Properties

Label 960.2.h.b.191.3
Level $960$
Weight $2$
Character 960.191
Analytic conductor $7.666$
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] = [960,2,Mod(191,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("960.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (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: \(7.66563859404\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.3
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 960.191
Dual form 960.2.h.b.191.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.292893 - 1.70711i) q^{3} -1.00000i q^{5} +3.41421i q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})\) \(q+(-0.292893 - 1.70711i) q^{3} -1.00000i q^{5} +3.41421i q^{7} +(-2.82843 + 1.00000i) q^{9} +2.82843 q^{11} +2.00000 q^{13} +(-1.70711 + 0.292893i) q^{15} -7.65685i q^{17} -2.82843i q^{19} +(5.82843 - 1.00000i) q^{21} +7.41421 q^{23} -1.00000 q^{25} +(2.53553 + 4.53553i) q^{27} -8.00000i q^{29} +5.65685i q^{31} +(-0.828427 - 4.82843i) q^{33} +3.41421 q^{35} +0.343146 q^{37} +(-0.585786 - 3.41421i) q^{39} -2.00000i q^{41} -7.89949i q^{43} +(1.00000 + 2.82843i) q^{45} -6.24264 q^{47} -4.65685 q^{49} +(-13.0711 + 2.24264i) q^{51} -3.65685i q^{53} -2.82843i q^{55} +(-4.82843 + 0.828427i) q^{57} -1.65685 q^{59} -5.65685 q^{61} +(-3.41421 - 9.65685i) q^{63} -2.00000i q^{65} +1.07107i q^{67} +(-2.17157 - 12.6569i) q^{69} +1.17157 q^{71} +15.6569 q^{73} +(0.292893 + 1.70711i) q^{75} +9.65685i q^{77} +4.48528i q^{79} +(7.00000 - 5.65685i) q^{81} -5.07107 q^{83} -7.65685 q^{85} +(-13.6569 + 2.34315i) q^{87} +7.31371i q^{89} +6.82843i q^{91} +(9.65685 - 1.65685i) q^{93} -2.82843 q^{95} +18.9706 q^{97} +(-8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{13} - 4 q^{15} + 12 q^{21} + 24 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{37} - 8 q^{39} + 4 q^{45} - 8 q^{47} + 4 q^{49} - 24 q^{51} - 8 q^{57} + 16 q^{59} - 8 q^{63} - 20 q^{69} + 16 q^{71} + 40 q^{73} + 4 q^{75} + 28 q^{81} + 8 q^{83} - 8 q^{85} - 32 q^{87} + 16 q^{93} + 8 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.292893 1.70711i −0.169102 0.985599i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.41421i 1.29045i 0.763992 + 0.645226i \(0.223237\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(8\) 0 0
\(9\) −2.82843 + 1.00000i −0.942809 + 0.333333i
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.70711 + 0.292893i −0.440773 + 0.0756247i
\(16\) 0 0
\(17\) 7.65685i 1.85706i −0.371257 0.928530i \(-0.621073\pi\)
0.371257 0.928530i \(-0.378927\pi\)
\(18\) 0 0
\(19\) 2.82843i 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 5.82843 1.00000i 1.27187 0.218218i
\(22\) 0 0
\(23\) 7.41421 1.54597 0.772985 0.634424i \(-0.218763\pi\)
0.772985 + 0.634424i \(0.218763\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 2.53553 + 4.53553i 0.487964 + 0.872864i
\(28\) 0 0
\(29\) 8.00000i 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 5.65685i 1.01600i 0.861357 + 0.508001i \(0.169615\pi\)
−0.861357 + 0.508001i \(0.830385\pi\)
\(32\) 0 0
\(33\) −0.828427 4.82843i −0.144211 0.840521i
\(34\) 0 0
\(35\) 3.41421 0.577107
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0 0
\(39\) −0.585786 3.41421i −0.0938009 0.546712i
\(40\) 0 0
\(41\) 2.00000i 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) 0 0
\(43\) 7.89949i 1.20466i −0.798247 0.602331i \(-0.794239\pi\)
0.798247 0.602331i \(-0.205761\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.82843i 0.149071 + 0.421637i
\(46\) 0 0
\(47\) −6.24264 −0.910583 −0.455291 0.890343i \(-0.650465\pi\)
−0.455291 + 0.890343i \(0.650465\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) −13.0711 + 2.24264i −1.83032 + 0.314033i
\(52\) 0 0
\(53\) 3.65685i 0.502308i −0.967947 0.251154i \(-0.919190\pi\)
0.967947 0.251154i \(-0.0808100\pi\)
\(54\) 0 0
\(55\) 2.82843i 0.381385i
\(56\) 0 0
\(57\) −4.82843 + 0.828427i −0.639541 + 0.109728i
\(58\) 0 0
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 0 0
\(63\) −3.41421 9.65685i −0.430150 1.21665i
\(64\) 0 0
\(65\) 2.00000i 0.248069i
\(66\) 0 0
\(67\) 1.07107i 0.130852i 0.997857 + 0.0654259i \(0.0208406\pi\)
−0.997857 + 0.0654259i \(0.979159\pi\)
\(68\) 0 0
\(69\) −2.17157 12.6569i −0.261427 1.52371i
\(70\) 0 0
\(71\) 1.17157 0.139040 0.0695201 0.997581i \(-0.477853\pi\)
0.0695201 + 0.997581i \(0.477853\pi\)
\(72\) 0 0
\(73\) 15.6569 1.83250 0.916248 0.400611i \(-0.131202\pi\)
0.916248 + 0.400611i \(0.131202\pi\)
\(74\) 0 0
\(75\) 0.292893 + 1.70711i 0.0338204 + 0.197120i
\(76\) 0 0
\(77\) 9.65685i 1.10050i
\(78\) 0 0
\(79\) 4.48528i 0.504634i 0.967645 + 0.252317i \(0.0811925\pi\)
