Properties

Label 325.2.a.g.1.2
Level $325$
Weight $2$
Character 325.1
Self dual yes
Analytic conductor $2.595$
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] = [325,2,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(2.59513806569\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} -4.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} -4.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +4.46410 q^{9} -4.73205 q^{11} -2.73205 q^{12} -1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} +7.73205 q^{18} -6.19615 q^{19} +5.46410 q^{21} -8.19615 q^{22} -1.26795 q^{23} +4.73205 q^{24} -1.73205 q^{26} -4.00000 q^{27} -2.00000 q^{28} -2.53590 q^{29} +10.1962 q^{31} -5.19615 q^{32} +12.9282 q^{33} +6.00000 q^{34} +4.46410 q^{36} +4.00000 q^{37} -10.7321 q^{38} +2.73205 q^{39} +3.46410 q^{41} +9.46410 q^{42} +0.196152 q^{43} -4.73205 q^{44} -2.19615 q^{46} -6.00000 q^{47} +13.6603 q^{48} -3.00000 q^{49} -9.46410 q^{51} -1.00000 q^{52} -10.3923 q^{53} -6.92820 q^{54} +3.46410 q^{56} +16.9282 q^{57} -4.39230 q^{58} +9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} -8.92820 q^{63} +1.00000 q^{64} +22.3923 q^{66} -6.39230 q^{67} +3.46410 q^{68} +3.46410 q^{69} +4.73205 q^{71} -7.73205 q^{72} +4.00000 q^{73} +6.92820 q^{74} -6.19615 q^{76} +9.46410 q^{77} +4.73205 q^{78} -8.39230 q^{79} -2.46410 q^{81} +6.00000 q^{82} +6.00000 q^{83} +5.46410 q^{84} +0.339746 q^{86} +6.92820 q^{87} +8.19615 q^{88} -12.9282 q^{89} +2.00000 q^{91} -1.26795 q^{92} -27.8564 q^{93} -10.3923 q^{94} +14.1962 q^{96} -2.00000 q^{97} -5.19615 q^{98} -21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{12} - 2 q^{13} - 10 q^{16} + 12 q^{18} - 2 q^{19} + 4 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{24} - 8 q^{27} - 4 q^{28} - 12 q^{29} + 10 q^{31} + 12 q^{33} + 12 q^{34} + 2 q^{36} + 8 q^{37} - 18 q^{38} + 2 q^{39} + 12 q^{42} - 10 q^{43} - 6 q^{44} + 6 q^{46} - 12 q^{47} + 10 q^{48} - 6 q^{49} - 12 q^{51} - 2 q^{52} + 20 q^{57} + 12 q^{58} - 6 q^{59} + 4 q^{61} + 18 q^{62} - 4 q^{63} + 2 q^{64} + 24 q^{66} + 8 q^{67} + 6 q^{71} - 12 q^{72} + 8 q^{73} - 2 q^{76} + 12 q^{77} + 6 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} + 12 q^{83} + 4 q^{84} + 18 q^{86} + 6 q^{88} - 12 q^{89} + 4 q^{91} - 6 q^{92} - 28 q^{93} + 18 q^{96} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −4.73205 −1.93185
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.73205 −0.612372
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) −2.73205 −0.788675
\(13\) −1.00000 −0.277350
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 7.73205 1.82246
\(19\) −6.19615 −1.42149 −0.710747 0.703447i \(-0.751643\pi\)
−0.710747 + 0.703447i \(0.751643\pi\)
\(20\) 0 0
\(21\) 5.46410 1.19236
\(22\) −8.19615 −1.74743
\(23\) −1.26795 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 4.73205 0.965926
\(25\) 0 0
\(26\) −1.73205 −0.339683
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) −2.53590 −0.470905 −0.235452 0.971886i \(-0.575657\pi\)
−0.235452 + 0.971886i \(0.575657\pi\)
\(30\) 0 0
\(31\) 10.1962 1.83128 0.915642 0.401996i \(-0.131683\pi\)
0.915642 + 0.401996i \(0.131683\pi\)
\(32\) −5.19615 −0.918559
\(33\) 12.9282 2.25051
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −10.7321 −1.74097
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 9.46410 1.46034
\(43\) 0.196152 0.0299130 0.0149565 0.999888i \(-0.495239\pi\)
0.0149565 + 0.999888i \(0.495239\pi\)
\(44\) −4.73205 −0.713384
\(45\) 0 0
\(46\) −2.19615 −0.323805
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 13.6603 1.97169
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −9.46410 −1.32524
\(52\) −1.00000 −0.138675
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) −6.92820 −0.942809
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 16.9282 2.24220
\(58\) −4.39230 −0.576738
\(59\) 9.12436 1.18789 0.593945 0.804506i \(-0.297570\pi\)
0.593945 + 0.804506i \(0.297570\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 17.6603 2.24285
\(63\) −8.92820 −1.12485
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 22.3923 2.75630
\(67\) −6.39230 −0.780944 −0.390472 0.920615i \(-0.627688\pi\)
−0.390472 + 0.920615i \(0.627688\pi\)
\(68\) 3.46410 0.420084
\(69\) 3.46410 0.417029
\(70\) 0 0
\(71\) 4.73205 0.561591 0.280796 0.959768i \(-0.409402\pi\)
0.280796 + 0.959768i \(0.409402\pi\)
\(72\) −7.73205 −0.911231
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 6.92820 0.805387
\(75\) 0 0
\(76\) −6.19615 −0.710747
\(77\) 9.46410 1.07853
\(78\) 4.73205 0.535799
