0.682065i \(-0.238918\pi\)
0.731291 + 0.682065i \(0.238918\pi\)
\(710\) −3.23607 −0.121447
\(711\) 0 0
\(712\) 31.7082 1.18832
\(713\) 15.8754 0.594538
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −7.05573 −0.263685
\(717\) 0 0
\(718\) 28.1803 1.05168
\(719\) 37.7771 1.40885 0.704424 0.709780i \(-0.251205\pi\)
0.704424 + 0.709780i \(0.251205\pi\)
\(720\) 0 0
\(721\) 15.2361 0.567421
\(722\) 27.2361 1.01362
\(723\) 0 0
\(724\) 5.09017 0.189175
\(725\) 5.70820 0.211997
\(726\) 0 0
\(727\) −15.0557 −0.558386 −0.279193 0.960235i \(-0.590067\pi\)
−0.279193 + 0.960235i \(0.590067\pi\)
\(728\) 14.4721 0.536373
\(729\) 0 0
\(730\) 13.7082 0.507363
\(731\) −0.111456 −0.00412236
\(732\) 0 0
\(733\) 25.1246 0.927999 0.463999 0.885836i \(-0.346414\pi\)
0.463999 + 0.885836i \(0.346414\pi\)
\(734\) −5.52786 −0.204037
\(735\) 0 0
\(736\) −5.96556 −0.219893
\(737\) −8.58359 −0.316181
\(738\) 0 0
\(739\) −10.2361 −0.376540 −0.188270 0.982117i \(-0.560288\pi\)
−0.188270 + 0.982117i \(0.560288\pi\)
\(740\) 2.29180 0.0842481
\(741\) 0 0
\(742\) −16.0902 −0.590689
\(743\) 6.11146 0.224208 0.112104 0.993697i \(-0.464241\pi\)
0.112104 + 0.993697i \(0.464241\pi\)
\(744\) 0 0
\(745\) −10.9443 −0.400967
\(746\) −24.9443 −0.913275
\(747\) 0 0
\(748\) −0.180340 −0.00659388
\(749\) 15.4164 0.563303
\(750\) 0 0
\(751\) −8.34752 −0.304605 −0.152303 0.988334i \(-0.548669\pi\)
−0.152303 + 0.988334i \(0.548669\pi\)
\(752\) 19.4164 0.708044
\(753\) 0 0
\(754\) 59.7771 2.17695
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) 1.41641 0.0514802 0.0257401 0.999669i \(-0.491806\pi\)
0.0257401 + 0.999669i \(0.491806\pi\)
\(758\) 54.4508 1.97774
\(759\) 0 0
\(760\) 3.29180 0.119406
\(761\) 2.47214 0.0896149 0.0448074 0.998996i \(-0.485733\pi\)
0.0448074 + 0.998996i \(0.485733\pi\)
\(762\) 0 0
\(763\) −1.00000 −0.0362024
\(764\) 15.5967 0.564271
\(765\) 0 0
\(766\) −16.2705 −0.587877
\(767\) 11.0557 0.399199
\(768\) 0 0
\(769\) 28.5967 1.03123 0.515613 0.856822i \(-0.327564\pi\)
0.515613 + 0.856822i \(0.327564\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) 14.4721 0.520864
\(773\) 47.1803 1.69696 0.848479 0.529228i \(-0.177519\pi\)
0.848479 + 0.529228i \(0.177519\pi\)
\(774\) 0 0
\(775\) −9.00000 −0.323290
\(776\) −1.70820 −0.0613209
\(777\) 0 0
\(778\) −39.8885 −1.43007
\(779\) −0.695048 −0.0249027
\(780\) 0 0
\(781\) 2.47214 0.0884600
\(782\) −0.673762 −0.0240937
\(783\) 0 0
\(784\) −4.85410 −0.173361
