Properties

Label 672.2.i.c.209.3
Level $672$
Weight $2$
Character 672.209
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(209,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("672.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.3
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 672.209
Dual form 672.2.i.c.209.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.41421i) q^{3} -1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(1.00000 + 1.41421i) q^{3} -1.41421i q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 + 2.82843i) q^{9} -2.44949 q^{11} +2.00000 q^{13} +(2.00000 - 1.41421i) q^{15} +7.34847 q^{17} +4.00000 q^{19} +(4.44949 + 1.09638i) q^{21} +1.41421i q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} -4.89898 q^{29} -6.92820i q^{31} +(-2.44949 - 3.46410i) q^{33} +(-2.44949 - 2.82843i) q^{35} +10.3923i q^{37} +(2.00000 + 2.82843i) q^{39} -2.44949 q^{41} -3.46410i q^{43} +(4.00000 + 1.41421i) q^{45} +4.89898 q^{47} +(1.00000 - 6.92820i) q^{49} +(7.34847 + 10.3923i) q^{51} +3.46410i q^{55} +(4.00000 + 5.65685i) q^{57} +5.65685i q^{59} -10.0000 q^{61} +(2.89898 + 7.38891i) q^{63} -2.82843i q^{65} +3.46410i q^{67} +(-2.00000 + 1.41421i) q^{69} +1.41421i q^{71} +(3.00000 + 4.24264i) q^{75} +(-4.89898 + 4.24264i) q^{77} -8.00000 q^{79} +(-7.00000 - 5.65685i) q^{81} -11.3137i q^{83} -10.3923i q^{85} +(-4.89898 - 6.92820i) q^{87} -7.34847 q^{89} +(4.00000 - 3.46410i) q^{91} +(9.79796 - 6.92820i) q^{93} -5.65685i q^{95} +13.8564i q^{97} +(2.44949 - 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{7} - 4 q^{9} + 8 q^{13} + 8 q^{15} + 16 q^{19} + 8 q^{21} + 12 q^{25} - 20 q^{27} + 8 q^{39} + 16 q^{45} + 4 q^{49} + 16 q^{57} - 40 q^{61} - 8 q^{63} - 8 q^{69} + 12 q^{75} - 32 q^{79} - 28 q^{81} + 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) 1.00000 + 1.41421i 0.577350 + 0.816497i
\(4\) 0 0
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 1.41421i 0.516398 0.365148i
\(16\) 0 0
\(17\) 7.34847 1.78227 0.891133 0.453743i \(-0.149911\pi\)
0.891133 + 0.453743i \(0.149911\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.44949 + 1.09638i 0.970958 + 0.239249i
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) −4.89898 −0.909718 −0.454859 0.890564i \(-0.650310\pi\)
−0.454859 + 0.890564i \(0.650310\pi\)
\(30\) 0 0
\(31\) 6.92820i 1.24434i −0.782881 0.622171i \(-0.786251\pi\)
0.782881 0.622171i \(-0.213749\pi\)
\(32\) 0 0
\(33\) −2.44949 3.46410i −0.426401 0.603023i
\(34\) 0 0
\(35\) −2.44949 2.82843i −0.414039 0.478091i
\(36\) 0 0
\(37\) 10.3923i 1.70848i 0.519875 + 0.854242i \(0.325978\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) 2.00000 + 2.82843i 0.320256 + 0.452911i
\(40\) 0 0
\(41\) −2.44949 −0.382546 −0.191273 0.981537i \(-0.561262\pi\)
−0.191273 + 0.981537i \(0.561262\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 4.00000 + 1.41421i 0.596285 + 0.210819i
\(46\) 0 0
\(47\) 4.89898 0.714590 0.357295 0.933992i \(-0.383699\pi\)
0.357295 + 0.933992i \(0.383699\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 7.34847 + 10.3923i 1.02899 + 1.45521i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 3.46410i 0.467099i
\(56\) 0 0
\(57\) 4.00000 + 5.65685i 0.529813 + 0.749269i
\(58\) 0 0
\(59\) 5.65685i 0.736460i 0.929735 + 0.368230i \(0.120036\pi\)
−0.929735 + 0.368230i \(0.879964\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 2.89898 + 7.38891i 0.365237 + 0.930915i
\(64\) 0 0
\(65\) 2.82843i 0.350823i
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) −2.00000 + 1.41421i −0.240772 + 0.170251i
\(70\) 0 0
\(71\) 1.41421i 0.167836i 0.996473 + 0.0839181i \(0.0267434\pi\)
−0.996473 + 0.0839181i \(0.973257\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.00000 + 4.24264i 0.346410 + 0.489898i
\(76\) 0 0
\(77\) −4.89898 + 4.24264i −0.558291 + 0.483494i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) 11.3137i 1.24184i −0.783874 0.620920i \(-0.786759\pi\)
0.783874 0.620920i \(-0.213241\pi\)
\(84\) 0 0
\(85\) 10.3923i 1.12720i
\(86\) 0 0
\(87\) −4.89898 6.92820i −0.525226 0.742781i
\(88\) 0 0
\(89\) −7.34847 −0.778936 −0.389468 0.921040i \(-0.627341\pi\)
−0.389468 + 0.921040i \(0.627341\pi\)
\(90\) 0 0
\(91\) 4.00000 3.46410i 0.419314 0.363137i