\(79\) −8.39230 −0.944208 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 5.46410 0.596182
\(85\) 0 0
\(86\) 0.339746 0.0366357
\(87\) 6.92820 0.742781
\(88\) 8.19615 0.873713
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −1.26795 −0.132193
\(93\) −27.8564 −2.88857
\(94\) −10.3923 −1.07188
\(95\) 0 0
\(96\) 14.1962 1.44889
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −5.19615 −0.524891
\(99\) −21.1244 −2.12308
\(100\) 0 0
\(101\) −0.928203 −0.0923597 −0.0461798 0.998933i \(-0.514705\pi\)
−0.0461798 + 0.998933i \(0.514705\pi\)
\(102\) −16.3923 −1.62308
\(103\) 0.196152 0.0193275 0.00966374 0.999953i \(-0.496924\pi\)
0.00966374 + 0.999953i \(0.496924\pi\)
\(104\) 1.73205 0.169842
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) −17.6603 −1.70728 −0.853641 0.520862i \(-0.825610\pi\)
−0.853641 + 0.520862i \(0.825610\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.9282 −1.03726
\(112\) 10.0000 0.944911
\(113\) −8.53590 −0.802990 −0.401495 0.915861i \(-0.631509\pi\)
−0.401495 + 0.915861i \(0.631509\pi\)
\(114\) 29.3205 2.74612
\(115\) 0 0
\(116\) −2.53590 −0.235452
\(117\) −4.46410 −0.412706
\(118\) 15.8038 1.45486
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) −14.5359 −1.31602
\(123\) −9.46410 −0.853349
\(124\) 10.1962 0.915642
\(125\) 0 0
\(126\) −15.4641 −1.37765
\(127\) −16.1962 −1.43718 −0.718588 0.695436i \(-0.755211\pi\)
−0.718588 + 0.695436i \(0.755211\pi\)
\(128\) 12.1244 1.07165
\(129\) −0.535898 −0.0471832
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 12.9282 1.12526
\(133\) 12.3923 1.07455
\(134\) −11.0718 −0.956458
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −0.928203 −0.0793018 −0.0396509 0.999214i \(-0.512625\pi\)
−0.0396509 + 0.999214i \(0.512625\pi\)
\(138\) 6.00000 0.510754
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) 16.3923 1.38048
\(142\) 8.19615 0.687806
\(143\) 4.73205 0.395714
\(144\) −22.3205 −1.86004
\(145\) 0 0
\(146\) 6.92820 0.573382
\(147\) 8.19615 0.676007
\(148\) 4.00000 0.328798
\(149\) −7.85641 −0.643622 −0.321811 0.946804i \(-0.604292\pi\)
−0.321811 + 0.946804i \(0.604292\pi\)
\(150\) 0 0
\(151\) −1.80385 −0.146795 −0.0733975 0.997303i \(-0.523384\pi\)
−0.0733975 + 0.997303i \(0.523384\pi\)
\(152\) 10.7321 0.870484
\(153\) 15.4641 1.25020
\(154\) 16.3923 1.32093
\(155\) 0 0
\(156\) 2.73205 0.218739
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −14.5359 −1.15641
\(159\) 28.3923 2.25166
\(160\) 0 0
\(161\) 2.53590 0.199857
\(162\) −4.26795 −0.335322
\(163\) 14.3923 1.12729 0.563646 0.826016i \(-0.309398\pi\)
0.563646 + 0.826016i \(0.309398\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) 10.3923 0.806599
\(167\) 0.928203 0.0718265 0.0359133 0.999355i \(-0.488566\pi\)
0.0359133 + 0.999355i \(0.488566\pi\)
\(168\) −9.46410 −0.730171
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −27.6603 −2.11523
\(172\) 0.196152 0.0149565
\(173\) −8.53590 −0.648972 −0.324486 0.945890i \(-0.605191\pi\)
−0.324486 + 0.945890i \(0.605191\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 23.6603 1.78346
\(177\) −24.9282 −1.87372
\(178\) −22.3923 −1.67837
\(179\) −18.9282 −1.41476 −0.707380 0.706833i \(-0.750123\pi\)
−0.707380 + 0.706833i \(0.750123\pi\)
\(180\) 0 0
\(181\) 0.392305 0.0291598 0.0145799 0.999894i \(-0.495359\pi\)
0.0145799 + 0.999894i \(0.495359\pi\)
\(182\) 3.46410 0.256776
\(183\) 22.9282 1.69490
\(184\) 2.19615 0.161903
\(185\) 0 0
\(186\) −48.2487 −3.53777
\(187\) −16.3923 −1.19872
\(188\) −6.00000 −0.437595
\(189\) 8.00000 0.581914
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) −2.73205 −0.197169
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −3.46410 −0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.9282 0.921096 0.460548 0.887635i \(-0.347653\pi\)
0.460548 + 0.887635i \(0.347653\pi\)
\(198\) −36.5885 −2.60023
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 17.4641 1.23182
\(202\) −1.60770 −0.113117
\(203\) 5.07180 0.355970
\(204\) −9.46410 −0.662620
\(205\) 0 0
\(206\) 0.339746 0.0236712
\(207\) −5.66025 −0.393415
\(208\) 5.00000 0.346688
\(209\) 29.3205 2.02814
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.3923 −0.713746
\(213\) −12.9282 −0.885826
\(214\) −30.5885 −2.09098
\(215\) 0 0
\(216\) 6.92820 0.471405
\(217\) −20.3923 −1.38432
\(218\) 3.46410 0.234619
\(219\) −10.9282 −0.738460
\(220\) 0 0
\(221\) −3.46410 −0.233021
\(222\) −18.9282 −1.27038
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −14.7846 −0.983458
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 16.9282 1.12110
\(229\) 6.39230 0.422415 0.211208 0.977441i \(-0.432260\pi\)