\(785\) −5.23607 −0.186883
\(786\) 0 0
\(787\) 34.1803 1.21840 0.609199 0.793018i \(-0.291491\pi\)
0.609199 + 0.793018i \(0.291491\pi\)
\(788\) 11.0902 0.395071
\(789\) 0 0
\(790\) −8.38197 −0.298217
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 82.2492 2.92076
\(794\) 18.1803 0.645196
\(795\) 0 0
\(796\) −10.8328 −0.383959
\(797\) −25.2918 −0.895881 −0.447941 0.894063i \(-0.647842\pi\)
−0.447941 + 0.894063i \(0.647842\pi\)
\(798\) 0 0
\(799\) 0.944272 0.0334059
\(800\) 3.38197 0.119571
\(801\) 0 0
\(802\) 25.1246 0.887181
\(803\) −10.4721 −0.369554
\(804\) 0 0
\(805\) 1.76393 0.0621704
\(806\) −94.2492 −3.31979
\(807\) 0 0
\(808\) −36.1803 −1.27282
\(809\) −29.0557 −1.02154 −0.510772 0.859716i \(-0.670641\pi\)
−0.510772 + 0.859716i \(0.670641\pi\)
\(810\) 0 0
\(811\) 35.0557 1.23097 0.615487 0.788147i \(-0.288960\pi\)
0.615487 + 0.788147i \(0.288960\pi\)
\(812\) −3.52786 −0.123804
\(813\) 0 0
\(814\) −7.41641 −0.259945
\(815\) 6.76393 0.236930
\(816\) 0 0
\(817\) 0.695048 0.0243167
\(818\) −6.85410 −0.239648
\(819\) 0 0
\(820\) −0.291796 −0.0101900
\(821\) −5.41641 −0.189034 −0.0945170 0.995523i \(-0.530131\pi\)
−0.0945170 + 0.995523i \(0.530131\pi\)
\(822\) 0 0
\(823\) 23.3050 0.812360 0.406180 0.913793i \(-0.366861\pi\)
0.406180 + 0.913793i \(0.366861\pi\)
\(824\) −34.0689 −1.18685
\(825\) 0 0
\(826\) −2.76393 −0.0961695
\(827\) 21.2918 0.740388 0.370194 0.928954i \(-0.379291\pi\)
0.370194 + 0.928954i \(0.379291\pi\)
\(828\) 0 0
\(829\) 28.4721 0.988878 0.494439 0.869212i \(-0.335374\pi\)
0.494439 + 0.869212i \(0.335374\pi\)
\(830\) 17.7984 0.617791
\(831\) 0 0
\(832\) −27.4164 −0.950493
\(833\) −0.236068 −0.00817927
\(834\) 0 0
\(835\) 11.9443 0.413349
\(836\) 1.12461 0.0388955
\(837\) 0 0
\(838\) 20.4721 0.707198
\(839\) 3.81966 0.131869 0.0659347 0.997824i \(-0.478997\pi\)
0.0659347 + 0.997824i \(0.478997\pi\)
\(840\) 0 0
\(841\) 3.58359 0.123572
\(842\) −41.7984 −1.44047
\(843\) 0 0
\(844\) −7.85410 −0.270349
\(845\) 28.8885 0.993796
\(846\) 0 0
\(847\) 9.47214 0.325466
\(848\) 48.2705 1.65762
\(849\) 0 0
\(850\) 0.381966 0.0131013
\(851\) −6.54102 −0.224223
\(852\) 0 0
\(853\) −42.2492 −1.44659 −0.723293 0.690541i \(-0.757372\pi\)
−0.723293 + 0.690541i \(0.757372\pi\)
\(854\) −20.5623 −0.703628
\(855\) 0 0
\(856\) −34.4721 −1.17823
\(857\) −0.819660 −0.0279991 −0.0139995 0.999902i \(-0.504456\pi\)
−0.0139995 + 0.999902i \(0.504456\pi\)
\(858\) 0 0
\(859\) −11.9443 −0.407533 −0.203767 0.979019i \(-0.565318\pi\)