−0.967645 + 0.252317i \(0.918807\pi\)
\(80\) 0 0
\(81\) 7.00000 5.65685i 0.777778 0.628539i
\(82\) 0 0
\(83\) −5.07107 −0.556622 −0.278311 0.960491i \(-0.589775\pi\)
−0.278311 + 0.960491i \(0.589775\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) −13.6569 + 2.34315i −1.46417 + 0.251212i
\(88\) 0 0
\(89\) 7.31371i 0.775252i 0.921817 + 0.387626i \(0.126705\pi\)
−0.921817 + 0.387626i \(0.873295\pi\)
\(90\) 0 0
\(91\) 6.82843i 0.715814i
\(92\) 0 0
\(93\) 9.65685 1.65685i 1.00137 0.171808i
\(94\) 0 0
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) 18.9706 1.92617 0.963084 0.269200i \(-0.0867591\pi\)
0.963084 + 0.269200i \(0.0867591\pi\)
\(98\) 0 0
\(99\) −8.00000 + 2.82843i −0.804030 + 0.284268i
\(100\) 0 0
\(101\) 7.31371i 0.727741i −0.931450 0.363871i \(-0.881455\pi\)
0.931450 0.363871i \(-0.118545\pi\)
\(102\) 0 0
\(103\) 2.24264i 0.220974i −0.993878 0.110487i \(-0.964759\pi\)
0.993878 0.110487i \(-0.0352410\pi\)
\(104\) 0 0
\(105\) −1.00000 5.82843i −0.0975900 0.568796i
\(106\) 0 0
\(107\) 15.4142 1.49015 0.745074 0.666982i \(-0.232414\pi\)
0.745074 + 0.666982i \(0.232414\pi\)
\(108\) 0 0
\(109\) −17.6569 −1.69122 −0.845610 0.533801i \(-0.820763\pi\)
−0.845610 + 0.533801i \(0.820763\pi\)
\(110\) 0 0
\(111\) −0.100505 0.585786i −0.00953952 0.0556004i
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 7.41421i 0.691379i
\(116\) 0 0
\(117\) −5.65685 + 2.00000i −0.522976 + 0.184900i
\(118\) 0 0
\(119\) 26.1421 2.39645
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) −3.41421 + 0.585786i −0.307849 + 0.0528186i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 17.0711i 1.51481i −0.652944 0.757406i \(-0.726466\pi\)
0.652944 0.757406i \(-0.273534\pi\)
\(128\) 0 0
\(129\) −13.4853 + 2.31371i −1.18731 + 0.203711i
\(130\) 0 0
\(131\) 10.8284 0.946084 0.473042 0.881040i \(-0.343156\pi\)
0.473042 + 0.881040i \(0.343156\pi\)
\(132\) 0 0
\(133\) 9.65685 0.837355
\(134\) 0 0
\(135\) 4.53553 2.53553i 0.390357 0.218224i
\(136\) 0 0
\(137\) 17.3137i 1.47921i −0.673041 0.739605i \(-0.735012\pi\)
0.673041 0.739605i \(-0.264988\pi\)
\(138\) 0 0
\(139\) 13.1716i 1.11720i 0.829438 + 0.558599i \(0.188661\pi\)
−0.829438 + 0.558599i \(0.811339\pi\)
\(140\) 0 0
\(141\) 1.82843 + 10.6569i 0.153981 + 0.897469i
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 1.36396 + 7.94975i 0.112498 + 0.655684i
\(148\) 0 0
\(149\) 13.3137i 1.09070i 0.838208 + 0.545351i \(0.183604\pi\)
−0.838208 + 0.545351i \(0.816396\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 7.65685 + 21.6569i 0.619020 + 1.75085i
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) 0 0
\(157\) 7.65685 0.611083 0.305542 0.952179i \(-0.401162\pi\)
0.305542 + 0.952179i \(0.401162\pi\)
\(158\) 0 0
\(159\) −6.24264 + 1.07107i −0.495074 + 0.0849412i
\(160\) 0 0
\(161\) 25.3137i 1.99500i
\(162\) 0 0
\(163\) 10.2426i 0.802266i 0.916020 + 0.401133i \(0.131383\pi\)
−0.916020 + 0.401133i \(0.868617\pi\)
\(164\) 0 0
\(165\) −4.82843 + 0.828427i −0.375893 + 0.0644930i
\(166\) 0 0
\(167\) 18.7279 1.44921 0.724605 0.689164i \(-0.242022\pi\)
0.724605 + 0.689164i \(0.242022\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.82843 + 8.00000i 0.216295 + 0.611775i
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 3.41421i 0.258090i
\(176\) 0 0
\(177\) 0.485281 + 2.82843i 0.0364760 + 0.212598i
\(178\) 0 0
\(179\) −17.6569 −1.31974 −0.659868 0.751382i \(-0.729388\pi\)
−0.659868 + 0.751382i \(0.729388\pi\)
\(180\) 0 0
\(181\) 9.31371 0.692283 0.346141 0.938182i \(-0.387492\pi\)
0.346141 + 0.938182i \(0.387492\pi\)
\(182\) 0 0
\(183\) 1.65685 + 9.65685i 0.122478 + 0.713855i
\(184\) 0 0
\(185\) 0.343146i 0.0252286i
\(186\) 0 0
\(187\) 21.6569i 1.58371i
\(188\) 0 0
\(189\) −15.4853 + 8.65685i −1.12639 + 0.629693i
\(190\) 0 0
\(191\) −18.1421 −1.31272 −0.656359 0.754448i \(-0.727904\pi\)
−0.656359 + 0.754448i \(0.727904\pi\)
\(192\) 0 0
\(193\) −7.65685 −0.551152 −0.275576 0.961279i \(-0.588869\pi\)
−0.275576 + 0.961279i \(0.588869\pi\)
\(194\) 0 0
\(195\) −3.41421 + 0.585786i −0.244497 + 0.0419490i
\(196\) 0 0
\(197\) 22.9706i 1.63658i 0.574802 + 0.818292i \(0.305079\pi\)
−0.574802 + 0.818292i \(0.694921\pi\)
\(198\) 0 0
\(199\) 22.8284i 1.61826i 0.587627 + 0.809132i \(0.300062\pi\)
−0.587627 + 0.809132i \(0.699938\pi\)
\(200\) 0 0
\(201\) 1.82843 0.313708i 0.128967 0.0221273i
\(202\) 0 0
\(203\) 27.3137 1.91705
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −20.9706 + 7.41421i −1.45755 + 0.515323i
\(208\) 0 0
\(209\) 8.00000i 0.553372i
\(210\) 0 0
\(211\) 12.0000i 0.826114i −0.910705 0.413057i \(-0.864461\pi\)
0.910705 0.413057i \(-0.135539\pi\)
\(212\) 0 0