0.211208 + 0.977441i \(0.432260\pi\)
\(230\) 0 0
\(231\) −25.8564 −1.70123
\(232\) 4.39230 0.288369
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −7.73205 −0.505460
\(235\) 0 0
\(236\) 9.12436 0.593945
\(237\) 22.9282 1.48935
\(238\) −12.0000 −0.777844
\(239\) −14.1962 −0.918273 −0.459136 0.888366i \(-0.651841\pi\)
−0.459136 + 0.888366i \(0.651841\pi\)
\(240\) 0 0
\(241\) −2.39230 −0.154102 −0.0770510 0.997027i \(-0.524550\pi\)
−0.0770510 + 0.997027i \(0.524550\pi\)
\(242\) 19.7321 1.26842
\(243\) 18.7321 1.20166
\(244\) −8.39230 −0.537262
\(245\) 0 0
\(246\) −16.3923 −1.04514
\(247\) 6.19615 0.394252
\(248\) −17.6603 −1.12143
\(249\) −16.3923 −1.03882
\(250\) 0 0
\(251\) −21.4641 −1.35480 −0.677401 0.735614i \(-0.736894\pi\)
−0.677401 + 0.735614i \(0.736894\pi\)
\(252\) −8.92820 −0.562424
\(253\) 6.00000 0.377217
\(254\) −28.0526 −1.76017
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −19.8564 −1.23861 −0.619304 0.785151i \(-0.712585\pi\)
−0.619304 + 0.785151i \(0.712585\pi\)
\(258\) −0.928203 −0.0577874
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −11.3205 −0.700722
\(262\) 0 0
\(263\) −1.26795 −0.0781851 −0.0390925 0.999236i \(-0.512447\pi\)
−0.0390925 + 0.999236i \(0.512447\pi\)
\(264\) −22.3923 −1.37815
\(265\) 0 0
\(266\) 21.4641 1.31605
\(267\) 35.3205 2.16158
\(268\) −6.39230 −0.390472
\(269\) −19.8564 −1.21067 −0.605333 0.795972i \(-0.706960\pi\)
−0.605333 + 0.795972i \(0.706960\pi\)
\(270\) 0 0
\(271\) 30.9808 1.88195 0.940974 0.338480i \(-0.109913\pi\)
0.940974 + 0.338480i \(0.109913\pi\)
\(272\) −17.3205 −1.05021
\(273\) −5.46410 −0.330702
\(274\) −1.60770 −0.0971244
\(275\) 0 0
\(276\) 3.46410 0.208514
\(277\) 26.3923 1.58576 0.792880 0.609378i \(-0.208581\pi\)
0.792880 + 0.609378i \(0.208581\pi\)
\(278\) 21.4641 1.28733
\(279\) 45.5167 2.72501
\(280\) 0 0
\(281\) 22.3923 1.33581 0.667906 0.744245i \(-0.267191\pi\)
0.667906 + 0.744245i \(0.267191\pi\)
\(282\) 28.3923 1.69074
\(283\) −32.5885 −1.93718 −0.968591 0.248658i \(-0.920010\pi\)
−0.968591 + 0.248658i \(0.920010\pi\)
\(284\) 4.73205 0.280796
\(285\) 0 0
\(286\) 8.19615 0.484649
\(287\) −6.92820 −0.408959
\(288\) −23.1962 −1.36685
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 5.46410 0.320311
\(292\) 4.00000 0.234082
\(293\) 5.07180 0.296298 0.148149 0.988965i \(-0.452669\pi\)
0.148149 + 0.988965i \(0.452669\pi\)
\(294\) 14.1962 0.827936
\(295\) 0 0
\(296\) −6.92820 −0.402694
\(297\) 18.9282 1.09833
\(298\) −13.6077 −0.788273
\(299\) 1.26795 0.0733274
\(300\) 0 0
\(301\) −0.392305 −0.0226121
\(302\) −3.12436 −0.179786
\(303\) 2.53590 0.145684
\(304\) 30.9808 1.77687
\(305\) 0 0
\(306\) 26.7846 1.53117
\(307\) 18.7846 1.07209 0.536047 0.844188i \(-0.319917\pi\)
0.536047 + 0.844188i \(0.319917\pi\)
\(308\) 9.46410 0.539267
\(309\) −0.535898 −0.0304862
\(310\) 0 0
\(311\) −16.3923 −0.929522 −0.464761 0.885436i \(-0.653860\pi\)
−0.464761 + 0.885436i \(0.653860\pi\)
\(312\) −4.73205 −0.267900
\(313\) 14.3923 0.813501 0.406751 0.913539i \(-0.366662\pi\)
0.406751 + 0.913539i \(0.366662\pi\)
\(314\) 17.3205 0.977453
\(315\) 0 0
\(316\) −8.39230 −0.472104
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 49.1769 2.75770
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 48.2487 2.69298
\(322\) 4.39230 0.244774
\(323\) −21.4641 −1.19429
\(324\) −2.46410 −0.136895
\(325\) 0 0
\(326\) 24.9282 1.38065
\(327\) −5.46410 −0.302166
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 2.58846 0.142274 0.0711372 0.997467i \(-0.477337\pi\)
0.0711372 + 0.997467i \(0.477337\pi\)
\(332\) 6.00000 0.329293
\(333\) 17.8564 0.978525
\(334\) 1.60770 0.0879692
\(335\) 0 0
\(336\) −27.3205 −1.49046
\(337\) 26.3923 1.43768 0.718840 0.695175i \(-0.244673\pi\)
0.718840 + 0.695175i \(0.244673\pi\)
\(338\) 1.73205 0.0942111
\(339\) 23.3205 1.26660
\(340\) 0 0
\(341\) −48.2487 −2.61281
\(342\) −47.9090 −2.59062
\(343\) 20.0000 1.07990
\(344\) −0.339746 −0.0183179
\(345\) 0 0
\(346\) −14.7846 −0.794826
\(347\) 5.66025 0.303858 0.151929 0.988391i \(-0.451451\pi\)
0.151929 + 0.988391i \(0.451451\pi\)
\(348\) 6.92820 0.371391
\(349\) −14.3923 −0.770402 −0.385201 0.922833i \(-0.625868\pi\)
−0.385201 + 0.922833i \(0.625868\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 24.5885 1.31057
\(353\) 27.7128 1.47500 0.737502 0.675345i \(-0.236005\pi\)
0.737502 + 0.675345i \(0.236005\pi\)
\(354\) −43.1769 −2.29483
\(355\) 0 0
\(356\) −12.9282 −0.685193
\(357\) 18.9282 1.00179
\(358\) −32.7846 −1.73272
\(359\) −2.19615 −0.115908 −0.0579542 0.998319i \(-0.518458\pi\)
−0.0579542 + 0.998319i \(0.518458\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) 0.679492 0.0357133
