Properties

Label 725.2.b.d.349.1
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.d.349.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.67513i q^{2} -0.806063i q^{3} -5.15633 q^{4} -2.15633 q^{6} +4.15633i q^{7} +8.44358i q^{8} +2.35026 q^{9} +O(q^{10})\) \(q-2.67513i q^{2} -0.806063i q^{3} -5.15633 q^{4} -2.15633 q^{6} +4.15633i q^{7} +8.44358i q^{8} +2.35026 q^{9} +2.80606 q^{11} +4.15633i q^{12} +1.35026i q^{13} +11.1187 q^{14} +12.2750 q^{16} +7.11871i q^{17} -6.28726i q^{18} -3.76845 q^{19} +3.35026 q^{21} -7.50659i q^{22} +4.80606i q^{23} +6.80606 q^{24} +3.61213 q^{26} -4.31265i q^{27} -21.4314i q^{28} -1.00000 q^{29} +0.231548 q^{31} -15.9502i q^{32} -2.26187i q^{33} +19.0435 q^{34} -12.1187 q^{36} +5.50659i q^{37} +10.0811i q^{38} +1.08840 q^{39} -6.96239 q^{41} -8.96239i q^{42} -3.19394i q^{43} -14.4690 q^{44} +12.8568 q^{46} -6.41819i q^{47} -9.89446i q^{48} -10.2750 q^{49} +5.73813 q^{51} -6.96239i q^{52} +6.96239i q^{53} -11.5369 q^{54} -35.0943 q^{56} +3.03761i q^{57} +2.67513i q^{58} +2.57452 q^{59} +5.35026 q^{61} -0.619421i q^{62} +9.76845i q^{63} -18.1187 q^{64} -6.05079 q^{66} +3.19394i q^{67} -36.7064i q^{68} +3.87399 q^{69} +11.3503 q^{71} +19.8446i q^{72} -11.2447i q^{73} +14.7308 q^{74} +19.4314 q^{76} +11.6629i q^{77} -2.91160i q^{78} +4.73084 q^{79} +3.57452 q^{81} +18.6253i q^{82} +2.54420i q^{83} -17.2750 q^{84} -8.54420 q^{86} +0.806063i q^{87} +23.6932i q^{88} -14.3127 q^{89} -5.61213 q^{91} -24.7816i q^{92} -0.186642i q^{93} -17.1695 q^{94} -12.8568 q^{96} +1.53102i q^{97} +27.4871i q^{98} +6.59498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 8 q^{6} - 6 q^{9} + 16 q^{11} + 24 q^{14} + 10 q^{16} + 40 q^{24} + 20 q^{26} - 6 q^{29} + 24 q^{31} + 28 q^{34} - 30 q^{36} - 32 q^{39} - 20 q^{41} - 24 q^{44} + 16 q^{46} + 2 q^{49} + 16 q^{51} - 24 q^{54} - 64 q^{56} - 8 q^{59} + 12 q^{61} - 66 q^{64} + 24 q^{66} + 40 q^{69} + 48 q^{71} + 44 q^{74} + 32 q^{76} - 16 q^{79} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 44 q^{89} - 32 q^{91} - 16 q^{96} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.67513i − 1.89160i −0.324745 0.945802i \(-0.605279\pi\)
0.324745 0.945802i \(-0.394721\pi\)
\(3\) − 0.806063i − 0.465381i −0.972551 0.232690i \(-0.925247\pi\)
0.972551 0.232690i \(-0.0747529\pi\)
\(4\) −5.15633 −2.57816
\(5\) 0 0
\(6\) −2.15633 −0.880316
\(7\) 4.15633i 1.57094i 0.618898 + 0.785472i \(0.287580\pi\)
−0.618898 + 0.785472i \(0.712420\pi\)
\(8\) 8.44358i 2.98526i
\(9\) 2.35026 0.783421
\(10\) 0 0
\(11\) 2.80606 0.846060 0.423030 0.906116i \(-0.360966\pi\)
0.423030 + 0.906116i \(0.360966\pi\)
\(12\) 4.15633i 1.19983i
\(13\) 1.35026i 0.374495i 0.982313 + 0.187248i \(0.0599567\pi\)
−0.982313 + 0.187248i \(0.940043\pi\)
\(14\) 11.1187 2.97160
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 7.11871i 1.72654i 0.504741 + 0.863271i \(0.331588\pi\)
−0.504741 + 0.863271i \(0.668412\pi\)
\(18\) − 6.28726i − 1.48192i
\(19\) −3.76845 −0.864542 −0.432271 0.901744i \(-0.642288\pi\)
−0.432271 + 0.901744i \(0.642288\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) − 7.50659i − 1.60041i
\(23\) 4.80606i 1.00213i 0.865409 + 0.501067i \(0.167059\pi\)
−0.865409 + 0.501067i \(0.832941\pi\)
\(24\) 6.80606 1.38928
\(25\) 0 0
\(26\) 3.61213 0.708396
\(27\) − 4.31265i − 0.829970i
\(28\) − 21.4314i − 4.05015i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 0.231548 0.0415872 0.0207936 0.999784i \(-0.493381\pi\)
0.0207936 + 0.999784i \(0.493381\pi\)
\(32\) − 15.9502i − 2.81962i
\(33\) − 2.26187i − 0.393740i
\(34\) 19.0435 3.26593
\(35\) 0 0
\(36\) −12.1187 −2.01979
\(37\) 5.50659i 0.905277i 0.891694 + 0.452639i \(0.149517\pi\)
−0.891694 + 0.452639i \(0.850483\pi\)
\(38\) 10.0811i 1.63537i
\(39\) 1.08840 0.174283
\(40\) 0 0
\(41\) −6.96239 −1.08734 −0.543671 0.839298i \(-0.682966\pi\)
−0.543671 + 0.839298i \(0.682966\pi\)
\(42\) − 8.96239i − 1.38293i
\(43\) − 3.19394i − 0.487071i −0.969892 0.243535i \(-0.921693\pi\)
0.969892 0.243535i \(-0.0783072\pi\)
\(44\) −14.4690 −2.18128
\(45\) 0 0
\(46\) 12.8568 1.89564
\(47\) − 6.41819i − 0.936189i −0.883678 0.468095i \(-0.844941\pi\)
0.883678 0.468095i \(-0.155059\pi\)
\(48\) − 9.89446i − 1.42814i
\(49\) −10.2750 −1.46786
\(50\) 0 0
\(51\) 5.73813 0.803500
\(52\) − 6.96239i − 0.965510i
\(53\) 6.96239i 0.956358i 0.878263 + 0.478179i \(0.158703\pi\)
−0.878263 + 0.478179i \(0.841297\pi\)
\(54\) −11.5369 −1.56997
\(55\) 0 0
\(56\) −35.0943 −4.68967
\(57\) 3.03761i 0.402341i
\(58\) 2.67513i 0.351262i
\(59\) 2.57452 0.335173 0.167587 0.985857i \(-0.446403\pi\)
0.167587 + 0.985857i \(0.446403\pi\)
\(60\) 0 0
\(61\) 5.35026 0.685031 0.342515 0.939512i \(-0.388721\pi\)
0.342515 + 0.939512i \(0.388721\pi\)
\(62\) − 0.619421i − 0.0786666i
\(63\) 9.76845i 1.23071i
\(64\) −18.1187 −2.26484
\(65\) 0 0
\(66\) −6.05079 −0.744800
\(67\) 3.19394i 0.390201i 0.980783 + 0.195101i \(0.0625034\pi\)
−0.980783 + 0.195101i \(0.937497\pi\)
