Properties

Label 725.2.b.c.349.4
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 349.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.c.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421i q^{2} -2.00000i q^{3} -3.82843 q^{4} +4.82843 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.41421i q^{2} -2.00000i q^{3} -3.82843 q^{4} +4.82843 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} -4.82843 q^{11} +7.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843i q^{17} -2.41421i q^{18} -0.828427 q^{19} -1.65685 q^{21} -11.6569i q^{22} -8.82843i q^{23} -8.82843 q^{24} +4.82843 q^{26} -4.00000i q^{27} +3.17157i q^{28} -1.00000 q^{29} -10.4853 q^{31} -1.58579i q^{32} +9.65685i q^{33} -6.82843 q^{34} +3.82843 q^{36} -8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +18.4853 q^{44} +21.3137 q^{46} +0.343146i q^{47} -6.00000i q^{48} +6.31371 q^{49} +5.65685 q^{51} +7.65685i q^{52} +7.65685i q^{53} +9.65685 q^{54} -3.65685 q^{56} +1.65685i q^{57} -2.41421i q^{58} +7.65685 q^{61} -25.3137i q^{62} +0.828427i q^{63} +9.82843 q^{64} -23.3137 q^{66} +10.4853i q^{67} -10.8284i q^{68} -17.6569 q^{69} +7.31371 q^{71} +4.41421i q^{72} -8.48528i q^{73} +20.4853 q^{74} +3.17157 q^{76} +4.00000i q^{77} -9.65685i q^{78} -14.4853 q^{79} -11.0000 q^{81} -14.4853i q^{82} +12.8284i q^{83} +6.34315 q^{84} +14.4853 q^{86} +2.00000i q^{87} +21.3137i q^{88} -3.65685 q^{89} -1.65685 q^{91} +33.7990i q^{92} +20.9706i q^{93} -0.828427 q^{94} -3.17157 q^{96} -4.48528i q^{97} +15.2426i q^{98} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46} - 20 q^{49} + 16 q^{54} + 8 q^{56} + 8 q^{61} + 28 q^{64} - 48 q^{66} - 48 q^{69} - 16 q^{71} + 48 q^{74} + 24 q^{76} - 24 q^{79} - 44 q^{81} + 48 q^{84} + 24 q^{86} + 8 q^{89} + 16 q^{91} + 8 q^{94} - 24 q^{96} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421i 1.70711i 0.521005 + 0.853553i \(0.325557\pi\)
−0.521005 + 0.853553i \(0.674443\pi\)
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) 4.82843 1.97120
\(7\) − 0.828427i − 0.313116i −0.987669 0.156558i \(-0.949960\pi\)
0.987669 0.156558i \(-0.0500398\pi\)
\(8\) − 4.41421i − 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 7.65685i 2.21034i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) − 2.41421i − 0.569036i
\(19\) −0.828427 −0.190054 −0.0950271 0.995475i \(-0.530294\pi\)
−0.0950271 + 0.995475i \(0.530294\pi\)
\(20\) 0 0
\(21\) −1.65685 −0.361555
\(22\) − 11.6569i − 2.48525i
\(23\) − 8.82843i − 1.84085i −0.390914 0.920427i \(-0.627841\pi\)
0.390914 0.920427i \(-0.372159\pi\)
\(24\) −8.82843 −1.80210
\(25\) 0 0
\(26\) 4.82843 0.946932
\(27\) − 4.00000i − 0.769800i
\(28\) 3.17157i 0.599371i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −10.4853 −1.88321 −0.941606 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) − 1.58579i − 0.280330i
\(33\) 9.65685i 1.68104i
\(34\) −6.82843 −1.17107
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) − 8.48528i − 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 18.4853 2.78676
\(45\) 0 0
\(46\) 21.3137 3.14253
\(47\) 0.343146i 0.0500530i 0.999687 + 0.0250265i \(0.00796701\pi\)
−0.999687 + 0.0250265i \(0.992033\pi\)
\(48\) − 6.00000i − 0.866025i
\(49\) 6.31371 0.901958
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 7.65685i 1.06181i
\(53\) 7.65685i 1.05175i 0.850562 + 0.525875i \(0.176262\pi\)
−0.850562 + 0.525875i \(0.823738\pi\)
\(54\) 9.65685 1.31413
\(55\) 0 0
\(56\) −3.65685 −0.488668
\(57\) 1.65685i 0.219456i
\(58\) − 2.41421i − 0.317002i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) − 25.3137i − 3.21484i
\(63\) 0.828427i 0.104372i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) −23.3137 −2.86972
\(67\) 10.4853i 1.28098i 0.767966 + 0.640490i \(0.221269\pi\)
−0.767966 + 0.640490i \(0.778731\pi\)
\(68\) − 10.8284i − 1.31314i
\(69\) −17.6569 −2.12564
\(70\) 0 0
\(71\) 7.31371 0.867978 0.433989 0.900918i \(-0.357106\pi\)
0.433989 + 0.900918i \(0.357106\pi\)
\(72\) 4.41421i 0.520220i
\(73\) − 8.48528i − 0.993127i −0.868000 0.496564i \(-0.834595\pi\)
0.868000 0.496564i \(-0.165405\pi\)
\(74\) 20.4853 2.38137
\(75\) 0 0
\(76\) 3.17157 0.363804
\(77\) 4.00000i 0.455842i
\(78\) − 9.65685i − 1.09342i
\(79\) −14.4853 −1.62972 −0.814861 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 14.4853i − 1.59963i
\(83\) 12.8284i 1.40810i 0.710149 + 0.704051i \(0.248628\pi\)
−0.710149 + 0.704051i \(0.751372\pi\)
\(84\) 6.34315 0.692094
\(85\) 0 0
\(86\) 14.4853 1.56199
\(87\) 2.00000i 0.214423i
\(88\) 21.3137i 2.27205i