\(92\) 0 0
\(93\) 9.79796 6.92820i 1.01600 0.718421i
\(94\) 0 0
\(95\) 5.65685i 0.580381i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 2.44949 6.92820i 0.246183 0.696311i
\(100\) 0 0
\(101\) 9.89949i 0.985037i −0.870302 0.492518i \(-0.836076\pi\)
0.870302 0.492518i \(-0.163924\pi\)
\(102\) 0 0
\(103\) 13.8564i 1.36531i 0.730740 + 0.682656i \(0.239175\pi\)
−0.730740 + 0.682656i \(0.760825\pi\)
\(104\) 0 0
\(105\) 1.55051 6.29253i 0.151314 0.614088i
\(106\) 0 0
\(107\) −7.34847 −0.710403 −0.355202 0.934790i \(-0.615588\pi\)
−0.355202 + 0.934790i \(0.615588\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −14.6969 + 10.3923i −1.39497 + 0.986394i
\(112\) 0 0
\(113\) 14.1421i 1.33038i −0.746674 0.665190i \(-0.768350\pi\)
0.746674 0.665190i \(-0.231650\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) −2.00000 + 5.65685i −0.184900 + 0.522976i
\(118\) 0 0
\(119\) 14.6969 12.7279i 1.34727 1.16677i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) −2.44949 3.46410i −0.220863 0.312348i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 4.89898 3.46410i 0.431331 0.304997i
\(130\) 0 0
\(131\) 11.3137i 0.988483i −0.869325 0.494242i \(-0.835446\pi\)
0.869325 0.494242i \(-0.164554\pi\)
\(132\) 0 0
\(133\) 8.00000 6.92820i 0.693688 0.600751i
\(134\) 0 0
\(135\) 2.00000 + 7.07107i 0.172133 + 0.608581i
\(136\) 0 0
\(137\) 11.3137i 0.966595i 0.875456 + 0.483298i \(0.160561\pi\)
−0.875456 + 0.483298i \(0.839439\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 4.89898 + 6.92820i 0.412568 + 0.583460i
\(142\) 0 0
\(143\) −4.89898 −0.409673
\(144\) 0 0
\(145\) 6.92820i 0.575356i
\(146\) 0 0
\(147\) 10.7980 5.51399i 0.890601 0.454786i
\(148\) 0 0
\(149\) −9.79796 −0.802680 −0.401340 0.915929i \(-0.631455\pi\)
−0.401340 + 0.915929i \(0.631455\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −7.34847 + 20.7846i −0.594089 + 1.68034i
\(154\) 0 0
\(155\) −9.79796 −0.786991
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.44949 + 2.82843i 0.193047 + 0.222911i
\(162\) 0 0
\(163\) 10.3923i 0.813988i −0.913431 0.406994i \(-0.866577\pi\)
0.913431 0.406994i \(-0.133423\pi\)
\(164\) 0 0
\(165\) −4.89898 + 3.46410i −0.381385 + 0.269680i
\(166\) 0 0
\(167\) −19.5959 −1.51638 −0.758189 0.652035i \(-0.773915\pi\)
−0.758189 + 0.652035i \(0.773915\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 + 11.3137i −0.305888 + 0.865181i
\(172\) 0 0
\(173\) 1.41421i 0.107521i −0.998554 0.0537603i \(-0.982879\pi\)
0.998554 0.0537603i \(-0.0171207\pi\)
\(174\) 0 0
\(175\) 6.00000 5.19615i 0.453557 0.392792i
\(176\) 0 0
\(177\) −8.00000 + 5.65685i −0.601317 + 0.425195i
\(178\) 0 0
\(179\) 22.0454 1.64775 0.823876 0.566771i \(-0.191807\pi\)
0.823876 + 0.566771i \(0.191807\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −10.0000 14.1421i −0.739221 1.04542i
\(184\) 0 0
\(185\) 14.6969 1.08054
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) −7.55051 + 11.4887i −0.549219 + 0.835679i
\(190\) 0 0
\(191\) 15.5563i 1.12562i −0.826587 0.562809i \(-0.809721\pi\)
0.826587 0.562809i \(-0.190279\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 4.00000 2.82843i 0.286446 0.202548i
\(196\) 0 0
\(197\) 14.6969 1.04711 0.523557 0.851991i \(-0.324605\pi\)
0.523557 + 0.851991i \(0.324605\pi\)
\(198\) 0 0
\(199\) 10.3923i 0.736691i −0.929689 0.368345i \(-0.879924\pi\)
0.929689 0.368345i \(-0.120076\pi\)
\(200\) 0 0
\(201\) −4.89898 + 3.46410i −0.345547 + 0.244339i
\(202\) 0 0
\(203\) −9.79796 + 8.48528i −0.687682 + 0.595550i
\(204\) 0 0
\(205\) 3.46410i 0.241943i
\(206\) 0 0
\(207\) −4.00000 1.41421i −0.278019 0.0982946i
\(208\) 0 0
\(209\) −9.79796 −0.677739
\(210\) 0 0
\(211\) 24.2487i 1.66935i 0.550743 + 0.834675i \(0.314345\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) −2.00000 + 1.41421i −0.137038 + 0.0969003i
\(214\) 0 0
\(215\) −4.89898 −0.334108
\(216\) 0 0
\(217\) −12.0000 13.8564i −0.814613 0.940634i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6969 0.988623
\(222\) 0 0
\(223\) 17.3205i 1.15987i 0.814664 + 0.579934i \(0.196921\pi\)
−0.814664 + 0.579934i \(0.803079\pi\)
\(224\) 0 0
\(225\) −3.00000 + 8.48528i −0.200000 + 0.565685i
\(226\) 0 0
