Properties

Label 784.2.p.g.607.2
Level $784$
Weight $2$
Character 784.607
Analytic conductor $6.260$
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] = [784,2,Mod(31,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.p (of order \(6\), degree \(2\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.2
Root \(1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 784.607
Dual form 784.2.p.g.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.32288 + 2.29129i) q^{3} +(1.50000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{9} +O(q^{10})\) \(q+(1.32288 + 2.29129i) q^{3} +(1.50000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{9} +(3.96863 - 2.29129i) q^{11} +3.46410i q^{13} +4.58258i q^{15} +(-4.50000 + 2.59808i) q^{17} +(1.32288 - 2.29129i) q^{19} +(3.96863 + 2.29129i) q^{23} +(-1.00000 - 1.73205i) q^{25} -2.64575 q^{27} +(-1.32288 - 2.29129i) q^{31} +(10.5000 + 6.06218i) q^{33} +(-3.50000 + 6.06218i) q^{37} +(-7.93725 + 4.58258i) q^{39} -3.46410i q^{41} -9.16515i q^{43} +(-6.00000 + 3.46410i) q^{45} +(-3.96863 + 6.87386i) q^{47} +(-11.9059 - 6.87386i) q^{51} +(-1.50000 - 2.59808i) q^{53} +7.93725 q^{55} +7.00000 q^{57} +(-3.96863 - 6.87386i) q^{59} +(-1.50000 - 0.866025i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(-3.96863 + 2.29129i) q^{67} +12.1244i q^{69} -9.16515i q^{71} +(4.50000 - 2.59808i) q^{73} +(2.64575 - 4.58258i) q^{75} +(3.96863 + 2.29129i) q^{79} +(2.50000 + 4.33013i) q^{81} -9.00000 q^{85} +(-1.50000 - 0.866025i) q^{89} +(3.50000 - 6.06218i) q^{93} +(3.96863 - 2.29129i) q^{95} -3.46410i q^{97} +18.3303i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{5} - 8 q^{9} - 18 q^{17} - 4 q^{25} + 42 q^{33} - 14 q^{37} - 24 q^{45} - 6 q^{53} + 28 q^{57} - 6 q^{61} - 12 q^{65} + 18 q^{73} + 10 q^{81} - 36 q^{85} - 6 q^{89} + 14 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 1.32288 + 2.29129i 0.763763 + 1.32288i 0.940898 + 0.338689i \(0.109984\pi\)
−0.177136 + 0.984186i \(0.556683\pi\)
\(4\) 0 0
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.00000 + 3.46410i −0.666667 + 1.15470i
\(10\) 0 0
\(11\) 3.96863 2.29129i 1.19659 0.690849i 0.236794 0.971560i \(-0.423903\pi\)
0.959792 + 0.280711i \(0.0905701\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 4.58258i 1.18322i
\(16\) 0 0
\(17\) −4.50000 + 2.59808i −1.09141 + 0.630126i −0.933952 0.357400i \(-0.883663\pi\)
−0.157459 + 0.987526i \(0.550330\pi\)
\(18\) 0 0
\(19\) 1.32288 2.29129i 0.303488 0.525657i −0.673435 0.739246i \(-0.735182\pi\)
0.976924 + 0.213589i \(0.0685153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.96863 + 2.29129i 0.827516 + 0.477767i 0.853001 0.521909i \(-0.174780\pi\)
−0.0254855 + 0.999675i \(0.508113\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) −2.64575 −0.509175
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.32288 2.29129i −0.237595 0.411527i 0.722428 0.691446i \(-0.243026\pi\)
−0.960024 + 0.279918i \(0.909693\pi\)
\(32\) 0 0
\(33\) 10.5000 + 6.06218i 1.82782 + 1.05529i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) −7.93725 + 4.58258i −1.27098 + 0.733799i