\(213\) −0.343146 2.00000i −0.0235120 0.137038i
\(214\) 0 0
\(215\) −7.89949 −0.538741
\(216\) 0 0
\(217\) −19.3137 −1.31110
\(218\) 0 0
\(219\) −4.58579 26.7279i −0.309879 1.80611i
\(220\) 0 0
\(221\) 15.3137i 1.03011i
\(222\) 0 0
\(223\) 7.89949i 0.528989i 0.964387 + 0.264495i \(0.0852052\pi\)
−0.964387 + 0.264495i \(0.914795\pi\)
\(224\) 0 0
\(225\) 2.82843 1.00000i 0.188562 0.0666667i
\(226\) 0 0
\(227\) −7.41421 −0.492099 −0.246049 0.969257i \(-0.579133\pi\)
−0.246049 + 0.969257i \(0.579133\pi\)
\(228\) 0 0
\(229\) −2.68629 −0.177515 −0.0887576 0.996053i \(-0.528290\pi\)
−0.0887576 + 0.996053i \(0.528290\pi\)
\(230\) 0 0
\(231\) 16.4853 2.82843i 1.08465 0.186097i
\(232\) 0 0
\(233\) 6.97056i 0.456657i −0.973584 0.228328i \(-0.926674\pi\)
0.973584 0.228328i \(-0.0733260\pi\)
\(234\) 0 0
\(235\) 6.24264i 0.407225i
\(236\) 0 0
\(237\) 7.65685 1.31371i 0.497366 0.0853345i
\(238\) 0 0
\(239\) 2.34315 0.151565 0.0757827 0.997124i \(-0.475854\pi\)
0.0757827 + 0.997124i \(0.475854\pi\)
\(240\) 0 0
\(241\) 9.65685 0.622053 0.311026 0.950401i \(-0.399327\pi\)
0.311026 + 0.950401i \(0.399327\pi\)
\(242\) 0 0
\(243\) −11.7071 10.2929i −0.751011 0.660289i
\(244\) 0 0
\(245\) 4.65685i 0.297516i
\(246\) 0 0
\(247\) 5.65685i 0.359937i
\(248\) 0 0
\(249\) 1.48528 + 8.65685i 0.0941259 + 0.548606i
\(250\) 0 0
\(251\) 0.485281 0.0306307 0.0153153 0.999883i \(-0.495125\pi\)
0.0153153 + 0.999883i \(0.495125\pi\)
\(252\) 0 0
\(253\) 20.9706 1.31841
\(254\) 0 0
\(255\) 2.24264 + 13.0711i 0.140440 + 0.818542i
\(256\) 0 0
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 0 0
\(259\) 1.17157i 0.0727980i
\(260\) 0 0
\(261\) 8.00000 + 22.6274i 0.495188 + 1.40060i
\(262\) 0 0
\(263\) −11.8995 −0.733754 −0.366877 0.930269i \(-0.619573\pi\)
−0.366877 + 0.930269i \(0.619573\pi\)
\(264\) 0 0
\(265\) −3.65685 −0.224639
\(266\) 0 0
\(267\) 12.4853 2.14214i 0.764087 0.131097i
\(268\) 0 0
\(269\) 18.0000i 1.09748i −0.835993 0.548740i \(-0.815108\pi\)
0.835993 0.548740i \(-0.184892\pi\)
\(270\) 0 0
\(271\) 2.34315i 0.142336i 0.997464 + 0.0711680i \(0.0226726\pi\)
−0.997464 + 0.0711680i \(0.977327\pi\)
\(272\) 0 0
\(273\) 11.6569 2.00000i 0.705505 0.121046i
\(274\) 0 0
\(275\) −2.82843 −0.170561
\(276\) 0 0
\(277\) −11.6569 −0.700392 −0.350196 0.936676i \(-0.613885\pi\)
−0.350196 + 0.936676i \(0.613885\pi\)
\(278\) 0 0
\(279\) −5.65685 16.0000i −0.338667 0.957895i
\(280\) 0 0
\(281\) 13.3137i 0.794229i 0.917769 + 0.397115i \(0.129988\pi\)
−0.917769 + 0.397115i \(0.870012\pi\)
\(282\) 0 0
\(283\) 13.5563i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(284\) 0 0
\(285\) 0.828427 + 4.82843i 0.0490718 + 0.286011i
\(286\) 0 0
\(287\) 6.82843 0.403069
\(288\) 0 0
\(289\) −41.6274 −2.44867
\(290\) 0 0
\(291\) −5.55635 32.3848i −0.325719 1.89843i
\(292\) 0 0
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 0 0
\(295\) 1.65685i 0.0964658i
\(296\) 0 0
\(297\) 7.17157 + 12.8284i 0.416137 + 0.744381i
\(298\) 0 0
\(299\) 14.8284 0.857550
\(300\) 0 0
\(301\) 26.9706 1.55456
\(302\) 0 0
\(303\) −12.4853 + 2.14214i −0.717261 + 0.123062i
\(304\) 0 0
\(305\) 5.65685i 0.323911i
\(306\) 0 0
\(307\) 7.89949i 0.450848i 0.974261 + 0.225424i \(0.0723767\pi\)
−0.974261 + 0.225424i \(0.927623\pi\)
\(308\) 0 0
\(309\) −3.82843 + 0.656854i −0.217792 + 0.0373671i
\(310\) 0 0
\(311\) 11.5147 0.652940 0.326470 0.945208i \(-0.394141\pi\)
0.326470 + 0.945208i \(0.394141\pi\)
\(312\) 0 0
\(313\) 17.3137 0.978629 0.489314 0.872107i \(-0.337247\pi\)
0.489314 + 0.872107i \(0.337247\pi\)
\(314\) 0 0
\(315\) −9.65685 + 3.41421i −0.544102 + 0.192369i
\(316\) 0 0
\(317\) 0.343146i 0.0192730i −0.999954 0.00963649i \(-0.996933\pi\)
0.999954 0.00963649i \(-0.00306744\pi\)
\(318\) 0 0
\(319\) 22.6274i 1.26689i
\(320\) 0 0
\(321\) −4.51472 26.3137i −0.251987 1.46869i
\(322\) 0 0
\(323\) −21.6569 −1.20502
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 5.17157 + 30.1421i 0.285989 + 1.66686i
\(328\) 0 0
\(329\) 21.3137i 1.17506i
\(330\) 0 0
\(331\) 8.68629i 0.477442i 0.971088 + 0.238721i \(0.0767281\pi\)
−0.971088 + 0.238721i \(0.923272\pi\)
\(332\) 0 0
\(333\) −0.970563 + 0.343146i −0.0531865 + 0.0188043i
\(334\) 0 0
\(335\) 1.07107 0.0585187
\(336\) 0 0
\(337\) −9.31371 −0.507350 −0.253675 0.967290i \(-0.581639\pi\)
−0.253675 + 0.967290i \(0.581639\pi\)
\(338\) 0 0
\(339\) 17.0711 2.92893i 0.927173 0.159078i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) −12.6569 + 2.17157i −0.681422 + 0.116914i
\(346\) 0 0
\(347\) −27.6985 −1.48693 −0.743466 0.668773i \(-0.766820\pi\)
−0.743466 + 0.668773i \(0.766820\pi\)
\(348\) 0 0
\(349\) −16.6274 −0.890045 −0.445023 0.895519i \(-0.646804\pi\)