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −1.65685 −0.173686
\(92\) 33.7990i 3.52379i
\(93\) 20.9706i 2.17455i
\(94\) −0.828427 −0.0854457
\(95\) 0 0
\(96\) −3.17157 −0.323697
\(97\) − 4.48528i − 0.455411i −0.973730 0.227706i \(-0.926878\pi\)
0.973730 0.227706i \(-0.0731224\pi\)
\(98\) 15.2426i 1.53974i
\(99\) 4.82843 0.485275
\(100\) 0 0
\(101\) 4.34315 0.432159 0.216080 0.976376i \(-0.430673\pi\)
0.216080 + 0.976376i \(0.430673\pi\)
\(102\) 13.6569i 1.35223i
\(103\) − 12.1421i − 1.19640i −0.801347 0.598200i \(-0.795883\pi\)
0.801347 0.598200i \(-0.204117\pi\)
\(104\) −8.82843 −0.865699
\(105\) 0 0
\(106\) −18.4853 −1.79545
\(107\) 8.14214i 0.787130i 0.919297 + 0.393565i \(0.128758\pi\)
−0.919297 + 0.393565i \(0.871242\pi\)
\(108\) 15.3137i 1.47356i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −16.9706 −1.61077
\(112\) − 2.48528i − 0.234837i
\(113\) 2.82843i 0.266076i 0.991111 + 0.133038i \(0.0424732\pi\)
−0.991111 + 0.133038i \(0.957527\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 3.82843 0.355461
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 18.4853i 1.67358i
\(123\) 12.0000i 1.08200i
\(124\) 40.1421 3.60487
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 20.5563i 1.81694i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 16.1421 1.41034 0.705172 0.709036i \(-0.250870\pi\)
0.705172 + 0.709036i \(0.250870\pi\)
\(132\) − 36.9706i − 3.21787i
\(133\) 0.686292i 0.0595090i
\(134\) −25.3137 −2.18677
\(135\) 0 0
\(136\) 12.4853 1.07060
\(137\) 10.8284i 0.925135i 0.886584 + 0.462567i \(0.153072\pi\)
−0.886584 + 0.462567i \(0.846928\pi\)
\(138\) − 42.6274i − 3.62869i
\(139\) −10.3431 −0.877294 −0.438647 0.898659i \(-0.644542\pi\)
−0.438647 + 0.898659i \(0.644542\pi\)
\(140\) 0 0
\(141\) 0.686292 0.0577962
\(142\) 17.6569i 1.48173i
\(143\) 9.65685i 0.807547i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 20.4853 1.69537
\(147\) − 12.6274i − 1.04149i
\(148\) 32.4853i 2.67027i
\(149\) 13.3137 1.09070 0.545351 0.838208i \(-0.316396\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 3.65685i 0.296610i
\(153\) − 2.82843i − 0.228665i
\(154\) −9.65685 −0.778171
\(155\) 0 0
\(156\) 15.3137 1.22608
\(157\) 16.4853i 1.31567i 0.753163 + 0.657834i \(0.228527\pi\)
−0.753163 + 0.657834i \(0.771473\pi\)
\(158\) − 34.9706i − 2.78211i
\(159\) 15.3137 1.21446
\(160\) 0 0
\(161\) −7.31371 −0.576401
\(162\) − 26.5563i − 2.08646i
\(163\) − 19.6569i − 1.53964i −0.638259 0.769822i \(-0.720345\pi\)
0.638259 0.769822i \(-0.279655\pi\)
\(164\) 22.9706 1.79370
\(165\) 0 0
\(166\) −30.9706 −2.40378
\(167\) − 14.4853i − 1.12090i −0.828187 0.560452i \(-0.810627\pi\)
0.828187 0.560452i \(-0.189373\pi\)
\(168\) 7.31371i 0.564265i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0.828427 0.0633514
\(172\) 22.9706i 1.75149i
\(173\) − 5.31371i − 0.403994i −0.979386 0.201997i \(-0.935257\pi\)
0.979386 0.201997i \(-0.0647431\pi\)
\(174\) −4.82843 −0.366042
\(175\) 0 0
\(176\) −14.4853 −1.09187
\(177\) 0 0
\(178\) − 8.82843i − 0.661719i
\(179\) 0.686292 0.0512958 0.0256479 0.999671i \(-0.491835\pi\)
0.0256479 + 0.999671i \(0.491835\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) − 15.3137i − 1.13202i
\(184\) −38.9706 −2.87295
\(185\) 0 0
\(186\) −50.6274 −3.71218
\(187\) − 13.6569i − 0.998688i
\(188\) − 1.31371i − 0.0958120i
\(189\) −3.31371 −0.241037
\(190\) 0 0
\(191\) −15.1716 −1.09778 −0.548888 0.835896i \(-0.684949\pi\)
−0.548888 + 0.835896i \(0.684949\pi\)
\(192\) − 19.6569i − 1.41861i
\(193\) − 12.4853i − 0.898710i −0.893353 0.449355i \(-0.851654\pi\)
0.893353 0.449355i \(-0.148346\pi\)
\(194\) 10.8284 0.777436
\(195\) 0 0
\(196\) −24.1716 −1.72654
\(197\) 8.34315i 0.594425i 0.954811 + 0.297212i \(0.0960569\pi\)
−0.954811 + 0.297212i \(0.903943\pi\)
\(198\) 11.6569i 0.828417i
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 20.9706 1.47915
\(202\) 10.4853i 0.737742i
\(203\) 0.828427i 0.0581442i
\(204\) −21.6569 −1.51628
\(205\) 0 0
\(206\) 29.3137 2.04238
\(207\) 8.82843i 0.613618i
\(208\) − 6.00000i − 0.416025i
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.82843 0.332403 0.166201 0.986092i \(-0.446850\pi\)
0.166201 + 0.986092i \(0.446850\pi\)
\(212\) − 29.3137i − 2.01327i
\(213\) − 14.6274i − 1.00225i
\(214\) −19.6569 −1.34371
\(215\) 0 0
\(216\) −17.6569 −1.20140
\(217\) 8.68629i 0.589664i
\(218\) − 4.82843i − 0.327022i
\(219\) −16.9706 −1.14676
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) − 40.9706i − 2.74976i
\(223\) 21.7990i 1.45977i 0.683571 + 0.729884i \(0.260426\pi\)
−0.683571 + 0.729884i \(0.739574\pi\)