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 9.16515i 1.39767i −0.715282 0.698836i \(-0.753702\pi\)
0.715282 0.698836i \(-0.246298\pi\)
\(44\) 0 0
\(45\) −6.00000 + 3.46410i −0.894427 + 0.516398i
\(46\) 0 0
\(47\) −3.96863 + 6.87386i −0.578884 + 1.00266i 0.416724 + 0.909033i \(0.363178\pi\)
−0.995608 + 0.0936230i \(0.970155\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −11.9059 6.87386i −1.66716 0.962533i
\(52\) 0 0
\(53\) −1.50000 2.59808i −0.206041 0.356873i 0.744423 0.667708i \(-0.232725\pi\)
−0.950464 + 0.310835i \(0.899391\pi\)
\(54\) 0 0
\(55\) 7.93725 1.07026
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) −3.96863 6.87386i −0.516671 0.894901i −0.999813 0.0193585i \(-0.993838\pi\)
0.483141 0.875542i \(-0.339496\pi\)
\(60\) 0 0
\(61\) −1.50000 0.866025i −0.192055 0.110883i 0.400889 0.916127i \(-0.368701\pi\)
−0.592944 + 0.805243i \(0.702035\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) −3.96863 + 2.29129i −0.484845 + 0.279925i −0.722433 0.691441i \(-0.756976\pi\)
0.237588 + 0.971366i \(0.423643\pi\)
\(68\) 0 0
\(69\) 12.1244i 1.45960i
\(70\) 0 0
\(71\) 9.16515i 1.08770i −0.839181 0.543852i \(-0.816965\pi\)
0.839181 0.543852i \(-0.183035\pi\)
\(72\) 0 0
\(73\) 4.50000 2.59808i 0.526685 0.304082i −0.212980 0.977056i \(-0.568317\pi\)
0.739666 + 0.672975i \(0.234984\pi\)
\(74\) 0 0
\(75\) 2.64575 4.58258i 0.305505 0.529150i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.96863 + 2.29129i 0.446505 + 0.257790i 0.706353 0.707860i \(-0.250339\pi\)
−0.259848 + 0.965650i \(0.583672\pi\)
\(80\) 0 0
\(81\) 2.50000 + 4.33013i 0.277778 + 0.481125i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 0.866025i −0.159000 0.0917985i 0.418389 0.908268i \(-0.362595\pi\)
−0.577389 + 0.816469i \(0.695928\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.50000 6.06218i 0.362933 0.628619i
\(94\) 0 0
\(95\) 3.96863 2.29129i 0.407173 0.235081i
\(96\) 0 0
\(97\) 3.46410i 0.351726i −0.984415 0.175863i \(-0.943728\pi\)
0.984415 0.175863i \(-0.0562716\pi\)
\(98\) 0 0
\(99\) 18.3303i 1.84226i
\(100\) 0 0
\(101\) −4.50000 + 2.59808i −0.447767 + 0.258518i −0.706887 0.707327i \(-0.749901\pi\)
0.259120 + 0.965845i \(0.416568\pi\)
\(102\) 0 0
\(103\) −1.32288 + 2.29129i −0.130347 + 0.225767i −0.923810 0.382851i \(-0.874942\pi\)
0.793463 + 0.608618i \(0.208276\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.9059 + 6.87386i 1.15098 + 0.664521i 0.949127 0.314893i \(-0.101969\pi\)
0.201858 + 0.979415i \(0.435302\pi\)
\(108\) 0 0
\(109\) 3.50000 + 6.06218i 0.335239 + 0.580651i 0.983531 0.180741i \(-0.0578495\pi\)
−0.648292 + 0.761392i \(0.724516\pi\)
\(110\) 0 0
\(111\) −18.5203 −1.75787
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 3.96863 + 6.87386i 0.370076 + 0.640991i
\(116\) 0 0
\(117\) −12.0000 6.92820i −1.10940 0.640513i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 8.66025i 0.454545 0.787296i
\(122\) 0 0
\(123\) 7.93725 4.58258i 0.715678 0.413197i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 9.16515i 0.813276i 0.913589 + 0.406638i \(0.133299\pi\)
−0.913589 + 0.406638i \(0.866701\pi\)
\(128\) 0 0
\(129\) 21.0000 12.1244i 1.84895 1.06749i