\(68\) − 36.7064i − 4.45131i
\(69\) 3.87399 0.466374
\(70\) 0 0
\(71\) 11.3503 1.34703 0.673514 0.739174i \(-0.264784\pi\)
0.673514 + 0.739174i \(0.264784\pi\)
\(72\) 19.8446i 2.33871i
\(73\) − 11.2447i − 1.31610i −0.752976 0.658048i \(-0.771383\pi\)
0.752976 0.658048i \(-0.228617\pi\)
\(74\) 14.7308 1.71243
\(75\) 0 0
\(76\) 19.4314 2.22893
\(77\) 11.6629i 1.32911i
\(78\) − 2.91160i − 0.329674i
\(79\) 4.73084 0.532261 0.266131 0.963937i \(-0.414255\pi\)
0.266131 + 0.963937i \(0.414255\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 18.6253i 2.05682i
\(83\) 2.54420i 0.279262i 0.990204 + 0.139631i \(0.0445916\pi\)
−0.990204 + 0.139631i \(0.955408\pi\)
\(84\) −17.2750 −1.88486
\(85\) 0 0
\(86\) −8.54420 −0.921345
\(87\) 0.806063i 0.0864191i
\(88\) 23.6932i 2.52571i
\(89\) −14.3127 −1.51714 −0.758569 0.651593i \(-0.774101\pi\)
−0.758569 + 0.651593i \(0.774101\pi\)
\(90\) 0 0
\(91\) −5.61213 −0.588311
\(92\) − 24.7816i − 2.58366i
\(93\) − 0.186642i − 0.0193539i
\(94\) −17.1695 −1.77090
\(95\) 0 0
\(96\) −12.8568 −1.31220
\(97\) 1.53102i 0.155452i 0.996975 + 0.0777260i \(0.0247659\pi\)
−0.996975 + 0.0777260i \(0.975234\pi\)
\(98\) 27.4871i 2.77661i
\(99\) 6.59498 0.662821
\(100\) 0 0
\(101\) 2.83638 0.282230 0.141115 0.989993i \(-0.454931\pi\)
0.141115 + 0.989993i \(0.454931\pi\)
\(102\) − 15.3503i − 1.51990i
\(103\) − 9.89446i − 0.974930i −0.873142 0.487465i \(-0.837922\pi\)
0.873142 0.487465i \(-0.162078\pi\)
\(104\) −11.4010 −1.11796
\(105\) 0 0
\(106\) 18.6253 1.80905
\(107\) 11.6932i 1.13043i 0.824945 + 0.565214i \(0.191206\pi\)
−0.824945 + 0.565214i \(0.808794\pi\)
\(108\) 22.2374i 2.13980i
\(109\) 14.4993 1.38878 0.694390 0.719599i \(-0.255674\pi\)
0.694390 + 0.719599i \(0.255674\pi\)
\(110\) 0 0
\(111\) 4.43866 0.421299
\(112\) 51.0191i 4.82085i
\(113\) − 16.3938i − 1.54219i −0.636717 0.771097i \(-0.719708\pi\)
0.636717 0.771097i \(-0.280292\pi\)
\(114\) 8.12601 0.761070
\(115\) 0 0
\(116\) 5.15633 0.478753
\(117\) 3.17347i 0.293387i
\(118\) − 6.88717i − 0.634015i
\(119\) −29.5877 −2.71230
\(120\) 0 0
\(121\) −3.12601 −0.284183
\(122\) − 14.3127i − 1.29581i
\(123\) 5.61213i 0.506028i
\(124\) −1.19394 −0.107219
\(125\) 0 0
\(126\) 26.1319 2.32801
\(127\) − 15.3561i − 1.36264i −0.731987 0.681319i \(-0.761407\pi\)
0.731987 0.681319i \(-0.238593\pi\)
\(128\) 16.5696i 1.46456i
\(129\) −2.57452 −0.226673
\(130\) 0 0
\(131\) 1.38058 0.120622 0.0603109 0.998180i \(-0.480791\pi\)
0.0603109 + 0.998180i \(0.480791\pi\)
\(132\) 11.6629i 1.01513i
\(133\) − 15.6629i − 1.35815i
\(134\) 8.54420 0.738106
\(135\) 0 0
\(136\) −60.1075 −5.15417
\(137\) 6.49341i 0.554770i 0.960759 + 0.277385i \(0.0894677\pi\)
−0.960759 + 0.277385i \(0.910532\pi\)
\(138\) − 10.3634i − 0.882194i
\(139\) 19.0132 1.61268 0.806338 0.591455i \(-0.201446\pi\)
0.806338 + 0.591455i \(0.201446\pi\)
\(140\) 0 0
\(141\) −5.17347 −0.435685
\(142\) − 30.3634i − 2.54804i
\(143\) 3.78892i 0.316845i
\(144\) 28.8496 2.40413
\(145\) 0 0
\(146\) −30.0811 −2.48953
\(147\) 8.28233i 0.683115i
\(148\) − 28.3938i − 2.33395i
\(149\) −6.62530 −0.542766 −0.271383 0.962471i \(-0.587481\pi\)
−0.271383 + 0.962471i \(0.587481\pi\)
\(150\) 0 0
\(151\) −9.27504 −0.754792 −0.377396 0.926052i \(-0.623180\pi\)
−0.377396 + 0.926052i \(0.623180\pi\)
\(152\) − 31.8192i − 2.58088i
\(153\) 16.7308i 1.35261i
\(154\) 31.1998 2.51415
\(155\) 0 0
\(156\) −5.61213 −0.449330
\(157\) 5.00729i 0.399626i 0.979834 + 0.199813i \(0.0640334\pi\)
−0.979834 + 0.199813i \(0.935967\pi\)
\(158\) − 12.6556i − 1.00683i
\(159\) 5.61213 0.445071
\(160\) 0 0
\(161\) −19.9756 −1.57429
\(162\) − 9.56230i − 0.751285i
\(163\) 7.50659i 0.587961i 0.955811 + 0.293981i \(0.0949801\pi\)
−0.955811 + 0.293981i \(0.905020\pi\)
\(164\) 35.9003 2.80335
\(165\) 0 0
\(166\) 6.80606 0.528253
\(167\) − 21.8945i − 1.69424i −0.531398 0.847122i \(-0.678333\pi\)
0.531398 0.847122i \(-0.321667\pi\)
\(168\) 28.2882i 2.18248i
\(169\) 11.1768 0.859753
\(170\) 0 0
\(171\) −8.85685 −0.677300
\(172\) 16.4690i 1.25575i
\(173\) 7.02302i 0.533951i 0.963703 + 0.266975i \(0.0860242\pi\)
−0.963703 + 0.266975i \(0.913976\pi\)
\(174\) 2.15633 0.163471
\(175\) 0 0
\(176\) 34.4445 2.59635
\(177\) − 2.07522i − 0.155983i
\(178\) 38.2882i 2.86982i
\(179\) −4.77575 −0.356956 −0.178478 0.983944i \(-0.557117\pi\)
−0.178478 + 0.983944i \(0.557117\pi\)
\(180\) 0 0
\(181\) 1.87399 0.139293 0.0696464 0.997572i \(-0.477813\pi\)
0.0696464 + 0.997572i \(0.477813\pi\)
\(182\) 15.0132i 1.11285i
\(183\) − 4.31265i − 0.318800i
\(184\) −40.5804 −2.99163
\(185\) 0 0
\(186\) −0.499293 −0.0366099
\(187\) 19.9756i 1.46076i
\(188\) 33.0943i 2.41365i
\(189\) 17.9248 1.30384
\(190\) 0 0
\(191\) 19.1187 1.38338 0.691691 0.722194i \(-0.256866\pi\)
0.691691 + 0.722194i \(0.256866\pi\)
\(192\) 14.6048i 1.05401i
\(193\) 19.8945i 1.43203i 0.698083 + 0.716017i \(0.254037\pi\)
−0.698083 + 0.716017i \(0.745963\pi\)
\(194\) 4.09569 0.294053