\(224\) −1.31371 −0.0877758
\(225\) 0 0
\(226\) −6.82843 −0.454220
\(227\) 8.14214i 0.540413i 0.962802 + 0.270206i \(0.0870919\pi\)
−0.962802 + 0.270206i \(0.912908\pi\)
\(228\) − 6.34315i − 0.420085i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 4.41421i 0.289807i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −4.82843 −0.315644
\(235\) 0 0
\(236\) 0 0
\(237\) 28.9706i 1.88184i
\(238\) 5.65685i 0.366679i
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 29.7279i 1.91098i
\(243\) 10.0000i 0.641500i
\(244\) −29.3137 −1.87662
\(245\) 0 0
\(246\) −28.9706 −1.84710
\(247\) 1.65685i 0.105423i
\(248\) 46.2843i 2.93905i
\(249\) 25.6569 1.62594
\(250\) 0 0
\(251\) 3.17157 0.200188 0.100094 0.994978i \(-0.468086\pi\)
0.100094 + 0.994978i \(0.468086\pi\)
\(252\) − 3.17157i − 0.199790i
\(253\) 42.6274i 2.67996i
\(254\) 14.4853 0.908887
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) − 29.3137i − 1.82854i −0.405107 0.914269i \(-0.632766\pi\)
0.405107 0.914269i \(-0.367234\pi\)
\(258\) − 28.9706i − 1.80363i
\(259\) −7.02944 −0.436788
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 38.9706i 2.40761i
\(263\) − 8.34315i − 0.514460i −0.966350 0.257230i \(-0.917190\pi\)
0.966350 0.257230i \(-0.0828099\pi\)
\(264\) 42.6274 2.62354
\(265\) 0 0
\(266\) −1.65685 −0.101588
\(267\) 7.31371i 0.447592i
\(268\) − 40.1421i − 2.45207i
\(269\) −1.31371 −0.0800982 −0.0400491 0.999198i \(-0.512751\pi\)
−0.0400491 + 0.999198i \(0.512751\pi\)
\(270\) 0 0
\(271\) 29.7990 1.81016 0.905080 0.425242i \(-0.139811\pi\)
0.905080 + 0.425242i \(0.139811\pi\)
\(272\) 8.48528i 0.514496i
\(273\) 3.31371i 0.200555i
\(274\) −26.1421 −1.57930
\(275\) 0 0
\(276\) 67.5980 4.06892
\(277\) − 7.65685i − 0.460056i −0.973184 0.230028i \(-0.926118\pi\)
0.973184 0.230028i \(-0.0738817\pi\)
\(278\) − 24.9706i − 1.49763i
\(279\) 10.4853 0.627737
\(280\) 0 0
\(281\) −6.68629 −0.398871 −0.199435 0.979911i \(-0.563911\pi\)
−0.199435 + 0.979911i \(0.563911\pi\)
\(282\) 1.65685i 0.0986642i
\(283\) − 0.828427i − 0.0492449i −0.999697 0.0246224i \(-0.992162\pi\)
0.999697 0.0246224i \(-0.00783836\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −23.3137 −1.37857
\(287\) 4.97056i 0.293403i
\(288\) 1.58579i 0.0934434i
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) −8.97056 −0.525864
\(292\) 32.4853i 1.90106i
\(293\) − 8.48528i − 0.495715i −0.968796 0.247858i \(-0.920273\pi\)
0.968796 0.247858i \(-0.0797265\pi\)
\(294\) 30.4853 1.77794
\(295\) 0 0
\(296\) −37.4558 −2.17708
\(297\) 19.3137i 1.12070i
\(298\) 32.1421i 1.86194i
\(299\) −17.6569 −1.02112
\(300\) 0 0
\(301\) −4.97056 −0.286498
\(302\) − 28.9706i − 1.66707i
\(303\) − 8.68629i − 0.499014i
\(304\) −2.48528 −0.142541
\(305\) 0 0
\(306\) 6.82843 0.390355
\(307\) − 10.9706i − 0.626123i −0.949733 0.313062i \(-0.898645\pi\)
0.949733 0.313062i \(-0.101355\pi\)
\(308\) − 15.3137i − 0.872580i
\(309\) −24.2843 −1.38148
\(310\) 0 0
\(311\) −2.48528 −0.140927 −0.0704637 0.997514i \(-0.522448\pi\)
−0.0704637 + 0.997514i \(0.522448\pi\)
\(312\) 17.6569i 0.999623i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) −39.7990 −2.24599
\(315\) 0 0
\(316\) 55.4558 3.11963
\(317\) 2.82843i 0.158860i 0.996840 + 0.0794301i \(0.0253101\pi\)
−0.996840 + 0.0794301i \(0.974690\pi\)
\(318\) 36.9706i 2.07321i
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) 16.2843 0.908899
\(322\) − 17.6569i − 0.983978i
\(323\) − 2.34315i − 0.130376i
\(324\) 42.1127 2.33959
\(325\) 0 0
\(326\) 47.4558 2.62834
\(327\) 4.00000i 0.221201i
\(328\) 26.4853i 1.46241i
\(329\) 0.284271 0.0156724
\(330\) 0 0
\(331\) −17.7990 −0.978321 −0.489160 0.872194i \(-0.662697\pi\)
−0.489160 + 0.872194i \(0.662697\pi\)
\(332\) − 49.1127i − 2.69541i
\(333\) 8.48528i 0.464991i
\(334\) 34.9706 1.91350
\(335\) 0 0
\(336\) −4.97056 −0.271166
\(337\) 6.82843i 0.371968i 0.982553 + 0.185984i \(0.0595473\pi\)
−0.982553 + 0.185984i \(0.940453\pi\)
\(338\) 21.7279i 1.18184i
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) 50.6274 2.74163
\(342\) 2.00000i 0.108148i
\(343\) − 11.0294i − 0.595534i
\(344\) −26.4853 −1.42799
\(345\) 0 0
\(346\) 12.8284 0.689661
\(347\) − 20.1421i − 1.08129i −0.841252 0.540643i \(-0.818181\pi\)
0.841252 0.540643i \(-0.181819\pi\)
\(348\) − 7.65685i − 0.410450i
\(349\) 24.6274 1.31828 0.659138 0.752022i \(-0.270921\pi\)
0.659138 + 0.752022i \(0.270921\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 7.65685i 0.408112i
\(353\) − 15.6569i − 0.833330i −0.909060 0.416665i \(-0.863199\pi\)
0.909060 0.416665i \(-0.136801\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 4.68629i − 0.248025i
