Properties

Label 1386.2.c.a.197.10
Level $1386$
Weight $2$
Character 1386.197
Analytic conductor $11.067$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(197,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.c (of order \(2\), degree \(1\), 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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.10
Root \(-0.0383715 + 1.73163i\) of defining polynomial
Character \(\chi\) \(=\) 1386.197
Dual form 1386.2.c.a.197.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.50315i q^{5} +1.00000i q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.50315i q^{5} +1.00000i q^{7} -1.00000 q^{8} -2.50315i q^{10} +(-3.23848 - 0.715732i) q^{11} -1.06863i q^{13} -1.00000i q^{14} +1.00000 q^{16} -1.11665 q^{17} -0.185274i q^{19} +2.50315i q^{20} +(3.23848 + 0.715732i) q^{22} +5.70439i q^{23} -1.26578 q^{25} +1.06863i q^{26} +1.00000i q^{28} -7.80144 q^{29} -8.60956 q^{31} -1.00000 q^{32} +1.11665 q^{34} -2.50315 q^{35} +4.67494 q^{37} +0.185274i q^{38} -2.50315i q^{40} +0.132606 q^{41} -8.83979i q^{43} +(-3.23848 - 0.715732i) q^{44} -5.70439i q^{46} -5.08738i q^{47} -1.00000 q^{49} +1.26578 q^{50} -1.06863i q^{52} +0.0980758i q^{53} +(1.79159 - 8.10640i) q^{55} -1.00000i q^{56} +7.80144 q^{58} +6.33137i q^{59} -3.93768i q^{61} +8.60956 q^{62} +1.00000 q^{64} +2.67494 q^{65} -4.49907 q^{67} -1.11665 q^{68} +2.50315 q^{70} +0.698081i q^{71} +0.169232i q^{73} -4.67494 q^{74} -0.185274i q^{76} +(0.715732 - 3.23848i) q^{77} -1.63002i q^{79} +2.50315i q^{80} -0.132606 q^{82} -6.27564 q^{83} -2.79514i q^{85} +8.83979i q^{86} +(3.23848 + 0.715732i) q^{88} -15.2423i q^{89} +1.06863 q^{91} +5.70439i q^{92} +5.08738i q^{94} +0.463769 q^{95} -2.71024 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{11} + 12 q^{16} - 16 q^{17} - 4 q^{22} - 4 q^{25} - 16 q^{29} - 12 q^{32} + 16 q^{34} + 8 q^{35} + 24 q^{37} - 16 q^{41} + 4 q^{44} - 12 q^{49} + 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} - 16 q^{68} - 8 q^{70} - 24 q^{74} - 8 q^{77} + 16 q^{82} - 16 q^{83} - 4 q^{88} + 48 q^{95} + 48 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.50315i 1.11944i 0.828680 + 0.559722i \(0.189092\pi\)
−0.828680 + 0.559722i \(0.810908\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.50315i 0.791567i
\(11\) −3.23848 0.715732i −0.976437 0.215801i
\(12\) 0 0
\(13\) 1.06863i 0.296384i −0.988959 0.148192i \(-0.952655\pi\)
0.988959 0.148192i \(-0.0473454\pi\)
\(14\) 1.00000i 0.267261i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.11665 −0.270826 −0.135413 0.990789i \(-0.543236\pi\)
−0.135413 + 0.990789i \(0.543236\pi\)
\(18\) 0 0
\(19\) 0.185274i 0.0425047i −0.999774 0.0212524i \(-0.993235\pi\)
0.999774 0.0212524i \(-0.00676535\pi\)
\(20\) 2.50315i 0.559722i
\(21\) 0 0
\(22\) 3.23848 + 0.715732i 0.690445 + 0.152595i
\(23\) 5.70439i 1.18945i 0.803930 + 0.594724i \(0.202739\pi\)
−0.803930 + 0.594724i \(0.797261\pi\)
\(24\) 0 0
\(25\) −1.26578 −0.253156
\(26\) 1.06863i 0.209575i
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −7.80144 −1.44869 −0.724346 0.689437i \(-0.757858\pi\)
−0.724346 + 0.689437i \(0.757858\pi\)
\(30\) 0 0
\(31\) −8.60956 −1.54632 −0.773161 0.634210i \(-0.781326\pi\)
−0.773161 + 0.634210i \(0.781326\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.11665 0.191503
\(35\) −2.50315 −0.423110
\(36\) 0 0
\(37\) 4.67494 0.768556 0.384278 0.923218i \(-0.374450\pi\)
0.384278 + 0.923218i \(0.374450\pi\)
\(38\) 0.185274i 0.0300554i
\(39\) 0 0
\(40\) 2.50315i 0.395783i
\(41\) 0.132606 0.0207096 0.0103548 0.999946i \(-0.496704\pi\)
0.0103548 + 0.999946i \(0.496704\pi\)
\(42\) 0 0
\(43\) 8.83979i 1.34805i −0.738706 0.674027i \(-0.764563\pi\)
0.738706 0.674027i \(-0.235437\pi\)
\(44\) −3.23848 0.715732i −0.488219 0.107901i
\(45\) 0 0
\(46\) 5.70439i 0.841066i
\(47\) 5.08738i 0.742071i −0.928619 0.371035i \(-0.879003\pi\)
0.928619 0.371035i \(-0.120997\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 1.26578 0.179008
\(51\) 0 0
\(52\) 1.06863i 0.148192i
\(53\) 0.0980758i 0.0134717i 0.999977 + 0.00673587i \(0.00214411\pi\)
−0.999977 + 0.00673587i \(0.997856\pi\)
\(54\) 0 0
\(55\) 1.79159 8.10640i 0.241578 1.09307i
\(56\) 1.00000i 0.133631i
\(57\) 0 0
\(58\) 7.80144 1.02438
\(59\) 6.33137i 0.824274i 0.911122 + 0.412137i \(0.135217\pi\)
−0.911122 + 0.412137i \(0.864783\pi\)
\(60\) 0 0
\(61\) 3.93768i 0.504168i −0.967705 0.252084i \(-0.918884\pi\)
0.967705 0.252084i \(-0.0811160\pi\)
\(62\) 8.60956 1.09342
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.67494 0.331786
\(66\) 0 0
\(67\) −4.49907 −0.549649 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(68\) −1.11665 −0.135413
\(69\) 0 0
\(70\) 2.50315 0.299184
\(71\) 0.698081i 0.0828469i 0.999142 + 0.0414235i \(0.0131893\pi\)