\(195\) 0 0
\(196\) 52.9814 3.78439
\(197\) 13.5369i 0.964464i 0.876043 + 0.482232i \(0.160174\pi\)
−0.876043 + 0.482232i \(0.839826\pi\)
\(198\) − 17.6424i − 1.25379i
\(199\) −2.57452 −0.182503 −0.0912513 0.995828i \(-0.529087\pi\)
−0.0912513 + 0.995828i \(0.529087\pi\)
\(200\) 0 0
\(201\) 2.57452 0.181592
\(202\) − 7.58769i − 0.533868i
\(203\) − 4.15633i − 0.291717i
\(204\) −29.5877 −2.07155
\(205\) 0 0
\(206\) −26.4690 −1.84418
\(207\) 11.2955i 0.785092i
\(208\) 16.5745i 1.14924i
\(209\) −10.5745 −0.731455
\(210\) 0 0
\(211\) 11.8945 0.818848 0.409424 0.912344i \(-0.365730\pi\)
0.409424 + 0.912344i \(0.365730\pi\)
\(212\) − 35.9003i − 2.46565i
\(213\) − 9.14903i − 0.626881i
\(214\) 31.2809 2.13832
\(215\) 0 0
\(216\) 36.4142 2.47767
\(217\) 0.962389i 0.0653312i
\(218\) − 38.7875i − 2.62702i
\(219\) −9.06396 −0.612486
\(220\) 0 0
\(221\) −9.61213 −0.646582
\(222\) − 11.8740i − 0.796930i
\(223\) − 1.11871i − 0.0749146i −0.999298 0.0374573i \(-0.988074\pi\)
0.999298 0.0374573i \(-0.0119258\pi\)
\(224\) 66.2941 4.42946
\(225\) 0 0
\(226\) −43.8554 −2.91722
\(227\) − 0.0303172i − 0.00201222i −0.999999 0.00100611i \(-0.999680\pi\)
0.999999 0.00100611i \(-0.000320255\pi\)
\(228\) − 15.6629i − 1.03730i
\(229\) 5.84955 0.386549 0.193275 0.981145i \(-0.438089\pi\)
0.193275 + 0.981145i \(0.438089\pi\)
\(230\) 0 0
\(231\) 9.40105 0.618543
\(232\) − 8.44358i − 0.554348i
\(233\) 26.1016i 1.70997i 0.518652 + 0.854985i \(0.326434\pi\)
−0.518652 + 0.854985i \(0.673566\pi\)
\(234\) 8.48944 0.554972
\(235\) 0 0
\(236\) −13.2750 −0.864131
\(237\) − 3.81336i − 0.247704i
\(238\) 79.1509i 5.13059i
\(239\) −1.42548 −0.0922069 −0.0461035 0.998937i \(-0.514680\pi\)
−0.0461035 + 0.998937i \(0.514680\pi\)
\(240\) 0 0
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) 8.36248i 0.537561i
\(243\) − 15.8192i − 1.01480i
\(244\) −27.5877 −1.76612
\(245\) 0 0
\(246\) 15.0132 0.957205
\(247\) − 5.08840i − 0.323767i
\(248\) 1.95509i 0.124149i
\(249\) 2.05079 0.129963
\(250\) 0 0
\(251\) 16.9829 1.07195 0.535974 0.844234i \(-0.319944\pi\)
0.535974 + 0.844234i \(0.319944\pi\)
\(252\) − 50.3693i − 3.17297i
\(253\) 13.4861i 0.847865i
\(254\) −41.0797 −2.57757
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) − 25.1998i − 1.57192i −0.618276 0.785961i \(-0.712169\pi\)
0.618276 0.785961i \(-0.287831\pi\)
\(258\) 6.88717i 0.428776i
\(259\) −22.8872 −1.42214
\(260\) 0 0
\(261\) −2.35026 −0.145478
\(262\) − 3.69323i − 0.228168i
\(263\) − 16.1319i − 0.994735i −0.867540 0.497367i \(-0.834300\pi\)
0.867540 0.497367i \(-0.165700\pi\)
\(264\) 19.0982 1.17542
\(265\) 0 0
\(266\) −41.9003 −2.56907
\(267\) 11.5369i 0.706047i
\(268\) − 16.4690i − 1.00600i
\(269\) −5.28963 −0.322514 −0.161257 0.986912i \(-0.551555\pi\)
−0.161257 + 0.986912i \(0.551555\pi\)
\(270\) 0 0
\(271\) 1.13330 0.0688432 0.0344216 0.999407i \(-0.489041\pi\)
0.0344216 + 0.999407i \(0.489041\pi\)
\(272\) 87.3825i 5.29834i
\(273\) 4.52373i 0.273789i
\(274\) 17.3707 1.04940
\(275\) 0 0
\(276\) −19.9756 −1.20239
\(277\) 16.3634i 0.983184i 0.870826 + 0.491592i \(0.163585\pi\)
−0.870826 + 0.491592i \(0.836415\pi\)
\(278\) − 50.8627i − 3.05054i
\(279\) 0.544198 0.0325803
\(280\) 0 0
\(281\) −24.8265 −1.48103 −0.740513 0.672042i \(-0.765418\pi\)
−0.740513 + 0.672042i \(0.765418\pi\)
\(282\) 13.8397i 0.824142i
\(283\) − 4.18076i − 0.248521i −0.992250 0.124260i \(-0.960344\pi\)
0.992250 0.124260i \(-0.0396558\pi\)
\(284\) −58.5256 −3.47286
\(285\) 0 0
\(286\) 10.1359 0.599346
\(287\) − 28.9380i − 1.70815i
\(288\) − 37.4871i − 2.20895i
\(289\) −33.6761 −1.98095
\(290\) 0 0
\(291\) 1.23410 0.0723444
\(292\) 57.9814i 3.39311i
\(293\) 23.6180i 1.37978i 0.723915 + 0.689889i \(0.242341\pi\)
−0.723915 + 0.689889i \(0.757659\pi\)
\(294\) 22.1563 1.29218
\(295\) 0 0
\(296\) −46.4953 −2.70249
\(297\) − 12.1016i − 0.702204i
\(298\) 17.7235i 1.02670i
\(299\) −6.48944 −0.375294
\(300\) 0 0
\(301\) 13.2750 0.765161
\(302\) 24.8119i 1.42777i
\(303\) − 2.28630i − 0.131345i
\(304\) −46.2579 −2.65307
\(305\) 0 0
\(306\) 44.7572 2.55860
\(307\) − 32.5052i − 1.85517i −0.373614 0.927584i \(-0.621882\pi\)
0.373614 0.927584i \(-0.378118\pi\)
\(308\) − 60.1378i − 3.42667i
\(309\) −7.97556 −0.453714
\(310\) 0 0
\(311\) −9.31994 −0.528486 −0.264243 0.964456i \(-0.585122\pi\)
−0.264243 + 0.964456i \(0.585122\pi\)
\(312\) 9.18997i 0.520279i
\(313\) 9.60228i 0.542753i 0.962473 + 0.271376i \(0.0874788\pi\)
−0.962473 + 0.271376i \(0.912521\pi\)
\(314\) 13.3952 0.755933
\(315\) 0 0
\(316\) −24.3938 −1.37226
\(317\) − 19.3707i − 1.08797i −0.839095 0.543984i \(-0.816915\pi\)
0.839095 0.543984i \(-0.183085\pi\)
\(318\) − 15.0132i − 0.841897i
\(319\) −2.80606 −0.157109
\(320\) 0 0
\(321\) 9.42548 0.526079
\(322\) 53.4372i 2.97794i
\(323\) − 26.8265i − 1.49267i
\(324\) −18.4314 −1.02396
\(325\) 0 0
\(326\) 20.0811 1.11219
\(327\) − 11.6873i − 0.646312i
\(328\) − 58.7875i − 3.24600i
\(329\) 26.6761 1.47070