−0.445023 + 0.895519i \(0.646804\pi\)
\(350\) 0 0
\(351\) 5.07107 + 9.07107i 0.270674 + 0.484178i
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 1.17157i 0.0621806i
\(356\) 0 0
\(357\) −7.65685 44.6274i −0.405244 2.36193i
\(358\) 0 0
\(359\) −29.6569 −1.56523 −0.782614 0.622507i \(-0.786114\pi\)
−0.782614 + 0.622507i \(0.786114\pi\)
\(360\) 0 0
\(361\) 11.0000 0.578947
\(362\) 0 0
\(363\) 0.878680 + 5.12132i 0.0461187 + 0.268800i
\(364\) 0 0
\(365\) 15.6569i 0.819517i
\(366\) 0 0
\(367\) 14.9289i 0.779284i −0.920966 0.389642i \(-0.872599\pi\)
0.920966 0.389642i \(-0.127401\pi\)
\(368\) 0 0
\(369\) 2.00000 + 5.65685i 0.104116 + 0.294484i
\(370\) 0 0
\(371\) 12.4853 0.648204
\(372\) 0 0
\(373\) 19.6569 1.01779 0.508897 0.860828i \(-0.330054\pi\)
0.508897 + 0.860828i \(0.330054\pi\)
\(374\) 0 0
\(375\) 1.70711 0.292893i 0.0881546 0.0151249i
\(376\) 0 0
\(377\) 16.0000i 0.824042i
\(378\) 0 0
\(379\) 14.1421i 0.726433i 0.931705 + 0.363216i \(0.118321\pi\)
−0.931705 + 0.363216i \(0.881679\pi\)
\(380\) 0 0
\(381\) −29.1421 + 5.00000i −1.49300 + 0.256158i
\(382\) 0 0
\(383\) 0.585786 0.0299323 0.0149661 0.999888i \(-0.495236\pi\)
0.0149661 + 0.999888i \(0.495236\pi\)
\(384\) 0 0
\(385\) 9.65685 0.492159
\(386\) 0 0
\(387\) 7.89949 + 22.3431i 0.401554 + 1.13577i
\(388\) 0 0
\(389\) 9.31371i 0.472224i 0.971726 + 0.236112i \(0.0758732\pi\)
−0.971726 + 0.236112i \(0.924127\pi\)
\(390\) 0 0
\(391\) 56.7696i 2.87096i
\(392\) 0 0
\(393\) −3.17157 18.4853i −0.159985 0.932459i
\(394\) 0 0
\(395\) 4.48528 0.225679
\(396\) 0 0
\(397\) 20.6274 1.03526 0.517630 0.855604i \(-0.326814\pi\)
0.517630 + 0.855604i \(0.326814\pi\)
\(398\) 0 0
\(399\) −2.82843 16.4853i −0.141598 0.825296i
\(400\) 0 0
\(401\) 8.00000i 0.399501i −0.979847 0.199750i \(-0.935987\pi\)
0.979847 0.199750i \(-0.0640132\pi\)
\(402\) 0 0
\(403\) 11.3137i 0.563576i
\(404\) 0 0
\(405\) −5.65685 7.00000i −0.281091 0.347833i
\(406\) 0 0
\(407\) 0.970563 0.0481090
\(408\) 0 0
\(409\) 5.65685 0.279713 0.139857 0.990172i \(-0.455336\pi\)
0.139857 + 0.990172i \(0.455336\pi\)
\(410\) 0 0
\(411\) −29.5563 + 5.07107i −1.45791 + 0.250137i
\(412\) 0 0
\(413\) 5.65685i 0.278356i
\(414\) 0 0
\(415\) 5.07107i 0.248929i
\(416\) 0 0
\(417\) 22.4853 3.85786i 1.10111 0.188920i
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −16.9706 −0.827095 −0.413547 0.910483i \(-0.635710\pi\)
−0.413547 + 0.910483i \(0.635710\pi\)
\(422\) 0 0
\(423\) 17.6569 6.24264i 0.858506 0.303528i
\(424\) 0 0
\(425\) 7.65685i 0.371412i
\(426\) 0 0
\(427\) 19.3137i 0.934656i
\(428\) 0 0
\(429\) −1.65685 9.65685i −0.0799937 0.466237i
\(430\) 0 0
\(431\) −11.5147 −0.554644 −0.277322 0.960777i \(-0.589447\pi\)
−0.277322 + 0.960777i \(0.589447\pi\)
\(432\) 0 0
\(433\) 15.6569 0.752420 0.376210 0.926534i \(-0.377227\pi\)
0.376210 + 0.926534i \(0.377227\pi\)
\(434\) 0 0
\(435\) 2.34315 + 13.6569i 0.112345 + 0.654796i
\(436\) 0 0
\(437\) 20.9706i 1.00316i
\(438\) 0 0
\(439\) 17.1716i 0.819554i 0.912186 + 0.409777i \(0.134394\pi\)
−0.912186 + 0.409777i \(0.865606\pi\)
\(440\) 0 0
\(441\) 13.1716 4.65685i 0.627218 0.221755i
\(442\) 0 0
\(443\) −35.8995 −1.70564 −0.852818 0.522208i \(-0.825109\pi\)
−0.852818 + 0.522208i \(0.825109\pi\)
\(444\) 0 0
\(445\) 7.31371 0.346703
\(446\) 0 0
\(447\) 22.7279 3.89949i 1.07499 0.184440i
\(448\) 0 0
\(449\) 2.00000i 0.0943858i 0.998886 + 0.0471929i \(0.0150276\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) 0 0
\(451\) 5.65685i 0.266371i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.82843 0.320122
\(456\) 0 0
\(457\) −5.31371 −0.248565 −0.124282 0.992247i \(-0.539663\pi\)
−0.124282 + 0.992247i \(0.539663\pi\)
\(458\) 0 0
\(459\) 34.7279 19.4142i 1.62096 0.906178i
\(460\) 0 0
\(461\) 26.6274i 1.24016i −0.784538 0.620081i \(-0.787100\pi\)
0.784538 0.620081i \(-0.212900\pi\)
\(462\) 0 0
\(463\) 20.5858i 0.956703i 0.878169 + 0.478351i \(0.158765\pi\)
−0.878169 + 0.478351i \(0.841235\pi\)
\(464\) 0 0
\(465\) −1.65685 9.65685i −0.0768348 0.447826i
\(466\) 0 0
\(467\) −35.8995 −1.66123 −0.830615 0.556847i \(-0.812011\pi\)
−0.830615 + 0.556847i \(0.812011\pi\)
\(468\) 0 0
\(469\) −3.65685 −0.168858
\(470\) 0 0
\(471\) −2.24264 13.0711i −0.103335 0.602283i
\(472\) 0 0
\(473\) 22.3431i 1.02734i
\(474\) 0 0
\(475\) 2.82843i 0.129777i
\(476\) 0 0
\(477\) 3.65685 + 10.3431i 0.167436 + 0.473580i
\(478\) 0 0
\(479\) 0.970563 0.0443461 0.0221731 0.999754i \(-0.492942\pi\)
0.0221731 + 0.999754i \(0.492942\pi\)
\(480\) 0 0
\(481\) 0.686292 0.0312922
\(482\) 0 0
\(483\) 43.2132 7.41421i 1.96627 0.337358i
\(484\) 0 0
\(485\) 18.9706i 0.861409i
\(486\) 0 0