−0.999142 + 0.0414235i \(0.986811\pi\)
\(72\) 0 0
\(73\) 0.169232i 0.0198071i 0.999951 + 0.00990357i \(0.00315246\pi\)
−0.999951 + 0.00990357i \(0.996848\pi\)
\(74\) −4.67494 −0.543451
\(75\) 0 0
\(76\) 0.185274i 0.0212524i
\(77\) 0.715732 3.23848i 0.0815652 0.369059i
\(78\) 0 0
\(79\) 1.63002i 0.183392i −0.995787 0.0916958i \(-0.970771\pi\)
0.995787 0.0916958i \(-0.0292287\pi\)
\(80\) 2.50315i 0.279861i
\(81\) 0 0
\(82\) −0.132606 −0.0146439
\(83\) −6.27564 −0.688841 −0.344420 0.938816i \(-0.611925\pi\)
−0.344420 + 0.938816i \(0.611925\pi\)
\(84\) 0 0
\(85\) 2.79514i 0.303175i
\(86\) 8.83979i 0.953219i
\(87\) 0 0
\(88\) 3.23848 + 0.715732i 0.345223 + 0.0762973i
\(89\) 15.2423i 1.61568i −0.589405 0.807838i \(-0.700638\pi\)
0.589405 0.807838i \(-0.299362\pi\)
\(90\) 0 0
\(91\) 1.06863 0.112023
\(92\) 5.70439i 0.594724i
\(93\) 0 0
\(94\) 5.08738i 0.524723i
\(95\) 0.463769 0.0475817
\(96\) 0 0
\(97\) −2.71024 −0.275184 −0.137592 0.990489i \(-0.543936\pi\)
−0.137592 + 0.990489i \(0.543936\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.26578 −0.126578
\(101\) −10.3811 −1.03296 −0.516478 0.856300i \(-0.672757\pi\)
−0.516478 + 0.856300i \(0.672757\pi\)
\(102\) 0 0
\(103\) −17.5234 −1.72663 −0.863316 0.504664i \(-0.831617\pi\)
−0.863316 + 0.504664i \(0.831617\pi\)
\(104\) 1.06863i 0.104788i
\(105\) 0 0
\(106\) 0.0980758i 0.00952596i
\(107\) −11.8692 −1.14744 −0.573721 0.819051i \(-0.694501\pi\)
−0.573721 + 0.819051i \(0.694501\pi\)
\(108\) 0 0
\(109\) 8.39312i 0.803915i 0.915658 + 0.401958i \(0.131670\pi\)
−0.915658 + 0.401958i \(0.868330\pi\)
\(110\) −1.79159 + 8.10640i −0.170821 + 0.772915i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 14.8006i 1.39232i −0.717885 0.696162i \(-0.754890\pi\)
0.717885 0.696162i \(-0.245110\pi\)
\(114\) 0 0
\(115\) −14.2790 −1.33152
\(116\) −7.80144 −0.724346
\(117\) 0 0
\(118\) 6.33137i 0.582850i
\(119\) 1.11665i 0.102363i
\(120\) 0 0
\(121\) 9.97546 + 4.63576i 0.906860 + 0.421433i
\(122\) 3.93768i 0.356501i
\(123\) 0 0
\(124\) −8.60956 −0.773161
\(125\) 9.34733i 0.836050i
\(126\) 0 0
\(127\) 1.09732i 0.0973718i 0.998814 + 0.0486859i \(0.0155033\pi\)
−0.998814 + 0.0486859i \(0.984497\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.67494 −0.234608
\(131\) 13.8373 1.20897 0.604483 0.796618i \(-0.293380\pi\)
0.604483 + 0.796618i \(0.293380\pi\)
\(132\) 0 0
\(133\) 0.185274 0.0160653
\(134\) 4.49907 0.388661
\(135\) 0 0
\(136\) 1.11665 0.0957516
\(137\) 4.53927i 0.387816i −0.981020 0.193908i \(-0.937884\pi\)
0.981020 0.193908i \(-0.0621164\pi\)
\(138\) 0 0
\(139\) 15.3511i 1.30206i 0.759050 + 0.651032i \(0.225664\pi\)
−0.759050 + 0.651032i \(0.774336\pi\)
\(140\) −2.50315 −0.211555
\(141\) 0 0
\(142\) 0.698081i 0.0585816i
\(143\) −0.764851 + 3.46073i −0.0639601 + 0.289401i
\(144\) 0 0
\(145\) 19.5282i 1.62173i
\(146\) 0.169232i 0.0140058i
\(147\) 0 0
\(148\) 4.67494 0.384278
\(149\) 8.97546 0.735298 0.367649 0.929965i \(-0.380163\pi\)
0.367649 + 0.929965i \(0.380163\pi\)
\(150\) 0 0
\(151\) 19.4441i 1.58234i 0.611598 + 0.791168i \(0.290527\pi\)
−0.611598 + 0.791168i \(0.709473\pi\)
\(152\) 0.185274i 0.0150277i
\(153\) 0 0
\(154\) −0.715732 + 3.23848i −0.0576753 + 0.260964i
\(155\) 21.5511i 1.73102i
\(156\) 0 0
\(157\) −6.21542 −0.496044 −0.248022 0.968754i \(-0.579781\pi\)
−0.248022 + 0.968754i \(0.579781\pi\)
\(158\) 1.63002i 0.129677i
\(159\) 0 0
\(160\) 2.50315i 0.197892i
\(161\) −5.70439 −0.449569
\(162\) 0 0
\(163\) −19.4877 −1.52640 −0.763198 0.646165i \(-0.776372\pi\)
−0.763198 + 0.646165i \(0.776372\pi\)
\(164\) 0.132606 0.0103548
\(165\) 0 0
\(166\) 6.27564 0.487084
\(167\) −18.1273 −1.40274 −0.701368 0.712799i \(-0.747427\pi\)
−0.701368 + 0.712799i \(0.747427\pi\)
\(168\) 0 0
\(169\) 11.8580 0.912156
\(170\) 2.79514i 0.214377i
\(171\) 0 0
\(172\) 8.83979i 0.674027i
\(173\) −10.0672 −0.765397 −0.382698 0.923873i \(-0.625005\pi\)
−0.382698 + 0.923873i \(0.625005\pi\)
\(174\) 0 0
\(175\) 1.26578i 0.0956840i
\(176\) −3.23848 0.715732i −0.244109 0.0539503i
\(177\) 0 0
\(178\) 15.2423i 1.14246i
\(179\) 20.8254i 1.55656i −0.627916 0.778281i \(-0.716092\pi\)
0.627916 0.778281i \(-0.283908\pi\)
\(180\) 0 0
\(181\) 2.54233 0.188970 0.0944851 0.995526i \(-0.469880\pi\)
0.0944851 + 0.995526i \(0.469880\pi\)
\(182\) −1.06863 −0.0792120
\(183\) 0 0
\(184\) 5.70439i 0.420533i
\(185\) 11.7021i 0.860355i
\(186\) 0 0
\(187\) 3.61623 + 0.799219i 0.264445 + 0.0584447i
\(188\) 5.08738i 0.371035i
\(189\) 0 0
\(190\) −0.463769 −0.0336453
\(191\) 10.0668i 0.728406i 0.931320 + 0.364203i \(0.118659\pi\)
−0.931320 + 0.364203i \(0.881341\pi\)
\(192\) 0 0
\(193\) 11.9537i 0.860448i 0.902722 + 0.430224i \(0.141565\pi\)
−0.902722 + 0.430224i \(0.858435\pi\)
\(194\) 2.71024 0.194584