\(363\) −31.1244 −1.63361
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 39.7128 2.07582
\(367\) −11.8038 −0.616156 −0.308078 0.951361i \(-0.599686\pi\)
−0.308078 + 0.951361i \(0.599686\pi\)
\(368\) 6.33975 0.330482
\(369\) 15.4641 0.805029
\(370\) 0 0
\(371\) 20.7846 1.07908
\(372\) −27.8564 −1.44429
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −28.3923 −1.46813
\(375\) 0 0
\(376\) 10.3923 0.535942
\(377\) 2.53590 0.130605
\(378\) 13.8564 0.712697
\(379\) 18.9808 0.974976 0.487488 0.873130i \(-0.337913\pi\)
0.487488 + 0.873130i \(0.337913\pi\)
\(380\) 0 0
\(381\) 44.2487 2.26693
\(382\) −8.78461 −0.449460
\(383\) −12.9282 −0.660600 −0.330300 0.943876i \(-0.607150\pi\)
−0.330300 + 0.943876i \(0.607150\pi\)
\(384\) −33.1244 −1.69037
\(385\) 0 0
\(386\) 17.3205 0.881591
\(387\) 0.875644 0.0445115
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −4.39230 −0.222128
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) 22.3923 1.12811
\(395\) 0 0
\(396\) −21.1244 −1.06154
\(397\) −28.7846 −1.44466 −0.722329 0.691549i \(-0.756928\pi\)
−0.722329 + 0.691549i \(0.756928\pi\)
\(398\) 34.6410 1.73640
\(399\) −33.8564 −1.69494
\(400\) 0 0
\(401\) −36.9282 −1.84411 −0.922053 0.387063i \(-0.873490\pi\)
−0.922053 + 0.387063i \(0.873490\pi\)
\(402\) 30.2487 1.50867
\(403\) −10.1962 −0.507907
\(404\) −0.928203 −0.0461798
\(405\) 0 0
\(406\) 8.78461 0.435973
\(407\) −18.9282 −0.938236
\(408\) 16.3923 0.811540
\(409\) −17.6077 −0.870644 −0.435322 0.900275i \(-0.643366\pi\)
−0.435322 + 0.900275i \(0.643366\pi\)
\(410\) 0 0
\(411\) 2.53590 0.125087
\(412\) 0.196152 0.00966374
\(413\) −18.2487 −0.897960
\(414\) −9.80385 −0.481833
\(415\) 0 0
\(416\) 5.19615 0.254762
\(417\) −33.8564 −1.65796
\(418\) 50.7846 2.48396
\(419\) −2.53590 −0.123887 −0.0619434 0.998080i \(-0.519730\pi\)
−0.0619434 + 0.998080i \(0.519730\pi\)
\(420\) 0 0
\(421\) −30.7846 −1.50035 −0.750175 0.661239i \(-0.770031\pi\)
−0.750175 + 0.661239i \(0.770031\pi\)
\(422\) 13.8564 0.674519
\(423\) −26.7846 −1.30231
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) −22.3923 −1.08491
\(427\) 16.7846 0.812264
\(428\) −17.6603 −0.853641
\(429\) −12.9282 −0.624180
\(430\) 0 0
\(431\) −25.5167 −1.22909 −0.614547 0.788880i \(-0.710661\pi\)
−0.614547 + 0.788880i \(0.710661\pi\)
\(432\) 20.0000 0.962250
\(433\) −34.7846 −1.67164 −0.835821 0.549002i \(-0.815008\pi\)
−0.835821 + 0.549002i \(0.815008\pi\)
\(434\) −35.3205 −1.69544
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 7.85641 0.375823
\(438\) −18.9282 −0.904425
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −13.3923 −0.637729
\(442\) −6.00000 −0.285391
\(443\) 16.9808 0.806780 0.403390 0.915028i \(-0.367832\pi\)
0.403390 + 0.915028i \(0.367832\pi\)
\(444\) −10.9282 −0.518630
\(445\) 0 0
\(446\) −3.46410 −0.164030
\(447\) 21.4641 1.01522
\(448\) −2.00000 −0.0944911
\(449\) 20.5359 0.969149 0.484574 0.874750i \(-0.338974\pi\)
0.484574 + 0.874750i \(0.338974\pi\)
\(450\) 0 0
\(451\) −16.3923 −0.771883
\(452\) −8.53590 −0.401495
\(453\) 4.92820 0.231547
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) −29.3205 −1.37306
\(457\) −10.7846 −0.504483 −0.252241 0.967664i \(-0.581168\pi\)
−0.252241 + 0.967664i \(0.581168\pi\)
\(458\) 11.0718 0.517351
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) −44.7846 −2.08357
\(463\) 2.39230 0.111180 0.0555899 0.998454i \(-0.482296\pi\)
0.0555899 + 0.998454i \(0.482296\pi\)
\(464\) 12.6795 0.588631
\(465\) 0 0
\(466\) 10.3923 0.481414
\(467\) −27.8038 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(468\) −4.46410 −0.206353
\(469\) 12.7846 0.590338
\(470\) 0 0
\(471\) −27.3205 −1.25886
\(472\) −15.8038 −0.727431
\(473\) −0.928203 −0.0426788
\(474\) 39.7128 1.82407
\(475\) 0 0
\(476\) −6.92820 −0.317554
\(477\) −46.3923 −2.12416
\(478\) −24.5885 −1.12465
\(479\) 35.6603 1.62936 0.814679 0.579912i \(-0.196913\pi\)
0.814679 + 0.579912i \(0.196913\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −4.14359 −0.188736
\(483\) −6.92820 −0.315244
\(484\) 11.3923 0.517832
\(485\) 0 0
\(486\) 32.4449 1.47173
\(487\) 26.3923 1.19595 0.597975 0.801515i \(-0.295972\pi\)
0.597975 + 0.801515i \(0.295972\pi\)
\(488\) 14.5359 0.658009
\(489\) −39.3205 −1.77813
\(490\) 0 0
\(491\) −2.53590 −0.114443 −0.0572217 0.998361i \(-0.518224\pi\)
−0.0572217 + 0.998361i \(0.518224\pi\)
\(492\) −9.46410 −0.426675
\(493\) −8.78461 −0.395639
\(494\) 10.7321 0.482858
\(495\) 0 0
\(496\) −50.9808 −2.28910
\(497\) −9.46410 −0.424523
\(498\) −28.3923 −1.27229