\(227\) 2.82843i 0.187729i −0.995585 0.0938647i \(-0.970078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −10.8990 2.68556i −0.717100 0.176697i
\(232\) 0 0
\(233\) 14.1421i 0.926482i −0.886232 0.463241i \(-0.846686\pi\)
0.886232 0.463241i \(-0.153314\pi\)
\(234\) 0 0
\(235\) 6.92820i 0.451946i
\(236\) 0 0
\(237\) −8.00000 11.3137i −0.519656 0.734904i
\(238\) 0 0
\(239\) 26.8701i 1.73808i 0.494742 + 0.869040i \(0.335262\pi\)
−0.494742 + 0.869040i \(0.664738\pi\)
\(240\) 0 0
\(241\) 6.92820i 0.446285i −0.974786 0.223142i \(-0.928369\pi\)
0.974786 0.223142i \(-0.0716315\pi\)
\(242\) 0 0
\(243\) 1.00000 15.5563i 0.0641500 0.997940i
\(244\) 0 0
\(245\) −9.79796 1.41421i −0.625969 0.0903508i
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 16.0000 11.3137i 1.01396 0.716977i
\(250\) 0 0
\(251\) 14.1421i 0.892644i 0.894873 + 0.446322i \(0.147266\pi\)
−0.894873 + 0.446322i \(0.852734\pi\)
\(252\) 0 0
\(253\) 3.46410i 0.217786i
\(254\) 0 0
\(255\) 14.6969 10.3923i 0.920358 0.650791i
\(256\) 0 0
\(257\) −2.44949 −0.152795 −0.0763975 0.997077i \(-0.524342\pi\)
−0.0763975 + 0.997077i \(0.524342\pi\)
\(258\) 0 0
\(259\) 18.0000 + 20.7846i 1.11847 + 1.29149i
\(260\) 0 0
\(261\) 4.89898 13.8564i 0.303239 0.857690i
\(262\) 0 0
\(263\) 9.89949i 0.610429i 0.952284 + 0.305215i \(0.0987282\pi\)
−0.952284 + 0.305215i \(0.901272\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −7.34847 10.3923i −0.449719 0.635999i
\(268\) 0 0
\(269\) 24.0416i 1.46584i 0.680313 + 0.732922i \(0.261844\pi\)
−0.680313 + 0.732922i \(0.738156\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 8.89898 + 2.19275i 0.538591 + 0.132711i
\(274\) 0 0
\(275\) −7.34847 −0.443129
\(276\) 0 0
\(277\) 3.46410i 0.208138i 0.994570 + 0.104069i \(0.0331862\pi\)
−0.994570 + 0.104069i \(0.966814\pi\)
\(278\) 0 0
\(279\) 19.5959 + 6.92820i 1.17318 + 0.414781i
\(280\) 0 0
\(281\) 2.82843i 0.168730i 0.996435 + 0.0843649i \(0.0268861\pi\)
−0.996435 + 0.0843649i \(0.973114\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 8.00000 5.65685i 0.473879 0.335083i
\(286\) 0 0
\(287\) −4.89898 + 4.24264i −0.289178 + 0.250435i
\(288\) 0 0
\(289\) 37.0000 2.17647
\(290\) 0 0
\(291\) −19.5959 + 13.8564i −1.14873 + 0.812277i
\(292\) 0 0
\(293\) 1.41421i 0.0826192i −0.999146 0.0413096i \(-0.986847\pi\)
0.999146 0.0413096i \(-0.0131530\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 12.2474 3.46410i 0.710669 0.201008i
\(298\) 0 0
\(299\) 2.82843i 0.163572i
\(300\) 0 0
\(301\) −6.00000 6.92820i −0.345834 0.399335i
\(302\) 0 0
\(303\) 14.0000 9.89949i 0.804279 0.568711i
\(304\) 0 0
\(305\) 14.1421i 0.809776i
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −19.5959 + 13.8564i −1.11477 + 0.788263i
\(310\) 0 0
\(311\) 24.4949 1.38898 0.694489 0.719503i \(-0.255630\pi\)
0.694489 + 0.719503i \(0.255630\pi\)
\(312\) 0 0
\(313\) 6.92820i 0.391605i −0.980643 0.195803i \(-0.937269\pi\)
0.980643 0.195803i \(-0.0627312\pi\)
\(314\) 0 0
\(315\) 10.4495 4.09978i 0.588762 0.230996i
\(316\) 0 0
\(317\) −19.5959 −1.10062 −0.550308 0.834962i \(-0.685490\pi\)
−0.550308 + 0.834962i \(0.685490\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −7.34847 10.3923i −0.410152 0.580042i
\(322\) 0 0
\(323\) 29.3939 1.63552
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.79796 8.48528i 0.540179 0.467809i
\(330\) 0 0
\(331\) 24.2487i 1.33283i −0.745581 0.666415i \(-0.767828\pi\)
0.745581 0.666415i \(-0.232172\pi\)
\(332\) 0 0
\(333\) −29.3939 10.3923i −1.61077 0.569495i
\(334\) 0 0
\(335\) 4.89898 0.267660
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 20.0000 14.1421i 1.08625 0.768095i
\(340\) 0 0
\(341\) 16.9706i 0.919007i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 2.00000 + 2.82843i 0.107676 + 0.152277i
\(346\) 0 0
\(347\) 2.44949 0.131495 0.0657477 0.997836i \(-0.479057\pi\)
0.0657477 + 0.997836i \(0.479057\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −10.0000 + 2.82843i −0.533761 + 0.150970i
\(352\) 0 0
\(353\) −12.2474 −0.651866 −0.325933 0.945393i \(-0.605678\pi\)
−0.325933 + 0.945393i \(0.605678\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) 32.6969 + 8.05669i 1.73051 + 0.426405i
\(358\) 0 0
\(359\) 26.8701i 1.41815i 0.705134 + 0.709074i \(0.250887\pi\)