\(358\) 1.65685i 0.0875675i
\(359\) 32.1421 1.69640 0.848199 0.529678i \(-0.177687\pi\)
0.848199 + 0.529678i \(0.177687\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) − 14.4853i − 0.761329i
\(363\) − 24.6274i − 1.29260i
\(364\) 6.34315 0.332471
\(365\) 0 0
\(366\) 36.9706 1.93248
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) − 26.4853i − 1.38064i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 6.34315 0.329320
\(372\) − 80.2843i − 4.16255i
\(373\) 26.9706i 1.39648i 0.715862 + 0.698241i \(0.246034\pi\)
−0.715862 + 0.698241i \(0.753966\pi\)
\(374\) 32.9706 1.70487
\(375\) 0 0
\(376\) 1.51472 0.0781156
\(377\) 2.00000i 0.103005i
\(378\) − 8.00000i − 0.411476i
\(379\) −5.51472 −0.283272 −0.141636 0.989919i \(-0.545236\pi\)
−0.141636 + 0.989919i \(0.545236\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) − 36.6274i − 1.87402i
\(383\) − 14.4853i − 0.740163i −0.928999 0.370082i \(-0.879330\pi\)
0.928999 0.370082i \(-0.120670\pi\)
\(384\) 41.1127 2.09802
\(385\) 0 0
\(386\) 30.1421 1.53419
\(387\) 6.00000i 0.304997i
\(388\) 17.1716i 0.871755i
\(389\) 6.68629 0.339008 0.169504 0.985529i \(-0.445783\pi\)
0.169504 + 0.985529i \(0.445783\pi\)
\(390\) 0 0
\(391\) 24.9706 1.26282
\(392\) − 27.8701i − 1.40765i
\(393\) − 32.2843i − 1.62853i
\(394\) −20.1421 −1.01475
\(395\) 0 0
\(396\) −18.4853 −0.928920
\(397\) 8.34315i 0.418730i 0.977838 + 0.209365i \(0.0671398\pi\)
−0.977838 + 0.209365i \(0.932860\pi\)
\(398\) − 28.9706i − 1.45216i
\(399\) 1.37258 0.0687151
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 50.6274i 2.52507i
\(403\) 20.9706i 1.04462i
\(404\) −16.6274 −0.827245
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 40.9706i 2.03084i
\(408\) − 24.9706i − 1.23623i
\(409\) −30.9706 −1.53140 −0.765698 0.643200i \(-0.777606\pi\)
−0.765698 + 0.643200i \(0.777606\pi\)
\(410\) 0 0
\(411\) 21.6569 1.06825
\(412\) 46.4853i 2.29017i
\(413\) 0 0
\(414\) −21.3137 −1.04751
\(415\) 0 0
\(416\) −3.17157 −0.155499
\(417\) 20.6863i 1.01301i
\(418\) 9.65685i 0.472332i
\(419\) 4.97056 0.242828 0.121414 0.992602i \(-0.461257\pi\)
0.121414 + 0.992602i \(0.461257\pi\)
\(420\) 0 0
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) 11.6569i 0.567447i
\(423\) − 0.343146i − 0.0166843i
\(424\) 33.7990 1.64142
\(425\) 0 0
\(426\) 35.3137 1.71095
\(427\) − 6.34315i − 0.306966i
\(428\) − 31.1716i − 1.50673i
\(429\) 19.3137 0.932475
\(430\) 0 0
\(431\) −19.3137 −0.930309 −0.465154 0.885230i \(-0.654001\pi\)
−0.465154 + 0.885230i \(0.654001\pi\)
\(432\) − 12.0000i − 0.577350i
\(433\) − 34.8284i − 1.67375i −0.547396 0.836874i \(-0.684381\pi\)
0.547396 0.836874i \(-0.315619\pi\)
\(434\) −20.9706 −1.00662
\(435\) 0 0
\(436\) 7.65685 0.366697
\(437\) 7.31371i 0.349862i
\(438\) − 40.9706i − 1.95765i
\(439\) 21.6569 1.03363 0.516813 0.856099i \(-0.327118\pi\)
0.516813 + 0.856099i \(0.327118\pi\)
\(440\) 0 0
\(441\) −6.31371 −0.300653
\(442\) 13.6569i 0.649590i
\(443\) 3.65685i 0.173742i 0.996220 + 0.0868712i \(0.0276869\pi\)
−0.996220 + 0.0868712i \(0.972313\pi\)
\(444\) 64.9706 3.08337
\(445\) 0 0
\(446\) −52.6274 −2.49198
\(447\) − 26.6274i − 1.25943i
\(448\) − 8.14214i − 0.384680i
\(449\) −0.343146 −0.0161940 −0.00809702 0.999967i \(-0.502577\pi\)
−0.00809702 + 0.999967i \(0.502577\pi\)
\(450\) 0 0
\(451\) 28.9706 1.36417
\(452\) − 10.8284i − 0.509326i
\(453\) 24.0000i 1.12762i
\(454\) −19.6569 −0.922542
\(455\) 0 0
\(456\) 7.31371 0.342496
\(457\) − 8.34315i − 0.390276i −0.980776 0.195138i \(-0.937485\pi\)
0.980776 0.195138i \(-0.0625155\pi\)
\(458\) 4.82843i 0.225618i
\(459\) 11.3137 0.528079
\(460\) 0 0
\(461\) −24.3431 −1.13377 −0.566887 0.823796i \(-0.691852\pi\)
−0.566887 + 0.823796i \(0.691852\pi\)
\(462\) 19.3137i 0.898555i
\(463\) − 17.7990i − 0.827189i −0.910461 0.413595i \(-0.864273\pi\)
0.910461 0.413595i \(-0.135727\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −43.4558 −2.01305
\(467\) 22.9706i 1.06295i 0.847074 + 0.531475i \(0.178362\pi\)
−0.847074 + 0.531475i \(0.821638\pi\)
\(468\) − 7.65685i − 0.353938i
\(469\) 8.68629 0.401096
\(470\) 0 0
\(471\) 32.9706 1.51920
\(472\) 0 0
\(473\) 28.9706i 1.33207i
\(474\) −69.9411 −3.21250
\(475\) 0 0
\(476\) −8.97056 −0.411165
\(477\) − 7.65685i − 0.350583i
\(478\) 56.2843i 2.57438i
\(479\) −12.8284 −0.586146 −0.293073 0.956090i \(-0.594678\pi\)
−0.293073 + 0.956090i \(0.594678\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 24.1421i 1.09964i
\(483\) 14.6274i 0.665571i
\(484\) −47.1421 −2.14282
\(485\) 0 0