\(499\) −38.9808 −1.74502 −0.872509 0.488598i \(-0.837509\pi\)
−0.872509 + 0.488598i \(0.837509\pi\)
\(500\) 0 0
\(501\) −2.53590 −0.113296
\(502\) −37.1769 −1.65929
\(503\) −19.5167 −0.870205 −0.435102 0.900381i \(-0.643288\pi\)
−0.435102 + 0.900381i \(0.643288\pi\)
\(504\) 15.4641 0.688826
\(505\) 0 0
\(506\) 10.3923 0.461994
\(507\) −2.73205 −0.121335
\(508\) −16.1962 −0.718588
\(509\) 39.4641 1.74922 0.874608 0.484831i \(-0.161119\pi\)
0.874608 + 0.484831i \(0.161119\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 8.66025 0.382733
\(513\) 24.7846 1.09427
\(514\) −34.3923 −1.51698
\(515\) 0 0
\(516\) −0.535898 −0.0235916
\(517\) 28.3923 1.24869
\(518\) −13.8564 −0.608816
\(519\) 23.3205 1.02366
\(520\) 0 0
\(521\) −28.3923 −1.24389 −0.621945 0.783061i \(-0.713657\pi\)
−0.621945 + 0.783061i \(0.713657\pi\)
\(522\) −19.6077 −0.858206
\(523\) 24.1962 1.05802 0.529012 0.848614i \(-0.322563\pi\)
0.529012 + 0.848614i \(0.322563\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.19615 −0.0957568
\(527\) 35.3205 1.53859
\(528\) −64.6410 −2.81314
\(529\) −21.3923 −0.930100
\(530\) 0 0
\(531\) 40.7321 1.76762
\(532\) 12.3923 0.537275
\(533\) −3.46410 −0.150047
\(534\) 61.1769 2.64738
\(535\) 0 0
\(536\) 11.0718 0.478229
\(537\) 51.7128 2.23157
\(538\) −34.3923 −1.48276
\(539\) 14.1962 0.611472
\(540\) 0 0
\(541\) −26.3923 −1.13469 −0.567347 0.823479i \(-0.692030\pi\)
−0.567347 + 0.823479i \(0.692030\pi\)
\(542\) 53.6603 2.30491
\(543\) −1.07180 −0.0459952
\(544\) −18.0000 −0.771744
\(545\) 0 0
\(546\) −9.46410 −0.405026
\(547\) 12.1962 0.521470 0.260735 0.965410i \(-0.416035\pi\)
0.260735 + 0.965410i \(0.416035\pi\)
\(548\) −0.928203 −0.0396509
\(549\) −37.4641 −1.59893
\(550\) 0 0
\(551\) 15.7128 0.669388
\(552\) −6.00000 −0.255377
\(553\) 16.7846 0.713754
\(554\) 45.7128 1.94215
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) −1.85641 −0.0786585 −0.0393292 0.999226i \(-0.512522\pi\)
−0.0393292 + 0.999226i \(0.512522\pi\)
\(558\) 78.8372 3.33744
\(559\) −0.196152 −0.00829636
\(560\) 0 0
\(561\) 44.7846 1.89081
\(562\) 38.7846 1.63603
\(563\) 22.0526 0.929405 0.464702 0.885467i \(-0.346161\pi\)
0.464702 + 0.885467i \(0.346161\pi\)
\(564\) 16.3923 0.690241
\(565\) 0 0
\(566\) −56.4449 −2.37255
\(567\) 4.92820 0.206965
\(568\) −8.19615 −0.343903
\(569\) −2.53590 −0.106310 −0.0531552 0.998586i \(-0.516928\pi\)
−0.0531552 + 0.998586i \(0.516928\pi\)
\(570\) 0 0
\(571\) 36.3923 1.52297 0.761485 0.648182i \(-0.224470\pi\)
0.761485 + 0.648182i \(0.224470\pi\)
\(572\) 4.73205 0.197857
\(573\) 13.8564 0.578860
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −8.66025 −0.360219
\(579\) −27.3205 −1.13540
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 9.46410 0.392300
\(583\) 49.1769 2.03670
\(584\) −6.92820 −0.286691
\(585\) 0 0
\(586\) 8.78461 0.362889
\(587\) 8.53590 0.352314 0.176157 0.984362i \(-0.443633\pi\)
0.176157 + 0.984362i \(0.443633\pi\)
\(588\) 8.19615 0.338004
\(589\) −63.1769 −2.60316
\(590\) 0 0
\(591\) −35.3205 −1.45289
\(592\) −20.0000 −0.821995
\(593\) −26.7846 −1.09991 −0.549956 0.835194i \(-0.685356\pi\)
−0.549956 + 0.835194i \(0.685356\pi\)
\(594\) 32.7846 1.34517
\(595\) 0 0
\(596\) −7.85641 −0.321811
\(597\) −54.6410 −2.23631
\(598\) 2.19615 0.0898074
\(599\) −7.60770 −0.310842 −0.155421 0.987848i \(-0.549673\pi\)
−0.155421 + 0.987848i \(0.549673\pi\)
\(600\) 0 0
\(601\) 43.5692 1.77723 0.888613 0.458658i \(-0.151670\pi\)
0.888613 + 0.458658i \(0.151670\pi\)
\(602\) −0.679492 −0.0276940
\(603\) −28.5359 −1.16207
\(604\) −1.80385 −0.0733975
\(605\) 0 0
\(606\) 4.39230 0.178425
\(607\) −24.9808 −1.01394 −0.506969 0.861964i \(-0.669234\pi\)
−0.506969 + 0.861964i \(0.669234\pi\)
\(608\) 32.1962 1.30573
\(609\) −13.8564 −0.561490
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 15.4641 0.625099
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 32.5359 1.31304
\(615\) 0 0
\(616\) −16.3923 −0.660465
\(617\) −33.7128 −1.35723 −0.678613 0.734496i \(-0.737419\pi\)
−0.678613 + 0.734496i \(0.737419\pi\)
\(618\) −0.928203 −0.0373378
\(619\) 6.98076 0.280581 0.140290 0.990110i \(-0.455196\pi\)
0.140290 + 0.990110i \(0.455196\pi\)
\(620\) 0 0
\(621\) 5.07180 0.203524
\(622\) −28.3923 −1.13843
\(623\) 25.8564 1.03592
\(624\) −13.6603 −0.546848
\(625\) 0 0
\(626\) 24.9282 0.996331
\(627\) −80.1051 −3.19909
\(628\) 10.0000 0.399043
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) 5.80385 0.231048 0.115524 0.993305i \(-0.463145\pi\)
0.115524 + 0.993305i \(0.463145\pi\)
\(632\) 14.5359 0.578207
\(633\) −21.8564 −0.868714
\(634\) −41.5692 −1.65092
\(635\) 0 0
\(636\) 28.3923 1.12583