\(487\) 8.10051i 0.367069i −0.983013 0.183534i \(-0.941246\pi\)
0.983013 0.183534i \(-0.0587539\pi\)
\(488\) 0 0
\(489\) 17.4853 3.00000i 0.790712 0.135665i
\(490\) 0 0
\(491\) 39.1127 1.76513 0.882566 0.470189i \(-0.155814\pi\)
0.882566 + 0.470189i \(0.155814\pi\)
\(492\) 0 0
\(493\) −61.2548 −2.75878
\(494\) 0 0
\(495\) 2.82843 + 8.00000i 0.127128 + 0.359573i
\(496\) 0 0
\(497\) 4.00000i 0.179425i
\(498\) 0 0
\(499\) 33.4558i 1.49769i 0.662746 + 0.748845i \(0.269391\pi\)
−0.662746 + 0.748845i \(0.730609\pi\)
\(500\) 0 0
\(501\) −5.48528 31.9706i −0.245064 1.42834i
\(502\) 0 0
\(503\) −0.786797 −0.0350815 −0.0175408 0.999846i \(-0.505584\pi\)
−0.0175408 + 0.999846i \(0.505584\pi\)
\(504\) 0 0
\(505\) −7.31371 −0.325456
\(506\) 0 0
\(507\) 2.63604 + 15.3640i 0.117071 + 0.682337i
\(508\) 0 0
\(509\) 22.6274i 1.00294i 0.865174 + 0.501471i \(0.167208\pi\)
−0.865174 + 0.501471i \(0.832792\pi\)
\(510\) 0 0
\(511\) 53.4558i 2.36475i
\(512\) 0 0
\(513\) 12.8284 7.17157i 0.566389 0.316633i
\(514\) 0 0
\(515\) −2.24264 −0.0988226
\(516\) 0 0
\(517\) −17.6569 −0.776548
\(518\) 0 0
\(519\) 3.41421 0.585786i 0.149867 0.0257132i
\(520\) 0 0
\(521\) 16.0000i 0.700973i −0.936568 0.350486i \(-0.886016\pi\)
0.936568 0.350486i \(-0.113984\pi\)
\(522\) 0 0
\(523\) 11.4142i 0.499109i 0.968361 + 0.249554i \(0.0802841\pi\)
−0.968361 + 0.249554i \(0.919716\pi\)
\(524\) 0 0
\(525\) −5.82843 + 1.00000i −0.254373 + 0.0436436i
\(526\) 0 0
\(527\) 43.3137 1.88677
\(528\) 0 0
\(529\) 31.9706 1.39002
\(530\) 0 0
\(531\) 4.68629 1.65685i 0.203368 0.0719014i
\(532\) 0 0
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) 15.4142i 0.666415i
\(536\) 0 0
\(537\) 5.17157 + 30.1421i 0.223170 + 1.30073i
\(538\) 0 0
\(539\) −13.1716 −0.567340
\(540\) 0 0
\(541\) 14.6863 0.631413 0.315706 0.948857i \(-0.397759\pi\)
0.315706 + 0.948857i \(0.397759\pi\)
\(542\) 0 0
\(543\) −2.72792 15.8995i −0.117066 0.682313i
\(544\) 0 0
\(545\) 17.6569i 0.756337i
\(546\) 0 0
\(547\) 18.0416i 0.771404i −0.922623 0.385702i \(-0.873959\pi\)
0.922623 0.385702i \(-0.126041\pi\)
\(548\) 0 0
\(549\) 16.0000 5.65685i 0.682863 0.241429i
\(550\) 0 0
\(551\) −22.6274 −0.963960
\(552\) 0 0
\(553\) −15.3137 −0.651205
\(554\) 0 0
\(555\) −0.585786 + 0.100505i −0.0248652 + 0.00426620i
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 15.7990i 0.668226i
\(560\) 0 0
\(561\) −36.9706 + 6.34315i −1.56090 + 0.267808i
\(562\) 0 0
\(563\) 25.5563 1.07707 0.538536 0.842603i \(-0.318978\pi\)
0.538536 + 0.842603i \(0.318978\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 19.3137 + 23.8995i 0.811100 + 1.00368i
\(568\) 0 0
\(569\) 28.6274i 1.20012i −0.799954 0.600062i \(-0.795143\pi\)
0.799954 0.600062i \(-0.204857\pi\)
\(570\) 0 0
\(571\) 16.2843i 0.681476i 0.940158 + 0.340738i \(0.110677\pi\)
−0.940158 + 0.340738i \(0.889323\pi\)
\(572\) 0 0
\(573\) 5.31371 + 30.9706i 0.221983 + 1.29381i
\(574\) 0 0
\(575\) −7.41421 −0.309194
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) 2.24264 + 13.0711i 0.0932010 + 0.543215i
\(580\) 0 0
\(581\) 17.3137i 0.718294i
\(582\) 0 0
\(583\) 10.3431i 0.428369i
\(584\) 0 0
\(585\) 2.00000 + 5.65685i 0.0826898 + 0.233882i
\(586\) 0 0
\(587\) −18.7279 −0.772984 −0.386492 0.922293i \(-0.626313\pi\)
−0.386492 + 0.922293i \(0.626313\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 39.2132 6.72792i 1.61302 0.276750i
\(592\) 0 0
\(593\) 32.6274i 1.33985i −0.742430 0.669924i \(-0.766327\pi\)
0.742430 0.669924i \(-0.233673\pi\)
\(594\) 0 0
\(595\) 26.1421i 1.07172i
\(596\) 0 0
\(597\) 38.9706 6.68629i 1.59496 0.273652i
\(598\) 0 0
\(599\) −30.6274 −1.25140 −0.625701 0.780063i \(-0.715187\pi\)
−0.625701 + 0.780063i \(0.715187\pi\)
\(600\) 0 0
\(601\) 4.97056 0.202753 0.101377 0.994848i \(-0.467675\pi\)
0.101377 + 0.994848i \(0.467675\pi\)
\(602\) 0 0
\(603\) −1.07107 3.02944i −0.0436173 0.123368i
\(604\) 0 0
\(605\) 3.00000i 0.121967i
\(606\) 0 0
\(607\) 41.0711i 1.66702i −0.552502 0.833512i \(-0.686327\pi\)
0.552502 0.833512i \(-0.313673\pi\)
\(608\) 0 0
\(609\) −8.00000 46.6274i −0.324176 1.88944i
\(610\) 0 0
\(611\) −12.4853 −0.505100
\(612\) 0 0
\(613\) 37.3137 1.50709 0.753543 0.657398i \(-0.228343\pi\)
0.753543 + 0.657398i \(0.228343\pi\)
\(614\) 0 0
\(615\) 0.585786 + 3.41421i 0.0236212 + 0.137674i
\(616\) 0 0
\(617\) 3.65685i 0.147219i 0.997287 + 0.0736097i \(0.0234519\pi\)
−0.997287 + 0.0736097i \(0.976548\pi\)
\(618\) 0 0
\(619\) 17.4558i 0.701610i −0.936449 0.350805i \(-0.885908\pi\)
0.936449 0.350805i \(-0.114092\pi\)
\(620\) 0 0
\(621\) 18.7990 + 33.6274i 0.754377 + 1.34942i
\(622\) 0 0
\(623\) −24.9706 −1.00042
