Properties

Label 2394.2.e.d.1063.13
Level $2394$
Weight $2$
Character 2394.1063
Analytic conductor $19.116$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2394,2,Mod(1063,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.1063");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.e (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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1063.13
Character \(\chi\) \(=\) 2394.1063
Dual form 2394.2.e.d.1063.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.63913i q^{5} +(1.91223 - 1.82849i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.63913i q^{5} +(1.91223 - 1.82849i) q^{7} +1.00000i q^{8} +1.63913 q^{10} -2.23105 q^{11} -0.801180 q^{13} +(-1.82849 - 1.91223i) q^{14} +1.00000 q^{16} -2.71542i q^{17} +(-3.13440 + 3.02912i) q^{19} -1.63913i q^{20} +2.23105i q^{22} -6.49732 q^{23} +2.31324 q^{25} +0.801180i q^{26} +(-1.91223 + 1.82849i) q^{28} -3.82446i q^{29} -5.19251 q^{31} -1.00000i q^{32} -2.71542 q^{34} +(2.99714 + 3.13440i) q^{35} -7.19617i q^{37} +(3.02912 + 3.13440i) q^{38} -1.63913 q^{40} -6.99299 q^{41} +7.13770 q^{43} +2.23105 q^{44} +6.49732i q^{46} +1.91424i q^{47} +(0.313241 - 6.99299i) q^{49} -2.31324i q^{50} +0.801180 q^{52} +1.19798i q^{53} -3.65698i q^{55} +(1.82849 + 1.91223i) q^{56} -3.82446 q^{58} -4.59174 q^{59} -0.934757i q^{61} +5.19251i q^{62} -1.00000 q^{64} -1.31324i q^{65} -15.2257i q^{67} +2.71542i q^{68} +(3.13440 - 2.99714i) q^{70} +1.19798i q^{71} -6.99299i q^{73} -7.19617 q^{74} +(3.13440 - 3.02912i) q^{76} +(-4.26627 + 4.07945i) q^{77} -0.503034i q^{79} +1.63913i q^{80} +6.99299i q^{82} +11.1494i q^{83} +4.45094 q^{85} -7.13770i q^{86} -2.23105i q^{88} -13.9860 q^{89} +(-1.53204 + 1.46495i) q^{91} +6.49732 q^{92} +1.91424 q^{94} +(-4.96513 - 5.13770i) q^{95} -17.9811 q^{97} +(-6.99299 - 0.313241i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.63913i 0.733043i 0.930410 + 0.366521i \(0.119451\pi\)
−0.930410 + 0.366521i \(0.880549\pi\)
\(6\) 0 0
\(7\) 1.91223 1.82849i 0.722755 0.691105i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.63913 0.518340
\(11\) −2.23105 −0.672686 −0.336343 0.941740i \(-0.609190\pi\)
−0.336343 + 0.941740i \(0.609190\pi\)
\(12\) 0 0
\(13\) −0.801180 −0.222207 −0.111104 0.993809i \(-0.535439\pi\)
−0.111104 + 0.993809i \(0.535439\pi\)
\(14\) −1.82849 1.91223i −0.488685 0.511065i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.71542i 0.658587i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(18\) 0 0
\(19\) −3.13440 + 3.02912i −0.719080 + 0.694927i
\(20\) 1.63913i 0.366521i
\(21\) 0 0
\(22\) 2.23105i 0.475661i
\(23\) −6.49732 −1.35478 −0.677392 0.735622i \(-0.736890\pi\)
−0.677392 + 0.735622i \(0.736890\pi\)
\(24\) 0 0
\(25\) 2.31324 0.462648
\(26\) 0.801180i 0.157124i
\(27\) 0 0
\(28\) −1.91223 + 1.82849i −0.361377 + 0.345552i
\(29\) 3.82446i 0.710184i −0.934831 0.355092i \(-0.884450\pi\)
0.934831 0.355092i \(-0.115550\pi\)
\(30\) 0 0
\(31\) −5.19251 −0.932602 −0.466301 0.884626i \(-0.654414\pi\)
−0.466301 + 0.884626i \(0.654414\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −2.71542 −0.465691
\(35\) 2.99714 + 3.13440i 0.506609 + 0.529810i
\(36\) 0 0
\(37\) 7.19617i 1.18304i −0.806289 0.591522i \(-0.798527\pi\)
0.806289 0.591522i \(-0.201473\pi\)
\(38\) 3.02912 + 3.13440i 0.491387 + 0.508467i
\(39\) 0 0
\(40\) −1.63913 −0.259170
\(41\) −6.99299 −1.09212 −0.546061 0.837746i \(-0.683873\pi\)
−0.546061 + 0.837746i \(0.683873\pi\)
\(42\) 0 0
\(43\) 7.13770 1.08849 0.544244 0.838927i \(-0.316816\pi\)
0.544244 + 0.838927i \(0.316816\pi\)
\(44\) 2.23105 0.336343
\(45\) 0 0
\(46\) 6.49732i 0.957977i
\(47\) 1.91424i 0.279221i 0.990207 + 0.139610i \(0.0445850\pi\)
−0.990207 + 0.139610i \(0.955415\pi\)
\(48\) 0 0
\(49\) 0.313241 6.99299i 0.0447487 0.998998i
\(50\) 2.31324i 0.327142i
\(51\) 0 0
\(52\) 0.801180 0.111104
\(53\) 1.19798i 0.164555i 0.996609 + 0.0822774i \(0.0262193\pi\)
−0.996609 + 0.0822774i \(0.973781\pi\)
\(54\) 0 0
\(55\) 3.65698i 0.493107i
\(56\) 1.82849 + 1.91223i 0.244342 + 0.255532i
\(57\) 0 0
\(58\) −3.82446 −0.502176
\(59\) −4.59174 −0.597793 −0.298897 0.954285i \(-0.596619\pi\)
−0.298897 + 0.954285i \(0.596619\pi\)
\(60\) 0 0
\(61\) 0.934757i 0.119683i −0.998208 0.0598417i \(-0.980940\pi\)
0.998208 0.0598417i \(-0.0190596\pi\)
\(62\) 5.19251i 0.659449i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.31324i 0.162887i
\(66\) 0 0
\(67\) 15.2257i 1.86011i −0.367417 0.930056i \(-0.619758\pi\)
0.367417 0.930056i \(-0.380242\pi\)
\(68\) 2.71542i 0.329293i
\(69\) 0 0
\(70\) 3.13440 2.99714i 0.374632 0.358227i
\(71\) 1.19798i 0.142174i 0.997470 + 0.0710868i \(0.0226467\pi\)
−0.997470 + 0.0710868i \(0.977353\pi\)
\(72\) 0 0
\(73\) 6.99299i 0.818467i −0.912430 0.409234i \(-0.865796\pi\)
0.912430 0.409234i \(-0.134204\pi\)
\(74\) −7.19617 −0.836538
\(75\) 0 0
\(76\) 3.13440 3.02912i 0.359540 0.347463i