−0.705134 + 0.709074i \(0.749113\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −5.00000 7.07107i −0.262432 0.371135i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.46410i 0.180825i −0.995904 0.0904123i \(-0.971182\pi\)
0.995904 0.0904123i \(-0.0288185\pi\)
\(368\) 0 0
\(369\) 2.44949 6.92820i 0.127515 0.360668i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.8564i 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) 16.0000 11.3137i 0.826236 0.584237i
\(376\) 0 0
\(377\) −9.79796 −0.504621
\(378\) 0 0
\(379\) 10.3923i 0.533817i 0.963722 + 0.266908i \(0.0860021\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(380\) 0 0
\(381\) 4.00000 + 5.65685i 0.204926 + 0.289809i
\(382\) 0 0
\(383\) −4.89898 −0.250326 −0.125163 0.992136i \(-0.539945\pi\)
−0.125163 + 0.992136i \(0.539945\pi\)
\(384\) 0 0
\(385\) 6.00000 + 6.92820i 0.305788 + 0.353094i
\(386\) 0 0
\(387\) 9.79796 + 3.46410i 0.498058 + 0.176090i
\(388\) 0 0
\(389\) 24.4949 1.24194 0.620970 0.783834i \(-0.286739\pi\)
0.620970 + 0.783834i \(0.286739\pi\)
\(390\) 0 0
\(391\) 10.3923i 0.525561i
\(392\) 0 0
\(393\) 16.0000 11.3137i 0.807093 0.570701i
\(394\) 0 0
\(395\) 11.3137i 0.569254i
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 17.7980 + 4.38551i 0.891012 + 0.219550i
\(400\) 0 0
\(401\) 11.3137i 0.564980i 0.959270 + 0.282490i \(0.0911603\pi\)
−0.959270 + 0.282490i \(0.908840\pi\)
\(402\) 0 0
\(403\) 13.8564i 0.690237i
\(404\) 0 0
\(405\) −8.00000 + 9.89949i −0.397523 + 0.491910i
\(406\) 0 0
\(407\) 25.4558i 1.26180i
\(408\) 0 0
\(409\) 34.6410i 1.71289i −0.516240 0.856444i \(-0.672669\pi\)
0.516240 0.856444i \(-0.327331\pi\)
\(410\) 0 0
\(411\) −16.0000 + 11.3137i −0.789222 + 0.558064i
\(412\) 0 0
\(413\) 9.79796 + 11.3137i 0.482126 + 0.556711i
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) −14.0000 19.7990i −0.685583 0.969561i
\(418\) 0 0
\(419\) 11.3137i 0.552711i −0.961056 0.276355i \(-0.910873\pi\)
0.961056 0.276355i \(-0.0891267\pi\)
\(420\) 0 0
\(421\) 24.2487i 1.18181i 0.806741 + 0.590905i \(0.201229\pi\)
−0.806741 + 0.590905i \(0.798771\pi\)
\(422\) 0 0
\(423\) −4.89898 + 13.8564i −0.238197 + 0.673722i
\(424\) 0 0
\(425\) 22.0454 1.06936
\(426\) 0 0
\(427\) −20.0000 + 17.3205i −0.967868 + 0.838198i
\(428\) 0 0
\(429\) −4.89898 6.92820i −0.236525 0.334497i
\(430\) 0 0
\(431\) 18.3848i 0.885564i 0.896629 + 0.442782i \(0.146008\pi\)
−0.896629 + 0.442782i \(0.853992\pi\)
\(432\) 0 0
\(433\) 20.7846i 0.998845i 0.866359 + 0.499422i \(0.166454\pi\)
−0.866359 + 0.499422i \(0.833546\pi\)
\(434\) 0 0
\(435\) −9.79796 + 6.92820i −0.469776 + 0.332182i
\(436\) 0 0
\(437\) 5.65685i 0.270604i
\(438\) 0 0
\(439\) 24.2487i 1.15733i −0.815566 0.578664i \(-0.803574\pi\)
0.815566 0.578664i \(-0.196426\pi\)
\(440\) 0 0
\(441\) 18.5959 + 9.75663i 0.885520 + 0.464601i
\(442\) 0 0
\(443\) −17.1464 −0.814651 −0.407326 0.913283i \(-0.633539\pi\)
−0.407326 + 0.913283i \(0.633539\pi\)
\(444\) 0 0
\(445\) 10.3923i 0.492642i
\(446\) 0 0
\(447\) −9.79796 13.8564i −0.463428 0.655386i
\(448\) 0 0
\(449\) 36.7696i 1.73526i 0.497208 + 0.867631i \(0.334358\pi\)
−0.497208 + 0.867631i \(0.665642\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 0 0
\(453\) −8.00000 11.3137i −0.375873 0.531564i
\(454\) 0 0
\(455\) −4.89898 5.65685i −0.229668 0.265197i
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −36.7423 + 10.3923i −1.71499 + 0.485071i
\(460\) 0 0
\(461\) 9.89949i 0.461065i −0.973065 0.230533i \(-0.925953\pi\)
0.973065 0.230533i \(-0.0740469\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) −9.79796 13.8564i −0.454369 0.642575i
\(466\) 0 0
\(467\) 31.1127i 1.43972i 0.694117 + 0.719862i \(0.255795\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 6.00000 + 6.92820i 0.277054 + 0.319915i
\(470\) 0 0
\(471\) 14.0000 + 19.7990i 0.645086 + 0.912289i
\(472\) 0 0
\(473\) 8.48528i 0.390154i
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.89898 0.223840 0.111920 0.993717i \(-0.464300\pi\)
0.111920 + 0.993717i \(0.464300\pi\)
\(480\) 0 0
\(481\) 20.7846i 0.947697i
\(482\) 0 0
\(483\) −1.55051 + 6.29253i −0.0705507 + 0.286320i
\(484\) 0 0
\(485\) 19.5959 0.889805
\(486\) 0 0
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 0 0