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −3.02214 −0.215318 −0.107659 0.994188i \(-0.534336\pi\)
−0.107659 + 0.994188i \(0.534336\pi\)
\(198\) 0 0
\(199\) −16.2611 −1.15272 −0.576359 0.817197i \(-0.695527\pi\)
−0.576359 + 0.817197i \(0.695527\pi\)
\(200\) 1.26578 0.0895042
\(201\) 0 0
\(202\) 10.3811 0.730410
\(203\) 7.80144i 0.547554i
\(204\) 0 0
\(205\) 0.331934i 0.0231833i
\(206\) 17.5234 1.22091
\(207\) 0 0
\(208\) 1.06863i 0.0740960i
\(209\) −0.132606 + 0.600005i −0.00917257 + 0.0415032i
\(210\) 0 0
\(211\) 4.79412i 0.330040i −0.986290 0.165020i \(-0.947231\pi\)
0.986290 0.165020i \(-0.0527689\pi\)
\(212\) 0.0980758i 0.00673587i
\(213\) 0 0
\(214\) 11.8692 0.811364
\(215\) 22.1273 1.50907
\(216\) 0 0
\(217\) 8.60956i 0.584455i
\(218\) 8.39312i 0.568454i
\(219\) 0 0
\(220\) 1.79159 8.10640i 0.120789 0.546534i
\(221\) 1.19328i 0.0802686i
\(222\) 0 0
\(223\) 0.562049 0.0376376 0.0188188 0.999823i \(-0.494009\pi\)
0.0188188 + 0.999823i \(0.494009\pi\)
\(224\) 1.00000i 0.0668153i
\(225\) 0 0
\(226\) 14.8006i 0.984522i
\(227\) 4.42647 0.293795 0.146897 0.989152i \(-0.453071\pi\)
0.146897 + 0.989152i \(0.453071\pi\)
\(228\) 0 0
\(229\) −6.64297 −0.438980 −0.219490 0.975615i \(-0.570439\pi\)
−0.219490 + 0.975615i \(0.570439\pi\)
\(230\) 14.2790 0.941527
\(231\) 0 0
\(232\) 7.80144 0.512190
\(233\) 7.25486 0.475282 0.237641 0.971353i \(-0.423626\pi\)
0.237641 + 0.971353i \(0.423626\pi\)
\(234\) 0 0
\(235\) 12.7345 0.830707
\(236\) 6.33137i 0.412137i
\(237\) 0 0
\(238\) 1.11665i 0.0723814i
\(239\) 20.6125 1.33331 0.666657 0.745365i \(-0.267725\pi\)
0.666657 + 0.745365i \(0.267725\pi\)
\(240\) 0 0
\(241\) 20.4491i 1.31724i 0.752475 + 0.658620i \(0.228860\pi\)
−0.752475 + 0.658620i \(0.771140\pi\)
\(242\) −9.97546 4.63576i −0.641247 0.297998i
\(243\) 0 0
\(244\) 3.93768i 0.252084i
\(245\) 2.50315i 0.159921i
\(246\) 0 0
\(247\) −0.197989 −0.0125977
\(248\) 8.60956 0.546708
\(249\) 0 0
\(250\) 9.34733i 0.591177i
\(251\) 23.7330i 1.49802i −0.662561 0.749008i \(-0.730530\pi\)
0.662561 0.749008i \(-0.269470\pi\)
\(252\) 0 0
\(253\) 4.08281 18.4735i 0.256684 1.16142i
\(254\) 1.09732i 0.0688523i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.6457i 1.28784i 0.765092 + 0.643921i \(0.222693\pi\)
−0.765092 + 0.643921i \(0.777307\pi\)
\(258\) 0 0
\(259\) 4.67494i 0.290487i
\(260\) 2.67494 0.165893
\(261\) 0 0
\(262\) −13.8373 −0.854869
\(263\) −0.605313 −0.0373252 −0.0186626 0.999826i \(-0.505941\pi\)
−0.0186626 + 0.999826i \(0.505941\pi\)
\(264\) 0 0
\(265\) −0.245499 −0.0150809
\(266\) −0.185274 −0.0113599
\(267\) 0 0
\(268\) −4.49907 −0.274825
\(269\) 21.8291i 1.33094i 0.746423 + 0.665472i \(0.231770\pi\)
−0.746423 + 0.665472i \(0.768230\pi\)
\(270\) 0 0
\(271\) 4.52103i 0.274633i −0.990527 0.137316i \(-0.956152\pi\)
0.990527 0.137316i \(-0.0438477\pi\)
\(272\) −1.11665 −0.0677066
\(273\) 0 0
\(274\) 4.53927i 0.274227i
\(275\) 4.09920 + 0.905959i 0.247191 + 0.0546314i
\(276\) 0 0
\(277\) 6.25586i 0.375878i −0.982181 0.187939i \(-0.939819\pi\)
0.982181 0.187939i \(-0.0601808\pi\)
\(278\) 15.3511i 0.920698i
\(279\) 0 0
\(280\) 2.50315 0.149592
\(281\) 31.9977 1.90882 0.954412 0.298493i \(-0.0964838\pi\)
0.954412 + 0.298493i \(0.0964838\pi\)
\(282\) 0 0
\(283\) 23.2203i 1.38031i 0.723663 + 0.690153i \(0.242457\pi\)
−0.723663 + 0.690153i \(0.757543\pi\)
\(284\) 0.698081i 0.0414235i
\(285\) 0 0
\(286\) 0.764851 3.46073i 0.0452266 0.204637i
\(287\) 0.132606i 0.00782751i
\(288\) 0 0
\(289\) −15.7531 −0.926653
\(290\) 19.5282i 1.14674i
\(291\) 0 0
\(292\) 0.169232i 0.00990357i
\(293\) 3.48219 0.203432 0.101716 0.994813i \(-0.467567\pi\)
0.101716 + 0.994813i \(0.467567\pi\)
\(294\) 0 0
\(295\) −15.8484 −0.922729
\(296\) −4.67494 −0.271725
\(297\) 0 0
\(298\) −8.97546 −0.519934
\(299\) 6.09587 0.352533
\(300\) 0 0
\(301\) 8.83979 0.509517
\(302\) 19.4441i 1.11888i
\(303\) 0 0
\(304\) 0.185274i 0.0106262i
\(305\) 9.85662 0.564388
\(306\) 0 0
\(307\) 9.61453i 0.548730i 0.961626 + 0.274365i \(0.0884677\pi\)
−0.961626 + 0.274365i \(0.911532\pi\)
\(308\) 0.715732 3.23848i 0.0407826 0.184529i
\(309\) 0 0
\(310\) 21.5511i 1.22402i
\(311\) 19.7571i 1.12032i 0.828383 + 0.560162i \(0.189261\pi\)
−0.828383 + 0.560162i \(0.810739\pi\)
\(312\) 0 0
\(313\) 14.0067 0.791703 0.395851 0.918315i \(-0.370450\pi\)
0.395851 + 0.918315i \(0.370450\pi\)
\(314\) 6.21542 0.350756
\(315\) 0 0
\(316\) 1.63002i 0.0916958i
\(317\) 7.26821i 0.408223i −0.978948 0.204112i \(-0.934569\pi\)
0.978948 0.204112i \(-0.0654305\pi\)
\(318\) 0 0
\(319\) 25.2648 + 5.58374i 1.41456 + 0.312629i
\(320\) 2.50315i 0.139931i
\(321\) 0 0
\(322\) 5.70439 0.317893
\(323\) 0.206885i 0.0115114i
\(324\) 0 0
\(325\) 1.35265i 0.0750314i
\(326\) 19.4877 1.07932
\(327\) 0 0