\(130\) 0 0
\(131\) 3.96863 6.87386i 0.346741 0.600572i −0.638928 0.769267i \(-0.720622\pi\)
0.985668 + 0.168694i \(0.0539551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.96863 2.29129i −0.341565 0.197203i
\(136\) 0 0
\(137\) −10.5000 18.1865i −0.897076 1.55378i −0.831215 0.555952i \(-0.812354\pi\)
−0.0658609 0.997829i \(-0.520979\pi\)
\(138\) 0 0
\(139\) 10.5830 0.897639 0.448819 0.893622i \(-0.351845\pi\)
0.448819 + 0.893622i \(0.351845\pi\)
\(140\) 0 0
\(141\) −21.0000 −1.76852
\(142\) 0 0
\(143\) 7.93725 + 13.7477i 0.663747 + 1.14964i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) 19.8431 11.4564i 1.61481 0.932312i 0.626578 0.779359i \(-0.284455\pi\)
0.988233 0.152953i \(-0.0488783\pi\)
\(152\) 0 0
\(153\) 20.7846i 1.68034i
\(154\) 0 0
\(155\) 4.58258i 0.368081i
\(156\) 0 0
\(157\) 4.50000 2.59808i 0.359139 0.207349i −0.309564 0.950879i \(-0.600183\pi\)
0.668703 + 0.743530i \(0.266850\pi\)
\(158\) 0 0
\(159\) 3.96863 6.87386i 0.314733 0.545133i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.96863 + 2.29129i 0.310847 + 0.179468i 0.647305 0.762231i \(-0.275896\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(164\) 0 0
\(165\) 10.5000 + 18.1865i 0.817424 + 1.41582i
\(166\) 0 0
\(167\) 15.8745 1.22841 0.614203 0.789148i \(-0.289478\pi\)
0.614203 + 0.789148i \(0.289478\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.29150 + 9.16515i 0.404651 + 0.700877i
\(172\) 0 0
\(173\) −19.5000 11.2583i −1.48256 0.855955i −0.482754 0.875756i \(-0.660363\pi\)
−0.999804 + 0.0198012i \(0.993697\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.5000 18.1865i 0.789228 1.36698i
\(178\) 0 0
\(179\) −11.9059 + 6.87386i −0.889887 + 0.513777i −0.873906 0.486096i \(-0.838421\pi\)
−0.0159817 + 0.999872i \(0.505087\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) 4.58258i 0.338754i
\(184\) 0 0
\(185\) −10.5000 + 6.06218i −0.771975 + 0.445700i
\(186\) 0 0
\(187\) −11.9059 + 20.6216i −0.870644 + 1.50800i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8431 11.4564i −1.43580 0.828959i −0.438245 0.898856i \(-0.644400\pi\)
−0.997554 + 0.0698969i \(0.977733\pi\)
\(192\) 0 0
\(193\) 5.50000 + 9.52628i 0.395899 + 0.685717i 0.993215 0.116289i \(-0.0370998\pi\)
−0.597317 + 0.802005i \(0.703766\pi\)
\(194\) 0 0
\(195\) −15.8745 −1.13680
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −6.61438 11.4564i −0.468881 0.812125i 0.530486 0.847693i \(-0.322009\pi\)
−0.999367 + 0.0355680i \(0.988676\pi\)
\(200\) 0 0
\(201\) −10.5000 6.06218i −0.740613 0.427593i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 5.19615i 0.209529 0.362915i
\(206\) 0 0
\(207\) −15.8745 + 9.16515i −1.10335 + 0.637022i
\(208\) 0 0
\(209\) 12.1244i 0.838659i
\(210\) 0 0
\(211\) 9.16515i 0.630955i −0.948933 0.315478i \(-0.897835\pi\)
0.948933 0.315478i \(-0.102165\pi\)
\(212\) 0 0
\(213\) 21.0000 12.1244i 1.43890 0.830747i
\(214\) 0 0
\(215\) 7.93725 13.7477i 0.541316 0.937587i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 11.9059 + 6.87386i 0.804525 + 0.464493i
\(220\) 0 0
\(221\) −9.00000 15.5885i −0.605406 1.04859i
\(222\) 0 0