\(486\) −24.1421 −1.09511
\(487\) 29.7990i 1.35032i 0.737671 + 0.675161i \(0.235926\pi\)
−0.737671 + 0.675161i \(0.764074\pi\)
\(488\) − 33.7990i − 1.53001i
\(489\) −39.3137 −1.77783
\(490\) 0 0
\(491\) 43.4558 1.96113 0.980567 0.196183i \(-0.0628545\pi\)
0.980567 + 0.196183i \(0.0628545\pi\)
\(492\) − 45.9411i − 2.07119i
\(493\) − 2.82843i − 0.127386i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −31.4558 −1.41241
\(497\) − 6.05887i − 0.271778i
\(498\) 61.9411i 2.77565i
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −28.9706 −1.29431
\(502\) 7.65685i 0.341742i
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 3.65685 0.162889
\(505\) 0 0
\(506\) −102.912 −4.57498
\(507\) − 18.0000i − 0.799408i
\(508\) 22.9706i 1.01915i
\(509\) 44.6274 1.97808 0.989038 0.147663i \(-0.0471751\pi\)
0.989038 + 0.147663i \(0.0471751\pi\)
\(510\) 0 0
\(511\) −7.02944 −0.310964
\(512\) − 31.2426i − 1.38074i
\(513\) 3.31371i 0.146304i
\(514\) 70.7696 3.12151
\(515\) 0 0
\(516\) 45.9411 2.02245
\(517\) − 1.65685i − 0.0728684i
\(518\) − 16.9706i − 0.745644i
\(519\) −10.6274 −0.466492
\(520\) 0 0
\(521\) 1.31371 0.0575546 0.0287773 0.999586i \(-0.490839\pi\)
0.0287773 + 0.999586i \(0.490839\pi\)
\(522\) 2.41421i 0.105667i
\(523\) − 14.4853i − 0.633397i −0.948526 0.316699i \(-0.897426\pi\)
0.948526 0.316699i \(-0.102574\pi\)
\(524\) −61.7990 −2.69970
\(525\) 0 0
\(526\) 20.1421 0.878239
\(527\) − 29.6569i − 1.29187i
\(528\) 28.9706i 1.26078i
\(529\) −54.9411 −2.38874
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.62742i − 0.113913i
\(533\) 12.0000i 0.519778i
\(534\) −17.6569 −0.764087
\(535\) 0 0
\(536\) 46.2843 1.99918
\(537\) − 1.37258i − 0.0592313i
\(538\) − 3.17157i − 0.136736i
\(539\) −30.4853 −1.31309
\(540\) 0 0
\(541\) −38.9706 −1.67548 −0.837738 0.546073i \(-0.816122\pi\)
−0.837738 + 0.546073i \(0.816122\pi\)
\(542\) 71.9411i 3.09014i
\(543\) 12.0000i 0.514969i
\(544\) 4.48528 0.192305
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 14.4853i − 0.619346i −0.950843 0.309673i \(-0.899780\pi\)
0.950843 0.309673i \(-0.100220\pi\)
\(548\) − 41.4558i − 1.77091i
\(549\) −7.65685 −0.326787
\(550\) 0 0
\(551\) 0.828427 0.0352922
\(552\) 77.9411i 3.31739i
\(553\) 12.0000i 0.510292i
\(554\) 18.4853 0.785364
\(555\) 0 0
\(556\) 39.5980 1.67933
\(557\) − 39.9411i − 1.69236i −0.532897 0.846180i \(-0.678897\pi\)
0.532897 0.846180i \(-0.321103\pi\)
\(558\) 25.3137i 1.07161i
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −27.3137 −1.15319
\(562\) − 16.1421i − 0.680915i
\(563\) − 3.65685i − 0.154118i −0.997027 0.0770590i \(-0.975447\pi\)
0.997027 0.0770590i \(-0.0245530\pi\)
\(564\) −2.62742 −0.110634
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) 9.11270i 0.382697i
\(568\) − 32.2843i − 1.35462i
\(569\) −16.3431 −0.685140 −0.342570 0.939492i \(-0.611297\pi\)
−0.342570 + 0.939492i \(0.611297\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) − 36.9706i − 1.54582i
\(573\) 30.3431i 1.26760i
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) 15.7990i 0.657721i 0.944379 + 0.328860i \(0.106665\pi\)
−0.944379 + 0.328860i \(0.893335\pi\)
\(578\) 21.7279i 0.903762i
\(579\) −24.9706 −1.03774
\(580\) 0 0
\(581\) 10.6274 0.440900
\(582\) − 21.6569i − 0.897705i
\(583\) − 36.9706i − 1.53116i
\(584\) −37.4558 −1.54993
\(585\) 0 0
\(586\) 20.4853 0.846239
\(587\) − 9.79899i − 0.404448i −0.979339 0.202224i \(-0.935183\pi\)
0.979339 0.202224i \(-0.0648168\pi\)
\(588\) 48.3431i 1.99364i
\(589\) 8.68629 0.357912
\(590\) 0 0
\(591\) 16.6863 0.686382
\(592\) − 25.4558i − 1.04623i
\(593\) 3.65685i 0.150169i 0.997177 + 0.0750845i \(0.0239227\pi\)
−0.997177 + 0.0750845i \(0.976077\pi\)
\(594\) −46.6274 −1.91315
\(595\) 0 0
\(596\) −50.9706 −2.08784
\(597\) 24.0000i 0.982255i
\(598\) − 42.6274i − 1.74316i
\(599\) −1.79899 −0.0735047 −0.0367524 0.999324i \(-0.511701\pi\)
−0.0367524 + 0.999324i \(0.511701\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) − 10.4853i − 0.426994i
\(604\) 45.9411 1.86932
\(605\) 0 0
\(606\) 20.9706 0.851871
\(607\) − 42.9706i − 1.74412i −0.489398 0.872061i \(-0.662783\pi\)
0.489398 0.872061i \(-0.337217\pi\)
\(608\) 1.31371i 0.0532779i
\(609\) 1.65685 0.0671391
\(610\) 0 0
\(611\) 0.686292 0.0277644
\(612\) 10.8284i 0.437713i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 26.4853 1.06886
\(615\) 0 0
\(616\) 17.6569 0.711415
\(617\) 14.8284i 0.596970i 0.954414 + 0.298485i \(0.0964813\pi\)
−0.954414 + 0.298485i \(0.903519\pi\)
\(618\) − 58.6274i − 2.35834i
\(619\) −29.7990 −1.19772 −0.598861 0.800853i \(-0.704380\pi\)