\(77\) −4.26627 + 4.07945i −0.486187 + 0.464896i
\(78\) 0 0
\(79\) 0.503034i 0.0565957i −0.999600 0.0282979i \(-0.990991\pi\)
0.999600 0.0282979i \(-0.00900869\pi\)
\(80\) 1.63913i 0.183261i
\(81\) 0 0
\(82\) 6.99299i 0.772246i
\(83\) 11.1494i 1.22381i 0.790932 + 0.611904i \(0.209596\pi\)
−0.790932 + 0.611904i \(0.790404\pi\)
\(84\) 0 0
\(85\) 4.45094 0.482772
\(86\) 7.13770i 0.769678i
\(87\) 0 0
\(88\) 2.23105i 0.237830i
\(89\) −13.9860 −1.48251 −0.741255 0.671223i \(-0.765769\pi\)
−0.741255 + 0.671223i \(0.765769\pi\)
\(90\) 0 0
\(91\) −1.53204 + 1.46495i −0.160601 + 0.153569i
\(92\) 6.49732 0.677392
\(93\) 0 0
\(94\) 1.91424 0.197439
\(95\) −4.96513 5.13770i −0.509411 0.527117i
\(96\) 0 0
\(97\) −17.9811 −1.82570 −0.912850 0.408294i \(-0.866124\pi\)
−0.912850 + 0.408294i \(0.866124\pi\)
\(98\) −6.99299 0.313241i −0.706398 0.0316421i
\(99\) 0 0
\(100\) −2.31324 −0.231324
\(101\) 9.29610i 0.924997i −0.886620 0.462498i \(-0.846953\pi\)
0.886620 0.462498i \(-0.153047\pi\)
\(102\) 0 0
\(103\) 12.0115 1.18353 0.591765 0.806110i \(-0.298431\pi\)
0.591765 + 0.806110i \(0.298431\pi\)
\(104\) 0.801180i 0.0785621i
\(105\) 0 0
\(106\) 1.19798 0.116358
\(107\) 2.62648i 0.253912i −0.991908 0.126956i \(-0.959479\pi\)
0.991908 0.126956i \(-0.0405206\pi\)
\(108\) 0 0
\(109\) 4.26627i 0.408635i −0.978905 0.204317i \(-0.934503\pi\)
0.978905 0.204317i \(-0.0654974\pi\)
\(110\) −3.65698 −0.348680
\(111\) 0 0
\(112\) 1.91223 1.82849i 0.180689 0.172776i
\(113\) 11.5886i 1.09017i −0.838382 0.545084i \(-0.816498\pi\)
0.838382 0.545084i \(-0.183502\pi\)
\(114\) 0 0
\(115\) 10.6500i 0.993115i
\(116\) 3.82446i 0.355092i
\(117\) 0 0
\(118\) 4.59174i 0.422704i
\(119\) −4.96513 5.19251i −0.455152 0.475997i
\(120\) 0 0
\(121\) −6.02243 −0.547494
\(122\) −0.934757 −0.0846289
\(123\) 0 0
\(124\) 5.19251 0.466301
\(125\) 11.9874i 1.07218i
\(126\) 0 0
\(127\) 0.503034i 0.0446370i 0.999751 + 0.0223185i \(0.00710480\pi\)
−0.999751 + 0.0223185i \(0.992895\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −1.31324 −0.115179
\(131\) 14.2135i 1.24184i 0.783874 + 0.620920i \(0.213241\pi\)
−0.783874 + 0.620920i \(0.786759\pi\)
\(132\) 0 0
\(133\) −0.454979 + 11.5236i −0.0394517 + 0.999221i
\(134\) −15.2257 −1.31530
\(135\) 0 0
\(136\) 2.71542 0.232846
\(137\) −1.53219 −0.130904 −0.0654520 0.997856i \(-0.520849\pi\)
−0.0654520 + 0.997856i \(0.520849\pi\)
\(138\) 0 0
\(139\) 3.33601i 0.282956i −0.989941 0.141478i \(-0.954815\pi\)
0.989941 0.141478i \(-0.0451855\pi\)
\(140\) −2.99714 3.13440i −0.253305 0.264905i
\(141\) 0 0
\(142\) 1.19798 0.100532
\(143\) 1.78747 0.149476
\(144\) 0 0
\(145\) 6.26880 0.520595
\(146\) −6.99299 −0.578744
\(147\) 0 0
\(148\) 7.19617i 0.591522i
\(149\) −20.1908 −1.65410 −0.827048 0.562132i \(-0.809981\pi\)
−0.827048 + 0.562132i \(0.809981\pi\)
\(150\) 0 0
\(151\) 8.02951i 0.653432i 0.945123 + 0.326716i \(0.105942\pi\)
−0.945123 + 0.326716i \(0.894058\pi\)
\(152\) −3.02912 3.13440i −0.245694 0.254233i
\(153\) 0 0
\(154\) 4.07945 + 4.26627i 0.328731 + 0.343786i
\(155\) 8.51122i 0.683637i
\(156\) 0 0
\(157\) 3.33601i 0.266242i −0.991100 0.133121i \(-0.957500\pi\)
0.991100 0.133121i \(-0.0424999\pi\)
\(158\) −0.503034 −0.0400192
\(159\) 0 0
\(160\) 1.63913 0.129585
\(161\) −12.4244 + 11.8803i −0.979177 + 0.936298i
\(162\) 0 0
\(163\) −2.62648 −0.205722 −0.102861 0.994696i \(-0.532800\pi\)
−0.102861 + 0.994696i \(0.532800\pi\)
\(164\) 6.99299 0.546061
\(165\) 0 0
\(166\) 11.1494 0.865363
\(167\) −10.9709 −0.848957 −0.424479 0.905438i \(-0.639543\pi\)
−0.424479 + 0.905438i \(0.639543\pi\)
\(168\) 0 0
\(169\) −12.3581 −0.950624
\(170\) 4.45094i 0.341372i
\(171\) 0 0
\(172\) −7.13770 −0.544244
\(173\) −20.3652 −1.54834 −0.774168 0.632980i \(-0.781832\pi\)
−0.774168 + 0.632980i \(0.781832\pi\)
\(174\) 0 0
\(175\) 4.42345 4.22974i 0.334381 0.319738i
\(176\) −2.23105 −0.168171
\(177\) 0 0
\(178\) 13.9860i 1.04829i
\(179\) 18.7866i 1.40418i −0.712089 0.702089i \(-0.752251\pi\)
0.712089 0.702089i \(-0.247749\pi\)
\(180\) 0 0
\(181\) 0.250962 0.0186539 0.00932694 0.999957i \(-0.497031\pi\)
0.00932694 + 0.999957i \(0.497031\pi\)
\(182\) 1.46495 + 1.53204i 0.108589 + 0.113562i
\(183\) 0 0
\(184\) 6.49732i 0.478989i
\(185\) 11.7955 0.867221
\(186\) 0 0
\(187\) 6.05823i 0.443022i
\(188\) 1.91424i 0.139610i
\(189\) 0 0
\(190\) −5.13770 + 4.96513i −0.372728 + 0.360208i
\(191\) 23.9540 1.73325 0.866627 0.498957i \(-0.166283\pi\)
0.866627 + 0.498957i \(0.166283\pi\)
\(192\) 0 0
\(193\) 10.4564i 0.752666i 0.926484 + 0.376333i \(0.122815\pi\)
−0.926484 + 0.376333i \(0.877185\pi\)
\(194\) 17.9811i 1.29097i
\(195\) 0 0
\(196\) −0.313241 + 6.99299i −0.0223743 + 0.499499i
\(197\) 0.195821 0.0139516 0.00697582 0.999976i \(-0.497780\pi\)
0.00697582 + 0.999976i \(0.497780\pi\)
\(198\) 0 0
\(199\) 13.9860i 0.991440i 0.868483 + 0.495720i \(0.165096\pi\)
−0.868483 + 0.495720i \(0.834904\pi\)
\(200\) 2.31324i 0.163571i
\(201\) 0 0