\(330\) 0 0
\(331\) −29.5428 −1.62382 −0.811909 0.583784i \(-0.801572\pi\)
−0.811909 + 0.583784i \(0.801572\pi\)
\(332\) − 13.1187i − 0.719983i
\(333\) 12.9419i 0.709213i
\(334\) −58.5705 −3.20484
\(335\) 0 0
\(336\) 41.1246 2.24353
\(337\) − 12.5442i − 0.683326i −0.939823 0.341663i \(-0.889010\pi\)
0.939823 0.341663i \(-0.110990\pi\)
\(338\) − 29.8994i − 1.62631i
\(339\) −13.2144 −0.717708
\(340\) 0 0
\(341\) 0.649738 0.0351853
\(342\) 23.6932i 1.28118i
\(343\) − 13.6121i − 0.734986i
\(344\) 26.9683 1.45403
\(345\) 0 0
\(346\) 18.7875 1.01002
\(347\) 25.0943i 1.34713i 0.739127 + 0.673566i \(0.235238\pi\)
−0.739127 + 0.673566i \(0.764762\pi\)
\(348\) − 4.15633i − 0.222802i
\(349\) 17.0738 0.913940 0.456970 0.889482i \(-0.348935\pi\)
0.456970 + 0.889482i \(0.348935\pi\)
\(350\) 0 0
\(351\) 5.82321 0.310820
\(352\) − 44.7572i − 2.38557i
\(353\) 5.66291i 0.301406i 0.988579 + 0.150703i \(0.0481538\pi\)
−0.988579 + 0.150703i \(0.951846\pi\)
\(354\) −5.55149 −0.295058
\(355\) 0 0
\(356\) 73.8007 3.91143
\(357\) 23.8496i 1.26225i
\(358\) 12.7757i 0.675219i
\(359\) −0.755278 −0.0398621 −0.0199310 0.999801i \(-0.506345\pi\)
−0.0199310 + 0.999801i \(0.506345\pi\)
\(360\) 0 0
\(361\) −4.79877 −0.252567
\(362\) − 5.01317i − 0.263487i
\(363\) 2.51976i 0.132253i
\(364\) 28.9380 1.51676
\(365\) 0 0
\(366\) −11.5369 −0.603044
\(367\) − 11.4460i − 0.597474i −0.954335 0.298737i \(-0.903435\pi\)
0.954335 0.298737i \(-0.0965653\pi\)
\(368\) 58.9946i 3.07531i
\(369\) −16.3634 −0.851846
\(370\) 0 0
\(371\) −28.9380 −1.50238
\(372\) 0.962389i 0.0498975i
\(373\) 3.86414i 0.200078i 0.994984 + 0.100039i \(0.0318967\pi\)
−0.994984 + 0.100039i \(0.968103\pi\)
\(374\) 53.4372 2.76317
\(375\) 0 0
\(376\) 54.1925 2.79477
\(377\) − 1.35026i − 0.0695420i
\(378\) − 47.9511i − 2.46634i
\(379\) −12.1055 −0.621820 −0.310910 0.950439i \(-0.600634\pi\)
−0.310910 + 0.950439i \(0.600634\pi\)
\(380\) 0 0
\(381\) −12.3780 −0.634145
\(382\) − 51.1451i − 2.61681i
\(383\) − 10.0205i − 0.512022i −0.966674 0.256011i \(-0.917592\pi\)
0.966674 0.256011i \(-0.0824083\pi\)
\(384\) 13.3561 0.681578
\(385\) 0 0
\(386\) 53.2203 2.70884
\(387\) − 7.50659i − 0.381581i
\(388\) − 7.89446i − 0.400780i
\(389\) 25.6629 1.30116 0.650581 0.759437i \(-0.274526\pi\)
0.650581 + 0.759437i \(0.274526\pi\)
\(390\) 0 0
\(391\) −34.2130 −1.73023
\(392\) − 86.7581i − 4.38195i
\(393\) − 1.11283i − 0.0561351i
\(394\) 36.2130 1.82438
\(395\) 0 0
\(396\) −34.0059 −1.70886
\(397\) − 27.7137i − 1.39091i −0.718569 0.695455i \(-0.755203\pi\)
0.718569 0.695455i \(-0.244797\pi\)
\(398\) 6.88717i 0.345222i
\(399\) −12.6253 −0.632056
\(400\) 0 0
\(401\) 7.42548 0.370811 0.185406 0.982662i \(-0.440640\pi\)
0.185406 + 0.982662i \(0.440640\pi\)
\(402\) − 6.88717i − 0.343501i
\(403\) 0.312650i 0.0155742i
\(404\) −14.6253 −0.727636
\(405\) 0 0
\(406\) −11.1187 −0.551812
\(407\) 15.4518i 0.765919i
\(408\) 48.4504i 2.39865i
\(409\) 33.1998 1.64163 0.820813 0.571198i \(-0.193521\pi\)
0.820813 + 0.571198i \(0.193521\pi\)
\(410\) 0 0
\(411\) 5.23410 0.258179
\(412\) 51.0191i 2.51353i
\(413\) 10.7005i 0.526538i
\(414\) 30.2170 1.48508
\(415\) 0 0
\(416\) 21.5369 1.05593
\(417\) − 15.3258i − 0.750509i
\(418\) 28.2882i 1.38362i
\(419\) −16.5599 −0.809005 −0.404503 0.914537i \(-0.632555\pi\)
−0.404503 + 0.914537i \(0.632555\pi\)
\(420\) 0 0
\(421\) −8.82653 −0.430179 −0.215089 0.976594i \(-0.569004\pi\)
−0.215089 + 0.976594i \(0.569004\pi\)
\(422\) − 31.8192i − 1.54894i
\(423\) − 15.0844i − 0.733430i
\(424\) −58.7875 −2.85497
\(425\) 0 0
\(426\) −24.4749 −1.18581
\(427\) 22.2374i 1.07614i
\(428\) − 60.2941i − 2.91442i
\(429\) 3.05411 0.147454
\(430\) 0 0
\(431\) 4.25202 0.204812 0.102406 0.994743i \(-0.467346\pi\)
0.102406 + 0.994743i \(0.467346\pi\)
\(432\) − 52.9380i − 2.54698i
\(433\) − 1.81924i − 0.0874270i −0.999044 0.0437135i \(-0.986081\pi\)
0.999044 0.0437135i \(-0.0139189\pi\)
\(434\) 2.57452 0.123581
\(435\) 0 0
\(436\) −74.7631 −3.58050
\(437\) − 18.1114i − 0.866387i
\(438\) 24.2473i 1.15858i
\(439\) −14.1114 −0.673501 −0.336751 0.941594i \(-0.609328\pi\)
−0.336751 + 0.941594i \(0.609328\pi\)
\(440\) 0 0
\(441\) −24.1490 −1.14995
\(442\) 25.7137i 1.22308i
\(443\) − 17.2809i − 0.821041i −0.911851 0.410521i \(-0.865347\pi\)
0.911851 0.410521i \(-0.134653\pi\)
\(444\) −22.8872 −1.08618
\(445\) 0 0
\(446\) −2.99271 −0.141709
\(447\) 5.34041i 0.252593i
\(448\) − 75.3073i − 3.55793i
\(449\) −9.35026 −0.441266 −0.220633 0.975357i \(-0.570812\pi\)
−0.220633 + 0.975357i \(0.570812\pi\)
\(450\) 0 0
\(451\) −19.5369 −0.919957
\(452\) 84.5315i 3.97603i
\(453\) 7.47627i 0.351266i
\(454\) −0.0811024 −0.00380632
\(455\) 0 0
\(456\) −25.6483 −1.20109
\(457\) 17.6629i 0.826236i 0.910677 + 0.413118i \(0.135560\pi\)
−0.910677 + 0.413118i \(0.864440\pi\)
\(458\) − 15.6483i − 0.731198i
\(459\) 30.7005 1.43298
\(460\) 0 0
\(461\) 15.5633 0.724853 0.362426 0.932012i \(-0.381948\pi\)