−0.598861 + 0.800853i \(0.704380\pi\)
\(620\) 0 0
\(621\) −35.3137 −1.41709
\(622\) − 6.00000i − 0.240578i
\(623\) 3.02944i 0.121372i
\(624\) −12.0000 −0.480384
\(625\) 0 0
\(626\) 14.4853 0.578948
\(627\) − 8.00000i − 0.319489i
\(628\) − 63.1127i − 2.51847i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 3.02944 0.120600 0.0603000 0.998180i \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) 63.9411i 2.54344i
\(633\) − 9.65685i − 0.383825i
\(634\) −6.82843 −0.271191
\(635\) 0 0
\(636\) −58.6274 −2.32473
\(637\) − 12.6274i − 0.500316i
\(638\) 11.6569i 0.461499i
\(639\) −7.31371 −0.289326
\(640\) 0 0
\(641\) −44.6274 −1.76268 −0.881338 0.472485i \(-0.843357\pi\)
−0.881338 + 0.472485i \(0.843357\pi\)
\(642\) 39.3137i 1.55159i
\(643\) − 31.4558i − 1.24050i −0.784405 0.620249i \(-0.787032\pi\)
0.784405 0.620249i \(-0.212968\pi\)
\(644\) 28.0000 1.10335
\(645\) 0 0
\(646\) 5.65685 0.222566
\(647\) − 21.1127i − 0.830026i −0.909816 0.415013i \(-0.863777\pi\)
0.909816 0.415013i \(-0.136223\pi\)
\(648\) 48.5563i 1.90747i
\(649\) 0 0
\(650\) 0 0
\(651\) 17.3726 0.680885
\(652\) 75.2548i 2.94721i
\(653\) − 22.8284i − 0.893345i −0.894697 0.446673i \(-0.852609\pi\)
0.894697 0.446673i \(-0.147391\pi\)
\(654\) −9.65685 −0.377613
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 8.48528i 0.331042i
\(658\) 0.686292i 0.0267544i
\(659\) 37.7990 1.47244 0.736220 0.676743i \(-0.236609\pi\)
0.736220 + 0.676743i \(0.236609\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) − 42.9706i − 1.67010i
\(663\) − 11.3137i − 0.439388i
\(664\) 56.6274 2.19757
\(665\) 0 0
\(666\) −20.4853 −0.793789
\(667\) 8.82843i 0.341838i
\(668\) 55.4558i 2.14565i
\(669\) 43.5980 1.68560
\(670\) 0 0
\(671\) −36.9706 −1.42723
\(672\) 2.62742i 0.101355i
\(673\) − 10.9706i − 0.422884i −0.977391 0.211442i \(-0.932184\pi\)
0.977391 0.211442i \(-0.0678160\pi\)
\(674\) −16.4853 −0.634989
\(675\) 0 0
\(676\) −34.4558 −1.32522
\(677\) − 36.7696i − 1.41317i −0.707629 0.706584i \(-0.750235\pi\)
0.707629 0.706584i \(-0.249765\pi\)
\(678\) 13.6569i 0.524488i
\(679\) −3.71573 −0.142597
\(680\) 0 0
\(681\) 16.2843 0.624015
\(682\) 122.225i 4.68025i
\(683\) 40.1421i 1.53600i 0.640452 + 0.767998i \(0.278747\pi\)
−0.640452 + 0.767998i \(0.721253\pi\)
\(684\) −3.17157 −0.121268
\(685\) 0 0
\(686\) 26.6274 1.01664
\(687\) − 4.00000i − 0.152610i
\(688\) − 18.0000i − 0.686244i
\(689\) 15.3137 0.583406
\(690\) 0 0
\(691\) −11.0294 −0.419580 −0.209790 0.977747i \(-0.567278\pi\)
−0.209790 + 0.977747i \(0.567278\pi\)
\(692\) 20.3431i 0.773330i
\(693\) − 4.00000i − 0.151947i
\(694\) 48.6274 1.84587
\(695\) 0 0
\(696\) 8.82843 0.334641
\(697\) − 16.9706i − 0.642806i
\(698\) 59.4558i 2.25044i
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) − 19.3137i − 0.728949i
\(703\) 7.02944i 0.265120i
\(704\) −47.4558 −1.78856
\(705\) 0 0
\(706\) 37.7990 1.42258
\(707\) − 3.59798i − 0.135316i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 14.4853 0.543240
\(712\) 16.1421i 0.604952i
\(713\) 92.5685i 3.46672i
\(714\) 11.3137 0.423405
\(715\) 0 0
\(716\) −2.62742 −0.0981912
\(717\) − 46.6274i − 1.74133i
\(718\) 77.5980i 2.89593i
\(719\) −10.6274 −0.396336 −0.198168 0.980168i \(-0.563499\pi\)
−0.198168 + 0.980168i \(0.563499\pi\)
\(720\) 0 0
\(721\) −10.0589 −0.374612
\(722\) − 44.2132i − 1.64545i
\(723\) − 20.0000i − 0.743808i
\(724\) 22.9706 0.853694
\(725\) 0 0
\(726\) 59.4558 2.20661
\(727\) − 43.9411i − 1.62969i −0.579682 0.814843i \(-0.696823\pi\)
0.579682 0.814843i \(-0.303177\pi\)
\(728\) 7.31371i 0.271064i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 58.6274i 2.16693i
\(733\) − 17.1716i − 0.634247i −0.948384 0.317123i \(-0.897283\pi\)
0.948384 0.317123i \(-0.102717\pi\)
\(734\) 43.4558 1.60398
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) − 50.6274i − 1.86488i
\(738\) 14.4853i 0.533211i
\(739\) 2.48528 0.0914226 0.0457113 0.998955i \(-0.485445\pi\)
0.0457113 + 0.998955i \(0.485445\pi\)
\(740\) 0 0
\(741\) 3.31371 0.121732
\(742\) 15.3137i 0.562184i
\(743\) − 7.37258i − 0.270474i −0.990813 0.135237i \(-0.956820\pi\)
0.990813 0.135237i \(-0.0431796\pi\)
\(744\) 92.5685 3.39373
\(745\) 0 0
\(746\) −65.1127 −2.38395
\(747\) − 12.8284i − 0.469368i
\(748\) 52.2843i 1.91170i
\(749\) 6.74517 0.246463
\(750\) 0 0
\(751\) −12.1421 −0.443073 −0.221536 0.975152i \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) 1.02944i 0.0375397i
\(753\) − 6.34315i − 0.231157i
\(754\) −4.82843 −0.175841
\(755\) 0 0
\(756\) 12.6863 0.461396
\(757\) − 36.4853i − 1.32608i −0.748584 0.663040i \(-0.769266\pi\)