\(328\) −0.132606 −0.00732196
\(329\) 5.08738 0.280476
\(330\) 0 0
\(331\) −13.6086 −0.747994 −0.373997 0.927430i \(-0.622013\pi\)
−0.373997 + 0.927430i \(0.622013\pi\)
\(332\) −6.27564 −0.344420
\(333\) 0 0
\(334\) 18.1273 0.991884
\(335\) 11.2619i 0.615302i
\(336\) 0 0
\(337\) 2.85239i 0.155380i −0.996978 0.0776899i \(-0.975246\pi\)
0.996978 0.0776899i \(-0.0247544\pi\)
\(338\) −11.8580 −0.644992
\(339\) 0 0
\(340\) 2.79514i 0.151588i
\(341\) 27.8819 + 6.16213i 1.50989 + 0.333698i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 8.83979i 0.476609i
\(345\) 0 0
\(346\) 10.0672 0.541217
\(347\) 7.05827 0.378908 0.189454 0.981890i \(-0.439328\pi\)
0.189454 + 0.981890i \(0.439328\pi\)
\(348\) 0 0
\(349\) 5.87266i 0.314356i 0.987570 + 0.157178i \(0.0502397\pi\)
−0.987570 + 0.157178i \(0.949760\pi\)
\(350\) 1.26578i 0.0676588i
\(351\) 0 0
\(352\) 3.23848 + 0.715732i 0.172611 + 0.0381486i
\(353\) 20.5326i 1.09284i 0.837512 + 0.546420i \(0.184010\pi\)
−0.837512 + 0.546420i \(0.815990\pi\)
\(354\) 0 0
\(355\) −1.74740 −0.0927425
\(356\) 15.2423i 0.807838i
\(357\) 0 0
\(358\) 20.8254i 1.10066i
\(359\) 9.66516 0.510108 0.255054 0.966927i \(-0.417907\pi\)
0.255054 + 0.966927i \(0.417907\pi\)
\(360\) 0 0
\(361\) 18.9657 0.998193
\(362\) −2.54233 −0.133622
\(363\) 0 0
\(364\) 1.06863 0.0560113
\(365\) −0.423615 −0.0221730
\(366\) 0 0
\(367\) 25.5991 1.33626 0.668132 0.744043i \(-0.267094\pi\)
0.668132 + 0.744043i \(0.267094\pi\)
\(368\) 5.70439i 0.297362i
\(369\) 0 0
\(370\) 11.7021i 0.608363i
\(371\) −0.0980758 −0.00509184
\(372\) 0 0
\(373\) 29.9547i 1.55100i 0.631350 + 0.775498i \(0.282501\pi\)
−0.631350 + 0.775498i \(0.717499\pi\)
\(374\) −3.61623 0.799219i −0.186991 0.0413266i
\(375\) 0 0
\(376\) 5.08738i 0.262362i
\(377\) 8.33684i 0.429369i
\(378\) 0 0
\(379\) 4.34578 0.223228 0.111614 0.993752i \(-0.464398\pi\)
0.111614 + 0.993752i \(0.464398\pi\)
\(380\) 0.463769 0.0237908
\(381\) 0 0
\(382\) 10.0668i 0.515061i
\(383\) 15.6402i 0.799179i 0.916694 + 0.399589i \(0.130847\pi\)
−0.916694 + 0.399589i \(0.869153\pi\)
\(384\) 0 0
\(385\) 8.10640 + 1.79159i 0.413141 + 0.0913077i
\(386\) 11.9537i 0.608428i
\(387\) 0 0
\(388\) −2.71024 −0.137592
\(389\) 25.6591i 1.30097i −0.759519 0.650485i \(-0.774566\pi\)
0.759519 0.650485i \(-0.225434\pi\)
\(390\) 0 0
\(391\) 6.36978i 0.322134i
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 3.02214 0.152253
\(395\) 4.08019 0.205297
\(396\) 0 0
\(397\) −26.3197 −1.32095 −0.660475 0.750848i \(-0.729645\pi\)
−0.660475 + 0.750848i \(0.729645\pi\)
\(398\) 16.2611 0.815095
\(399\) 0 0
\(400\) −1.26578 −0.0632890
\(401\) 25.8351i 1.29014i 0.764122 + 0.645072i \(0.223173\pi\)
−0.764122 + 0.645072i \(0.776827\pi\)
\(402\) 0 0
\(403\) 9.20042i 0.458305i
\(404\) −10.3811 −0.516478
\(405\) 0 0
\(406\) 7.80144i 0.387179i
\(407\) −15.1397 3.34600i −0.750446 0.165855i
\(408\) 0 0
\(409\) 5.55129i 0.274494i −0.990537 0.137247i \(-0.956175\pi\)
0.990537 0.137247i \(-0.0438254\pi\)
\(410\) 0.331934i 0.0163931i
\(411\) 0 0
\(412\) −17.5234 −0.863316
\(413\) −6.33137 −0.311546
\(414\) 0 0
\(415\) 15.7089i 0.771119i
\(416\) 1.06863i 0.0523938i
\(417\) 0 0
\(418\) 0.132606 0.600005i 0.00648599 0.0293472i
\(419\) 25.8709i 1.26388i −0.775019 0.631938i \(-0.782260\pi\)
0.775019 0.631938i \(-0.217740\pi\)
\(420\) 0 0
\(421\) 32.6798 1.59271 0.796357 0.604827i \(-0.206758\pi\)
0.796357 + 0.604827i \(0.206758\pi\)
\(422\) 4.79412i 0.233374i
\(423\) 0 0
\(424\) 0.0980758i 0.00476298i
\(425\) 1.41343 0.0685613
\(426\) 0 0
\(427\) 3.93768 0.190558
\(428\) −11.8692 −0.573721
\(429\) 0 0
\(430\) −22.1273 −1.06708
\(431\) −33.2571 −1.60194 −0.800970 0.598705i \(-0.795682\pi\)
−0.800970 + 0.598705i \(0.795682\pi\)
\(432\) 0 0
\(433\) 37.8532 1.81911 0.909554 0.415585i \(-0.136423\pi\)
0.909554 + 0.415585i \(0.136423\pi\)
\(434\) 8.60956i 0.413272i
\(435\) 0 0
\(436\) 8.39312i 0.401958i
\(437\) 1.05687 0.0505571
\(438\) 0 0
\(439\) 28.8480i 1.37684i 0.725313 + 0.688419i \(0.241695\pi\)
−0.725313 + 0.688419i \(0.758305\pi\)
\(440\) −1.79159 + 8.10640i −0.0854105 + 0.386458i
\(441\) 0 0
\(442\) 1.19328i 0.0567585i
\(443\) 8.29505i 0.394110i −0.980392 0.197055i \(-0.936862\pi\)
0.980392 0.197055i \(-0.0631377\pi\)
\(444\) 0 0
\(445\) 38.1537 1.80866
\(446\) −0.562049 −0.0266138
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 14.7839i 0.697698i −0.937179 0.348849i \(-0.886573\pi\)
0.937179 0.348849i \(-0.113427\pi\)
\(450\) 0 0
\(451\) −0.429443 0.0949106i −0.0202217 0.00446916i
\(452\) 14.8006i 0.696162i
\(453\) 0 0
\(454\) −4.42647 −0.207744
\(455\) 2.67494i 0.125403i
\(456\) 0 0
\(457\) 3.70169i 0.173158i 0.996245 + 0.0865789i \(0.0275935\pi\)
−0.996245 + 0.0865789i \(0.972407\pi\)
\(458\) 6.64297 0.310405