\(202\) −9.29610 −0.654072
\(203\) −6.99299 7.31324i −0.490812 0.513289i
\(204\) 0 0
\(205\) 11.4624i 0.800572i
\(206\) 12.0115i 0.836883i
\(207\) 0 0
\(208\) −0.801180 −0.0555518
\(209\) 6.99299 6.75810i 0.483715 0.467467i
\(210\) 0 0
\(211\) 6.69314i 0.460775i −0.973099 0.230387i \(-0.926001\pi\)
0.973099 0.230387i \(-0.0739993\pi\)
\(212\) 1.19798i 0.0822774i
\(213\) 0 0
\(214\) −2.62648 −0.179543
\(215\) 11.6996i 0.797909i
\(216\) 0 0
\(217\) −9.92927 + 9.49446i −0.674043 + 0.644526i
\(218\) −4.26627 −0.288948
\(219\) 0 0
\(220\) 3.65698i 0.246554i
\(221\) 2.17554i 0.146343i
\(222\) 0 0
\(223\) 3.03993 0.203569 0.101784 0.994806i \(-0.467545\pi\)
0.101784 + 0.994806i \(0.467545\pi\)
\(224\) −1.82849 1.91223i −0.122171 0.127766i
\(225\) 0 0
\(226\) −11.5886 −0.770865
\(227\) 27.3582 1.81583 0.907913 0.419159i \(-0.137675\pi\)
0.907913 + 0.419159i \(0.137675\pi\)
\(228\) 0 0
\(229\) 5.84747i 0.386412i −0.981158 0.193206i \(-0.938111\pi\)
0.981158 0.193206i \(-0.0618885\pi\)
\(230\) −10.6500 −0.702238
\(231\) 0 0
\(232\) 3.82446 0.251088
\(233\) 3.06438 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(234\) 0 0
\(235\) −3.13770 −0.204681
\(236\) 4.59174 0.298897
\(237\) 0 0
\(238\) −5.19251 + 4.96513i −0.336580 + 0.321841i
\(239\) −15.4215 −0.997534 −0.498767 0.866736i \(-0.666214\pi\)
−0.498767 + 0.866736i \(0.666214\pi\)
\(240\) 0 0
\(241\) −2.42769 −0.156381 −0.0781905 0.996938i \(-0.524914\pi\)
−0.0781905 + 0.996938i \(0.524914\pi\)
\(242\) 6.02243i 0.387137i
\(243\) 0 0
\(244\) 0.934757i 0.0598417i
\(245\) 11.4624 + 0.513444i 0.732309 + 0.0328027i
\(246\) 0 0
\(247\) 2.51122 2.42687i 0.159785 0.154418i
\(248\) 5.19251i 0.329725i
\(249\) 0 0
\(250\) 11.9874 0.758148
\(251\) 9.90835i 0.625409i −0.949850 0.312705i \(-0.898765\pi\)
0.949850 0.312705i \(-0.101235\pi\)
\(252\) 0 0
\(253\) 14.4958 0.911344
\(254\) 0.503034 0.0315632
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.80250 −0.299572 −0.149786 0.988718i \(-0.547858\pi\)
−0.149786 + 0.988718i \(0.547858\pi\)
\(258\) 0 0
\(259\) −13.1581 13.7607i −0.817607 0.855050i
\(260\) 1.31324i 0.0814437i
\(261\) 0 0
\(262\) 14.2135 0.878113
\(263\) 7.89503 0.486828 0.243414 0.969922i \(-0.421733\pi\)
0.243414 + 0.969922i \(0.421733\pi\)
\(264\) 0 0
\(265\) −1.96364 −0.120626
\(266\) 11.5236 + 0.454979i 0.706556 + 0.0278965i
\(267\) 0 0
\(268\) 15.2257i 0.930056i
\(269\) 8.78046 0.535354 0.267677 0.963509i \(-0.413744\pi\)
0.267677 + 0.963509i \(0.413744\pi\)
\(270\) 0 0
\(271\) 24.9569i 1.51603i 0.652240 + 0.758013i \(0.273829\pi\)
−0.652240 + 0.758013i \(0.726171\pi\)
\(272\) 2.71542i 0.164647i
\(273\) 0 0
\(274\) 1.53219i 0.0925630i
\(275\) −5.16095 −0.311217
\(276\) 0 0
\(277\) 8.62648 0.518315 0.259158 0.965835i \(-0.416555\pi\)
0.259158 + 0.965835i \(0.416555\pi\)
\(278\) −3.33601 −0.200080
\(279\) 0 0
\(280\) −3.13440 + 2.99714i −0.187316 + 0.179113i
\(281\) 26.8865i 1.60391i 0.597383 + 0.801956i \(0.296207\pi\)
−0.597383 + 0.801956i \(0.703793\pi\)
\(282\) 0 0
\(283\) 3.97796i 0.236465i −0.992986 0.118233i \(-0.962277\pi\)
0.992986 0.118233i \(-0.0377228\pi\)
\(284\) 1.19798i 0.0710868i
\(285\) 0 0
\(286\) 1.78747i 0.105695i
\(287\) −13.3722 + 12.7866i −0.789336 + 0.754770i
\(288\) 0 0
\(289\) 9.62648 0.566264
\(290\) 6.26880i 0.368117i
\(291\) 0 0
\(292\) 6.99299i 0.409234i
\(293\) 3.97796 0.232395 0.116197 0.993226i \(-0.462929\pi\)
0.116197 + 0.993226i \(0.462929\pi\)
\(294\) 0 0
\(295\) 7.52647i 0.438208i
\(296\) 7.19617 0.418269
\(297\) 0 0
\(298\) 20.1908i 1.16962i
\(299\) 5.20552 0.301043
\(300\) 0 0
\(301\) 13.6489 13.0512i 0.786710 0.752260i
\(302\) 8.02951 0.462046
\(303\) 0 0
\(304\) −3.13440 + 3.02912i −0.179770 + 0.173732i
\(305\) 1.53219 0.0877330
\(306\) 0 0
\(307\) 19.8585 1.13339 0.566693 0.823929i \(-0.308222\pi\)
0.566693 + 0.823929i \(0.308222\pi\)
\(308\) 4.26627 4.07945i 0.243093 0.232448i
\(309\) 0 0
\(310\) −8.51122 −0.483405
\(311\) 22.3230i 1.26582i −0.774225 0.632911i \(-0.781860\pi\)
0.774225 0.632911i \(-0.218140\pi\)
\(312\) 0 0
\(313\) 5.12347i 0.289596i 0.989461 + 0.144798i \(0.0462532\pi\)
−0.989461 + 0.144798i \(0.953747\pi\)
\(314\) −3.33601 −0.188262
\(315\) 0 0
\(316\) 0.503034i 0.0282979i
\(317\) 13.8794i 0.779547i 0.920911 + 0.389774i \(0.127447\pi\)
−0.920911 + 0.389774i \(0.872553\pi\)
\(318\) 0 0
\(319\) 8.53254i 0.477731i
\(320\) 1.63913i 0.0916304i
\(321\) 0 0
\(322\) 11.8803 + 12.4244i 0.662062 + 0.692382i
\(323\) 8.22533 + 8.51122i 0.457670 + 0.473577i
\(324\) 0 0
\(325\) −1.85332 −0.102804
\(326\) 2.62648i 0.145467i
\(327\) 0 0
\(328\) 6.99299i 0.386123i
\(329\) 3.50018 + 3.66047i 0.192971 + 0.201808i
\(330\) 0 0
\(331\) 25.6821i 1.41161i 0.708405 + 0.705807i \(0.249415\pi\)
−0.708405 + 0.705807i \(0.750585\pi\)
\(332\) 11.1494i 0.611904i
\(333\) 0 0
\(334\) 10.9709i 0.600303i
\(335\) 24.9569 1.36354
\(336\) 0 0
\(337\) 1.92383i 0.104798i 0.998626 + 0.0523989i \(0.0166867\pi\)