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −13.6569 + 2.34315i −0.545402 + 0.0935762i
\(628\) 0 0
\(629\) 2.62742i 0.104762i
\(630\) 0 0
\(631\) 3.31371i 0.131917i 0.997822 + 0.0659583i \(0.0210104\pi\)
−0.997822 + 0.0659583i \(0.978990\pi\)
\(632\) 0 0
\(633\) −20.4853 + 3.51472i −0.814217 + 0.139698i
\(634\) 0 0
\(635\) −17.0711 −0.677445
\(636\) 0 0
\(637\) −9.31371 −0.369023
\(638\) 0 0
\(639\) −3.31371 + 1.17157i −0.131088 + 0.0463467i
\(640\) 0 0
\(641\) 5.31371i 0.209879i −0.994479 0.104939i \(-0.966535\pi\)
0.994479 0.104939i \(-0.0334649\pi\)
\(642\) 0 0
\(643\) 16.1005i 0.634942i 0.948268 + 0.317471i \(0.102834\pi\)
−0.948268 + 0.317471i \(0.897166\pi\)
\(644\) 0 0
\(645\) 2.31371 + 13.4853i 0.0911022 + 0.530982i
\(646\) 0 0
\(647\) −33.7574 −1.32714 −0.663569 0.748115i \(-0.730959\pi\)
−0.663569 + 0.748115i \(0.730959\pi\)
\(648\) 0 0
\(649\) −4.68629 −0.183953
\(650\) 0 0
\(651\) 5.65685 + 32.9706i 0.221710 + 1.29222i
\(652\) 0 0
\(653\) 32.3431i 1.26569i −0.774281 0.632843i \(-0.781888\pi\)
0.774281 0.632843i \(-0.218112\pi\)
\(654\) 0 0
\(655\) 10.8284i 0.423102i
\(656\) 0 0
\(657\) −44.2843 + 15.6569i −1.72769 + 0.610832i
\(658\) 0 0
\(659\) −22.3431 −0.870365 −0.435183 0.900342i \(-0.643316\pi\)
−0.435183 + 0.900342i \(0.643316\pi\)
\(660\) 0 0
\(661\) −10.3431 −0.402302 −0.201151 0.979560i \(-0.564468\pi\)
−0.201151 + 0.979560i \(0.564468\pi\)
\(662\) 0 0
\(663\) −26.1421 + 4.48528i −1.01528 + 0.174194i
\(664\) 0 0
\(665\) 9.65685i 0.374477i
\(666\) 0 0
\(667\) 59.3137i 2.29664i
\(668\) 0 0
\(669\) 13.4853 2.31371i 0.521371 0.0894531i
\(670\) 0 0
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) −24.3431 −0.938359 −0.469180 0.883103i \(-0.655450\pi\)
−0.469180 + 0.883103i \(0.655450\pi\)
\(674\) 0 0
\(675\) −2.53553 4.53553i −0.0975927 0.174573i
\(676\) 0 0
\(677\) 0.343146i 0.0131882i 0.999978 + 0.00659408i \(0.00209898\pi\)
−0.999978 + 0.00659408i \(0.997901\pi\)
\(678\) 0 0
\(679\) 64.7696i 2.48563i
\(680\) 0 0
\(681\) 2.17157 + 12.6569i 0.0832149 + 0.485012i
\(682\) 0 0
\(683\) −9.55635 −0.365664 −0.182832 0.983144i \(-0.558526\pi\)
−0.182832 + 0.983144i \(0.558526\pi\)
\(684\) 0 0
\(685\) −17.3137 −0.661523
\(686\) 0 0
\(687\) 0.786797 + 4.58579i 0.0300182 + 0.174959i
\(688\) 0 0
\(689\) 7.31371i 0.278630i
\(690\) 0 0
\(691\) 40.2843i 1.53249i −0.642551 0.766243i \(-0.722124\pi\)
0.642551 0.766243i \(-0.277876\pi\)
\(692\) 0 0
\(693\) −9.65685 27.3137i −0.366834 1.03756i
\(694\) 0 0
\(695\) 13.1716 0.499626
\(696\) 0 0
\(697\) −15.3137 −0.580048
\(698\) 0 0
\(699\) −11.8995 + 2.04163i −0.450080 + 0.0772216i
\(700\) 0 0
\(701\) 15.9411i 0.602088i 0.953610 + 0.301044i \(0.0973351\pi\)
−0.953610 + 0.301044i \(0.902665\pi\)
\(702\) 0 0
\(703\) 0.970563i 0.0366055i
\(704\) 0 0
\(705\) 10.6569 1.82843i 0.401360 0.0688625i
\(706\) 0 0
\(707\) 24.9706 0.939115
\(708\) 0 0
\(709\) 2.68629 0.100886 0.0504429 0.998727i \(-0.483937\pi\)
0.0504429 + 0.998727i \(0.483937\pi\)
\(710\) 0 0
\(711\) −4.48528 12.6863i −0.168211 0.475773i
\(712\) 0 0
\(713\) 41.9411i 1.57071i
\(714\) 0 0
\(715\) 5.65685i 0.211554i
\(716\) 0 0
\(717\) −0.686292 4.00000i −0.0256300 0.149383i
\(718\) 0 0
\(719\) −12.2843 −0.458126 −0.229063 0.973412i \(-0.573566\pi\)
−0.229063 + 0.973412i \(0.573566\pi\)
\(720\) 0 0
\(721\) 7.65685 0.285156
\(722\) 0 0
\(723\) −2.82843 16.4853i −0.105190 0.613094i
\(724\) 0 0
\(725\) 8.00000i 0.297113i
\(726\) 0 0
\(727\) 28.1838i 1.04528i 0.852554 + 0.522639i \(0.175052\pi\)
−0.852554 + 0.522639i \(0.824948\pi\)
\(728\) 0 0
\(729\) −14.1421 + 23.0000i −0.523783 + 0.851852i
\(730\) 0 0
\(731\) −60.4853 −2.23713
\(732\) 0 0
\(733\) −9.02944 −0.333510 −0.166755 0.985998i \(-0.553329\pi\)
−0.166755 + 0.985998i \(0.553329\pi\)
\(734\) 0 0
\(735\) 7.94975 1.36396i 0.293231 0.0503105i
\(736\) 0 0
\(737\) 3.02944i 0.111591i
\(738\) 0 0
\(739\) 15.5147i 0.570718i 0.958421 + 0.285359i \(0.0921129\pi\)
−0.958421 + 0.285359i \(0.907887\pi\)
\(740\) 0 0
\(741\) −9.65685 + 1.65685i −0.354753 + 0.0608661i
\(742\) 0 0
\(743\) 5.27208 0.193414 0.0967069 0.995313i \(-0.469169\pi\)
0.0967069 + 0.995313i \(0.469169\pi\)
\(744\) 0 0
\(745\) 13.3137 0.487777
\(746\) 0 0
\(747\) 14.3431 5.07107i 0.524788 0.185541i
\(748\) 0 0
\(749\) 52.6274i 1.92296i
\(750\) 0 0
\(751\) 38.6274i 1.40953i 0.709439 + 0.704767i \(0.248949\pi\)
−0.709439 + 0.704767i \(0.751051\pi\)
\(752\) 0 0
\(753\) −0.142136 0.828427i −0.00517971 0.0301896i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.3431 −0.594002 −0.297001 0.954877i \(-0.595986\pi\)
−0.297001 + 0.954877i \(0.595986\pi\)
\(758\) 0 0
\(759\) −6.14214 35.7990i −0.222945 1.29942i