\(223\) −10.5830 −0.708690 −0.354345 0.935115i \(-0.615296\pi\)
−0.354345 + 0.935115i \(0.615296\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) −11.9059 20.6216i −0.790221 1.36870i −0.925830 0.377941i \(-0.876632\pi\)
0.135609 0.990762i \(-0.456701\pi\)
\(228\) 0 0
\(229\) 1.50000 + 0.866025i 0.0991228 + 0.0572286i 0.548742 0.835992i \(-0.315107\pi\)
−0.449619 + 0.893220i \(0.648440\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.5000 + 18.1865i −0.687878 + 1.19144i 0.284645 + 0.958633i \(0.408124\pi\)
−0.972523 + 0.232806i \(0.925209\pi\)
\(234\) 0 0
\(235\) −11.9059 + 6.87386i −0.776654 + 0.448401i
\(236\) 0 0
\(237\) 12.1244i 0.787562i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 16.5000 9.52628i 1.06286 0.613642i 0.136637 0.990621i \(-0.456371\pi\)
0.926222 + 0.376980i \(0.123037\pi\)
\(242\) 0 0
\(243\) −10.5830 + 18.3303i −0.678900 + 1.17589i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.93725 + 4.58258i 0.505035 + 0.291582i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.8745 1.00199 0.500995 0.865450i \(-0.332967\pi\)
0.500995 + 0.865450i \(0.332967\pi\)
\(252\) 0 0
\(253\) 21.0000 1.32026
\(254\) 0 0
\(255\) −11.9059 20.6216i −0.745575 1.29137i
\(256\) 0 0
\(257\) 1.50000 + 0.866025i 0.0935674 + 0.0540212i 0.546054 0.837750i \(-0.316129\pi\)
−0.452486 + 0.891771i \(0.649463\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.96863 + 2.29129i −0.244716 + 0.141287i −0.617342 0.786695i \(-0.711791\pi\)
0.372626 + 0.927981i \(0.378457\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 4.58258i 0.280449i
\(268\) 0 0
\(269\) −25.5000 + 14.7224i −1.55476 + 0.897643i −0.557019 + 0.830500i \(0.688055\pi\)
−0.997743 + 0.0671428i \(0.978612\pi\)
\(270\) 0 0
\(271\) 14.5516 25.2042i 0.883949 1.53104i 0.0370348 0.999314i \(-0.488209\pi\)
0.846914 0.531730i \(-0.178458\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.93725 4.58258i −0.478634 0.276340i
\(276\) 0 0
\(277\) 8.50000 + 14.7224i 0.510716 + 0.884585i 0.999923 + 0.0124177i \(0.00395278\pi\)
−0.489207 + 0.872167i \(0.662714\pi\)
\(278\) 0 0
\(279\) 10.5830 0.633588
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −6.61438 11.4564i −0.393184 0.681015i 0.599684 0.800237i \(-0.295293\pi\)
−0.992868 + 0.119223i \(0.961960\pi\)
\(284\) 0 0
\(285\) 10.5000 + 6.06218i 0.621966 + 0.359092i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.00000 8.66025i 0.294118 0.509427i
\(290\) 0 0
\(291\) 7.93725 4.58258i 0.465290 0.268635i
\(292\) 0 0
\(293\) 20.7846i 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) 13.7477i 0.800424i
\(296\) 0 0
\(297\) −10.5000 + 6.06218i −0.609272 + 0.351763i
\(298\) 0 0
\(299\) −7.93725 + 13.7477i −0.459023 + 0.795052i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −11.9059 6.87386i −0.683975 0.394893i
\(304\) 0 0
\(305\) −1.50000 2.59808i −0.0858898 0.148765i
\(306\) 0 0
\(307\) −10.5830 −0.604004 −0.302002 0.953307i \(-0.597655\pi\)
−0.302002 + 0.953307i \(0.597655\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 11.9059 + 20.6216i 0.675121 + 1.16934i 0.976434 + 0.215818i \(0.0692417\pi\)
−0.301313 + 0.953525i \(0.597425\pi\)