−0.998626 + 0.0523989i \(0.983313\pi\)
\(338\) 12.3581i 0.672193i
\(339\) 0 0
\(340\) −4.45094 −0.241386
\(341\) 11.5847 0.627348
\(342\) 0 0
\(343\) −12.1876 13.9450i −0.658070 0.752957i
\(344\) 7.13770i 0.384839i
\(345\) 0 0
\(346\) 20.3652i 1.09484i
\(347\) 34.0801 1.82952 0.914758 0.404002i \(-0.132381\pi\)
0.914758 + 0.404002i \(0.132381\pi\)
\(348\) 0 0
\(349\) 31.9499i 1.71024i −0.518431 0.855120i \(-0.673484\pi\)
0.518431 0.855120i \(-0.326516\pi\)
\(350\) −4.22974 4.42345i −0.226089 0.236443i
\(351\) 0 0
\(352\) 2.23105i 0.118915i
\(353\) 19.2957i 1.02701i −0.858088 0.513503i \(-0.828348\pi\)
0.858088 0.513503i \(-0.171652\pi\)
\(354\) 0 0
\(355\) −1.96364 −0.104219
\(356\) 13.9860 0.741255
\(357\) 0 0
\(358\) −18.7866 −0.992904
\(359\) −9.56170 −0.504647 −0.252324 0.967643i \(-0.581195\pi\)
−0.252324 + 0.967643i \(0.581195\pi\)
\(360\) 0 0
\(361\) 0.648917 18.9889i 0.0341535 0.999417i
\(362\) 0.250962i 0.0131903i
\(363\) 0 0
\(364\) 1.53204 1.46495i 0.0803007 0.0767843i
\(365\) 11.4624 0.599972
\(366\) 0 0
\(367\) 10.9709i 0.572679i 0.958128 + 0.286339i \(0.0924385\pi\)
−0.958128 + 0.286339i \(0.907562\pi\)
\(368\) −6.49732 −0.338696
\(369\) 0 0
\(370\) 11.7955i 0.613218i
\(371\) 2.19049 + 2.29081i 0.113725 + 0.118933i
\(372\) 0 0
\(373\) 33.7116i 1.74552i −0.488151 0.872759i \(-0.662328\pi\)
0.488151 0.872759i \(-0.337672\pi\)
\(374\) 6.05823 0.313264
\(375\) 0 0
\(376\) −1.91424 −0.0987195
\(377\) 3.06408i 0.157808i
\(378\) 0 0
\(379\) 28.6120i 1.46970i 0.678231 + 0.734849i \(0.262747\pi\)
−0.678231 + 0.734849i \(0.737253\pi\)
\(380\) 4.96513 + 5.13770i 0.254706 + 0.263558i
\(381\) 0 0
\(382\) 23.9540i 1.22560i
\(383\) −18.7885 −0.960046 −0.480023 0.877256i \(-0.659372\pi\)
−0.480023 + 0.877256i \(0.659372\pi\)
\(384\) 0 0
\(385\) −6.68676 6.99299i −0.340789 0.356396i
\(386\) 10.4564 0.532215
\(387\) 0 0
\(388\) 17.9811 0.912850
\(389\) 17.1264 0.868344 0.434172 0.900830i \(-0.357041\pi\)
0.434172 + 0.900830i \(0.357041\pi\)
\(390\) 0 0
\(391\) 17.6430i 0.892243i
\(392\) 6.99299 + 0.313241i 0.353199 + 0.0158211i
\(393\) 0 0
\(394\) 0.195821i 0.00986531i
\(395\) 0.824540 0.0414871
\(396\) 0 0
\(397\) 1.57671i 0.0791328i −0.999217 0.0395664i \(-0.987402\pi\)
0.999217 0.0395664i \(-0.0125977\pi\)
\(398\) 13.9860 0.701054
\(399\) 0 0
\(400\) 2.31324 0.115662
\(401\) 18.2600i 0.911860i 0.890016 + 0.455930i \(0.150693\pi\)
−0.890016 + 0.455930i \(0.849307\pi\)
\(402\) 0 0
\(403\) 4.16013 0.207231
\(404\) 9.29610i 0.462498i
\(405\) 0 0
\(406\) −7.31324 + 6.99299i −0.362950 + 0.347056i
\(407\) 16.0550i 0.795816i
\(408\) 0 0
\(409\) −27.3139 −1.35059 −0.675294 0.737549i \(-0.735983\pi\)
−0.675294 + 0.737549i \(0.735983\pi\)
\(410\) −11.4624 −0.566090
\(411\) 0 0
\(412\) −12.0115 −0.591765
\(413\) −8.78046 + 8.39595i −0.432058 + 0.413138i
\(414\) 0 0
\(415\) −18.2754 −0.897104
\(416\) 0.801180i 0.0392811i
\(417\) 0 0
\(418\) −6.75810 6.99299i −0.330549 0.342038i
\(419\) 2.72805i 0.133274i 0.997777 + 0.0666370i \(0.0212269\pi\)
−0.997777 + 0.0666370i \(0.978773\pi\)
\(420\) 0 0
\(421\) 10.1261i 0.493515i −0.969077 0.246757i \(-0.920635\pi\)
0.969077 0.246757i \(-0.0793650\pi\)
\(422\) −6.69314 −0.325817
\(423\) 0 0
\(424\) −1.19798 −0.0581789
\(425\) 6.28143i 0.304694i
\(426\) 0 0
\(427\) −1.70919 1.78747i −0.0827137 0.0865017i
\(428\) 2.62648i 0.126956i
\(429\) 0 0
\(430\) 11.6996 0.564207
\(431\) 5.02243i 0.241922i −0.992657 0.120961i \(-0.961402\pi\)
0.992657 0.120961i \(-0.0385976\pi\)
\(432\) 0 0
\(433\) 18.5313 0.890557 0.445278 0.895392i \(-0.353105\pi\)
0.445278 + 0.895392i \(0.353105\pi\)
\(434\) 9.49446 + 9.92927i 0.455749 + 0.476620i
\(435\) 0 0
\(436\) 4.26627i 0.204317i
\(437\) 20.3652 19.6811i 0.974199 0.941476i
\(438\) 0 0
\(439\) 26.6534 1.27210 0.636049 0.771649i \(-0.280568\pi\)
0.636049 + 0.771649i \(0.280568\pi\)
\(440\) 3.65698 0.174340
\(441\) 0 0
\(442\) 2.17554 0.103480
\(443\) 0.698855 0.0332036 0.0166018 0.999862i \(-0.494715\pi\)
0.0166018 + 0.999862i \(0.494715\pi\)
\(444\) 0 0
\(445\) 22.9249i 1.08674i
\(446\) 3.03993i 0.143945i
\(447\) 0 0
\(448\) −1.91223 + 1.82849i −0.0903443 + 0.0863881i
\(449\) 26.3203i 1.24213i −0.783759 0.621065i \(-0.786700\pi\)
0.783759 0.621065i \(-0.213300\pi\)
\(450\) 0 0
\(451\) 15.6017 0.734654
\(452\) 11.5886i 0.545084i
\(453\) 0 0
\(454\) 27.3582i 1.28398i
\(455\) −2.40125 2.51122i −0.112572 0.117728i
\(456\) 0 0
\(457\) 2.97757 0.139285 0.0696423 0.997572i \(-0.477814\pi\)
0.0696423 + 0.997572i \(0.477814\pi\)
\(458\) −5.84747 −0.273234
\(459\) 0 0
\(460\) 10.6500i 0.496557i
\(461\) 38.0779i 1.77347i −0.462281 0.886733i \(-0.652969\pi\)
0.462281 0.886733i \(-0.347031\pi\)
\(462\) 0 0
\(463\) −8.72634 −0.405547 −0.202774 0.979226i \(-0.564996\pi\)
−0.202774 + 0.979226i \(0.564996\pi\)
\(464\) 3.82446i 0.177546i
\(465\) 0 0
\(466\) 3.06438i 0.141955i
\(467\) 3.63955i 0.168418i −0.996448 0.0842092i \(-0.973164\pi\)