0.748584 0.663040i \(-0.230734\pi\)
\(758\) − 13.3137i − 0.483576i
\(759\) 85.2548 3.09455
\(760\) 0 0
\(761\) −36.6274 −1.32774 −0.663871 0.747847i \(-0.731088\pi\)
−0.663871 + 0.747847i \(0.731088\pi\)
\(762\) − 28.9706i − 1.04949i
\(763\) 1.65685i 0.0599822i
\(764\) 58.0833 2.10138
\(765\) 0 0
\(766\) 34.9706 1.26354
\(767\) 0 0
\(768\) 59.9411i 2.16294i
\(769\) 4.34315 0.156618 0.0783089 0.996929i \(-0.475048\pi\)
0.0783089 + 0.996929i \(0.475048\pi\)
\(770\) 0 0
\(771\) −58.6274 −2.11141
\(772\) 47.7990i 1.72032i
\(773\) 8.48528i 0.305194i 0.988288 + 0.152597i \(0.0487637\pi\)
−0.988288 + 0.152597i \(0.951236\pi\)
\(774\) −14.4853 −0.520663
\(775\) 0 0
\(776\) −19.7990 −0.710742
\(777\) 14.0589i 0.504359i
\(778\) 16.1421i 0.578724i
\(779\) 4.97056 0.178089
\(780\) 0 0
\(781\) −35.3137 −1.26362
\(782\) 60.2843i 2.15576i
\(783\) 4.00000i 0.142948i
\(784\) 18.9411 0.676469
\(785\) 0 0
\(786\) 77.9411 2.78007
\(787\) 21.7990i 0.777050i 0.921438 + 0.388525i \(0.127015\pi\)
−0.921438 + 0.388525i \(0.872985\pi\)
\(788\) − 31.9411i − 1.13786i
\(789\) −16.6863 −0.594048
\(790\) 0 0
\(791\) 2.34315 0.0833127
\(792\) − 21.3137i − 0.757350i
\(793\) − 15.3137i − 0.543806i
\(794\) −20.1421 −0.714818
\(795\) 0 0
\(796\) 45.9411 1.62834
\(797\) 34.1421i 1.20938i 0.796462 + 0.604688i \(0.206702\pi\)
−0.796462 + 0.604688i \(0.793298\pi\)
\(798\) 3.31371i 0.117304i
\(799\) −0.970563 −0.0343360
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) − 70.7696i − 2.49896i
\(803\) 40.9706i 1.44582i
\(804\) −80.2843 −2.83141
\(805\) 0 0
\(806\) −50.6274 −1.78327
\(807\) 2.62742i 0.0924895i
\(808\) − 19.1716i − 0.674454i
\(809\) 14.2843 0.502208 0.251104 0.967960i \(-0.419206\pi\)
0.251104 + 0.967960i \(0.419206\pi\)
\(810\) 0 0
\(811\) 26.3431 0.925033 0.462516 0.886611i \(-0.346947\pi\)
0.462516 + 0.886611i \(0.346947\pi\)
\(812\) − 3.17157i − 0.111300i
\(813\) − 59.5980i − 2.09019i
\(814\) −98.9117 −3.46685
\(815\) 0 0
\(816\) 16.9706 0.594089
\(817\) 4.97056i 0.173898i
\(818\) − 74.7696i − 2.61426i
\(819\) 1.65685 0.0578952
\(820\) 0 0
\(821\) −45.3137 −1.58146 −0.790730 0.612165i \(-0.790299\pi\)
−0.790730 + 0.612165i \(0.790299\pi\)
\(822\) 52.2843i 1.82362i
\(823\) 2.97056i 0.103547i 0.998659 + 0.0517737i \(0.0164874\pi\)
−0.998659 + 0.0517737i \(0.983513\pi\)
\(824\) −53.5980 −1.86717
\(825\) 0 0
\(826\) 0 0
\(827\) 5.31371i 0.184776i 0.995723 + 0.0923879i \(0.0294500\pi\)
−0.995723 + 0.0923879i \(0.970550\pi\)
\(828\) − 33.7990i − 1.17460i
\(829\) 24.6274 0.855346 0.427673 0.903934i \(-0.359334\pi\)
0.427673 + 0.903934i \(0.359334\pi\)
\(830\) 0 0
\(831\) −15.3137 −0.531227
\(832\) − 19.6569i − 0.681479i
\(833\) 17.8579i 0.618738i
\(834\) −49.9411 −1.72932
\(835\) 0 0
\(836\) −15.3137 −0.529636
\(837\) 41.9411i 1.44970i
\(838\) 12.0000i 0.414533i
\(839\) 14.4853 0.500087 0.250044 0.968235i \(-0.419555\pi\)
0.250044 + 0.968235i \(0.419555\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 36.1421i − 1.24554i
\(843\) 13.3726i 0.460576i
\(844\) −18.4853 −0.636290
\(845\) 0 0
\(846\) 0.828427 0.0284819
\(847\) − 10.2010i − 0.350511i
\(848\) 22.9706i 0.788812i
\(849\) −1.65685 −0.0568631
\(850\) 0 0
\(851\) −74.9117 −2.56794
\(852\) 56.0000i 1.91853i
\(853\) − 11.1127i − 0.380492i −0.981736 0.190246i \(-0.939072\pi\)
0.981736 0.190246i \(-0.0609285\pi\)
\(854\) 15.3137 0.524024
\(855\) 0 0
\(856\) 35.9411 1.22844
\(857\) − 48.6274i − 1.66108i −0.556958 0.830540i \(-0.688032\pi\)
0.556958 0.830540i \(-0.311968\pi\)
\(858\) 46.6274i 1.59183i
\(859\) −28.4264 −0.969896 −0.484948 0.874543i \(-0.661162\pi\)
−0.484948 + 0.874543i \(0.661162\pi\)
\(860\) 0 0
\(861\) 9.94113 0.338793
\(862\) − 46.6274i − 1.58814i
\(863\) − 7.85786i − 0.267485i −0.991016 0.133742i \(-0.957301\pi\)
0.991016 0.133742i \(-0.0426995\pi\)
\(864\) −6.34315 −0.215798
\(865\) 0 0
\(866\) 84.0833 2.85727
\(867\) − 18.0000i − 0.611312i
\(868\) − 33.2548i − 1.12874i
\(869\) 69.9411 2.37259
\(870\) 0 0
\(871\) 20.9706 0.710560
\(872\) 8.82843i 0.298968i
\(873\) 4.48528i 0.151804i
\(874\) −17.6569 −0.597252
\(875\) 0 0
\(876\) 64.9706 2.19515
\(877\) 18.2843i 0.617416i 0.951157 + 0.308708i \(0.0998966\pi\)
−0.951157 + 0.308708i \(0.900103\pi\)
\(878\) 52.2843i 1.76451i
\(879\) −16.9706 −0.572403
\(880\) 0 0
\(881\) 6.68629 0.225267 0.112633 0.993637i \(-0.464071\pi\)
0.112633 + 0.993637i \(0.464071\pi\)
\(882\) − 15.2426i − 0.513246i
\(883\) 2.48528i 0.0836364i 0.999125 + 0.0418182i \(0.0133150\pi\)
−0.999125 + 0.0418182i \(0.986685\pi\)
\(884\) −21.6569 −0.728399