\(489\) 14.6969 10.3923i 0.664619 0.469956i
\(490\) 0 0
\(491\) 17.1464 0.773807 0.386904 0.922120i \(-0.373545\pi\)
0.386904 + 0.922120i \(0.373545\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −9.79796 3.46410i −0.440386 0.155700i
\(496\) 0 0
\(497\) 2.44949 + 2.82843i 0.109875 + 0.126872i
\(498\) 0 0
\(499\) 3.46410i 0.155074i 0.996989 + 0.0775372i \(0.0247057\pi\)
−0.996989 + 0.0775372i \(0.975294\pi\)
\(500\) 0 0
\(501\) −19.5959 27.7128i −0.875481 1.23812i
\(502\) 0 0
\(503\) −14.6969 −0.655304 −0.327652 0.944798i \(-0.606257\pi\)
−0.327652 + 0.944798i \(0.606257\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) −9.00000 12.7279i −0.399704 0.565267i
\(508\) 0 0
\(509\) 18.3848i 0.814891i −0.913230 0.407445i \(-0.866420\pi\)
0.913230 0.407445i \(-0.133580\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −20.0000 + 5.65685i −0.883022 + 0.249756i
\(514\) 0 0
\(515\) 19.5959 0.863499
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 2.00000 1.41421i 0.0877903 0.0620771i
\(520\) 0 0
\(521\) −36.7423 −1.60971 −0.804856 0.593471i \(-0.797757\pi\)
−0.804856 + 0.593471i \(0.797757\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 13.3485 + 3.28913i 0.582575 + 0.143549i
\(526\) 0 0
\(527\) 50.9117i 2.21775i
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) −16.0000 5.65685i −0.694341 0.245487i
\(532\) 0 0
\(533\) −4.89898 −0.212198
\(534\) 0 0
\(535\) 10.3923i 0.449299i
\(536\) 0 0
\(537\) 22.0454 + 31.1769i 0.951330 + 1.34538i
\(538\) 0 0
\(539\) −2.44949 + 16.9706i −0.105507 + 0.730974i
\(540\) 0 0
\(541\) 31.1769i 1.34040i 0.742180 + 0.670200i \(0.233792\pi\)
−0.742180 + 0.670200i \(0.766208\pi\)
\(542\) 0 0
\(543\) −22.0000 31.1127i −0.944110 1.33517i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24.2487i 1.03680i −0.855138 0.518400i \(-0.826528\pi\)
0.855138 0.518400i \(-0.173472\pi\)
\(548\) 0 0
\(549\) 10.0000 28.2843i 0.426790 1.20714i
\(550\) 0 0
\(551\) −19.5959 −0.834814
\(552\) 0 0
\(553\) −16.0000 + 13.8564i −0.680389 + 0.589234i
\(554\) 0 0
\(555\) 14.6969 + 20.7846i 0.623850 + 0.882258i
\(556\) 0 0
\(557\) 29.3939 1.24546 0.622729 0.782437i \(-0.286024\pi\)
0.622729 + 0.782437i \(0.286024\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) 0 0
\(561\) −18.0000 25.4558i −0.759961 1.07475i
\(562\) 0 0
\(563\) 11.3137i 0.476816i −0.971165 0.238408i \(-0.923374\pi\)
0.971165 0.238408i \(-0.0766255\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 0 0
\(567\) −23.7980 + 0.810647i −0.999420 + 0.0340440i
\(568\) 0 0
\(569\) 14.1421i 0.592869i −0.955053 0.296435i \(-0.904202\pi\)
0.955053 0.296435i \(-0.0957977\pi\)
\(570\) 0 0
\(571\) 17.3205i 0.724841i −0.932015 0.362420i \(-0.881950\pi\)
0.932015 0.362420i \(-0.118050\pi\)
\(572\) 0 0
\(573\) 22.0000 15.5563i 0.919063 0.649876i
\(574\) 0 0
\(575\) 4.24264i 0.176930i
\(576\) 0 0
\(577\) 20.7846i 0.865275i 0.901568 + 0.432637i \(0.142417\pi\)
−0.901568 + 0.432637i \(0.857583\pi\)
\(578\) 0 0
\(579\) −4.00000 5.65685i −0.166234 0.235091i
\(580\) 0 0
\(581\) −19.5959 22.6274i −0.812976 0.938743i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.00000 + 2.82843i 0.330759 + 0.116941i
\(586\) 0 0
\(587\) 2.82843i 0.116742i −0.998295 0.0583708i \(-0.981409\pi\)
0.998295 0.0583708i \(-0.0185906\pi\)
\(588\) 0 0
\(589\) 27.7128i 1.14189i
\(590\) 0 0
\(591\) 14.6969 + 20.7846i 0.604551 + 0.854965i
\(592\) 0 0
\(593\) −7.34847 −0.301765 −0.150883 0.988552i \(-0.548212\pi\)
−0.150883 + 0.988552i \(0.548212\pi\)
\(594\) 0 0
\(595\) −18.0000 20.7846i −0.737928 0.852086i
\(596\) 0 0
\(597\) 14.6969 10.3923i 0.601506 0.425329i
\(598\) 0 0
\(599\) 18.3848i 0.751182i 0.926786 + 0.375591i \(0.122560\pi\)
−0.926786 + 0.375591i \(0.877440\pi\)
\(600\) 0 0
\(601\) 6.92820i 0.282607i −0.989966 0.141304i \(-0.954871\pi\)
0.989966 0.141304i \(-0.0451294\pi\)
\(602\) 0 0
\(603\) −9.79796 3.46410i −0.399004 0.141069i
\(604\) 0 0
\(605\) 7.07107i 0.287480i
\(606\) 0 0
\(607\) 3.46410i 0.140604i 0.997526 + 0.0703018i \(0.0223962\pi\)
−0.997526 + 0.0703018i \(0.977604\pi\)
\(608\) 0 0
\(609\) −21.7980 5.37113i −0.883298 0.217649i
\(610\) 0 0
\(611\) 9.79796 0.396383
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −4.89898 + 3.46410i −0.197546 + 0.139686i