\(760\) 0 0
\(761\) 11.3137i 0.410122i 0.978749 + 0.205061i \(0.0657392\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 0 0
\(763\) 60.2843i 2.18244i
\(764\) 0 0
\(765\) 21.6569 7.65685i 0.783005 0.276834i
\(766\) 0 0
\(767\) −3.31371 −0.119651
\(768\) 0 0
\(769\) 25.3137 0.912836 0.456418 0.889766i \(-0.349132\pi\)
0.456418 + 0.889766i \(0.349132\pi\)
\(770\) 0 0
\(771\) 37.5563 6.44365i 1.35256 0.232062i
\(772\) 0 0
\(773\) 26.2843i 0.945380i 0.881229 + 0.472690i \(0.156717\pi\)
−0.881229 + 0.472690i \(0.843283\pi\)
\(774\) 0 0
\(775\) 5.65685i 0.203200i
\(776\) 0 0
\(777\) 2.00000 0.343146i 0.0717496 0.0123103i
\(778\) 0 0
\(779\) −5.65685 −0.202678
\(780\) 0 0
\(781\) 3.31371 0.118574
\(782\) 0 0
\(783\) 36.2843 20.2843i 1.29669 0.724901i
\(784\) 0 0
\(785\) 7.65685i 0.273285i
\(786\) 0 0
\(787\) 21.5563i 0.768401i 0.923250 + 0.384200i \(0.125523\pi\)
−0.923250 + 0.384200i \(0.874477\pi\)
\(788\) 0 0
\(789\) 3.48528 + 20.3137i 0.124079 + 0.723187i
\(790\) 0 0
\(791\) −34.1421 −1.21395
\(792\) 0 0
\(793\) −11.3137 −0.401762
\(794\) 0 0
\(795\) 1.07107 + 6.24264i 0.0379869 + 0.221404i
\(796\) 0 0
\(797\) 30.9706i 1.09703i −0.836140 0.548517i \(-0.815193\pi\)
0.836140 0.548517i \(-0.184807\pi\)
\(798\) 0 0
\(799\) 47.7990i 1.69101i
\(800\) 0 0
\(801\) −7.31371 20.6863i −0.258417 0.730914i
\(802\) 0 0
\(803\) 44.2843 1.56276
\(804\) 0 0
\(805\) 25.3137 0.892191
\(806\) 0 0
\(807\) −30.7279 + 5.27208i −1.08167 + 0.185586i
\(808\) 0 0
\(809\) 26.6274i 0.936170i 0.883683 + 0.468085i \(0.155056\pi\)
−0.883683 + 0.468085i \(0.844944\pi\)
\(810\) 0 0
\(811\) 42.6274i 1.49685i 0.663219 + 0.748426i \(0.269190\pi\)
−0.663219 + 0.748426i \(0.730810\pi\)
\(812\) 0 0
\(813\) 4.00000 0.686292i 0.140286 0.0240693i
\(814\) 0 0
\(815\) 10.2426 0.358784
\(816\) 0 0
\(817\) −22.3431 −0.781688
\(818\) 0 0
\(819\) −6.82843 19.3137i −0.238605 0.674876i
\(820\) 0 0
\(821\) 28.6274i 0.999104i 0.866284 + 0.499552i \(0.166502\pi\)
−0.866284 + 0.499552i \(0.833498\pi\)
\(822\) 0 0
\(823\) 16.1005i 0.561228i 0.959821 + 0.280614i \(0.0905381\pi\)
−0.959821 + 0.280614i \(0.909462\pi\)
\(824\) 0 0
\(825\) 0.828427 + 4.82843i 0.0288421 + 0.168104i
\(826\) 0 0
\(827\) −24.5858 −0.854932 −0.427466 0.904032i \(-0.640594\pi\)
−0.427466 + 0.904032i \(0.640594\pi\)
\(828\) 0 0
\(829\) 12.9706 0.450486 0.225243 0.974303i \(-0.427682\pi\)
0.225243 + 0.974303i \(0.427682\pi\)
\(830\) 0 0
\(831\) 3.41421 + 19.8995i 0.118438 + 0.690306i
\(832\) 0 0
\(833\) 35.6569i 1.23544i
\(834\) 0 0
\(835\) 18.7279i 0.648106i
\(836\) 0 0
\(837\) −25.6569 + 14.3431i −0.886831 + 0.495772i
\(838\) 0 0
\(839\) 17.9411 0.619396 0.309698 0.950835i \(-0.399772\pi\)
0.309698 + 0.950835i \(0.399772\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) 22.7279 3.89949i 0.782791 0.134306i
\(844\) 0 0
\(845\) 9.00000i 0.309609i
\(846\) 0 0
\(847\) 10.2426i 0.351941i
\(848\) 0 0
\(849\) 23.1421 3.97056i 0.794236 0.136269i
\(850\) 0 0
\(851\) 2.54416 0.0872125
\(852\) 0 0
\(853\) −47.9411 −1.64147 −0.820736 0.571307i \(-0.806437\pi\)
−0.820736 + 0.571307i \(0.806437\pi\)
\(854\) 0 0
\(855\) 8.00000 2.82843i 0.273594 0.0967302i
\(856\) 0 0
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 0 0
\(859\) 7.51472i 0.256399i −0.991748 0.128199i \(-0.959080\pi\)
0.991748 0.128199i \(-0.0409198\pi\)
\(860\) 0 0
\(861\) −2.00000 11.6569i −0.0681598 0.397265i
\(862\) 0 0
\(863\) 17.5563 0.597625 0.298813 0.954312i \(-0.403409\pi\)
0.298813 + 0.954312i \(0.403409\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 12.1924 + 71.0624i 0.414075 + 2.41341i
\(868\) 0 0
\(869\) 12.6863i 0.430353i
\(870\) 0 0
\(871\) 2.14214i 0.0725835i
\(872\) 0 0
\(873\) −53.6569 + 18.9706i −1.81601 + 0.642056i
\(874\) 0 0
\(875\) −3.41421 −0.115421
\(876\) 0 0
\(877\) 23.6569 0.798835 0.399418 0.916769i \(-0.369212\pi\)
0.399418 + 0.916769i \(0.369212\pi\)
\(878\) 0 0
\(879\) 37.5563 6.44365i 1.26674 0.217339i
\(880\) 0 0
\(881\) 28.6274i 0.964482i 0.876039 + 0.482241i \(0.160177\pi\)
−0.876039 + 0.482241i \(0.839823\pi\)
\(882\) 0 0
\(883\) 1.27208i 0.0428088i 0.999771 + 0.0214044i \(0.00681376\pi\)
−0.999771 + 0.0214044i \(0.993186\pi\)
\(884\) 0 0
\(885\) 2.82843 0.485281i 0.0950765 0.0163126i
\(886\) 0 0
\(887\) 11.8995 0.399546 0.199773 0.979842i \(-0.435980\pi\)
0.199773 + 0.979842i \(0.435980\pi\)
\(888\) 0 0
\(889\) 58.2843 1.95479
\(890\) 0 0
\(891\) 19.7990 16.0000i 0.663291 0.536020i
\(892\) 0 0
\(893\) 17.6569i 0.590864i
\(894\) 0 0
\(895\) 17.6569i 0.590204i
\(896\) 0 0
\(897\) −4.34315 25.3137i −0.145013 0.845200i
\(898\) 0 0
\(899\) 45.2548 1.50933
\(900\) 0 0
\(901\) −28.0000 −0.932815