\(637\) 3.00000 0.118864
\(638\) 20.7846 0.822871
\(639\) 21.1244 0.835667
\(640\) 0 0
\(641\) 12.9282 0.510633 0.255317 0.966857i \(-0.417820\pi\)
0.255317 + 0.966857i \(0.417820\pi\)
\(642\) 83.5692 3.29821
\(643\) 6.78461 0.267559 0.133779 0.991011i \(-0.457289\pi\)
0.133779 + 0.991011i \(0.457289\pi\)
\(644\) 2.53590 0.0999284
\(645\) 0 0
\(646\) −37.1769 −1.46271
\(647\) −22.0526 −0.866976 −0.433488 0.901159i \(-0.642717\pi\)
−0.433488 + 0.901159i \(0.642717\pi\)
\(648\) 4.26795 0.167661
\(649\) −43.1769 −1.69484
\(650\) 0 0
\(651\) 55.7128 2.18356
\(652\) 14.3923 0.563646
\(653\) −7.85641 −0.307445 −0.153722 0.988114i \(-0.549126\pi\)
−0.153722 + 0.988114i \(0.549126\pi\)
\(654\) −9.46410 −0.370076
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 17.8564 0.696645
\(658\) 20.7846 0.810268
\(659\) −21.4641 −0.836123 −0.418061 0.908419i \(-0.637290\pi\)
−0.418061 + 0.908419i \(0.637290\pi\)
\(660\) 0 0
\(661\) 10.7846 0.419473 0.209736 0.977758i \(-0.432739\pi\)
0.209736 + 0.977758i \(0.432739\pi\)
\(662\) 4.48334 0.174250
\(663\) 9.46410 0.367555
\(664\) −10.3923 −0.403300
\(665\) 0 0
\(666\) 30.9282 1.19844
\(667\) 3.21539 0.124500
\(668\) 0.928203 0.0359133
\(669\) 5.46410 0.211254
\(670\) 0 0
\(671\) 39.7128 1.53310
\(672\) −28.3923 −1.09526
\(673\) 14.3923 0.554783 0.277391 0.960757i \(-0.410530\pi\)
0.277391 + 0.960757i \(0.410530\pi\)
\(674\) 45.7128 1.76079
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −10.3923 −0.399409 −0.199704 0.979856i \(-0.563998\pi\)
−0.199704 + 0.979856i \(0.563998\pi\)
\(678\) 40.3923 1.55126
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −9.46410 −0.362665
\(682\) −83.5692 −3.20003
\(683\) −32.5359 −1.24495 −0.622476 0.782639i \(-0.713873\pi\)
−0.622476 + 0.782639i \(0.713873\pi\)
\(684\) −27.6603 −1.05762
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) −17.4641 −0.666297
\(688\) −0.980762 −0.0373912
\(689\) 10.3923 0.395915
\(690\) 0 0
\(691\) −47.7654 −1.81708 −0.908540 0.417797i \(-0.862802\pi\)
−0.908540 + 0.417797i \(0.862802\pi\)
\(692\) −8.53590 −0.324486
\(693\) 42.2487 1.60490
\(694\) 9.80385 0.372149
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) 12.0000 0.454532
\(698\) −24.9282 −0.943546
\(699\) −16.3923 −0.620014
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 6.92820 0.261488
\(703\) −24.7846 −0.934769
\(704\) −4.73205 −0.178346
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) 1.85641 0.0698174
\(708\) −24.9282 −0.936859
\(709\) 30.3923 1.14141 0.570703 0.821156i \(-0.306671\pi\)
0.570703 + 0.821156i \(0.306671\pi\)
\(710\) 0 0
\(711\) −37.4641 −1.40501
\(712\) 22.3923 0.839187
\(713\) −12.9282 −0.484165
\(714\) 32.7846 1.22693
\(715\) 0 0
\(716\) −18.9282 −0.707380
\(717\) 38.7846 1.44844
\(718\) −3.80385 −0.141958
\(719\) −25.8564 −0.964281 −0.482141 0.876094i \(-0.660141\pi\)
−0.482141 + 0.876094i \(0.660141\pi\)
\(720\) 0 0
\(721\) −0.392305 −0.0146102
\(722\) 33.5885 1.25003
\(723\) 6.53590 0.243073
\(724\) 0.392305 0.0145799
\(725\) 0 0
\(726\) −53.9090 −2.00075
\(727\) −44.5885 −1.65369 −0.826847 0.562427i \(-0.809868\pi\)
−0.826847 + 0.562427i \(0.809868\pi\)
\(728\) −3.46410 −0.128388
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) 0.679492 0.0251319
\(732\) 22.9282 0.847451
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) −20.4449 −0.754634
\(735\) 0 0
\(736\) 6.58846 0.242854
\(737\) 30.2487 1.11423
\(738\) 26.7846 0.985955
\(739\) −18.1962 −0.669356 −0.334678 0.942332i \(-0.608628\pi\)
−0.334678 + 0.942332i \(0.608628\pi\)
\(740\) 0 0
\(741\) −16.9282 −0.621873
\(742\) 36.0000 1.32160
\(743\) 16.1436 0.592251 0.296126 0.955149i \(-0.404305\pi\)
0.296126 + 0.955149i \(0.404305\pi\)
\(744\) 48.2487 1.76888
\(745\) 0 0
\(746\) 17.3205 0.634149
\(747\) 26.7846 0.979998
\(748\) −16.3923 −0.599362
\(749\) 35.3205 1.29058
\(750\) 0 0
\(751\) 36.3923 1.32797 0.663987 0.747744i \(-0.268863\pi\)
0.663987 + 0.747744i \(0.268863\pi\)
\(752\) 30.0000 1.09399
\(753\) 58.6410 2.13700
\(754\) 4.39230 0.159958
\(755\) 0 0
\(756\) 8.00000 0.290957
\(757\) 2.39230 0.0869498 0.0434749 0.999055i \(-0.486157\pi\)
0.0434749 + 0.999055i \(0.486157\pi\)
\(758\) 32.8756 1.19410
\(759\) −16.3923 −0.595003
\(760\) 0 0
\(761\) −19.8564 −0.719794 −0.359897 0.932992i \(-0.617188\pi\)
−0.359897 + 0.932992i \(0.617188\pi\)
\(762\) 76.6410 2.77641
\(763\) −4.00000 −0.144810
\(764\) −5.07180 −0.183491
\(765\) 0 0
\(766\) −22.3923 −0.809067
\(767\) −9.12436 −0.329461
\(768\) −51.9090 −1.87310
\(769\) 34.7846 1.25437 0.627183 0.778872i \(-0.284208\pi\)
0.627183 + 0.778872i \(0.284208\pi\)