\(459\) 0 0
\(460\) −14.2790 −0.665760
\(461\) 22.9799 1.07028 0.535140 0.844764i \(-0.320259\pi\)
0.535140 + 0.844764i \(0.320259\pi\)
\(462\) 0 0
\(463\) −13.2666 −0.616552 −0.308276 0.951297i \(-0.599752\pi\)
−0.308276 + 0.951297i \(0.599752\pi\)
\(464\) −7.80144 −0.362173
\(465\) 0 0
\(466\) −7.25486 −0.336075
\(467\) 3.25013i 0.150398i −0.997169 0.0751991i \(-0.976041\pi\)
0.997169 0.0751991i \(-0.0239592\pi\)
\(468\) 0 0
\(469\) 4.49907i 0.207748i
\(470\) −12.7345 −0.587399
\(471\) 0 0
\(472\) 6.33137i 0.291425i
\(473\) −6.32692 + 28.6274i −0.290912 + 1.31629i
\(474\) 0 0
\(475\) 0.234516i 0.0107603i
\(476\) 1.11665i 0.0511814i
\(477\) 0 0
\(478\) −20.6125 −0.942795
\(479\) 30.7002 1.40273 0.701363 0.712804i \(-0.252575\pi\)
0.701363 + 0.712804i \(0.252575\pi\)
\(480\) 0 0
\(481\) 4.99577i 0.227788i
\(482\) 20.4491i 0.931430i
\(483\) 0 0
\(484\) 9.97546 + 4.63576i 0.453430 + 0.210716i
\(485\) 6.78416i 0.308053i
\(486\) 0 0
\(487\) −12.4008 −0.561934 −0.280967 0.959717i \(-0.590655\pi\)
−0.280967 + 0.959717i \(0.590655\pi\)
\(488\) 3.93768i 0.178250i
\(489\) 0 0
\(490\) 2.50315i 0.113081i
\(491\) −40.0902 −1.80925 −0.904623 0.426213i \(-0.859847\pi\)
−0.904623 + 0.426213i \(0.859847\pi\)
\(492\) 0 0
\(493\) 8.71145 0.392344
\(494\) 0.197989 0.00890794
\(495\) 0 0
\(496\) −8.60956 −0.386581
\(497\) −0.698081 −0.0313132
\(498\) 0 0
\(499\) −23.2350 −1.04014 −0.520070 0.854123i \(-0.674094\pi\)
−0.520070 + 0.854123i \(0.674094\pi\)
\(500\) 9.34733i 0.418025i
\(501\) 0 0
\(502\) 23.7330i 1.05926i
\(503\) −28.0211 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(504\) 0 0
\(505\) 25.9854i 1.15634i
\(506\) −4.08281 + 18.4735i −0.181503 + 0.821248i
\(507\) 0 0
\(508\) 1.09732i 0.0486859i
\(509\) 29.5301i 1.30890i 0.756107 + 0.654448i \(0.227099\pi\)
−0.756107 + 0.654448i \(0.772901\pi\)
\(510\) 0 0
\(511\) −0.169232 −0.00748640
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 20.6457i 0.910642i
\(515\) 43.8638i 1.93287i
\(516\) 0 0
\(517\) −3.64120 + 16.4754i −0.160140 + 0.724586i
\(518\) 4.67494i 0.205405i
\(519\) 0 0
\(520\) −2.67494 −0.117304
\(521\) 30.2772i 1.32647i 0.748412 + 0.663234i \(0.230817\pi\)
−0.748412 + 0.663234i \(0.769183\pi\)
\(522\) 0 0
\(523\) 41.4615i 1.81299i −0.422221 0.906493i \(-0.638749\pi\)
0.422221 0.906493i \(-0.361251\pi\)
\(524\) 13.8373 0.604483
\(525\) 0 0
\(526\) 0.605313 0.0263929
\(527\) 9.61383 0.418785
\(528\) 0 0
\(529\) −9.54005 −0.414785
\(530\) 0.245499 0.0106638
\(531\) 0 0
\(532\) 0.185274 0.00803264
\(533\) 0.141707i 0.00613801i
\(534\) 0 0
\(535\) 29.7105i 1.28450i
\(536\) 4.49907 0.194330
\(537\) 0 0
\(538\) 21.8291i 0.941119i
\(539\) 3.23848 + 0.715732i 0.139491 + 0.0308287i
\(540\) 0 0
\(541\) 13.2080i 0.567855i −0.958846 0.283927i \(-0.908363\pi\)
0.958846 0.283927i \(-0.0916375\pi\)
\(542\) 4.52103i 0.194195i
\(543\) 0 0
\(544\) 1.11665 0.0478758
\(545\) −21.0093 −0.899938
\(546\) 0 0
\(547\) 24.4642i 1.04601i 0.852328 + 0.523007i \(0.175190\pi\)
−0.852328 + 0.523007i \(0.824810\pi\)
\(548\) 4.53927i 0.193908i
\(549\) 0 0
\(550\) −4.09920 0.905959i −0.174790 0.0386302i
\(551\) 1.44540i 0.0615763i
\(552\) 0 0
\(553\) 1.63002 0.0693155
\(554\) 6.25586i 0.265786i
\(555\) 0 0
\(556\) 15.3511i 0.651032i
\(557\) −38.1747 −1.61751 −0.808757 0.588143i \(-0.799859\pi\)
−0.808757 + 0.588143i \(0.799859\pi\)
\(558\) 0 0
\(559\) −9.44645 −0.399542
\(560\) −2.50315 −0.105778
\(561\) 0 0
\(562\) −31.9977 −1.34974
\(563\) −5.14634 −0.216892 −0.108446 0.994102i \(-0.534587\pi\)
−0.108446 + 0.994102i \(0.534587\pi\)
\(564\) 0 0
\(565\) 37.0482 1.55863
\(566\) 23.2203i 0.976024i
\(567\) 0 0
\(568\) 0.698081i 0.0292908i
\(569\) 12.4086 0.520195 0.260098 0.965582i \(-0.416245\pi\)
0.260098 + 0.965582i \(0.416245\pi\)
\(570\) 0 0
\(571\) 31.2262i 1.30678i −0.757023 0.653388i \(-0.773347\pi\)
0.757023 0.653388i \(-0.226653\pi\)
\(572\) −0.764851 + 3.46073i −0.0319800 + 0.144700i
\(573\) 0 0
\(574\) 0.132606i 0.00553488i
\(575\) 7.22050i 0.301116i
\(576\) 0 0
\(577\) −24.6322 −1.02545 −0.512727 0.858552i \(-0.671365\pi\)
−0.512727 + 0.858552i \(0.671365\pi\)
\(578\) 15.7531 0.655243
\(579\) 0 0
\(580\) 19.5282i 0.810865i
\(581\) 6.27564i 0.260357i
\(582\) 0 0
\(583\) 0.0701959 0.317616i 0.00290722 0.0131543i
\(584\) 0.169232i 0.00700288i
\(585\) 0 0
\(586\) −3.48219 −0.143848
\(587\) 8.06347i 0.332815i −0.986057 0.166407i \(-0.946783\pi\)
0.986057 0.166407i \(-0.0532167\pi\)
\(588\) 0 0
\(589\) 1.59513i 0.0657260i
\(590\) 15.8484 0.652468
\(591\) 0 0
\(592\) 4.67494 0.192139
\(593\) 10.5472 0.433120 0.216560 0.976269i \(-0.430516\pi\)
0.216560 + 0.976269i \(0.430516\pi\)
\(594\) 0 0
\(595\) 2.79514 0.114589
\(596\) 8.97546 0.367649
\(597\) 0 0
\(598\) −6.09587 −0.249279