0.362426 + 0.932012i \(0.381948\pi\)
\(462\) − 25.1490i − 1.17004i
\(463\) 2.98286i 0.138625i 0.997595 + 0.0693126i \(0.0220806\pi\)
−0.997595 + 0.0693126i \(0.977919\pi\)
\(464\) −12.2750 −0.569854
\(465\) 0 0
\(466\) 69.8251 3.23459
\(467\) − 34.5804i − 1.60019i −0.599873 0.800095i \(-0.704782\pi\)
0.599873 0.800095i \(-0.295218\pi\)
\(468\) − 16.3634i − 0.756400i
\(469\) −13.2750 −0.612984
\(470\) 0 0
\(471\) 4.03620 0.185978
\(472\) 21.7381i 1.00058i
\(473\) − 8.96239i − 0.412091i
\(474\) −10.2012 −0.468558
\(475\) 0 0
\(476\) 152.564 6.99275
\(477\) 16.3634i 0.749230i
\(478\) 3.81336i 0.174419i
\(479\) 34.1925 1.56230 0.781148 0.624346i \(-0.214634\pi\)
0.781148 + 0.624346i \(0.214634\pi\)
\(480\) 0 0
\(481\) −7.43533 −0.339022
\(482\) 0.201231i 0.00916581i
\(483\) 16.1016i 0.732647i
\(484\) 16.1187 0.732669
\(485\) 0 0
\(486\) −42.3185 −1.91961
\(487\) − 38.4953i − 1.74439i −0.489159 0.872195i \(-0.662696\pi\)
0.489159 0.872195i \(-0.337304\pi\)
\(488\) 45.1754i 2.04499i
\(489\) 6.05079 0.273626
\(490\) 0 0
\(491\) −27.4676 −1.23959 −0.619797 0.784762i \(-0.712785\pi\)
−0.619797 + 0.784762i \(0.712785\pi\)
\(492\) − 28.9380i − 1.30462i
\(493\) − 7.11871i − 0.320611i
\(494\) −13.6121 −0.612439
\(495\) 0 0
\(496\) 2.84226 0.127621
\(497\) 47.1754i 2.11610i
\(498\) − 5.48612i − 0.245839i
\(499\) 32.1016 1.43706 0.718532 0.695494i \(-0.244814\pi\)
0.718532 + 0.695494i \(0.244814\pi\)
\(500\) 0 0
\(501\) −17.6483 −0.788469
\(502\) − 45.4314i − 2.02770i
\(503\) 9.74401i 0.434464i 0.976120 + 0.217232i \(0.0697028\pi\)
−0.976120 + 0.217232i \(0.930297\pi\)
\(504\) −82.4807 −3.67398
\(505\) 0 0
\(506\) 36.0771 1.60382
\(507\) − 9.00920i − 0.400113i
\(508\) 79.1813i 3.51310i
\(509\) 8.57452 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(510\) 0 0
\(511\) 46.7367 2.06751
\(512\) 11.5017i 0.508306i
\(513\) 16.2520i 0.717544i
\(514\) −67.4128 −2.97345
\(515\) 0 0
\(516\) 13.2750 0.584401
\(517\) − 18.0098i − 0.792072i
\(518\) 61.2262i 2.69012i
\(519\) 5.66100 0.248490
\(520\) 0 0
\(521\) −37.8251 −1.65715 −0.828574 0.559879i \(-0.810848\pi\)
−0.828574 + 0.559879i \(0.810848\pi\)
\(522\) 6.28726i 0.275186i
\(523\) − 13.5818i − 0.593891i −0.954894 0.296946i \(-0.904032\pi\)
0.954894 0.296946i \(-0.0959680\pi\)
\(524\) −7.11871 −0.310982
\(525\) 0 0
\(526\) −43.1549 −1.88164
\(527\) 1.64832i 0.0718021i
\(528\) − 27.7645i − 1.20829i
\(529\) −0.0982457 −0.00427155
\(530\) 0 0
\(531\) 6.05079 0.262582
\(532\) 80.7631i 3.50152i
\(533\) − 9.40105i − 0.407205i
\(534\) 30.8627 1.33556
\(535\) 0 0
\(536\) −26.9683 −1.16485
\(537\) 3.84955i 0.166121i
\(538\) 14.1504i 0.610069i
\(539\) −28.8324 −1.24190
\(540\) 0 0
\(541\) 42.3127 1.81916 0.909581 0.415526i \(-0.136402\pi\)
0.909581 + 0.415526i \(0.136402\pi\)
\(542\) − 3.03173i − 0.130224i
\(543\) − 1.51056i − 0.0648242i
\(544\) 113.545 4.86819
\(545\) 0 0
\(546\) 12.1016 0.517899
\(547\) 36.4690i 1.55930i 0.626215 + 0.779650i \(0.284603\pi\)
−0.626215 + 0.779650i \(0.715397\pi\)
\(548\) − 33.4821i − 1.43029i
\(549\) 12.5745 0.536667
\(550\) 0 0
\(551\) 3.76845 0.160541
\(552\) 32.7104i 1.39225i
\(553\) 19.6629i 0.836152i
\(554\) 43.7743 1.85979
\(555\) 0 0
\(556\) −98.0381 −4.15774
\(557\) − 19.5223i − 0.827187i −0.910462 0.413594i \(-0.864273\pi\)
0.910462 0.413594i \(-0.135727\pi\)
\(558\) − 1.45580i − 0.0616290i
\(559\) 4.31265 0.182406
\(560\) 0 0
\(561\) 16.1016 0.679809
\(562\) 66.4142i 2.80151i
\(563\) − 2.94192i − 0.123987i −0.998077 0.0619936i \(-0.980254\pi\)
0.998077 0.0619936i \(-0.0197458\pi\)
\(564\) 26.6761 1.12327
\(565\) 0 0
\(566\) −11.1841 −0.470102
\(567\) 14.8568i 0.623929i
\(568\) 95.8369i 4.02123i
\(569\) −2.49929 −0.104776 −0.0523879 0.998627i \(-0.516683\pi\)
−0.0523879 + 0.998627i \(0.516683\pi\)
\(570\) 0 0
\(571\) 43.8007 1.83300 0.916501 0.400033i \(-0.131001\pi\)
0.916501 + 0.400033i \(0.131001\pi\)
\(572\) − 19.5369i − 0.816879i
\(573\) − 15.4109i − 0.643799i
\(574\) −77.4128 −3.23115
\(575\) 0 0
\(576\) −42.5837 −1.77432
\(577\) 23.9062i 0.995229i 0.867398 + 0.497614i \(0.165791\pi\)
−0.867398 + 0.497614i \(0.834209\pi\)
\(578\) 90.0879i 3.74716i
\(579\) 16.0362 0.666442
\(580\) 0 0
\(581\) −10.5745 −0.438705
\(582\) − 3.30139i − 0.136847i
\(583\) 19.5369i 0.809136i
\(584\) 94.9457 3.92888
\(585\) 0 0
\(586\) 63.1813 2.60999
\(587\) 3.71767i 0.153445i 0.997053 + 0.0767223i \(0.0244455\pi\)
−0.997053 + 0.0767223i \(0.975555\pi\)
\(588\) − 42.7064i − 1.76118i
\(589\) −0.872577 −0.0359539
\(590\) 0 0
\(591\) 10.9116 0.448843
\(592\) 67.5936i 2.77808i
\(593\) − 25.5125i − 1.04767i −0.851819 0.523836i \(-0.824501\pi\)
0.851819 0.523836i \(-0.175499\pi\)
\(594\) −32.3733 −1.32829
\(595\) 0 0
\(596\) 34.1622 1.39934
\(597\) 2.07522i 0.0849332i
\(598\) 17.3601i 0.709908i
\(599\) 15.6834 0.640806 0.320403 0.947281i \(-0.396182\pi\)
0.320403 + 0.947281i \(0.396182\pi\)
\(600\) 0 0
\(601\) 13.1392 0.535958 0.267979 0.963425i \(-0.413644\pi\)