\(885\) 0 0
\(886\) −8.82843 −0.296597
\(887\) 29.3137i 0.984258i 0.870522 + 0.492129i \(0.163781\pi\)
−0.870522 + 0.492129i \(0.836219\pi\)
\(888\) 74.9117i 2.51387i
\(889\) −4.97056 −0.166707
\(890\) 0 0
\(891\) 53.1127 1.77934
\(892\) − 83.4558i − 2.79431i
\(893\) − 0.284271i − 0.00951277i
\(894\) 64.2843 2.14999
\(895\) 0 0
\(896\) 17.0294 0.568914
\(897\) 35.3137i 1.17909i
\(898\) − 0.828427i − 0.0276450i
\(899\) 10.4853 0.349704
\(900\) 0 0
\(901\) −21.6569 −0.721494
\(902\) 69.9411i 2.32878i
\(903\) 9.94113i 0.330820i
\(904\) 12.4853 0.415254
\(905\) 0 0
\(906\) −57.9411 −1.92496
\(907\) 10.0000i 0.332045i 0.986122 + 0.166022i \(0.0530924\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) − 31.1716i − 1.03446i
\(909\) −4.34315 −0.144053
\(910\) 0 0
\(911\) 3.85786 0.127817 0.0639084 0.997956i \(-0.479643\pi\)
0.0639084 + 0.997956i \(0.479643\pi\)
\(912\) 4.97056i 0.164592i
\(913\) − 61.9411i − 2.04995i
\(914\) 20.1421 0.666243
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) − 13.3726i − 0.441602i
\(918\) 27.3137i 0.901487i
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) −21.9411 −0.722985
\(922\) − 58.7696i − 1.93547i
\(923\) − 14.6274i − 0.481467i
\(924\) −30.6274 −1.00757
\(925\) 0 0
\(926\) 42.9706 1.41210
\(927\) 12.1421i 0.398800i
\(928\) 1.58579i 0.0520560i
\(929\) −40.6274 −1.33294 −0.666471 0.745531i \(-0.732196\pi\)
−0.666471 + 0.745531i \(0.732196\pi\)
\(930\) 0 0
\(931\) −5.23045 −0.171421
\(932\) − 68.9117i − 2.25728i
\(933\) 4.97056i 0.162729i
\(934\) −55.4558 −1.81457
\(935\) 0 0
\(936\) 8.82843 0.288566
\(937\) − 8.34315i − 0.272559i −0.990670 0.136279i \(-0.956486\pi\)
0.990670 0.136279i \(-0.0435145\pi\)
\(938\) 20.9706i 0.684713i
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 39.9411 1.30204 0.651022 0.759059i \(-0.274341\pi\)
0.651022 + 0.759059i \(0.274341\pi\)
\(942\) 79.5980i 2.59344i
\(943\) 52.9706i 1.72496i
\(944\) 0 0
\(945\) 0 0
\(946\) −69.9411 −2.27398
\(947\) 56.9117i 1.84938i 0.380719 + 0.924691i \(0.375676\pi\)
−0.380719 + 0.924691i \(0.624324\pi\)
\(948\) − 110.912i − 3.60224i
\(949\) −16.9706 −0.550888
\(950\) 0 0
\(951\) 5.65685 0.183436
\(952\) − 10.3431i − 0.335223i
\(953\) 6.68629i 0.216590i 0.994119 + 0.108295i \(0.0345391\pi\)
−0.994119 + 0.108295i \(0.965461\pi\)
\(954\) 18.4853 0.598483
\(955\) 0 0
\(956\) −89.2548 −2.88671
\(957\) − 9.65685i − 0.312162i
\(958\) − 30.9706i − 1.00061i
\(959\) 8.97056 0.289675
\(960\) 0 0
\(961\) 78.9411 2.54649
\(962\) − 40.9706i − 1.32094i
\(963\) − 8.14214i − 0.262377i
\(964\) −38.2843 −1.23305
\(965\) 0 0
\(966\) −35.3137 −1.13620
\(967\) 18.9706i 0.610052i 0.952344 + 0.305026i \(0.0986652\pi\)
−0.952344 + 0.305026i \(0.901335\pi\)
\(968\) − 54.3553i − 1.74705i
\(969\) −4.68629 −0.150545
\(970\) 0 0
\(971\) −0.142136 −0.00456135 −0.00228067 0.999997i \(-0.500726\pi\)
−0.00228067 + 0.999997i \(0.500726\pi\)
\(972\) − 38.2843i − 1.22797i
\(973\) 8.56854i 0.274695i
\(974\) −71.9411 −2.30514
\(975\) 0 0
\(976\) 22.9706 0.735270
\(977\) 25.3137i 0.809857i 0.914348 + 0.404929i \(0.132704\pi\)
−0.914348 + 0.404929i \(0.867296\pi\)
\(978\) − 94.9117i − 3.03494i
\(979\) 17.6569 0.564316
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 104.912i 3.34787i
\(983\) 13.3137i 0.424641i 0.977200 + 0.212321i \(0.0681021\pi\)
−0.977200 + 0.212321i \(0.931898\pi\)
\(984\) 52.9706 1.68864
\(985\) 0 0
\(986\) 6.82843 0.217461
\(987\) − 0.568542i − 0.0180969i
\(988\) − 6.34315i − 0.201802i
\(989\) −52.9706 −1.68437
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 16.6274i 0.527921i
\(993\) 35.5980i 1.12967i
\(994\) 14.6274 0.463953
\(995\) 0 0
\(996\) −98.2254 −3.11239
\(997\) − 1.17157i − 0.0371041i −0.999828 0.0185520i \(-0.994094\pi\)
0.999828 0.0185520i \(-0.00590564\pi\)
\(998\) − 86.9117i − 2.75114i
\(999\) −33.9411 −1.07385
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.c.349.4 4
5.2 odd 4 145.2.a.b.1.1 2
5.3 odd 4 725.2.a.c.1.2 2
5.4 even 2 inner 725.2.b.c.349.1 4
15.2 even 4 1305.2.a.n.1.2 2
15.8 even 4 6525.2.a.p.1.1 2
20.7 even 4 2320.2.a.k.1.1 2
35.27 even 4 7105.2.a.e.1.1 2
40.27 even 4 9280.2.a.w.1.1 2
40.37 odd 4 9280.2.a.be.1.2 2
145.57 odd 4 4205.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 5.2 odd 4
725.2.a.c.1.2 2 5.3 odd 4
725.2.b.c.349.1 4 5.4 even 2 inner
725.2.b.c.349.4 4 1.1 even 1 trivial
1305.2.a.n.1.2 2 15.2 even 4
2320.2.a.k.1.1 2 20.7 even 4
4205.2.a.d.1.2 2 145.57 odd 4
6525.2.a.p.1.1 2 15.8 even 4
7105.2.a.e.1.1 2 35.27 even 4
9280.2.a.w.1.1 2 40.27 even 4
9280.2.a.be.1.2 2 40.37 odd 4