\(599\) 21.5467i 0.880372i 0.897907 + 0.440186i \(0.145088\pi\)
−0.897907 + 0.440186i \(0.854912\pi\)
\(600\) 0 0
\(601\) 25.2592i 1.03035i 0.857086 + 0.515173i \(0.172272\pi\)
−0.857086 + 0.515173i \(0.827728\pi\)
\(602\) −8.83979 −0.360283
\(603\) 0 0
\(604\) 19.4441i 0.791168i
\(605\) −11.6040 + 24.9701i −0.471771 + 1.01518i
\(606\) 0 0
\(607\) 9.79879i 0.397721i −0.980028 0.198860i \(-0.936276\pi\)
0.980028 0.198860i \(-0.0637240\pi\)
\(608\) 0.185274i 0.00751385i
\(609\) 0 0
\(610\) −9.85662 −0.399083
\(611\) −5.43652 −0.219938
\(612\) 0 0
\(613\) 43.9777i 1.77624i −0.459610 0.888121i \(-0.652011\pi\)
0.459610 0.888121i \(-0.347989\pi\)
\(614\) 9.61453i 0.388011i
\(615\) 0 0
\(616\) −0.715732 + 3.23848i −0.0288377 + 0.130482i
\(617\) 4.41152i 0.177601i −0.996049 0.0888005i \(-0.971697\pi\)
0.996049 0.0888005i \(-0.0283034\pi\)
\(618\) 0 0
\(619\) −33.6155 −1.35112 −0.675561 0.737304i \(-0.736098\pi\)
−0.675561 + 0.737304i \(0.736098\pi\)
\(620\) 21.5511i 0.865511i
\(621\) 0 0
\(622\) 19.7571i 0.792188i
\(623\) 15.2423 0.610668
\(624\) 0 0
\(625\) −29.7267 −1.18907
\(626\) −14.0067 −0.559819
\(627\) 0 0
\(628\) −6.21542 −0.248022
\(629\) −5.22025 −0.208145
\(630\) 0 0
\(631\) 33.0266 1.31477 0.657385 0.753555i \(-0.271663\pi\)
0.657385 + 0.753555i \(0.271663\pi\)
\(632\) 1.63002i 0.0648387i
\(633\) 0 0
\(634\) 7.26821i 0.288657i
\(635\) −2.74677 −0.109002
\(636\) 0 0
\(637\) 1.06863i 0.0423406i
\(638\) −25.2648 5.58374i −1.00024 0.221062i
\(639\) 0 0
\(640\) 2.50315i 0.0989459i
\(641\) 2.69865i 0.106590i 0.998579 + 0.0532951i \(0.0169724\pi\)
−0.998579 + 0.0532951i \(0.983028\pi\)
\(642\) 0 0
\(643\) 41.6860 1.64394 0.821968 0.569534i \(-0.192876\pi\)
0.821968 + 0.569534i \(0.192876\pi\)
\(644\) −5.70439 −0.224784
\(645\) 0 0
\(646\) 0.206885i 0.00813979i
\(647\) 3.68211i 0.144759i 0.997377 + 0.0723793i \(0.0230592\pi\)
−0.997377 + 0.0723793i \(0.976941\pi\)
\(648\) 0 0
\(649\) 4.53156 20.5040i 0.177879 0.804852i
\(650\) 1.35265i 0.0530552i
\(651\) 0 0
\(652\) −19.4877 −0.763198
\(653\) 30.4080i 1.18996i 0.803742 + 0.594978i \(0.202839\pi\)
−0.803742 + 0.594978i \(0.797161\pi\)
\(654\) 0 0
\(655\) 34.6368i 1.35337i
\(656\) 0.132606 0.00517741
\(657\) 0 0
\(658\) −5.08738 −0.198327
\(659\) −18.6322 −0.725809 −0.362905 0.931826i \(-0.618215\pi\)
−0.362905 + 0.931826i \(0.618215\pi\)
\(660\) 0 0
\(661\) −16.7097 −0.649934 −0.324967 0.945725i \(-0.605353\pi\)
−0.324967 + 0.945725i \(0.605353\pi\)
\(662\) 13.6086 0.528912
\(663\) 0 0
\(664\) 6.27564 0.243542
\(665\) 0.463769i 0.0179842i
\(666\) 0 0
\(667\) 44.5025i 1.72314i
\(668\) −18.1273 −0.701368
\(669\) 0 0
\(670\) 11.2619i 0.435084i
\(671\) −2.81832 + 12.7521i −0.108800 + 0.492289i
\(672\) 0 0
\(673\) 39.0503i 1.50528i −0.658433 0.752639i \(-0.728780\pi\)
0.658433 0.752639i \(-0.271220\pi\)
\(674\) 2.85239i 0.109870i
\(675\) 0 0
\(676\) 11.8580 0.456078
\(677\) −24.9257 −0.957971 −0.478986 0.877823i \(-0.658995\pi\)
−0.478986 + 0.877823i \(0.658995\pi\)
\(678\) 0 0
\(679\) 2.71024i 0.104010i
\(680\) 2.79514i 0.107189i
\(681\) 0 0
\(682\) −27.8819 6.16213i −1.06765 0.235960i
\(683\) 23.5527i 0.901220i −0.892721 0.450610i \(-0.851207\pi\)
0.892721 0.450610i \(-0.148793\pi\)
\(684\) 0 0
\(685\) 11.3625 0.434139
\(686\) 1.00000i 0.0381802i
\(687\) 0 0
\(688\) 8.83979i 0.337014i
\(689\) 0.104807 0.00399281
\(690\) 0 0
\(691\) −13.5706 −0.516248 −0.258124 0.966112i \(-0.583104\pi\)
−0.258124 + 0.966112i \(0.583104\pi\)
\(692\) −10.0672 −0.382698
\(693\) 0 0
\(694\) −7.05827 −0.267928
\(695\) −38.4262 −1.45759
\(696\) 0 0
\(697\) −0.148074 −0.00560871
\(698\) 5.87266i 0.222283i
\(699\) 0 0
\(700\) 1.26578i 0.0478420i
\(701\) 32.7927 1.23856 0.619281 0.785170i \(-0.287424\pi\)
0.619281 + 0.785170i \(0.287424\pi\)
\(702\) 0 0
\(703\) 0.866144i 0.0326673i
\(704\) −3.23848 0.715732i −0.122055 0.0269752i
\(705\) 0 0
\(706\) 20.5326i 0.772754i
\(707\) 10.3811i 0.390421i
\(708\) 0 0
\(709\) 35.7358 1.34209 0.671043 0.741419i \(-0.265847\pi\)
0.671043 + 0.741419i \(0.265847\pi\)
\(710\) 1.74740 0.0655789
\(711\) 0 0
\(712\) 15.2423i 0.571228i
\(713\) 49.1123i 1.83927i
\(714\) 0 0
\(715\) −8.66273 1.91454i −0.323968 0.0715997i
\(716\) 20.8254i 0.778281i
\(717\) 0 0
\(718\) −9.66516 −0.360701
\(719\) 10.2851i 0.383571i −0.981437 0.191785i \(-0.938572\pi\)
0.981437 0.191785i \(-0.0614278\pi\)
\(720\) 0 0
\(721\) 17.5234i 0.652606i
\(722\) −18.9657 −0.705829
\(723\) 0 0
\(724\) 2.54233 0.0944851
\(725\) 9.87491 0.366745
\(726\) 0 0
\(727\) −27.1407 −1.00659 −0.503296 0.864114i \(-0.667879\pi\)
−0.503296 + 0.864114i \(0.667879\pi\)
\(728\) −1.06863 −0.0396060
\(729\) 0 0
\(730\) 0.423615 0.0156787
\(731\) 9.87091i 0.365089i
\(732\) 0 0
\(733\) 38.5878i 1.42527i 0.701533 + 0.712637i \(0.252499\pi\)