0.996448 0.0842092i \(-0.0268364\pi\)
\(468\) 0 0
\(469\) −27.8400 29.1150i −1.28553 1.34441i
\(470\) 3.13770i 0.144731i
\(471\) 0 0
\(472\) 4.59174i 0.211352i
\(473\) −15.9245 −0.732211
\(474\) 0 0
\(475\) −7.25062 + 7.00707i −0.332681 + 0.321507i
\(476\) 4.96513 + 5.19251i 0.227576 + 0.237998i
\(477\) 0 0
\(478\) 15.4215i 0.705363i
\(479\) 25.5277i 1.16639i −0.812332 0.583196i \(-0.801802\pi\)
0.812332 0.583196i \(-0.198198\pi\)
\(480\) 0 0
\(481\) 5.76543i 0.262881i
\(482\) 2.42769i 0.110578i
\(483\) 0 0
\(484\) 6.02243 0.273747
\(485\) 29.4734i 1.33832i
\(486\) 0 0
\(487\) 31.9605i 1.44827i 0.689660 + 0.724133i \(0.257760\pi\)
−0.689660 + 0.724133i \(0.742240\pi\)
\(488\) 0.934757 0.0423144
\(489\) 0 0
\(490\) 0.513444 11.4624i 0.0231950 0.517820i
\(491\) 0.307213 0.0138643 0.00693217 0.999976i \(-0.497793\pi\)
0.00693217 + 0.999976i \(0.497793\pi\)
\(492\) 0 0
\(493\) −10.3850 −0.467718
\(494\) −2.42687 2.51122i −0.109190 0.112985i
\(495\) 0 0
\(496\) −5.19251 −0.233151
\(497\) 2.19049 + 2.29081i 0.0982569 + 0.102757i
\(498\) 0 0
\(499\) 14.4355 0.646223 0.323112 0.946361i \(-0.395271\pi\)
0.323112 + 0.946361i \(0.395271\pi\)
\(500\) 11.9874i 0.536092i
\(501\) 0 0
\(502\) −9.90835 −0.442231
\(503\) 10.5751i 0.471519i 0.971811 + 0.235759i \(0.0757577\pi\)
−0.971811 + 0.235759i \(0.924242\pi\)
\(504\) 0 0
\(505\) 15.2376 0.678062
\(506\) 14.4958i 0.644417i
\(507\) 0 0
\(508\) 0.503034i 0.0223185i
\(509\) −30.7223 −1.36174 −0.680872 0.732402i \(-0.738399\pi\)
−0.680872 + 0.732402i \(0.738399\pi\)
\(510\) 0 0
\(511\) −12.7866 13.3722i −0.565647 0.591551i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 4.80250i 0.211829i
\(515\) 19.6885i 0.867579i
\(516\) 0 0
\(517\) 4.27076i 0.187828i
\(518\) −13.7607 + 13.1581i −0.604612 + 0.578135i
\(519\) 0 0
\(520\) 1.31324 0.0575894
\(521\) 40.7304 1.78443 0.892215 0.451611i \(-0.149151\pi\)
0.892215 + 0.451611i \(0.149151\pi\)
\(522\) 0 0
\(523\) −8.42138 −0.368241 −0.184121 0.982904i \(-0.558944\pi\)
−0.184121 + 0.982904i \(0.558944\pi\)
\(524\) 14.2135i 0.620920i
\(525\) 0 0
\(526\) 7.89503i 0.344239i
\(527\) 14.0999i 0.614199i
\(528\) 0 0
\(529\) 19.2151 0.835440
\(530\) 1.96364i 0.0852952i
\(531\) 0 0
\(532\) 0.454979 11.5236i 0.0197258 0.499611i
\(533\) 5.60264 0.242677
\(534\) 0 0
\(535\) 4.30515 0.186128
\(536\) 15.2257 0.657649
\(537\) 0 0
\(538\) 8.78046i 0.378552i
\(539\) −0.698855 + 15.6017i −0.0301018 + 0.672012i
\(540\) 0 0
\(541\) 11.8245 0.508373 0.254187 0.967155i \(-0.418192\pi\)
0.254187 + 0.967155i \(0.418192\pi\)
\(542\) 24.9569 1.07199
\(543\) 0 0
\(544\) −2.71542 −0.116423
\(545\) 6.99299 0.299547
\(546\) 0 0
\(547\) 10.6291i 0.454468i −0.973840 0.227234i \(-0.927032\pi\)
0.973840 0.227234i \(-0.0729682\pi\)
\(548\) 1.53219 0.0654520
\(549\) 0 0
\(550\) 5.16095i 0.220063i
\(551\) 11.5847 + 11.9874i 0.493526 + 0.510680i
\(552\) 0 0
\(553\) −0.919793 0.961916i −0.0391136 0.0409048i
\(554\) 8.62648i 0.366504i
\(555\) 0 0
\(556\) 3.33601i 0.141478i
\(557\) −15.1143 −0.640413 −0.320206 0.947348i \(-0.603752\pi\)
−0.320206 + 0.947348i \(0.603752\pi\)
\(558\) 0 0
\(559\) −5.71858 −0.241870
\(560\) 2.99714 + 3.13440i 0.126652 + 0.132453i
\(561\) 0 0
\(562\) 26.8865 1.13414
\(563\) 36.1386 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(564\) 0 0
\(565\) 18.9953 0.799139
\(566\) −3.97796 −0.167206
\(567\) 0 0
\(568\) −1.19798 −0.0502660
\(569\) 17.2530i 0.723282i 0.932317 + 0.361641i \(0.117783\pi\)
−0.932317 + 0.361641i \(0.882217\pi\)
\(570\) 0 0
\(571\) −39.4580 −1.65126 −0.825632 0.564209i \(-0.809181\pi\)
−0.825632 + 0.564209i \(0.809181\pi\)
\(572\) −1.78747 −0.0747378
\(573\) 0 0
\(574\) 12.7866 + 13.3722i 0.533703 + 0.558145i
\(575\) −15.0299 −0.626788
\(576\) 0 0
\(577\) 28.6139i 1.19121i −0.803277 0.595606i \(-0.796912\pi\)
0.803277 0.595606i \(-0.203088\pi\)
\(578\) 9.62648i 0.400409i
\(579\) 0 0
\(580\) −6.26880 −0.260298
\(581\) 20.3866 + 21.3203i 0.845780 + 0.884513i
\(582\) 0 0
\(583\) 2.67274i 0.110694i
\(584\) 6.99299 0.289372
\(585\) 0 0
\(586\) 3.97796i 0.164328i
\(587\) 42.3045i 1.74609i −0.487637 0.873046i \(-0.662141\pi\)
0.487637 0.873046i \(-0.337859\pi\)
\(588\) 0 0
\(589\) 16.2754 15.7287i 0.670616 0.648090i
\(590\) −7.52647 −0.309860
\(591\) 0 0
\(592\) 7.19617i 0.295761i
\(593\) 40.4270i 1.66014i 0.557660 + 0.830069i \(0.311699\pi\)
−0.557660 + 0.830069i \(0.688301\pi\)
\(594\) 0 0
\(595\) 8.51122 8.13850i 0.348926 0.333646i
\(596\) 20.1908 0.827048
\(597\) 0 0
\(598\) 5.20552i 0.212870i
\(599\) 1.72460i 0.0704653i 0.999379 + 0.0352327i \(0.0112172\pi\)
−0.999379 + 0.0352327i \(0.988783\pi\)
\(600\) 0 0
\(601\) −23.3384 −0.951992 −0.475996 0.879448i \(-0.657912\pi\)
−0.475996 + 0.879448i \(0.657912\pi\)
\(602\) −13.0512 13.6489i −0.531928 0.556288i
\(603\) 0 0
\(604\) 8.02951i 0.326716i
\(605\) 9.87158i 0.401337i
\(606\) 0 0
\(607\) 15.9388 0.646936 0.323468 0.946239i \(-0.395151\pi\)