\(616\) 0 0
\(617\) 5.65685i 0.227736i −0.993496 0.113868i \(-0.963676\pi\)
0.993496 0.113868i \(-0.0363242\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −2.00000 7.07107i −0.0802572 0.283752i
\(622\) 0 0
\(623\) −14.6969 + 12.7279i −0.588820 + 0.509933i
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) −9.79796 13.8564i −0.391293 0.553372i
\(628\) 0 0
\(629\) 76.3675i 3.04497i
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) −34.2929 + 24.2487i −1.36302 + 0.963800i
\(634\) 0 0
\(635\) 5.65685i 0.224485i
\(636\) 0 0
\(637\) 2.00000 13.8564i 0.0792429 0.549011i
\(638\) 0 0
\(639\) −4.00000 1.41421i −0.158238 0.0559454i
\(640\) 0 0
\(641\) 22.6274i 0.893729i −0.894602 0.446865i \(-0.852541\pi\)
0.894602 0.446865i \(-0.147459\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −4.89898 6.92820i −0.192897 0.272798i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 13.8564i 0.543912i
\(650\) 0 0
\(651\) 7.59592 30.8270i 0.297707 1.20820i
\(652\) 0 0
\(653\) −9.79796 −0.383424 −0.191712 0.981451i \(-0.561404\pi\)
−0.191712 + 0.981451i \(0.561404\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.8434 −1.24044 −0.620221 0.784427i \(-0.712957\pi\)
−0.620221 + 0.784427i \(0.712957\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 14.6969 + 20.7846i 0.570782 + 0.807207i
\(664\) 0 0
\(665\) −9.79796 11.3137i −0.379949 0.438727i
\(666\) 0 0
\(667\) 6.92820i 0.268261i
\(668\) 0 0
\(669\) −24.4949 + 17.3205i −0.947027 + 0.669650i
\(670\) 0 0
\(671\) 24.4949 0.945615
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) −15.0000 + 4.24264i −0.577350 + 0.163299i
\(676\) 0 0
\(677\) 26.8701i 1.03270i −0.856378 0.516350i \(-0.827290\pi\)
0.856378 0.516350i \(-0.172710\pi\)
\(678\) 0 0
\(679\) 24.0000 + 27.7128i 0.921035 + 1.06352i
\(680\) 0 0
\(681\) 4.00000 2.82843i 0.153280 0.108386i
\(682\) 0 0
\(683\) 51.4393 1.96827 0.984135 0.177423i \(-0.0567759\pi\)
0.984135 + 0.177423i \(0.0567759\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 14.0000 + 19.7990i 0.534133 + 0.755379i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) −7.10102 18.0990i −0.269745 0.687526i
\(694\) 0 0
\(695\) 19.7990i 0.751018i
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 20.0000 14.1421i 0.756469 0.534905i
\(700\) 0 0
\(701\) 29.3939 1.11019 0.555096 0.831786i \(-0.312682\pi\)
0.555096 + 0.831786i \(0.312682\pi\)
\(702\) 0 0
\(703\) 41.5692i 1.56781i
\(704\) 0 0
\(705\) 9.79796 6.92820i 0.369012 0.260931i
\(706\) 0 0
\(707\) −17.1464 19.7990i −0.644858 0.744618i
\(708\) 0 0
\(709\) 13.8564i 0.520388i 0.965556 + 0.260194i \(0.0837866\pi\)
−0.965556 + 0.260194i \(0.916213\pi\)
\(710\) 0 0
\(711\) 8.00000 22.6274i 0.300023 0.848594i
\(712\) 0 0
\(713\) 9.79796 0.366936
\(714\) 0 0
\(715\) 6.92820i 0.259100i
\(716\) 0 0
\(717\) −38.0000 + 26.8701i −1.41914 + 1.00348i
\(718\) 0 0
\(719\) −14.6969 −0.548103 −0.274052 0.961715i \(-0.588364\pi\)
−0.274052 + 0.961715i \(0.588364\pi\)
\(720\) 0 0
\(721\) 24.0000 + 27.7128i 0.893807 + 1.03208i
\(722\) 0 0
\(723\) 9.79796 6.92820i 0.364390 0.257663i
\(724\) 0 0
\(725\) −14.6969 −0.545831
\(726\) 0 0
\(727\) 13.8564i 0.513906i −0.966424 0.256953i \(-0.917281\pi\)
0.966424 0.256953i \(-0.0827185\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 25.4558i 0.941518i
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) −7.79796 15.2706i −0.287632 0.563265i
\(736\) 0 0
\(737\) 8.48528i 0.312559i
\(738\) 0 0
\(739\) 31.1769i 1.14686i 0.819254 + 0.573431i \(0.194388\pi\)
−0.819254 + 0.573431i \(0.805612\pi\)
\(740\) 0 0
\(741\) 8.00000 + 11.3137i 0.293887 + 0.415619i
\(742\) 0 0
\(743\) 7.07107i 0.259412i −0.991552 0.129706i \(-0.958597\pi\)
0.991552 0.129706i \(-0.0414034\pi\)
\(744\) 0 0
\(745\) 13.8564i 0.507659i
\(746\) 0 0
\(747\) 32.0000 + 11.3137i 1.17082 + 0.413947i
\(748\) 0 0
\(749\) −14.6969 + 12.7279i −0.537014 + 0.465068i
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −20.0000 + 14.1421i −0.728841 + 0.515368i
\(754\) 0 0
\(755\) 11.3137i 0.411748i
\(756\) 0 0
\(757\) 41.5692i 1.51086i −0.655230 0.755429i \(-0.727428\pi\)
0.655230 0.755429i \(-0.272572\pi\)
\(758\) 0 0
\(759\) 4.89898 3.46410i 0.177822 0.125739i
\(760\) 0 0