−0.203767 + 0.979019i \(0.565318\pi\)
\(860\) 0.291796 0.00995016
\(861\) 0 0
\(862\) 15.4164 0.525085
\(863\) −4.81966 −0.164063 −0.0820316 0.996630i \(-0.526141\pi\)
−0.0820316 + 0.996630i \(0.526141\pi\)
\(864\) 0 0
\(865\) 9.18034 0.312141
\(866\) −1.23607 −0.0420033
\(867\) 0 0
\(868\) 5.56231 0.188797
\(869\) 6.40325 0.217215
\(870\) 0 0
\(871\) 44.9443 1.52288
\(872\) 2.23607 0.0757228
\(873\) 0 0
\(874\) 4.20163 0.142122
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 52.6525 1.77795 0.888974 0.457958i \(-0.151419\pi\)
0.888974 + 0.457958i \(0.151419\pi\)
\(878\) 33.9787 1.14673
\(879\) 0 0
\(880\) −6.00000 −0.202260
\(881\) −37.4853 −1.26291 −0.631456 0.775412i \(-0.717542\pi\)
−0.631456 + 0.775412i \(0.717542\pi\)
\(882\) 0 0
\(883\) −34.6525 −1.16615 −0.583074 0.812419i \(-0.698150\pi\)
−0.583074 + 0.812419i \(0.698150\pi\)
\(884\) 0.944272 0.0317593
\(885\) 0 0
\(886\) −49.6869 −1.66926
\(887\) −37.7214 −1.26656 −0.633280 0.773923i \(-0.718292\pi\)
−0.633280 + 0.773923i \(0.718292\pi\)
\(888\) 0 0
\(889\) 6.18034 0.207282
\(890\) −22.9443 −0.769094
\(891\) 0 0
\(892\) −7.81966 −0.261822
\(893\) −5.88854 −0.197053
\(894\) 0 0
\(895\) −11.4164 −0.381608
\(896\) 13.6180 0.454947
\(897\) 0 0
\(898\) −11.4164 −0.380970
\(899\) −51.3738 −1.71341
\(900\) 0 0
\(901\) 2.34752 0.0782074
\(902\) 0.944272 0.0314408
\(903\) 0 0
\(904\) −4.47214 −0.148741
\(905\) 8.23607 0.273776
\(906\) 0 0
\(907\) −13.1246 −0.435796 −0.217898 0.975972i \(-0.569920\pi\)
−0.217898 + 0.975972i \(0.569920\pi\)
\(908\) −6.79837 −0.225612
\(909\) 0 0
\(910\) −10.4721 −0.347148
\(911\) −59.2361 −1.96258 −0.981289 0.192539i \(-0.938328\pi\)
−0.981289 + 0.192539i \(0.938328\pi\)
\(912\) 0 0
\(913\) −13.5967 −0.449987
\(914\) 29.2361 0.967043
\(915\) 0 0
\(916\) 2.32624 0.0768611
\(917\) −21.2361 −0.701277
\(918\) 0 0
\(919\) 13.3050 0.438890 0.219445 0.975625i \(-0.429575\pi\)
0.219445 + 0.975625i \(0.429575\pi\)
\(920\) −3.94427 −0.130039
\(921\) 0 0
\(922\) −46.0689 −1.51720
\(923\) −12.9443 −0.426066
\(924\) 0 0
\(925\) 3.70820 0.121925
\(926\) 52.5410 1.72661
\(927\) 0 0
\(928\) 19.3050 0.633716
\(929\) −35.1246 −1.15240 −0.576201 0.817308i \(-0.695465\pi\)
−0.576201 + 0.817308i \(0.695465\pi\)
\(930\) 0 0
\(931\) 1.47214 0.0482473
\(932\) 16.2918 0.533656
\(933\) 0 0
\(934\) 67.1033 2.19569
\(935\) −0.291796 −0.00954275
\(936\) 0 0
\(937\) 13.5279 0.441936 0.220968 0.975281i \(-0.429078\pi\)