0.323468 + 0.946239i \(0.395151\pi\)
\(608\) 3.02912 + 3.13440i 0.122847 + 0.127117i
\(609\) 0 0
\(610\) 1.53219i 0.0620366i
\(611\) 1.53365i 0.0620449i
\(612\) 0 0
\(613\) 23.9243 0.966294 0.483147 0.875539i \(-0.339494\pi\)
0.483147 + 0.875539i \(0.339494\pi\)
\(614\) 19.8585i 0.801425i
\(615\) 0 0
\(616\) −4.07945 4.26627i −0.164366 0.171893i
\(617\) 31.8491 1.28220 0.641098 0.767459i \(-0.278479\pi\)
0.641098 + 0.767459i \(0.278479\pi\)
\(618\) 0 0
\(619\) 6.67201i 0.268171i −0.990970 0.134085i \(-0.957190\pi\)
0.990970 0.134085i \(-0.0428096\pi\)
\(620\) 8.51122i 0.341819i
\(621\) 0 0
\(622\) −22.3230 −0.895071
\(623\) −26.7444 + 25.5732i −1.07149 + 1.02457i
\(624\) 0 0
\(625\) −8.08271 −0.323308
\(626\) 5.12347 0.204775
\(627\) 0 0
\(628\) 3.33601i 0.133121i
\(629\) −19.5406 −0.779136
\(630\) 0 0
\(631\) −21.7488 −0.865805 −0.432903 0.901441i \(-0.642511\pi\)
−0.432903 + 0.901441i \(0.642511\pi\)
\(632\) 0.503034 0.0200096
\(633\) 0 0
\(634\) 13.8794 0.551223
\(635\) −0.824540 −0.0327209
\(636\) 0 0
\(637\) −0.250962 + 5.60264i −0.00994349 + 0.221985i
\(638\) 8.53254 0.337807
\(639\) 0 0
\(640\) −1.63913 −0.0647924
\(641\) 18.9019i 0.746579i 0.927715 + 0.373290i \(0.121770\pi\)
−0.927715 + 0.373290i \(0.878230\pi\)
\(642\) 0 0
\(643\) 23.3802i 0.922026i 0.887393 + 0.461013i \(0.152514\pi\)
−0.887393 + 0.461013i \(0.847486\pi\)
\(644\) 12.4244 11.8803i 0.489588 0.468149i
\(645\) 0 0
\(646\) 8.51122 8.22533i 0.334869 0.323621i
\(647\) 45.2456i 1.77879i −0.457141 0.889394i \(-0.651126\pi\)
0.457141 0.889394i \(-0.348874\pi\)
\(648\) 0 0
\(649\) 10.2444 0.402127
\(650\) 1.85332i 0.0726933i
\(651\) 0 0
\(652\) 2.62648 0.102861
\(653\) 12.1844 0.476812 0.238406 0.971166i \(-0.423375\pi\)
0.238406 + 0.971166i \(0.423375\pi\)
\(654\) 0 0
\(655\) −23.2978 −0.910322
\(656\) −6.99299 −0.273030
\(657\) 0 0
\(658\) 3.66047 3.50018i 0.142700 0.136451i
\(659\) 11.8091i 0.460015i 0.973189 + 0.230008i \(0.0738751\pi\)
−0.973189 + 0.230008i \(0.926125\pi\)
\(660\) 0 0
\(661\) −29.0811 −1.13112 −0.565562 0.824706i \(-0.691341\pi\)
−0.565562 + 0.824706i \(0.691341\pi\)
\(662\) 25.6821 0.998161
\(663\) 0 0
\(664\) −11.1494 −0.432682
\(665\) −18.8887 0.745771i −0.732472 0.0289198i
\(666\) 0 0
\(667\) 24.8487i 0.962146i
\(668\) 10.9709 0.424479
\(669\) 0 0
\(670\) 24.9569i 0.964170i
\(671\) 2.08548i 0.0805092i
\(672\) 0 0
\(673\) 10.4564i 0.403064i 0.979482 + 0.201532i \(0.0645919\pi\)
−0.979482 + 0.201532i \(0.935408\pi\)
\(674\) 1.92383 0.0741033
\(675\) 0 0
\(676\) 12.3581 0.475312
\(677\) 24.7462 0.951072 0.475536 0.879696i \(-0.342254\pi\)
0.475536 + 0.879696i \(0.342254\pi\)
\(678\) 0 0
\(679\) −34.3839 + 32.8782i −1.31953 + 1.26175i
\(680\) 4.45094i 0.170686i
\(681\) 0 0
\(682\) 11.5847i 0.443602i
\(683\) 8.74175i 0.334494i −0.985915 0.167247i \(-0.946512\pi\)
0.985915 0.167247i \(-0.0534877\pi\)
\(684\) 0 0
\(685\) 2.51147i 0.0959582i
\(686\) −13.9450 + 12.1876i −0.532421 + 0.465326i
\(687\) 0 0
\(688\) 7.13770 0.272122
\(689\) 0.959795i 0.0365653i
\(690\) 0 0
\(691\) 22.5557i 0.858058i −0.903291 0.429029i \(-0.858856\pi\)
0.903291 0.429029i \(-0.141144\pi\)
\(692\) 20.3652 0.774168
\(693\) 0 0
\(694\) 34.0801i 1.29366i
\(695\) 5.46816 0.207419
\(696\) 0 0
\(697\) 18.9889i 0.719256i
\(698\) −31.9499 −1.20932
\(699\) 0 0
\(700\) −4.42345 + 4.22974i −0.167191 + 0.159869i
\(701\) −5.79846 −0.219005 −0.109502 0.993987i \(-0.534926\pi\)
−0.109502 + 0.993987i \(0.534926\pi\)
\(702\) 0 0
\(703\) 21.7980 + 22.5557i 0.822128 + 0.850703i
\(704\) 2.23105 0.0840857
\(705\) 0 0
\(706\) −19.2957 −0.726203
\(707\) −16.9978 17.7763i −0.639270 0.668546i
\(708\) 0 0
\(709\) 41.0466 1.54154 0.770769 0.637115i \(-0.219872\pi\)
0.770769 + 0.637115i \(0.219872\pi\)
\(710\) 1.96364i 0.0736942i
\(711\) 0 0
\(712\) 13.9860i 0.524147i
\(713\) 33.7374 1.26347
\(714\) 0 0
\(715\) 2.92990i 0.109572i
\(716\) 18.7866i 0.702089i
\(717\) 0 0
\(718\) 9.56170i 0.356839i
\(719\) 14.1894i 0.529174i 0.964362 + 0.264587i \(0.0852356\pi\)
−0.964362 + 0.264587i \(0.914764\pi\)
\(720\) 0 0
\(721\) 22.9688 21.9630i 0.855402 0.817944i
\(722\) −18.9889 0.648917i −0.706694 0.0241502i
\(723\) 0 0
\(724\) −0.250962 −0.00932694
\(725\) 8.84689i 0.328565i
\(726\) 0 0
\(727\) 38.9429i 1.44431i 0.691730 + 0.722156i \(0.256849\pi\)
−0.691730 + 0.722156i \(0.743151\pi\)
\(728\) −1.46495 1.53204i −0.0542947 0.0567812i
\(729\) 0 0
\(730\) 11.4624i 0.424244i
\(731\) 19.3819i 0.716864i
\(732\) 0 0
\(733\) 36.7524i 1.35748i 0.734378 + 0.678741i \(0.237474\pi\)
−0.734378 + 0.678741i \(0.762526\pi\)
\(734\) 10.9709 0.404945
\(735\) 0 0
\(736\) 6.49732i 0.239494i
\(737\) 33.9692i 1.25127i
\(738\) 0 0
\(739\) −3.29783 −0.121313 −0.0606564 0.998159i \(-0.519319\pi\)
−0.0606564 + 0.998159i \(0.519319\pi\)
\(740\) −11.7955 −0.433611
\(741\) 0 0
\(742\) 2.29081 2.19049i 0.0840981 0.0804154i
\(743\) 37.8693i 1.38929i 0.719352 + 0.694645i \(0.244439\pi\)