\(902\) 0 0
\(903\) −7.89949 46.0416i −0.262879 1.53217i
\(904\) 0 0
\(905\) 9.31371i 0.309598i
\(906\) 0 0
\(907\) 1.07107i 0.0355642i 0.999842 + 0.0177821i \(0.00566052\pi\)
−0.999842 + 0.0177821i \(0.994339\pi\)
\(908\) 0 0
\(909\) 7.31371 + 20.6863i 0.242580 + 0.686121i
\(910\) 0 0
\(911\) 11.1127 0.368180 0.184090 0.982909i \(-0.441066\pi\)
0.184090 + 0.982909i \(0.441066\pi\)
\(912\) 0 0
\(913\) −14.3431 −0.474689
\(914\) 0 0
\(915\) 9.65685 1.65685i 0.319246 0.0547739i
\(916\) 0 0
\(917\) 36.9706i 1.22088i
\(918\) 0 0
\(919\) 44.4853i 1.46743i −0.679455 0.733717i \(-0.737784\pi\)
0.679455 0.733717i \(-0.262216\pi\)
\(920\) 0 0
\(921\) 13.4853 2.31371i 0.444355 0.0762393i
\(922\) 0 0
\(923\) 2.34315 0.0771256
\(924\) 0 0
\(925\) −0.343146 −0.0112826
\(926\) 0 0
\(927\) 2.24264 + 6.34315i 0.0736580 + 0.208336i
\(928\) 0 0
\(929\) 46.0000i 1.50921i 0.656179 + 0.754606i \(0.272172\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) 13.1716i 0.431681i
\(932\) 0 0
\(933\) −3.37258 19.6569i −0.110413 0.643537i
\(934\) 0 0
\(935\) −21.6569 −0.708255
\(936\) 0 0
\(937\) 37.5980 1.22827 0.614136 0.789200i \(-0.289505\pi\)
0.614136 + 0.789200i \(0.289505\pi\)
\(938\) 0 0
\(939\) −5.07107 29.5563i −0.165488 0.964535i
\(940\) 0 0
\(941\) 12.0000i 0.391189i 0.980685 + 0.195594i \(0.0626636\pi\)
−0.980685 + 0.195594i \(0.937336\pi\)
\(942\) 0 0
\(943\) 14.8284i 0.482880i
\(944\) 0 0
\(945\) 8.65685 + 15.4853i 0.281607 + 0.503736i
\(946\) 0 0
\(947\) 23.2132 0.754328 0.377164 0.926147i \(-0.376899\pi\)
0.377164 + 0.926147i \(0.376899\pi\)
\(948\) 0 0
\(949\) 31.3137 1.01649
\(950\) 0 0
\(951\) −0.585786 + 0.100505i −0.0189954 + 0.00325910i
\(952\) 0 0
\(953\) 25.3137i 0.819991i −0.912087 0.409996i \(-0.865530\pi\)
0.912087 0.409996i \(-0.134470\pi\)
\(954\) 0 0
\(955\) 18.1421i 0.587066i
\(956\) 0 0
\(957\) −38.6274 + 6.62742i −1.24865 + 0.214234i
\(958\) 0 0
\(959\) 59.1127 1.90885
\(960\) 0 0
\(961\) −1.00000 −0.0322581
\(962\) 0 0
\(963\) −43.5980 + 15.4142i −1.40493 + 0.496716i
\(964\) 0 0
\(965\) 7.65685i 0.246483i
\(966\) 0 0
\(967\) 11.4142i 0.367056i 0.983014 + 0.183528i \(0.0587519\pi\)
−0.983014 + 0.183528i \(0.941248\pi\)
\(968\) 0 0
\(969\) 6.34315 + 36.9706i 0.203771 + 1.18767i
\(970\) 0 0
\(971\) −32.4853 −1.04250 −0.521251 0.853403i \(-0.674535\pi\)
−0.521251 + 0.853403i \(0.674535\pi\)
\(972\) 0 0
\(973\) −44.9706 −1.44169
\(974\) 0 0
\(975\) 0.585786 + 3.41421i 0.0187602 + 0.109342i
\(976\) 0 0
\(977\) 21.0294i 0.672791i −0.941721 0.336396i \(-0.890792\pi\)
0.941721 0.336396i \(-0.109208\pi\)
\(978\) 0 0
\(979\) 20.6863i 0.661137i
\(980\) 0 0
\(981\) 49.9411 17.6569i 1.59450 0.563740i
\(982\) 0 0
\(983\) −45.0711 −1.43754 −0.718772 0.695246i \(-0.755295\pi\)
−0.718772 + 0.695246i \(0.755295\pi\)
\(984\) 0 0
\(985\) 22.9706 0.731903
\(986\) 0 0
\(987\) −36.3848 + 6.24264i −1.15814 + 0.198705i
\(988\) 0 0
\(989\) 58.5685i 1.86237i
\(990\) 0 0
\(991\) 28.2843i 0.898479i −0.893411 0.449240i \(-0.851695\pi\)
0.893411 0.449240i \(-0.148305\pi\)
\(992\) 0 0
\(993\) 14.8284 2.54416i 0.470566 0.0807363i
\(994\) 0 0
\(995\) 22.8284 0.723710
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 0 0
\(999\) 0.870058 + 1.55635i 0.0275274 + 0.0492407i
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 960.2.h.b.191.3 4
3.2 odd 2 960.2.h.f.191.1 4
4.3 odd 2 960.2.h.f.191.2 4
8.3 odd 2 480.2.h.b.191.3 4
8.5 even 2 480.2.h.d.191.2 yes 4
12.11 even 2 inner 960.2.h.b.191.4 4
24.5 odd 2 480.2.h.b.191.4 yes 4
24.11 even 2 480.2.h.d.191.1 yes 4
40.3 even 4 2400.2.o.i.2399.3 4
40.13 odd 4 2400.2.o.b.2399.2 4
40.19 odd 2 2400.2.h.e.1151.2 4
40.27 even 4 2400.2.o.c.2399.2 4
40.29 even 2 2400.2.h.b.1151.3 4
40.37 odd 4 2400.2.o.j.2399.3 4
120.29 odd 2 2400.2.h.e.1151.1 4
120.53 even 4 2400.2.o.c.2399.1 4
120.59 even 2 2400.2.h.b.1151.4 4
120.77 even 4 2400.2.o.i.2399.4 4
120.83 odd 4 2400.2.o.j.2399.4 4
120.107 odd 4 2400.2.o.b.2399.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.h.b.191.3 4 8.3 odd 2
480.2.h.b.191.4 yes 4 24.5 odd 2
480.2.h.d.191.1 yes 4 24.11 even 2
480.2.h.d.191.2 yes 4 8.5 even 2
960.2.h.b.191.3 4 1.1 even 1 trivial
960.2.h.b.191.4 4 12.11 even 2 inner
960.2.h.f.191.1 4 3.2 odd 2
960.2.h.f.191.2 4 4.3 odd 2
2400.2.h.b.1151.3 4 40.29 even 2
2400.2.h.b.1151.4 4 120.59 even 2
2400.2.h.e.1151.1 4 120.29 odd 2
2400.2.h.e.1151.2 4 40.19 odd 2
2400.2.o.b.2399.1 4 120.107 odd 4
2400.2.o.b.2399.2 4 40.13 odd 4
2400.2.o.c.2399.1 4 120.53 even 4
2400.2.o.c.2399.2 4 40.27 even 4
2400.2.o.i.2399.3 4 40.3 even 4
2400.2.o.i.2399.4 4 120.77 even 4
2400.2.o.j.2399.3 4 40.37 odd 4
2400.2.o.j.2399.4 4 120.83 odd 4