−0.701533 + 0.712637i \(0.747501\pi\)
\(734\) −25.5991 −0.944881
\(735\) 0 0
\(736\) 5.70439i 0.210267i
\(737\) 14.5701 + 3.22013i 0.536698 + 0.118615i
\(738\) 0 0
\(739\) 22.7365i 0.836377i 0.908360 + 0.418188i \(0.137335\pi\)
−0.908360 + 0.418188i \(0.862665\pi\)
\(740\) 11.7021i 0.430178i
\(741\) 0 0
\(742\) 0.0980758 0.00360047
\(743\) 34.6798 1.27228 0.636138 0.771575i \(-0.280531\pi\)
0.636138 + 0.771575i \(0.280531\pi\)
\(744\) 0 0
\(745\) 22.4669i 0.823125i
\(746\) 29.9547i 1.09672i
\(747\) 0 0
\(748\) 3.61623 + 0.799219i 0.132222 + 0.0292223i
\(749\) 11.8692i 0.433692i
\(750\) 0 0
\(751\) −10.8291 −0.395160 −0.197580 0.980287i \(-0.563308\pi\)
−0.197580 + 0.980287i \(0.563308\pi\)
\(752\) 5.08738i 0.185518i
\(753\) 0 0
\(754\) 8.33684i 0.303610i
\(755\) −48.6715 −1.77134
\(756\) 0 0
\(757\) −26.4465 −0.961213 −0.480607 0.876936i \(-0.659583\pi\)
−0.480607 + 0.876936i \(0.659583\pi\)
\(758\) −4.34578 −0.157846
\(759\) 0 0
\(760\) −0.463769 −0.0168227
\(761\) −34.7276 −1.25887 −0.629437 0.777052i \(-0.716714\pi\)
−0.629437 + 0.777052i \(0.716714\pi\)
\(762\) 0 0
\(763\) −8.39312 −0.303851
\(764\) 10.0668i 0.364203i
\(765\) 0 0
\(766\) 15.6402i 0.565105i
\(767\) 6.76588 0.244302
\(768\) 0 0
\(769\) 26.7806i 0.965734i 0.875694 + 0.482867i \(0.160405\pi\)
−0.875694 + 0.482867i \(0.839595\pi\)
\(770\) −8.10640 1.79159i −0.292135 0.0645643i
\(771\) 0 0
\(772\) 11.9537i 0.430224i
\(773\) 39.2042i 1.41008i 0.709169 + 0.705039i \(0.249070\pi\)
−0.709169 + 0.705039i \(0.750930\pi\)
\(774\) 0 0
\(775\) 10.8978 0.391461
\(776\) 2.71024 0.0972921
\(777\) 0 0
\(778\) 25.6591i 0.919925i
\(779\) 0.0245685i 0.000880257i
\(780\) 0 0
\(781\) 0.499638 2.26072i 0.0178785 0.0808948i
\(782\) 6.36978i 0.227783i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 15.5581i 0.555294i
\(786\) 0 0
\(787\) 17.2615i 0.615308i 0.951498 + 0.307654i \(0.0995439\pi\)
−0.951498 + 0.307654i \(0.900456\pi\)
\(788\) −3.02214 −0.107659
\(789\) 0 0
\(790\) −4.08019 −0.145167
\(791\) 14.8006 0.526249
\(792\) 0 0
\(793\) −4.20792 −0.149427
\(794\) 26.3197 0.934052
\(795\) 0 0
\(796\) −16.2611 −0.576359
\(797\) 2.24242i 0.0794305i 0.999211 + 0.0397153i \(0.0126451\pi\)
−0.999211 + 0.0397153i \(0.987355\pi\)
\(798\) 0 0
\(799\) 5.68080i 0.200972i
\(800\) 1.26578 0.0447521
\(801\) 0 0
\(802\) 25.8351i 0.912270i
\(803\) 0.121125 0.548055i 0.00427441 0.0193404i
\(804\) 0 0
\(805\) 14.2790i 0.503267i
\(806\) 9.20042i 0.324071i
\(807\) 0 0
\(808\) 10.3811 0.365205
\(809\) −34.3573 −1.20794 −0.603968 0.797008i \(-0.706415\pi\)
−0.603968 + 0.797008i \(0.706415\pi\)
\(810\) 0 0
\(811\) 16.8512i 0.591725i 0.955231 + 0.295862i \(0.0956069\pi\)
−0.955231 + 0.295862i \(0.904393\pi\)
\(812\) 7.80144i 0.273777i
\(813\) 0 0
\(814\) 15.1397 + 3.34600i 0.530646 + 0.117277i
\(815\) 48.7807i 1.70871i
\(816\) 0 0
\(817\) −1.63778 −0.0572987
\(818\) 5.55129i 0.194096i
\(819\) 0 0
\(820\) 0.331934i 0.0115916i
\(821\) 30.1344 1.05170 0.525850 0.850577i \(-0.323747\pi\)
0.525850 + 0.850577i \(0.323747\pi\)
\(822\) 0 0
\(823\) 30.2285 1.05370 0.526849 0.849959i \(-0.323373\pi\)
0.526849 + 0.849959i \(0.323373\pi\)
\(824\) 17.5234 0.610457
\(825\) 0 0
\(826\) 6.33137 0.220296
\(827\) −20.0319 −0.696577 −0.348288 0.937387i \(-0.613237\pi\)
−0.348288 + 0.937387i \(0.613237\pi\)
\(828\) 0 0
\(829\) 0.420503 0.0146047 0.00730234 0.999973i \(-0.497676\pi\)
0.00730234 + 0.999973i \(0.497676\pi\)
\(830\) 15.7089i 0.545263i
\(831\) 0 0
\(832\) 1.06863i 0.0370480i
\(833\) 1.11665 0.0386895
\(834\) 0 0
\(835\) 45.3755i 1.57029i
\(836\) −0.132606 + 0.600005i −0.00458629 + 0.0207516i
\(837\) 0 0
\(838\) 25.8709i 0.893696i
\(839\) 49.3044i 1.70218i 0.525023 + 0.851088i \(0.324057\pi\)
−0.525023 + 0.851088i \(0.675943\pi\)
\(840\) 0 0
\(841\) 31.8625 1.09871
\(842\) −32.6798 −1.12622
\(843\) 0 0
\(844\) 4.79412i 0.165020i
\(845\) 29.6825i 1.02111i
\(846\) 0 0
\(847\) −4.63576 + 9.97546i −0.159287 + 0.342761i
\(848\) 0.0980758i 0.00336794i
\(849\) 0 0
\(850\) −1.41343 −0.0484802
\(851\) 26.6677i 0.914156i
\(852\) 0 0
\(853\) 34.3388i 1.17574i 0.808956 + 0.587870i \(0.200033\pi\)
−0.808956 + 0.587870i \(0.799967\pi\)
\(854\) −3.93768 −0.134745
\(855\) 0 0
\(856\) 11.8692 0.405682
\(857\) −52.2385 −1.78443 −0.892216 0.451609i \(-0.850850\pi\)
−0.892216 + 0.451609i \(0.850850\pi\)
\(858\) 0 0
\(859\) −51.7082 −1.76426 −0.882131 0.471004i \(-0.843892\pi\)
−0.882131 + 0.471004i \(0.843892\pi\)
\(860\) 22.1273 0.754536
\(861\) 0 0
\(862\) 33.2571 1.13274
\(863\) 37.3394i 1.27105i 0.772082 + 0.635523i \(0.219216\pi\)
−0.772082 + 0.635523i \(0.780784\pi\)
\(864\) 0 0
\(865\) 25.1998i 0.856819i
\(866\) −37.8532 −1.28630
\(867\) 0 0
\(868\) 8.60956i 0.292227i
\(869\) −1.16666 + 5.27878i −0.0395761 + 0.179070i