0.267979 + 0.963425i \(0.413644\pi\)
\(602\) − 35.5125i − 1.44738i
\(603\) 7.50659i 0.305692i
\(604\) 47.8251 1.94598
\(605\) 0 0
\(606\) −6.11616 −0.248452
\(607\) − 18.1465i − 0.736543i −0.929718 0.368271i \(-0.879950\pi\)
0.929718 0.368271i \(-0.120050\pi\)
\(608\) 60.1075i 2.43768i
\(609\) −3.35026 −0.135759
\(610\) 0 0
\(611\) 8.66624 0.350598
\(612\) − 86.2697i − 3.48724i
\(613\) − 32.5501i − 1.31469i −0.753592 0.657343i \(-0.771680\pi\)
0.753592 0.657343i \(-0.228320\pi\)
\(614\) −86.9556 −3.50924
\(615\) 0 0
\(616\) −98.4768 −3.96774
\(617\) − 20.2433i − 0.814965i −0.913213 0.407482i \(-0.866407\pi\)
0.913213 0.407482i \(-0.133593\pi\)
\(618\) 21.3357i 0.858247i
\(619\) 16.2071 0.651419 0.325709 0.945470i \(-0.394397\pi\)
0.325709 + 0.945470i \(0.394397\pi\)
\(620\) 0 0
\(621\) 20.7269 0.831741
\(622\) 24.9321i 0.999685i
\(623\) − 59.4880i − 2.38334i
\(624\) 13.3601 0.534832
\(625\) 0 0
\(626\) 25.6873 1.02667
\(627\) 8.52373i 0.340405i
\(628\) − 25.8192i − 1.03030i
\(629\) −39.1998 −1.56300
\(630\) 0 0
\(631\) 2.13586 0.0850271 0.0425136 0.999096i \(-0.486463\pi\)
0.0425136 + 0.999096i \(0.486463\pi\)
\(632\) 39.9452i 1.58894i
\(633\) − 9.58769i − 0.381076i
\(634\) −51.8192 −2.05800
\(635\) 0 0
\(636\) −28.9380 −1.14746
\(637\) − 13.8740i − 0.549708i
\(638\) 7.50659i 0.297189i
\(639\) 26.6761 1.05529
\(640\) 0 0
\(641\) −14.0362 −0.554396 −0.277198 0.960813i \(-0.589406\pi\)
−0.277198 + 0.960813i \(0.589406\pi\)
\(642\) − 25.2144i − 0.995133i
\(643\) − 43.8799i − 1.73045i −0.501381 0.865227i \(-0.667174\pi\)
0.501381 0.865227i \(-0.332826\pi\)
\(644\) 103.000 4.05879
\(645\) 0 0
\(646\) −71.7645 −2.82354
\(647\) 22.5560i 0.886766i 0.896332 + 0.443383i \(0.146222\pi\)
−0.896332 + 0.443383i \(0.853778\pi\)
\(648\) 30.1817i 1.18565i
\(649\) 7.22425 0.283577
\(650\) 0 0
\(651\) 0.775746 0.0304039
\(652\) − 38.7064i − 1.51586i
\(653\) 27.3054i 1.06854i 0.845314 + 0.534271i \(0.179414\pi\)
−0.845314 + 0.534271i \(0.820586\pi\)
\(654\) −31.2652 −1.22257
\(655\) 0 0
\(656\) −85.4636 −3.33679
\(657\) − 26.4280i − 1.03106i
\(658\) − 71.3620i − 2.78198i
\(659\) −14.0665 −0.547954 −0.273977 0.961736i \(-0.588339\pi\)
−0.273977 + 0.961736i \(0.588339\pi\)
\(660\) 0 0
\(661\) 38.9741 1.51592 0.757959 0.652302i \(-0.226197\pi\)
0.757959 + 0.652302i \(0.226197\pi\)
\(662\) 79.0308i 3.07162i
\(663\) 7.74798i 0.300907i
\(664\) −21.4821 −0.833669
\(665\) 0 0
\(666\) 34.6213 1.34155
\(667\) − 4.80606i − 0.186092i
\(668\) 112.895i 4.36804i
\(669\) −0.901754 −0.0348638
\(670\) 0 0
\(671\) 15.0132 0.579577
\(672\) − 53.4372i − 2.06139i
\(673\) 20.3390i 0.784011i 0.919963 + 0.392005i \(0.128219\pi\)
−0.919963 + 0.392005i \(0.871781\pi\)
\(674\) −33.5574 −1.29258
\(675\) 0 0
\(676\) −57.6312 −2.21658
\(677\) 19.1841i 0.737304i 0.929567 + 0.368652i \(0.120181\pi\)
−0.929567 + 0.368652i \(0.879819\pi\)
\(678\) 35.3503i 1.35762i
\(679\) −6.36344 −0.244206
\(680\) 0 0
\(681\) −0.0244376 −0.000936449 0
\(682\) − 1.73813i − 0.0665566i
\(683\) − 24.1319i − 0.923381i −0.887041 0.461691i \(-0.847243\pi\)
0.887041 0.461691i \(-0.152757\pi\)
\(684\) 45.6688 1.74619
\(685\) 0 0
\(686\) −36.4142 −1.39030
\(687\) − 4.71511i − 0.179893i
\(688\) − 39.2057i − 1.49470i
\(689\) −9.40105 −0.358151
\(690\) 0 0
\(691\) −4.28821 −0.163131 −0.0815657 0.996668i \(-0.525992\pi\)
−0.0815657 + 0.996668i \(0.525992\pi\)
\(692\) − 36.2130i − 1.37661i
\(693\) 27.4109i 1.04125i
\(694\) 67.1305 2.54824
\(695\) 0 0
\(696\) −6.80606 −0.257983
\(697\) − 49.5633i − 1.87734i
\(698\) − 45.6747i − 1.72881i
\(699\) 21.0395 0.795788
\(700\) 0 0
\(701\) −14.1260 −0.533532 −0.266766 0.963761i \(-0.585955\pi\)
−0.266766 + 0.963761i \(0.585955\pi\)
\(702\) − 15.5778i − 0.587948i
\(703\) − 20.7513i − 0.782650i
\(704\) −50.8423 −1.91619
\(705\) 0 0
\(706\) 15.1490 0.570141
\(707\) 11.7889i 0.443368i
\(708\) 10.7005i 0.402150i
\(709\) −6.75131 −0.253551 −0.126775 0.991931i \(-0.540463\pi\)
−0.126775 + 0.991931i \(0.540463\pi\)
\(710\) 0 0
\(711\) 11.1187 0.416984
\(712\) − 120.850i − 4.52905i
\(713\) 1.11283i 0.0416760i
\(714\) 63.8007 2.38768
\(715\) 0 0
\(716\) 24.6253 0.920291
\(717\) 1.14903i 0.0429113i
\(718\) 2.02047i 0.0754032i
\(719\) 43.8251 1.63440 0.817201 0.576353i \(-0.195525\pi\)
0.817201 + 0.576353i \(0.195525\pi\)
\(720\) 0 0
\(721\) 41.1246 1.53156
\(722\) 12.8373i 0.477756i
\(723\) 0.0606343i 0.00225502i
\(724\) −9.66291 −0.359119
\(725\) 0 0
\(726\) 6.74069 0.250170
\(727\) 14.8813i 0.551916i 0.961170 + 0.275958i \(0.0889951\pi\)
−0.961170 + 0.275958i \(0.911005\pi\)
\(728\) − 47.3865i − 1.75626i
\(729\) −2.02776 −0.0751023
\(730\) 0 0
\(731\) 22.7367 0.840948
\(732\) 22.2374i 0.821919i
\(733\) 7.17935i 0.265175i 0.991171 + 0.132588i \(0.0423286\pi\)
−0.991171 + 0.132588i \(0.957671\pi\)
\(734\) −30.6194 −1.13018
\(735\) 0 0
\(736\) 76.6575 2.82563
\(737\) 8.96239i 0.330134i
\(738\) 43.7743i 1.61136i