\(312\) 0 0
\(313\) 22.5000 + 12.9904i 1.27178 + 0.734260i 0.975322 0.220788i \(-0.0708628\pi\)
0.296453 + 0.955047i \(0.404196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.50000 + 12.9904i −0.421242 + 0.729612i −0.996061 0.0886679i \(-0.971739\pi\)
0.574819 + 0.818280i \(0.305072\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 36.3731i 2.03015i
\(322\) 0 0
\(323\) 13.7477i 0.764944i
\(324\) 0 0
\(325\) 6.00000 3.46410i 0.332820 0.192154i
\(326\) 0 0
\(327\) −9.26013 + 16.0390i −0.512086 + 0.886960i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −11.9059 6.87386i −0.654406 0.377822i 0.135736 0.990745i \(-0.456660\pi\)
−0.790142 + 0.612923i \(0.789993\pi\)
\(332\) 0 0
\(333\) −14.0000 24.2487i −0.767195 1.32882i
\(334\) 0 0
\(335\) −7.93725 −0.433659
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 0 0
\(339\) 15.8745 + 27.4955i 0.862185 + 1.49335i
\(340\) 0 0
\(341\) −10.5000 6.06218i −0.568607 0.328285i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −10.5000 + 18.1865i −0.565301 + 0.979130i
\(346\) 0 0
\(347\) −3.96863 + 2.29129i −0.213047 + 0.123003i −0.602727 0.797948i \(-0.705919\pi\)
0.389680 + 0.920950i \(0.372586\pi\)
\(348\) 0 0
\(349\) 3.46410i 0.185429i −0.995693 0.0927146i \(-0.970446\pi\)
0.995693 0.0927146i \(-0.0295544\pi\)
\(350\) 0 0
\(351\) 9.16515i 0.489200i
\(352\) 0 0
\(353\) −16.5000 + 9.52628i −0.878206 + 0.507033i −0.870067 0.492934i \(-0.835924\pi\)
−0.00813978 + 0.999967i \(0.502591\pi\)
\(354\) 0 0
\(355\) 7.93725 13.7477i 0.421266 0.729654i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.9059 + 6.87386i 0.628368 + 0.362789i 0.780120 0.625630i \(-0.215158\pi\)
−0.151752 + 0.988419i \(0.548491\pi\)
\(360\) 0 0
\(361\) 6.00000 + 10.3923i 0.315789 + 0.546963i
\(362\) 0 0
\(363\) 26.4575 1.38866
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) 17.1974 + 29.7867i 0.897696 + 1.55486i 0.830432 + 0.557120i \(0.188094\pi\)
0.0672642 + 0.997735i \(0.478573\pi\)
\(368\) 0 0
\(369\) 12.0000 + 6.92820i 0.624695 + 0.360668i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.500000 0.866025i 0.0258890 0.0448411i −0.852791 0.522253i \(-0.825092\pi\)
0.878680 + 0.477412i \(0.158425\pi\)
\(374\) 0 0
\(375\) 27.7804 16.0390i 1.43457 0.828251i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 18.3303i 0.941564i 0.882249 + 0.470782i \(0.156028\pi\)
−0.882249 + 0.470782i \(0.843972\pi\)
\(380\) 0 0
\(381\) −21.0000 + 12.1244i −1.07586 + 0.621150i
\(382\) 0 0
\(383\) −3.96863 + 6.87386i −0.202787 + 0.351238i −0.949425 0.313992i \(-0.898333\pi\)
0.746638 + 0.665230i \(0.231667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 31.7490 + 18.3303i 1.61389 + 0.931782i
\(388\) 0 0
\(389\) −10.5000 18.1865i −0.532371 0.922094i −0.999286 0.0377914i \(-0.987968\pi\)
0.466915 0.884302i \(-0.345366\pi\)
\(390\) 0 0
\(391\) −23.8118 −1.20421
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) 0 0
\(395\) 3.96863 + 6.87386i 0.199683 + 0.345862i
\(396\) 0 0
\(397\) 19.5000 + 11.2583i 0.978677 + 0.565039i 0.901870 0.432007i \(-0.142194\pi\)