\(761\) −17.1464 −0.621558 −0.310779 0.950482i \(-0.600590\pi\)
−0.310779 + 0.950482i \(0.600590\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 29.3939 + 10.3923i 1.06274 + 0.375735i
\(766\) 0 0
\(767\) 11.3137i 0.408514i
\(768\) 0 0
\(769\) 34.6410i 1.24919i −0.780950 0.624593i \(-0.785265\pi\)
0.780950 0.624593i \(-0.214735\pi\)
\(770\) 0 0
\(771\) −2.44949 3.46410i −0.0882162 0.124757i
\(772\) 0 0
\(773\) 26.8701i 0.966449i −0.875497 0.483224i \(-0.839466\pi\)
0.875497 0.483224i \(-0.160534\pi\)
\(774\) 0 0
\(775\) 20.7846i 0.746605i
\(776\) 0 0
\(777\) −11.3939 + 46.2405i −0.408753 + 1.65887i
\(778\) 0 0
\(779\) −9.79796 −0.351048
\(780\) 0 0
\(781\) 3.46410i 0.123955i
\(782\) 0 0
\(783\) 24.4949 6.92820i 0.875376 0.247594i
\(784\) 0 0
\(785\) 19.7990i 0.706656i
\(786\) 0 0
\(787\) −2.00000 −0.0712923 −0.0356462 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) 0 0
\(789\) −14.0000 + 9.89949i −0.498413 + 0.352431i
\(790\) 0 0
\(791\) −24.4949 28.2843i −0.870938 1.00567i
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 52.3259i 1.85348i −0.375705 0.926739i \(-0.622599\pi\)
0.375705 0.926739i \(-0.377401\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 7.34847 20.7846i 0.259645 0.734388i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 4.00000 3.46410i 0.140981 0.122094i
\(806\) 0 0
\(807\) −34.0000 + 24.0416i −1.19686 + 0.846305i
\(808\) 0 0
\(809\) 19.7990i 0.696095i 0.937477 + 0.348048i \(0.113155\pi\)
−0.937477 + 0.348048i \(0.886845\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −14.6969 −0.514811
\(816\) 0 0
\(817\) 13.8564i 0.484774i
\(818\) 0 0
\(819\) 5.79796 + 14.7778i 0.202597 + 0.516378i
\(820\) 0 0
\(821\) −48.9898 −1.70976 −0.854878 0.518829i \(-0.826368\pi\)
−0.854878 + 0.518829i \(0.826368\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) −7.34847 10.3923i −0.255841 0.361814i
\(826\) 0 0
\(827\) 7.34847 0.255531 0.127766 0.991804i \(-0.459219\pi\)
0.127766 + 0.991804i \(0.459219\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −4.89898 + 3.46410i −0.169944 + 0.120168i
\(832\) 0 0
\(833\) 7.34847 50.9117i 0.254609 1.76399i
\(834\) 0 0
\(835\) 27.7128i 0.959041i
\(836\) 0 0
\(837\) 9.79796 + 34.6410i 0.338667 + 1.19737i
\(838\) 0 0
\(839\) 48.9898 1.69132 0.845658 0.533726i \(-0.179208\pi\)
0.845658 + 0.533726i \(0.179208\pi\)
\(840\) 0 0
\(841\) −5.00000 −0.172414
\(842\) 0 0
\(843\) −4.00000 + 2.82843i −0.137767 + 0.0974162i
\(844\) 0 0
\(845\) 12.7279i 0.437854i
\(846\) 0 0
\(847\) −10.0000 + 8.66025i −0.343604 + 0.297570i
\(848\) 0 0
\(849\) 4.00000 + 5.65685i 0.137280 + 0.194143i
\(850\) 0 0
\(851\) −14.6969 −0.503805
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 16.0000 + 5.65685i 0.547188 + 0.193460i
\(856\) 0 0
\(857\) 31.8434 1.08775 0.543874 0.839167i \(-0.316957\pi\)
0.543874 + 0.839167i \(0.316957\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −10.8990 2.68556i −0.371436 0.0915237i
\(862\) 0 0
\(863\) 1.41421i 0.0481404i 0.999710 + 0.0240702i \(0.00766252\pi\)
−0.999710 + 0.0240702i \(0.992337\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 37.0000 + 52.3259i 1.25659 + 1.77708i
\(868\) 0 0
\(869\) 19.5959 0.664746
\(870\) 0 0
\(871\) 6.92820i 0.234753i
\(872\) 0 0
\(873\) −39.1918 13.8564i −1.32644 0.468968i
\(874\) 0 0
\(875\) −19.5959 22.6274i −0.662463 0.764946i
\(876\) 0 0
\(877\) 13.8564i 0.467898i −0.972249 0.233949i \(-0.924835\pi\)
0.972249 0.233949i \(-0.0751648\pi\)
\(878\) 0 0
\(879\) 2.00000 1.41421i 0.0674583 0.0477002i
\(880\) 0 0
\(881\) 22.0454 0.742729 0.371364 0.928487i \(-0.378890\pi\)
0.371364 + 0.928487i \(0.378890\pi\)
\(882\) 0 0
\(883\) 10.3923i 0.349729i −0.984593 0.174864i \(-0.944051\pi\)
0.984593 0.174864i \(-0.0559487\pi\)
\(884\) 0 0
\(885\) 8.00000 + 11.3137i 0.268917 + 0.380306i
\(886\) 0 0
\(887\) 39.1918 1.31593 0.657967 0.753047i \(-0.271417\pi\)
0.657967 + 0.753047i \(0.271417\pi\)
\(888\) 0 0
\(889\) 8.00000 6.92820i 0.268311 0.232364i
\(890\) 0 0
\(891\) 17.1464 + 13.8564i 0.574427 + 0.464207i
\(892\) 0 0
\(893\) 19.5959 0.655752
\(894\) 0 0
\(895\) 31.1769i 1.04213i
\(896\) 0 0
\(897\) −4.00000 + 2.82843i −0.133556 + 0.0944384i
\(898\) 0 0