\(770\) 0 0
\(771\) 54.2487 1.95372
\(772\) 10.0000 0.359908
\(773\) −6.92820 −0.249190 −0.124595 0.992208i \(-0.539763\pi\)
−0.124595 + 0.992208i \(0.539763\pi\)
\(774\) 1.51666 0.0545152
\(775\) 0 0
\(776\) 3.46410 0.124354
\(777\) 21.8564 0.784094
\(778\) 10.3923 0.372582
\(779\) −21.4641 −0.769031
\(780\) 0 0
\(781\) −22.3923 −0.801260
\(782\) −7.60770 −0.272051
\(783\) 10.1436 0.362502
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) −31.5692 −1.12532 −0.562661 0.826688i \(-0.690222\pi\)
−0.562661 + 0.826688i \(0.690222\pi\)
\(788\) 12.9282 0.460548
\(789\) 3.46410 0.123325
\(790\) 0 0
\(791\) 17.0718 0.607003
\(792\) 36.5885 1.30011
\(793\) 8.39230 0.298019
\(794\) −49.8564 −1.76934
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 40.6410 1.43958 0.719789 0.694193i \(-0.244238\pi\)
0.719789 + 0.694193i \(0.244238\pi\)
\(798\) −58.6410 −2.07587
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) −57.7128 −2.03918
\(802\) −63.9615 −2.25856
\(803\) −18.9282 −0.667962
\(804\) 17.4641 0.615911
\(805\) 0 0
\(806\) −17.6603 −0.622056
\(807\) 54.2487 1.90965
\(808\) 1.60770 0.0565585
\(809\) −2.53590 −0.0891574 −0.0445787 0.999006i \(-0.514195\pi\)
−0.0445787 + 0.999006i \(0.514195\pi\)
\(810\) 0 0
\(811\) 17.8038 0.625178 0.312589 0.949889i \(-0.398804\pi\)
0.312589 + 0.949889i \(0.398804\pi\)
\(812\) 5.07180 0.177985
\(813\) −84.6410 −2.96849
\(814\) −32.7846 −1.14910
\(815\) 0 0
\(816\) 47.3205 1.65655
\(817\) −1.21539 −0.0425211
\(818\) −30.4974 −1.06632
\(819\) 8.92820 0.311977
\(820\) 0 0
\(821\) 28.6410 0.999578 0.499789 0.866147i \(-0.333411\pi\)
0.499789 + 0.866147i \(0.333411\pi\)
\(822\) 4.39230 0.153199
\(823\) 15.4115 0.537213 0.268606 0.963250i \(-0.413437\pi\)
0.268606 + 0.963250i \(0.413437\pi\)
\(824\) −0.339746 −0.0118356
\(825\) 0 0
\(826\) −31.6077 −1.09977
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −5.66025 −0.196707
\(829\) 0.392305 0.0136253 0.00681266 0.999977i \(-0.497831\pi\)
0.00681266 + 0.999977i \(0.497831\pi\)
\(830\) 0 0
\(831\) −72.1051 −2.50130
\(832\) −1.00000 −0.0346688
\(833\) −10.3923 −0.360072
\(834\) −58.6410 −2.03057
\(835\) 0 0
\(836\) 29.3205 1.01407
\(837\) −40.7846 −1.40972
\(838\) −4.39230 −0.151730
\(839\) 0.339746 0.0117293 0.00586467 0.999983i \(-0.498133\pi\)
0.00586467 + 0.999983i \(0.498133\pi\)
\(840\) 0 0
\(841\) −22.5692 −0.778249
\(842\) −53.3205 −1.83755
\(843\) −61.1769 −2.10704
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) −46.3923 −1.59500
\(847\) −22.7846 −0.782888
\(848\) 51.9615 1.78437
\(849\) 89.0333 3.05562
\(850\) 0 0
\(851\) −5.07180 −0.173859
\(852\) −12.9282 −0.442913
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 29.0718 0.994816
\(855\) 0 0
\(856\) 30.5885 1.04549
\(857\) 35.5692 1.21502 0.607511 0.794311i \(-0.292168\pi\)
0.607511 + 0.794311i \(0.292168\pi\)
\(858\) −22.3923 −0.764461
\(859\) −17.1769 −0.586069 −0.293034 0.956102i \(-0.594665\pi\)
−0.293034 + 0.956102i \(0.594665\pi\)
\(860\) 0 0
\(861\) 18.9282 0.645071
\(862\) −44.1962 −1.50533
\(863\) 38.7846 1.32024 0.660122 0.751159i \(-0.270505\pi\)
0.660122 + 0.751159i \(0.270505\pi\)
\(864\) 20.7846 0.707107
\(865\) 0 0
\(866\) −60.2487 −2.04733
\(867\) 13.6603 0.463927
\(868\) −20.3923 −0.692160
\(869\) 39.7128 1.34716
\(870\) 0 0
\(871\) 6.39230 0.216595
\(872\) −3.46410 −0.117309
\(873\) −8.92820 −0.302174
\(874\) 13.6077 0.460287
\(875\) 0 0
\(876\) −10.9282 −0.369230
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 55.4256 1.87052
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) −47.3205 −1.59427 −0.797134 0.603802i \(-0.793652\pi\)
−0.797134 + 0.603802i \(0.793652\pi\)
\(882\) −23.1962 −0.781055
\(883\) −23.8038 −0.801063 −0.400532 0.916283i \(-0.631175\pi\)
−0.400532 + 0.916283i \(0.631175\pi\)
\(884\) −3.46410 −0.116510
\(885\) 0 0
\(886\) 29.4115 0.988100
\(887\) 47.9090 1.60863 0.804313 0.594206i \(-0.202534\pi\)
0.804313 + 0.594206i \(0.202534\pi\)
\(888\) 18.9282 0.635189
\(889\) 32.3923 1.08640
\(890\) 0 0
\(891\) 11.6603 0.390633
\(892\) −2.00000 −0.0669650
\(893\) 37.1769 1.24408
\(894\) 37.1769 1.24338
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) −3.46410 −0.115663
\(898\) 35.5692 1.18696
\(899\) −25.8564 −0.862359
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −28.3923 −0.945360
\(903\) 1.07180 0.0356672
\(904\) 14.7846 0.491729
\(905\) 0 0
\(906\) 8.53590 0.283586
\(907\) 53.7654 1.78525 0.892625 0.450800i \(-0.148861\pi\)
0.892625 + 0.450800i \(0.148861\pi\)
\(908\) 3.46410 0.114960
\(909\) −4.14359 −0.137434
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −84.6410 −2.80274