−0.719352 + 0.694645i \(0.755561\pi\)
\(744\) 0 0
\(745\) 33.0954i 1.21252i
\(746\) −33.7116 −1.23427
\(747\) 0 0
\(748\) 6.05823i 0.221511i
\(749\) −4.80250 5.02243i −0.175479 0.183516i
\(750\) 0 0
\(751\) 45.3467i 1.65473i 0.561668 + 0.827363i \(0.310160\pi\)
−0.561668 + 0.827363i \(0.689840\pi\)
\(752\) 1.91424i 0.0698052i
\(753\) 0 0
\(754\) 3.06408 0.111587
\(755\) −13.1614 −0.478994
\(756\) 0 0
\(757\) −21.2978 −0.774083 −0.387041 0.922062i \(-0.626503\pi\)
−0.387041 + 0.922062i \(0.626503\pi\)
\(758\) 28.6120 1.03923
\(759\) 0 0
\(760\) 5.13770 4.96513i 0.186364 0.180104i
\(761\) 7.57079i 0.274441i 0.990541 + 0.137221i \(0.0438169\pi\)
−0.990541 + 0.137221i \(0.956183\pi\)
\(762\) 0 0
\(763\) −7.80084 8.15809i −0.282409 0.295343i
\(764\) −23.9540 −0.866627
\(765\) 0 0
\(766\) 18.7885i 0.678855i
\(767\) 3.67881 0.132834
\(768\) 0 0
\(769\) 9.50445i 0.342739i −0.985207 0.171370i \(-0.945181\pi\)
0.985207 0.171370i \(-0.0548193\pi\)
\(770\) −6.99299 + 6.68676i −0.252010 + 0.240974i
\(771\) 0 0
\(772\) 10.4564i 0.376333i
\(773\) −0.403021 −0.0144956 −0.00724782 0.999974i \(-0.502307\pi\)
−0.00724782 + 0.999974i \(0.502307\pi\)
\(774\) 0 0
\(775\) −12.0115 −0.431467
\(776\) 17.9811i 0.645483i
\(777\) 0 0
\(778\) 17.1264i 0.614012i
\(779\) 21.9188 21.1826i 0.785323 0.758944i
\(780\) 0 0
\(781\) 2.67274i 0.0956382i
\(782\) 17.6430 0.630911
\(783\) 0 0
\(784\) 0.313241 6.99299i 0.0111872 0.249750i
\(785\) 5.46816 0.195167
\(786\) 0 0
\(787\) 34.3115 1.22307 0.611536 0.791216i \(-0.290552\pi\)
0.611536 + 0.791216i \(0.290552\pi\)
\(788\) −0.195821 −0.00697582
\(789\) 0 0
\(790\) 0.824540i 0.0293358i
\(791\) −21.1897 22.1601i −0.753420 0.787924i
\(792\) 0 0
\(793\) 0.748908i 0.0265945i
\(794\) −1.57671 −0.0559553
\(795\) 0 0
\(796\) 13.9860i 0.495720i
\(797\) 6.37921 0.225963 0.112982 0.993597i \(-0.463960\pi\)
0.112982 + 0.993597i \(0.463960\pi\)
\(798\) 0 0
\(799\) 5.19798 0.183891
\(800\) 2.31324i 0.0817854i
\(801\) 0 0
\(802\) 18.2600 0.644783
\(803\) 15.6017i 0.550571i
\(804\) 0 0
\(805\) −19.4734 20.3652i −0.686346 0.717778i
\(806\) 4.16013i 0.146534i
\(807\) 0 0
\(808\) 9.29610 0.327036
\(809\) 22.4449 0.789122 0.394561 0.918870i \(-0.370897\pi\)
0.394561 + 0.918870i \(0.370897\pi\)
\(810\) 0 0
\(811\) 10.3850 0.364667 0.182334 0.983237i \(-0.441635\pi\)
0.182334 + 0.983237i \(0.441635\pi\)
\(812\) 6.99299 + 7.31324i 0.245406 + 0.256644i
\(813\) 0 0
\(814\) 16.0550 0.562727
\(815\) 4.30515i 0.150803i
\(816\) 0 0
\(817\) −22.3724 + 21.6209i −0.782711 + 0.756420i
\(818\) 27.3139i 0.955010i
\(819\) 0 0
\(820\) 11.4624i 0.400286i
\(821\) 29.2495 1.02081 0.510407 0.859933i \(-0.329495\pi\)
0.510407 + 0.859933i \(0.329495\pi\)
\(822\) 0 0
\(823\) 21.8143 0.760400 0.380200 0.924904i \(-0.375855\pi\)
0.380200 + 0.924904i \(0.375855\pi\)
\(824\) 12.0115i 0.418441i
\(825\) 0 0
\(826\) 8.39595 + 8.78046i 0.292133 + 0.305511i
\(827\) 13.5337i 0.470611i −0.971921 0.235306i \(-0.924391\pi\)
0.971921 0.235306i \(-0.0756091\pi\)
\(828\) 0 0
\(829\) −26.5989 −0.923819 −0.461910 0.886927i \(-0.652836\pi\)
−0.461910 + 0.886927i \(0.652836\pi\)
\(830\) 18.2754i 0.634348i
\(831\) 0 0
\(832\) 0.801180 0.0277759
\(833\) −18.9889 0.850581i −0.657927 0.0294709i
\(834\) 0 0
\(835\) 17.9828i 0.622322i
\(836\) −6.99299 + 6.75810i −0.241858 + 0.233734i
\(837\) 0 0
\(838\) 2.72805 0.0942389
\(839\) 51.7013 1.78493 0.892464 0.451119i \(-0.148975\pi\)
0.892464 + 0.451119i \(0.148975\pi\)
\(840\) 0 0
\(841\) 14.3735 0.495639
\(842\) −10.1261 −0.348967
\(843\) 0 0
\(844\) 6.69314i 0.230387i
\(845\) 20.2566i 0.696848i
\(846\) 0 0
\(847\) −11.5163 + 11.0120i −0.395704 + 0.378376i
\(848\) 1.19798i 0.0411387i
\(849\) 0 0
\(850\) −6.28143 −0.215451
\(851\) 46.7558i 1.60277i
\(852\) 0 0
\(853\) 12.4093i 0.424885i 0.977174 + 0.212443i \(0.0681419\pi\)
−0.977174 + 0.212443i \(0.931858\pi\)
\(854\) −1.78747 + 1.70919i −0.0611659 + 0.0584874i
\(855\) 0 0
\(856\) 2.62648 0.0897713
\(857\) 6.03006 0.205983 0.102991 0.994682i \(-0.467159\pi\)
0.102991 + 0.994682i \(0.467159\pi\)
\(858\) 0 0
\(859\) 21.2999i 0.726745i 0.931644 + 0.363372i \(0.118375\pi\)
−0.931644 + 0.363372i \(0.881625\pi\)
\(860\) 11.6996i 0.398955i
\(861\) 0 0
\(862\) −5.02243 −0.171065
\(863\) 44.9261i 1.52930i −0.644445 0.764650i \(-0.722912\pi\)
0.644445 0.764650i \(-0.277088\pi\)
\(864\) 0 0
\(865\) 33.3813i 1.13500i
\(866\) 18.5313i 0.629719i
\(867\) 0 0
\(868\) 9.92927 9.49446i 0.337021 0.322263i
\(869\) 1.12229i 0.0380711i
\(870\) 0 0
\(871\) 12.1985i 0.413331i
\(872\) 4.26627 0.144474
\(873\) 0 0
\(874\) −19.6811 20.3652i −0.665724 0.688863i
\(875\) 21.9188 + 22.9226i 0.740991 + 0.774926i
\(876\) 0 0
\(877\) 42.5013i 1.43517i 0.696473 + 0.717583i \(0.254752\pi\)
−0.696473 + 0.717583i \(0.745248\pi\)
\(878\) 26.6534i 0.899509i
\(879\) 0 0
\(880\) 3.65698i 0.123277i
\(881\) 28.6286i 0.964521i −0.876028 0.482261i \(-0.839816\pi\)