\(739\) −16.1709 −0.594857 −0.297428 0.954744i \(-0.596129\pi\)
−0.297428 + 0.954744i \(0.596129\pi\)
\(740\) 0 0
\(741\) −4.10157 −0.150675
\(742\) 77.4128i 2.84191i
\(743\) 27.8192i 1.02059i 0.860000 + 0.510294i \(0.170464\pi\)
−0.860000 + 0.510294i \(0.829536\pi\)
\(744\) 1.57593 0.0577764
\(745\) 0 0
\(746\) 10.3371 0.378468
\(747\) 5.97953i 0.218780i
\(748\) − 103.000i − 3.76607i
\(749\) −48.6009 −1.77584
\(750\) 0 0
\(751\) −17.6326 −0.643423 −0.321711 0.946838i \(-0.604258\pi\)
−0.321711 + 0.946838i \(0.604258\pi\)
\(752\) − 78.7835i − 2.87294i
\(753\) − 13.6893i − 0.498864i
\(754\) −3.61213 −0.131546
\(755\) 0 0
\(756\) −92.4260 −3.36150
\(757\) − 1.53102i − 0.0556460i −0.999613 0.0278230i \(-0.991143\pi\)
0.999613 0.0278230i \(-0.00885748\pi\)
\(758\) 32.3839i 1.17624i
\(759\) 10.8707 0.394580
\(760\) 0 0
\(761\) −34.4749 −1.24971 −0.624856 0.780740i \(-0.714842\pi\)
−0.624856 + 0.780740i \(0.714842\pi\)
\(762\) 33.1128i 1.19955i
\(763\) 60.2638i 2.18170i
\(764\) −98.5823 −3.56658
\(765\) 0 0
\(766\) −26.8061 −0.968542
\(767\) 3.47627i 0.125521i
\(768\) − 6.51976i − 0.235262i
\(769\) 19.3404 0.697433 0.348717 0.937228i \(-0.386618\pi\)
0.348717 + 0.937228i \(0.386618\pi\)
\(770\) 0 0
\(771\) −20.3127 −0.731542
\(772\) − 102.582i − 3.69202i
\(773\) 36.2677i 1.30446i 0.758021 + 0.652230i \(0.226166\pi\)
−0.758021 + 0.652230i \(0.773834\pi\)
\(774\) −20.0811 −0.721800
\(775\) 0 0
\(776\) −12.9273 −0.464064
\(777\) 18.4485i 0.661837i
\(778\) − 68.6516i − 2.46128i
\(779\) 26.2374 0.940053
\(780\) 0 0
\(781\) 31.8496 1.13967
\(782\) 91.5242i 3.27290i
\(783\) 4.31265i 0.154122i
\(784\) −126.127 −4.50452
\(785\) 0 0
\(786\) −2.97698 −0.106185
\(787\) − 32.0059i − 1.14089i −0.821337 0.570443i \(-0.806771\pi\)
0.821337 0.570443i \(-0.193229\pi\)
\(788\) − 69.8007i − 2.48655i
\(789\) −13.0033 −0.462931
\(790\) 0 0
\(791\) 68.1378 2.42270
\(792\) 55.6853i 1.97869i
\(793\) 7.22425i 0.256541i
\(794\) −74.1378 −2.63105
\(795\) 0 0
\(796\) 13.2750 0.470521
\(797\) 41.6932i 1.47685i 0.674336 + 0.738425i \(0.264430\pi\)
−0.674336 + 0.738425i \(0.735570\pi\)
\(798\) 33.7743i 1.19560i
\(799\) 45.6893 1.61637
\(800\) 0 0
\(801\) −33.6385 −1.18856
\(802\) − 19.8641i − 0.701427i
\(803\) − 31.5534i − 1.11350i
\(804\) −13.2750 −0.468175
\(805\) 0 0
\(806\) 0.836381 0.0294603
\(807\) 4.26378i 0.150092i
\(808\) 23.9492i 0.842530i
\(809\) −30.6371 −1.07714 −0.538571 0.842580i \(-0.681036\pi\)
−0.538571 + 0.842580i \(0.681036\pi\)
\(810\) 0 0
\(811\) 25.4617 0.894081 0.447040 0.894514i \(-0.352478\pi\)
0.447040 + 0.894514i \(0.352478\pi\)
\(812\) 21.4314i 0.752093i
\(813\) − 0.913513i − 0.0320383i
\(814\) 41.3357 1.44881
\(815\) 0 0
\(816\) 70.4358 2.46575
\(817\) 12.0362i 0.421093i
\(818\) − 88.8139i − 3.10530i
\(819\) −13.1900 −0.460895
\(820\) 0 0
\(821\) −32.7005 −1.14126 −0.570628 0.821209i \(-0.693300\pi\)
−0.570628 + 0.821209i \(0.693300\pi\)
\(822\) − 14.0019i − 0.488373i
\(823\) − 31.1041i − 1.08422i −0.840307 0.542111i \(-0.817625\pi\)
0.840307 0.542111i \(-0.182375\pi\)
\(824\) 83.5447 2.91042
\(825\) 0 0
\(826\) 28.6253 0.996002
\(827\) 1.58181i 0.0550049i 0.999622 + 0.0275025i \(0.00875541\pi\)
−0.999622 + 0.0275025i \(0.991245\pi\)
\(828\) − 58.2433i − 2.02409i
\(829\) 0.111420 0.00386976 0.00193488 0.999998i \(-0.499384\pi\)
0.00193488 + 0.999998i \(0.499384\pi\)
\(830\) 0 0
\(831\) 13.1900 0.457555
\(832\) − 24.4650i − 0.848171i
\(833\) − 73.1451i − 2.53433i
\(834\) −40.9986 −1.41966
\(835\) 0 0
\(836\) 54.5256 1.88581
\(837\) − 0.998585i − 0.0345162i
\(838\) 44.3000i 1.53032i
\(839\) −28.9829 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 23.6121i 0.813728i
\(843\) 20.0118i 0.689242i
\(844\) −61.3317 −2.11112
\(845\) 0 0
\(846\) −40.3528 −1.38736
\(847\) − 12.9927i − 0.446435i
\(848\) 85.4636i 2.93483i
\(849\) −3.36996 −0.115657
\(850\) 0 0
\(851\) −26.4650 −0.907209
\(852\) 47.1754i 1.61620i
\(853\) − 7.77319i − 0.266149i −0.991106 0.133075i \(-0.957515\pi\)
0.991106 0.133075i \(-0.0424850\pi\)
\(854\) 59.4880 2.03564
\(855\) 0 0
\(856\) −98.7328 −3.37462
\(857\) 13.8740i 0.473927i 0.971519 + 0.236963i \(0.0761521\pi\)
−0.971519 + 0.236963i \(0.923848\pi\)
\(858\) − 8.17014i − 0.278924i
\(859\) 15.2809 0.521378 0.260689 0.965423i \(-0.416050\pi\)
0.260689 + 0.965423i \(0.416050\pi\)
\(860\) 0 0
\(861\) −23.3258 −0.794942
\(862\) − 11.3747i − 0.387424i
\(863\) 31.2301i 1.06309i 0.847031 + 0.531543i \(0.178388\pi\)
−0.847031 + 0.531543i \(0.821612\pi\)
\(864\) −68.7875 −2.34020
\(865\) 0 0
\(866\) −4.86670 −0.165377
\(867\) 27.1451i 0.921895i
\(868\) − 4.96239i − 0.168434i
\(869\) 13.2750 0.450325
\(870\) 0 0
\(871\) −4.31265 −0.146129
\(872\) 122.426i 4.14587i
\(873\) 3.59831i 0.121784i
\(874\) −48.4504 −1.63886
\(875\) 0 0
\(876\) 46.7367 1.57909
\(877\) 4.26187i 0.143913i 0.997408 + 0.0719565i \(0.0229243\pi\)
−0.997408 + 0.0719565i \(0.977076\pi\)
\(878\) 37.7499i 1.27400i
\(879\) 19.0376 0.642123