\(913\) −28.3923 −0.939648
\(914\) −18.6795 −0.617863
\(915\) 0 0
\(916\) 6.39230 0.211208
\(917\) 0 0
\(918\) −24.0000 −0.792118
\(919\) 9.17691 0.302718 0.151359 0.988479i \(-0.451635\pi\)
0.151359 + 0.988479i \(0.451635\pi\)
\(920\) 0 0
\(921\) −51.3205 −1.69107
\(922\) −6.00000 −0.197599
\(923\) −4.73205 −0.155757
\(924\) −25.8564 −0.850613
\(925\) 0 0
\(926\) 4.14359 0.136167
\(927\) 0.875644 0.0287599
\(928\) 13.1769 0.432553
\(929\) 44.5359 1.46118 0.730588 0.682819i \(-0.239246\pi\)
0.730588 + 0.682819i \(0.239246\pi\)
\(930\) 0 0
\(931\) 18.5885 0.609212
\(932\) 6.00000 0.196537
\(933\) 44.7846 1.46618
\(934\) −48.1577 −1.57577
\(935\) 0 0
\(936\) 7.73205 0.252730
\(937\) −34.7846 −1.13636 −0.568182 0.822903i \(-0.692353\pi\)
−0.568182 + 0.822903i \(0.692353\pi\)
\(938\) 22.1436 0.723014
\(939\) −39.3205 −1.28318
\(940\) 0 0
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) −47.3205 −1.54179
\(943\) −4.39230 −0.143033
\(944\) −45.6218 −1.48486
\(945\) 0 0
\(946\) −1.60770 −0.0522707
\(947\) −40.6410 −1.32066 −0.660328 0.750978i \(-0.729583\pi\)
−0.660328 + 0.750978i \(0.729583\pi\)
\(948\) 22.9282 0.744673
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 65.5692 2.12623
\(952\) 12.0000 0.388922
\(953\) −0.928203 −0.0300675 −0.0150337 0.999887i \(-0.504786\pi\)
−0.0150337 + 0.999887i \(0.504786\pi\)
\(954\) −80.3538 −2.60155
\(955\) 0 0
\(956\) −14.1962 −0.459136
\(957\) −32.7846 −1.05978
\(958\) 61.7654 1.99555
\(959\) 1.85641 0.0599465
\(960\) 0 0
\(961\) 72.9615 2.35360
\(962\) −6.92820 −0.223374
\(963\) −78.8372 −2.54049
\(964\) −2.39230 −0.0770510
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 50.3923 1.62051 0.810254 0.586079i \(-0.199329\pi\)
0.810254 + 0.586079i \(0.199329\pi\)
\(968\) −19.7321 −0.634212
\(969\) 58.6410 1.88382
\(970\) 0 0
\(971\) 18.9282 0.607435 0.303717 0.952762i \(-0.401772\pi\)
0.303717 + 0.952762i \(0.401772\pi\)
\(972\) 18.7321 0.600831
\(973\) −24.7846 −0.794558
\(974\) 45.7128 1.46473
\(975\) 0 0
\(976\) 41.9615 1.34316
\(977\) 15.7128 0.502697 0.251349 0.967897i \(-0.419126\pi\)
0.251349 + 0.967897i \(0.419126\pi\)
\(978\) −68.1051 −2.17776
\(979\) 61.1769 1.95522
\(980\) 0 0
\(981\) 8.92820 0.285056
\(982\) −4.39230 −0.140164
\(983\) 34.3923 1.09694 0.548472 0.836169i \(-0.315210\pi\)
0.548472 + 0.836169i \(0.315210\pi\)
\(984\) 16.3923 0.522568
\(985\) 0 0
\(986\) −15.2154 −0.484557
\(987\) −32.7846 −1.04355
\(988\) 6.19615 0.197126
\(989\) −0.248711 −0.00790856
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −52.9808 −1.68214
\(993\) −7.07180 −0.224417
\(994\) −16.3923 −0.519932
\(995\) 0 0
\(996\) −16.3923 −0.519410
\(997\) −33.6077 −1.06437 −0.532183 0.846629i \(-0.678628\pi\)
−0.532183 + 0.846629i \(0.678628\pi\)
\(998\) −67.5167 −2.13720
\(999\) −16.0000 −0.506218
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 325.2.a.g.1.2 2
3.2 odd 2 2925.2.a.z.1.1 2
4.3 odd 2 5200.2.a.ca.1.2 2
5.2 odd 4 325.2.b.e.274.4 4
5.3 odd 4 325.2.b.e.274.1 4
5.4 even 2 65.2.a.c.1.1 2
13.12 even 2 4225.2.a.w.1.1 2
15.2 even 4 2925.2.c.v.2224.1 4
15.8 even 4 2925.2.c.v.2224.4 4
15.14 odd 2 585.2.a.k.1.2 2
20.19 odd 2 1040.2.a.h.1.1 2
35.34 odd 2 3185.2.a.k.1.1 2
40.19 odd 2 4160.2.a.bj.1.2 2
40.29 even 2 4160.2.a.y.1.1 2
55.54 odd 2 7865.2.a.h.1.2 2
60.59 even 2 9360.2.a.cm.1.1 2
65.4 even 6 845.2.e.f.146.1 4
65.9 even 6 845.2.e.e.146.2 4
65.19 odd 12 845.2.m.c.361.1 4
65.24 odd 12 845.2.m.c.316.1 4
65.29 even 6 845.2.e.e.191.2 4
65.34 odd 4 845.2.c.e.506.2 4
65.44 odd 4 845.2.c.e.506.4 4
65.49 even 6 845.2.e.f.191.1 4
65.54 odd 12 845.2.m.a.316.1 4
65.59 odd 12 845.2.m.a.361.1 4
65.64 even 2 845.2.a.d.1.2 2
195.194 odd 2 7605.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.1 2 5.4 even 2
325.2.a.g.1.2 2 1.1 even 1 trivial
325.2.b.e.274.1 4 5.3 odd 4
325.2.b.e.274.4 4 5.2 odd 4
585.2.a.k.1.2 2 15.14 odd 2
845.2.a.d.1.2 2 65.64 even 2
845.2.c.e.506.2 4 65.34 odd 4
845.2.c.e.506.4 4 65.44 odd 4
845.2.e.e.146.2 4 65.9 even 6
845.2.e.e.191.2 4 65.29 even 6
845.2.e.f.146.1 4 65.4 even 6
845.2.e.f.191.1 4 65.49 even 6
845.2.m.a.316.1 4 65.54 odd 12
845.2.m.a.361.1 4 65.59 odd 12
845.2.m.c.316.1 4 65.24 odd 12
845.2.m.c.361.1 4 65.19 odd 12
1040.2.a.h.1.1 2 20.19 odd 2
2925.2.a.z.1.1 2 3.2 odd 2
2925.2.c.v.2224.1 4 15.2 even 4
2925.2.c.v.2224.4 4 15.8 even 4
3185.2.a.k.1.1 2 35.34 odd 2
4160.2.a.y.1.1 2 40.29 even 2
4160.2.a.bj.1.2 2 40.19 odd 2
4225.2.a.w.1.1 2 13.12 even 2
5200.2.a.ca.1.2 2 4.3 odd 2
7605.2.a.be.1.1 2 195.194 odd 2
7865.2.a.h.1.2 2 55.54 odd 2
9360.2.a.cm.1.1 2 60.59 even 2