0.876028 0.482261i \(-0.160184\pi\)
\(882\) 0 0
\(883\) −36.0396 −1.21283 −0.606414 0.795149i \(-0.707393\pi\)
−0.606414 + 0.795149i \(0.707393\pi\)
\(884\) 2.17554i 0.0731714i
\(885\) 0 0
\(886\) 0.698855i 0.0234785i
\(887\) −17.5609 −0.589638 −0.294819 0.955553i \(-0.595259\pi\)
−0.294819 + 0.955553i \(0.595259\pi\)
\(888\) 0 0
\(889\) 0.919793 + 0.961916i 0.0308489 + 0.0322616i
\(890\) −22.9249 −0.768444
\(891\) 0 0
\(892\) −3.03993 −0.101784
\(893\) −5.79846 6.00000i −0.194038 0.200782i
\(894\) 0 0
\(895\) 30.7938 1.02932
\(896\) 1.82849 + 1.91223i 0.0610856 + 0.0638831i
\(897\) 0 0
\(898\) −26.3203 −0.878319
\(899\) 19.8585i 0.662319i
\(900\) 0 0
\(901\) 3.25301 0.108374
\(902\) 15.6017i 0.519479i
\(903\) 0 0
\(904\) 11.5886 0.385432
\(905\) 0.411361i 0.0136741i
\(906\) 0 0
\(907\) 28.6120i 0.950044i −0.879974 0.475022i \(-0.842440\pi\)
0.879974 0.475022i \(-0.157560\pi\)
\(908\) −27.3582 −0.907913
\(909\) 0 0
\(910\) −2.51122 + 2.40125i −0.0832460 + 0.0796006i
\(911\) 0.230528i 0.00763775i 0.999993 + 0.00381887i \(0.00121559\pi\)
−0.999993 + 0.00381887i \(0.998784\pi\)
\(912\) 0 0
\(913\) 24.8749i 0.823238i
\(914\) 2.97757i 0.0984891i
\(915\) 0 0
\(916\) 5.84747i 0.193206i
\(917\) 25.9893 + 27.1795i 0.858241 + 0.897545i
\(918\) 0 0
\(919\) −37.8038 −1.24703 −0.623515 0.781811i \(-0.714296\pi\)
−0.623515 + 0.781811i \(0.714296\pi\)
\(920\) 10.6500 0.351119
\(921\) 0 0
\(922\) −38.0779 −1.25403
\(923\) 0.959795i 0.0315920i
\(924\) 0 0
\(925\) 16.6465i 0.547333i
\(926\) 8.72634i 0.286765i
\(927\) 0 0
\(928\) −3.82446 −0.125544
\(929\) 32.3352i 1.06088i 0.847721 + 0.530442i \(0.177974\pi\)
−0.847721 + 0.530442i \(0.822026\pi\)
\(930\) 0 0
\(931\) 20.2007 + 22.8677i 0.662053 + 0.749457i
\(932\) −3.06438 −0.100377
\(933\) 0 0
\(934\) −3.63955 −0.119090
\(935\) −9.93025 −0.324754
\(936\) 0 0
\(937\) 41.6933i 1.36206i 0.732255 + 0.681030i \(0.238468\pi\)
−0.732255 + 0.681030i \(0.761532\pi\)
\(938\) −29.1150 + 27.8400i −0.950638 + 0.909009i
\(939\) 0 0
\(940\) 3.13770 0.102340
\(941\) 36.7524 1.19809 0.599047 0.800714i \(-0.295546\pi\)
0.599047 + 0.800714i \(0.295546\pi\)
\(942\) 0 0
\(943\) 45.4357 1.47959
\(944\) −4.59174 −0.149448
\(945\) 0 0
\(946\) 15.9245i 0.517751i
\(947\) 21.6999 0.705152 0.352576 0.935783i \(-0.385306\pi\)
0.352576 + 0.935783i \(0.385306\pi\)
\(948\) 0 0
\(949\) 5.60264i 0.181869i
\(950\) 7.00707 + 7.25062i 0.227340 + 0.235241i
\(951\) 0 0
\(952\) 5.19251 4.96513i 0.168290 0.160921i
\(953\) 59.4373i 1.92536i −0.270639 0.962681i \(-0.587235\pi\)
0.270639 0.962681i \(-0.412765\pi\)
\(954\) 0 0
\(955\) 39.2639i 1.27055i
\(956\) 15.4215 0.498767
\(957\) 0 0
\(958\) −25.5277 −0.824763
\(959\) −2.92990 + 2.80160i −0.0946114 + 0.0904683i
\(960\) 0 0
\(961\) −4.03784 −0.130253
\(962\) 5.76543 0.185885
\(963\) 0 0
\(964\) 2.42769 0.0781905
\(965\) −17.1394 −0.551737
\(966\) 0 0
\(967\) −43.0567 −1.38461 −0.692305 0.721605i \(-0.743405\pi\)
−0.692305 + 0.721605i \(0.743405\pi\)
\(968\) 6.02243i 0.193568i
\(969\) 0 0
\(970\) −29.4734 −0.946333
\(971\) −44.4976 −1.42799 −0.713997 0.700148i \(-0.753117\pi\)
−0.713997 + 0.700148i \(0.753117\pi\)
\(972\) 0 0
\(973\) −6.09986 6.37921i −0.195552 0.204508i
\(974\) 31.9605 1.02408
\(975\) 0 0
\(976\) 0.934757i 0.0299208i
\(977\) 37.0070i 1.18396i −0.805953 0.591980i \(-0.798346\pi\)
0.805953 0.591980i \(-0.201654\pi\)
\(978\) 0 0
\(979\) 31.2034 0.997263
\(980\) −11.4624 0.513444i −0.366154 0.0164014i
\(981\) 0 0
\(982\) 0.307213i 0.00980357i
\(983\) −34.1404 −1.08891 −0.544455 0.838790i \(-0.683263\pi\)
−0.544455 + 0.838790i \(0.683263\pi\)
\(984\) 0 0
\(985\) 0.320976i 0.0102272i
\(986\) 10.3850i 0.330726i
\(987\) 0 0
\(988\) −2.51122 + 2.42687i −0.0798925 + 0.0772089i
\(989\) −46.3759 −1.47467
\(990\) 0 0
\(991\) 36.8142i 1.16944i 0.811235 + 0.584721i \(0.198796\pi\)
−0.811235 + 0.584721i \(0.801204\pi\)
\(992\) 5.19251i 0.164862i
\(993\) 0 0
\(994\) 2.29081 2.19049i 0.0726600 0.0694781i
\(995\) −22.9249 −0.726768
\(996\) 0 0
\(997\) 42.1969i 1.33639i −0.743987 0.668194i \(-0.767068\pi\)
0.743987 0.668194i \(-0.232932\pi\)
\(998\) 14.4355i 0.456949i
\(999\) 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 2394.2.e.d.1063.13 yes 24
3.2 odd 2 inner 2394.2.e.d.1063.8 yes 24
7.6 odd 2 inner 2394.2.e.d.1063.5 24
19.18 odd 2 inner 2394.2.e.d.1063.7 yes 24
21.20 even 2 inner 2394.2.e.d.1063.16 yes 24
57.56 even 2 inner 2394.2.e.d.1063.14 yes 24
133.132 even 2 inner 2394.2.e.d.1063.15 yes 24
399.398 odd 2 inner 2394.2.e.d.1063.6 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2394.2.e.d.1063.5 24 7.6 odd 2 inner
2394.2.e.d.1063.6 yes 24 399.398 odd 2 inner
2394.2.e.d.1063.7 yes 24 19.18 odd 2 inner
2394.2.e.d.1063.8 yes 24 3.2 odd 2 inner
2394.2.e.d.1063.13 yes 24 1.1 even 1 trivial
2394.2.e.d.1063.14 yes 24 57.56 even 2 inner
2394.2.e.d.1063.15 yes 24 133.132 even 2 inner
2394.2.e.d.1063.16 yes 24 21.20 even 2 inner