0.220968 + 0.975281i \(0.429078\pi\)
\(938\) −11.2361 −0.366871
\(939\) 0 0
\(940\) −2.47214 −0.0806322
\(941\) −30.7639 −1.00288 −0.501438 0.865194i \(-0.667195\pi\)
−0.501438 + 0.865194i \(0.667195\pi\)
\(942\) 0 0
\(943\) 0.832816 0.0271202
\(944\) 8.29180 0.269875
\(945\) 0 0
\(946\) −0.944272 −0.0307009
\(947\) 25.5410 0.829972 0.414986 0.909828i \(-0.363787\pi\)
0.414986 + 0.909828i \(0.363787\pi\)
\(948\) 0 0
\(949\) 54.8328 1.77995
\(950\) −2.38197 −0.0772812
\(951\) 0 0
\(952\) 0.527864 0.0171082
\(953\) −13.7771 −0.446284 −0.223142 0.974786i \(-0.571631\pi\)
−0.223142 + 0.974786i \(0.571631\pi\)
\(954\) 0 0
\(955\) 25.2361 0.816620
\(956\) 2.18034 0.0705172
\(957\) 0 0
\(958\) −34.9443 −1.12900
\(959\) 15.9443 0.514867
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 38.8328 1.25202
\(963\) 0 0
\(964\) 15.8541 0.510626
\(965\) 23.4164 0.753801
\(966\) 0 0
\(967\) 29.5967 0.951767 0.475884 0.879508i \(-0.342128\pi\)
0.475884 + 0.879508i \(0.342128\pi\)
\(968\) −21.1803 −0.680762
\(969\) 0 0
\(970\) 1.23607 0.0396878
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) −14.4721 −0.463955
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 61.6869 1.97455
\(977\) −55.8885 −1.78803 −0.894016 0.448034i \(-0.852124\pi\)
−0.894016 + 0.448034i \(0.852124\pi\)
\(978\) 0 0
\(979\) 17.5279 0.560193
\(980\) 0.618034 0.0197424
\(981\) 0 0
\(982\) −6.29180 −0.200779
\(983\) −41.0000 −1.30770 −0.653848 0.756626i \(-0.726847\pi\)
−0.653848 + 0.756626i \(0.726847\pi\)
\(984\) 0 0
\(985\) 17.9443 0.571752
\(986\) 2.18034 0.0694362
\(987\) 0 0
\(988\) −5.88854 −0.187340
\(989\) −0.832816 −0.0264820
\(990\) 0 0
\(991\) −42.0132 −1.33459 −0.667296 0.744793i \(-0.732548\pi\)
−0.667296 + 0.744793i \(0.732548\pi\)
\(992\) −30.4377 −0.966398
\(993\) 0 0
\(994\) 3.23607 0.102642
\(995\) −17.5279 −0.555671
\(996\) 0 0
\(997\) 33.4164 1.05831 0.529154 0.848526i \(-0.322509\pi\)
0.529154 + 0.848526i \(0.322509\pi\)
\(998\) −16.5623 −0.524271
\(999\) 0 0
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 945.2.a.c.1.1 2
3.2 odd 2 945.2.a.h.1.2 yes 2
5.4 even 2 4725.2.a.bd.1.2 2
7.6 odd 2 6615.2.a.n.1.1 2
15.14 odd 2 4725.2.a.y.1.1 2
21.20 even 2 6615.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
945.2.a.c.1.1 2 1.1 even 1 trivial
945.2.a.h.1.2 yes 2 3.2 odd 2
4725.2.a.y.1.1 2 15.14 odd 2
4725.2.a.bd.1.2 2 5.4 even 2
6615.2.a.n.1.1 2 7.6 odd 2
6615.2.a.t.1.2 2 21.20 even 2