\(899\) 33.9411i 1.13200i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 3.79796 15.4135i 0.126388 0.512929i
\(904\) 0 0
\(905\) 31.1127i 1.03422i
\(906\) 0 0
\(907\) 38.1051i 1.26526i 0.774454 + 0.632630i \(0.218025\pi\)
−0.774454 + 0.632630i \(0.781975\pi\)
\(908\) 0 0
\(909\) 28.0000 + 9.89949i 0.928701 + 0.328346i
\(910\) 0 0
\(911\) 15.5563i 0.515405i −0.966224 0.257702i \(-0.917035\pi\)
0.966224 0.257702i \(-0.0829654\pi\)
\(912\) 0 0
\(913\) 27.7128i 0.917160i
\(914\) 0 0
\(915\) −20.0000 + 14.1421i −0.661180 + 0.467525i
\(916\) 0 0
\(917\) −19.5959 22.6274i −0.647114 0.747223i
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 4.00000 + 5.65685i 0.131804 + 0.186400i
\(922\) 0 0
\(923\) 2.82843i 0.0930988i
\(924\) 0 0
\(925\) 31.1769i 1.02509i
\(926\) 0 0
\(927\) −39.1918 13.8564i −1.28723 0.455104i
\(928\) 0 0
\(929\) 31.8434 1.04475 0.522373 0.852717i \(-0.325047\pi\)
0.522373 + 0.852717i \(0.325047\pi\)
\(930\) 0 0
\(931\) 4.00000 27.7128i 0.131095 0.908251i
\(932\) 0 0
\(933\) 24.4949 + 34.6410i 0.801927 + 1.13410i
\(934\) 0 0
\(935\) 25.4558i 0.832495i
\(936\) 0 0
\(937\) 41.5692i 1.35801i −0.734135 0.679004i \(-0.762412\pi\)
0.734135 0.679004i \(-0.237588\pi\)
\(938\) 0 0
\(939\) 9.79796 6.92820i 0.319744 0.226093i
\(940\) 0 0
\(941\) 1.41421i 0.0461020i −0.999734 0.0230510i \(-0.992662\pi\)
0.999734 0.0230510i \(-0.00733802\pi\)
\(942\) 0 0
\(943\) 3.46410i 0.112807i
\(944\) 0 0
\(945\) 16.2474 + 10.6780i 0.528530 + 0.347356i
\(946\) 0 0
\(947\) −46.5403 −1.51236 −0.756178 0.654366i \(-0.772936\pi\)
−0.756178 + 0.654366i \(0.772936\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −19.5959 27.7128i −0.635441 0.898650i
\(952\) 0 0
\(953\) 45.2548i 1.46595i 0.680257 + 0.732974i \(0.261868\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) −22.0000 −0.711903
\(956\) 0 0
\(957\) 12.0000 + 16.9706i 0.387905 + 0.548580i
\(958\) 0 0
\(959\) 19.5959 + 22.6274i 0.632785 + 0.730677i
\(960\) 0 0
\(961\) −17.0000 −0.548387
\(962\) 0 0
\(963\) 7.34847 20.7846i 0.236801 0.669775i
\(964\) 0 0
\(965\) 5.65685i 0.182101i
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) 29.3939 + 41.5692i 0.944267 + 1.33540i
\(970\) 0 0
\(971\) 36.7696i 1.17999i −0.807406 0.589996i \(-0.799129\pi\)
0.807406 0.589996i \(-0.200871\pi\)
\(972\) 0 0
\(973\) −28.0000 + 24.2487i −0.897639 + 0.777378i
\(974\) 0 0
\(975\) 6.00000 + 8.48528i 0.192154 + 0.271746i
\(976\) 0 0
\(977\) 19.7990i 0.633426i 0.948521 + 0.316713i \(0.102579\pi\)
−0.948521 + 0.316713i \(0.897421\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.1918 −1.25003 −0.625013 0.780615i \(-0.714906\pi\)
−0.625013 + 0.780615i \(0.714906\pi\)
\(984\) 0 0
\(985\) 20.7846i 0.662253i
\(986\) 0 0
\(987\) 21.7980 + 5.37113i 0.693837 + 0.170965i
\(988\) 0 0
\(989\) 4.89898 0.155778
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 34.2929 24.2487i 1.08825 0.769510i
\(994\) 0 0
\(995\) −14.6969 −0.465924
\(996\) 0 0
\(997\) 62.0000 1.96356 0.981780 0.190022i \(-0.0608559\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) 0 0
\(999\) −14.6969 51.9615i −0.464991 1.64399i
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 672.2.i.c.209.3 4
3.2 odd 2 inner 672.2.i.c.209.1 4
4.3 odd 2 168.2.i.a.125.3 yes 4
7.6 odd 2 672.2.i.a.209.2 4
8.3 odd 2 168.2.i.c.125.1 yes 4
8.5 even 2 672.2.i.a.209.1 4
12.11 even 2 168.2.i.a.125.2 yes 4
21.20 even 2 672.2.i.a.209.4 4
24.5 odd 2 672.2.i.a.209.3 4
24.11 even 2 168.2.i.c.125.4 yes 4
28.27 even 2 168.2.i.c.125.3 yes 4
56.13 odd 2 inner 672.2.i.c.209.4 4
56.27 even 2 168.2.i.a.125.1 4
84.83 odd 2 168.2.i.c.125.2 yes 4
168.83 odd 2 168.2.i.a.125.4 yes 4
168.125 even 2 inner 672.2.i.c.209.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.i.a.125.1 4 56.27 even 2
168.2.i.a.125.2 yes 4 12.11 even 2
168.2.i.a.125.3 yes 4 4.3 odd 2
168.2.i.a.125.4 yes 4 168.83 odd 2
168.2.i.c.125.1 yes 4 8.3 odd 2
168.2.i.c.125.2 yes 4 84.83 odd 2
168.2.i.c.125.3 yes 4 28.27 even 2
168.2.i.c.125.4 yes 4 24.11 even 2
672.2.i.a.209.1 4 8.5 even 2
672.2.i.a.209.2 4 7.6 odd 2
672.2.i.a.209.3 4 24.5 odd 2
672.2.i.a.209.4 4 21.20 even 2
672.2.i.c.209.1 4 3.2 odd 2 inner
672.2.i.c.209.2 4 168.125 even 2 inner
672.2.i.c.209.3 4 1.1 even 1 trivial
672.2.i.c.209.4 4 56.13 odd 2 inner