\(870\) 0 0
\(871\) 4.80783i 0.162907i
\(872\) 8.39312i 0.284227i
\(873\) 0 0
\(874\) −1.05687 −0.0357493
\(875\) −9.34733 −0.315997
\(876\) 0 0
\(877\) 26.8942i 0.908152i −0.890963 0.454076i \(-0.849969\pi\)
0.890963 0.454076i \(-0.150031\pi\)
\(878\) 28.8480i 0.973572i
\(879\) 0 0
\(880\) 1.79159 8.10640i 0.0603944 0.273267i
\(881\) 48.7635i 1.64288i 0.570292 + 0.821442i \(0.306830\pi\)
−0.570292 + 0.821442i \(0.693170\pi\)
\(882\) 0 0
\(883\) 10.1578 0.341838 0.170919 0.985285i \(-0.445326\pi\)
0.170919 + 0.985285i \(0.445326\pi\)
\(884\) 1.19328i 0.0401343i
\(885\) 0 0
\(886\) 8.29505i 0.278678i
\(887\) −42.3535 −1.42209 −0.711046 0.703145i \(-0.751778\pi\)
−0.711046 + 0.703145i \(0.751778\pi\)
\(888\) 0 0
\(889\) −1.09732 −0.0368031
\(890\) −38.1537 −1.27892
\(891\) 0 0
\(892\) 0.562049 0.0188188
\(893\) −0.942559 −0.0315415
\(894\) 0 0
\(895\) 52.1291 1.74248
\(896\) 1.00000i 0.0334077i
\(897\) 0 0
\(898\) 14.7839i 0.493347i
\(899\) 67.1670 2.24014
\(900\) 0 0
\(901\) 0.109516i 0.00364850i
\(902\) 0.429443 + 0.0949106i 0.0142989 + 0.00316018i
\(903\) 0 0
\(904\) 14.8006i 0.492261i
\(905\) 6.36386i 0.211542i
\(906\) 0 0
\(907\) −24.5243 −0.814316 −0.407158 0.913358i \(-0.633480\pi\)
−0.407158 + 0.913358i \(0.633480\pi\)
\(908\) 4.42647 0.146897
\(909\) 0 0
\(910\) 2.67494i 0.0886734i
\(911\) 9.61511i 0.318563i 0.987233 + 0.159281i \(0.0509177\pi\)
−0.987233 + 0.159281i \(0.949082\pi\)
\(912\) 0 0
\(913\) 20.3235 + 4.49167i 0.672610 + 0.148653i
\(914\) 3.70169i 0.122441i
\(915\) 0 0
\(916\) −6.64297 −0.219490
\(917\) 13.8373i 0.456947i
\(918\) 0 0
\(919\) 24.8699i 0.820381i −0.912000 0.410191i \(-0.865462\pi\)
0.912000 0.410191i \(-0.134538\pi\)
\(920\) 14.2790 0.470763
\(921\) 0 0
\(922\) −22.9799 −0.756802
\(923\) 0.745989 0.0245545
\(924\) 0 0
\(925\) −5.91745 −0.194564
\(926\) 13.2666 0.435968
\(927\) 0 0
\(928\) 7.80144 0.256095
\(929\) 3.89344i 0.127740i −0.997958 0.0638698i \(-0.979656\pi\)
0.997958 0.0638698i \(-0.0203443\pi\)
\(930\) 0 0
\(931\) 0.185274i 0.00607210i
\(932\) 7.25486 0.237641
\(933\) 0 0
\(934\) 3.25013i 0.106348i
\(935\) −2.00057 + 9.05198i −0.0654255 + 0.296031i
\(936\) 0 0
\(937\) 39.8350i 1.30135i −0.759355 0.650676i \(-0.774486\pi\)
0.759355 0.650676i \(-0.225514\pi\)
\(938\) 4.49907i 0.146900i
\(939\) 0 0
\(940\) 12.7345 0.415354
\(941\) −19.7807 −0.644833 −0.322417 0.946598i \(-0.604495\pi\)
−0.322417 + 0.946598i \(0.604495\pi\)
\(942\) 0 0
\(943\) 0.756438i 0.0246330i
\(944\) 6.33137i 0.206068i
\(945\) 0 0
\(946\) 6.32692 28.6274i 0.205706 0.930758i
\(947\) 9.25486i 0.300743i 0.988630 + 0.150371i \(0.0480469\pi\)
−0.988630 + 0.150371i \(0.951953\pi\)
\(948\) 0 0
\(949\) 0.180846 0.00587053
\(950\) 0.234516i 0.00760870i
\(951\) 0 0
\(952\) 1.11665i 0.0361907i
\(953\) 33.1689 1.07445 0.537223 0.843440i \(-0.319473\pi\)
0.537223 + 0.843440i \(0.319473\pi\)
\(954\) 0 0
\(955\) −25.1987 −0.815410
\(956\) 20.6125 0.666657
\(957\) 0 0
\(958\) −30.7002 −0.991878
\(959\) 4.53927 0.146581
\(960\) 0 0
\(961\) 43.1245 1.39111
\(962\) 4.99577i 0.161070i
\(963\) 0 0
\(964\) 20.4491i 0.658620i
\(965\) −29.9220 −0.963223
\(966\) 0 0
\(967\) 22.3192i 0.717737i −0.933388 0.358869i \(-0.883163\pi\)
0.933388 0.358869i \(-0.116837\pi\)
\(968\) −9.97546 4.63576i −0.320623 0.148999i
\(969\) 0 0
\(970\) 6.78416i 0.217826i
\(971\) 15.7712i 0.506121i −0.967451 0.253060i \(-0.918563\pi\)
0.967451 0.253060i \(-0.0814371\pi\)
\(972\) 0 0
\(973\) −15.3511 −0.492134
\(974\) 12.4008 0.397347
\(975\) 0 0
\(976\) 3.93768i 0.126042i
\(977\) 9.68067i 0.309712i 0.987937 + 0.154856i \(0.0494914\pi\)
−0.987937 + 0.154856i \(0.950509\pi\)
\(978\) 0 0
\(979\) −10.9094 + 49.3617i −0.348665 + 1.57761i
\(980\) 2.50315i 0.0799603i
\(981\) 0 0
\(982\) 40.0902 1.27933
\(983\) 6.18360i 0.197226i 0.995126 + 0.0986131i \(0.0314406\pi\)
−0.995126 + 0.0986131i \(0.968559\pi\)
\(984\) 0 0
\(985\) 7.56488i 0.241037i
\(986\) −8.71145 −0.277429
\(987\) 0 0
\(988\) −0.197989 −0.00629886
\(989\) 50.4256 1.60344
\(990\) 0 0
\(991\) −33.9326 −1.07791 −0.538953 0.842336i \(-0.681180\pi\)
−0.538953 + 0.842336i \(0.681180\pi\)
\(992\) 8.60956 0.273354
\(993\) 0 0
\(994\) 0.698081 0.0221418
\(995\) 40.7040i 1.29040i
\(996\) 0 0
\(997\) 33.2430i 1.05282i −0.850232 0.526409i \(-0.823538\pi\)
0.850232 0.526409i \(-0.176462\pi\)
\(998\) 23.2350 0.735491
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1386.2.c.a.197.10 yes 12
3.2 odd 2 1386.2.c.b.197.3 yes 12
11.10 odd 2 1386.2.c.b.197.10 yes 12
33.32 even 2 inner 1386.2.c.a.197.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.c.a.197.3 12 33.32 even 2 inner
1386.2.c.a.197.10 yes 12 1.1 even 1 trivial
1386.2.c.b.197.3 yes 12 3.2 odd 2
1386.2.c.b.197.10 yes 12 11.10 odd 2