\(880\) 0 0
\(881\) 15.2144 0.512586 0.256293 0.966599i \(-0.417499\pi\)
0.256293 + 0.966599i \(0.417499\pi\)
\(882\) 64.6018i 2.17526i
\(883\) − 13.7078i − 0.461305i −0.973036 0.230652i \(-0.925914\pi\)
0.973036 0.230652i \(-0.0740860\pi\)
\(884\) 49.5633 1.66699
\(885\) 0 0
\(886\) −46.2287 −1.55308
\(887\) − 12.6556i − 0.424934i −0.977168 0.212467i \(-0.931850\pi\)
0.977168 0.212467i \(-0.0681498\pi\)
\(888\) 37.4782i 1.25769i
\(889\) 63.8251 2.14063
\(890\) 0 0
\(891\) 10.0303 0.336028
\(892\) 5.76845i 0.193142i
\(893\) 24.1866i 0.809375i
\(894\) 14.2863 0.477805
\(895\) 0 0
\(896\) −68.8686 −2.30074
\(897\) 5.23090i 0.174655i
\(898\) 25.0132i 0.834700i
\(899\) −0.231548 −0.00772256
\(900\) 0 0
\(901\) −49.5633 −1.65119
\(902\) 52.2638i 1.74019i
\(903\) − 10.7005i − 0.356091i
\(904\) 138.422 4.60385
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) 12.5540i 0.416850i 0.978038 + 0.208425i \(0.0668338\pi\)
−0.978038 + 0.208425i \(0.933166\pi\)
\(908\) 0.156325i 0.00518783i
\(909\) 6.66624 0.221105
\(910\) 0 0
\(911\) 22.8714 0.757765 0.378882 0.925445i \(-0.376309\pi\)
0.378882 + 0.925445i \(0.376309\pi\)
\(912\) 37.2868i 1.23469i
\(913\) 7.13918i 0.236272i
\(914\) 47.2506 1.56291
\(915\) 0 0
\(916\) −30.1622 −0.996587
\(917\) 5.73813i 0.189490i
\(918\) − 82.1279i − 2.71063i
\(919\) −9.67750 −0.319231 −0.159616 0.987179i \(-0.551025\pi\)
−0.159616 + 0.987179i \(0.551025\pi\)
\(920\) 0 0
\(921\) −26.2012 −0.863360
\(922\) − 41.6337i − 1.37113i
\(923\) 15.3258i 0.504456i
\(924\) −48.4749 −1.59471
\(925\) 0 0
\(926\) 7.97953 0.262224
\(927\) − 23.2546i − 0.763780i
\(928\) 15.9502i 0.523590i
\(929\) −51.9248 −1.70360 −0.851798 0.523870i \(-0.824488\pi\)
−0.851798 + 0.523870i \(0.824488\pi\)
\(930\) 0 0
\(931\) 38.7210 1.26903
\(932\) − 134.588i − 4.40858i
\(933\) 7.51247i 0.245947i
\(934\) −92.5071 −3.02692
\(935\) 0 0
\(936\) −26.7954 −0.875837
\(937\) − 3.58769i − 0.117205i −0.998281 0.0586024i \(-0.981336\pi\)
0.998281 0.0586024i \(-0.0186644\pi\)
\(938\) 35.5125i 1.15952i
\(939\) 7.74004 0.252587
\(940\) 0 0
\(941\) −18.6253 −0.607167 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(942\) − 10.7974i − 0.351797i
\(943\) − 33.4617i − 1.08966i
\(944\) 31.6023 1.02857
\(945\) 0 0
\(946\) −23.9756 −0.779513
\(947\) 16.5950i 0.539265i 0.962963 + 0.269632i \(0.0869021\pi\)
−0.962963 + 0.269632i \(0.913098\pi\)
\(948\) 19.6629i 0.638622i
\(949\) 15.1833 0.492871
\(950\) 0 0
\(951\) −15.6140 −0.506320
\(952\) − 249.826i − 8.09691i
\(953\) − 12.7005i − 0.411410i −0.978614 0.205705i \(-0.934051\pi\)
0.978614 0.205705i \(-0.0659488\pi\)
\(954\) 43.7743 1.41725
\(955\) 0 0
\(956\) 7.35026 0.237724
\(957\) 2.26187i 0.0731157i
\(958\) − 91.4695i − 2.95524i
\(959\) −26.9887 −0.871512
\(960\) 0 0
\(961\) −30.9464 −0.998271
\(962\) 19.8905i 0.641295i
\(963\) 27.4821i 0.885600i
\(964\) 0.387873 0.0124926
\(965\) 0 0
\(966\) 43.0738 1.38588
\(967\) − 37.4314i − 1.20371i −0.798605 0.601856i \(-0.794428\pi\)
0.798605 0.601856i \(-0.205572\pi\)
\(968\) − 26.3947i − 0.848358i
\(969\) −21.6239 −0.694659
\(970\) 0 0
\(971\) 7.51644 0.241214 0.120607 0.992700i \(-0.461516\pi\)
0.120607 + 0.992700i \(0.461516\pi\)
\(972\) 81.5691i 2.61633i
\(973\) 79.0249i 2.53342i
\(974\) −102.980 −3.29969
\(975\) 0 0
\(976\) 65.6747 2.10220
\(977\) 2.52847i 0.0808929i 0.999182 + 0.0404465i \(0.0128780\pi\)
−0.999182 + 0.0404465i \(0.987122\pi\)
\(978\) − 16.1866i − 0.517592i
\(979\) −40.1622 −1.28359
\(980\) 0 0
\(981\) 34.0771 1.08800
\(982\) 73.4793i 2.34482i
\(983\) − 9.32979i − 0.297574i −0.988869 0.148787i \(-0.952463\pi\)
0.988869 0.148787i \(-0.0475369\pi\)
\(984\) −47.3865 −1.51063
\(985\) 0 0
\(986\) −19.0435 −0.606468
\(987\) − 21.5026i − 0.684436i
\(988\) 26.2374i 0.834724i
\(989\) 15.3503 0.488110
\(990\) 0 0
\(991\) 38.4241 1.22058 0.610290 0.792178i \(-0.291053\pi\)
0.610290 + 0.792178i \(0.291053\pi\)
\(992\) − 3.69323i − 0.117260i
\(993\) 23.8134i 0.755694i
\(994\) 126.200 4.00283
\(995\) 0 0
\(996\) −10.5745 −0.335066
\(997\) 18.8423i 0.596740i 0.954450 + 0.298370i \(0.0964430\pi\)
−0.954450 + 0.298370i \(0.903557\pi\)
\(998\) − 85.8759i − 2.71835i
\(999\) 23.7480 0.751353
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 725.2.b.d.349.1 6
5.2 odd 4 145.2.a.d.1.3 3
5.3 odd 4 725.2.a.d.1.1 3
5.4 even 2 inner 725.2.b.d.349.6 6
15.2 even 4 1305.2.a.o.1.1 3
15.8 even 4 6525.2.a.bh.1.3 3
20.7 even 4 2320.2.a.s.1.2 3
35.27 even 4 7105.2.a.p.1.3 3
40.27 even 4 9280.2.a.bm.1.2 3
40.37 odd 4 9280.2.a.bu.1.2 3
145.57 odd 4 4205.2.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.3 3 5.2 odd 4
725.2.a.d.1.1 3 5.3 odd 4
725.2.b.d.349.1 6 1.1 even 1 trivial
725.2.b.d.349.6 6 5.4 even 2 inner
1305.2.a.o.1.1 3 15.2 even 4
2320.2.a.s.1.2 3 20.7 even 4
4205.2.a.e.1.1 3 145.57 odd 4
6525.2.a.bh.1.3 3 15.8 even 4
7105.2.a.p.1.3 3 35.27 even 4
9280.2.a.bm.1.2 3 40.27 even 4
9280.2.a.bu.1.2 3 40.37 odd 4