0.0768065 + 0.997046i \(0.475528\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.5000 + 18.1865i −0.524345 + 0.908192i 0.475253 + 0.879849i \(0.342356\pi\)
−0.999598 + 0.0283431i \(0.990977\pi\)
\(402\) 0 0
\(403\) 7.93725 4.58258i 0.395383 0.228274i
\(404\) 0 0
\(405\) 8.66025i 0.430331i
\(406\) 0 0
\(407\) 32.0780i 1.59005i
\(408\) 0 0
\(409\) −16.5000 + 9.52628i −0.815872 + 0.471044i −0.848991 0.528407i \(-0.822789\pi\)
0.0331186 + 0.999451i \(0.489456\pi\)
\(410\) 0 0
\(411\) 27.7804 48.1170i 1.37031 2.37344i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000 + 24.2487i 0.685583 + 1.18746i
\(418\) 0 0
\(419\) −31.7490 −1.55104 −0.775520 0.631322i \(-0.782512\pi\)
−0.775520 + 0.631322i \(0.782512\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 0 0
\(423\) −15.8745 27.4955i −0.771845 1.33687i
\(424\) 0 0
\(425\) 9.00000 + 5.19615i 0.436564 + 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −21.0000 + 36.3731i −1.01389 + 1.75611i
\(430\) 0 0
\(431\) 19.8431 11.4564i 0.955810 0.551837i 0.0609292 0.998142i \(-0.480594\pi\)
0.894881 + 0.446305i \(0.147260\pi\)
\(432\) 0 0
\(433\) 3.46410i 0.166474i 0.996530 + 0.0832370i \(0.0265259\pi\)
−0.996530 + 0.0832370i \(0.973474\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.5000 6.06218i 0.502283 0.289993i
\(438\) 0 0
\(439\) −14.5516 + 25.2042i −0.694512 + 1.20293i 0.275834 + 0.961205i \(0.411046\pi\)
−0.970345 + 0.241724i \(0.922287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.96863 2.29129i −0.188555 0.108862i 0.402751 0.915310i \(-0.368054\pi\)
−0.591306 + 0.806447i \(0.701387\pi\)
\(444\) 0 0
\(445\) −1.50000 2.59808i −0.0711068 0.123161i
\(446\) 0 0
\(447\) 39.6863 1.87710
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −7.93725 13.7477i −0.373751 0.647355i
\(452\) 0 0
\(453\) 52.5000 + 30.3109i 2.46667 + 1.42413i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.0233890 0.0405110i −0.854094 0.520119i \(-0.825888\pi\)
0.877483 + 0.479608i \(0.159221\pi\)
\(458\) 0 0
\(459\) 11.9059 6.87386i 0.555719 0.320844i
\(460\) 0 0
\(461\) 3.46410i 0.161339i −0.996741 0.0806696i \(-0.974294\pi\)
0.996741 0.0806696i \(-0.0257059\pi\)
\(462\) 0 0
\(463\) 27.4955i 1.27782i 0.769281 + 0.638911i \(0.220615\pi\)
−0.769281 + 0.638911i \(0.779385\pi\)
\(464\) 0 0
\(465\) 10.5000 6.06218i 0.486926 0.281127i
\(466\) 0 0
\(467\) 3.96863 6.87386i 0.183646 0.318084i −0.759473 0.650538i \(-0.774543\pi\)
0.943119 + 0.332454i \(0.107877\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 11.9059 + 6.87386i 0.548594 + 0.316731i
\(472\) 0 0
\(473\) −21.0000 36.3731i −0.965581 1.67244i
\(474\) 0 0
\(475\) −5.29150 −0.242791
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −3.96863 6.87386i −0.181331 0.314075i 0.761003 0.648748i \(-0.224707\pi\)
−0.942334 + 0.334674i \(0.891374\pi\)
\(480\) 0 0
\(481\) −21.0000 12.1244i −0.957518 0.552823i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.00000 5.19615i 0.136223 0.235945i
\(486\) 0 0
\(487\) −35.7176 + 20.6216i −1.61852 + 0.934453i −0.631218 + 0.775606i \(0.717445\pi\)
−0.987303 + 0.158848i \(0.949222\pi\)
\(488\) 0 0
\(489\) 12.1244i 0.548282i
\(490\) 0 0
\(491\) 18.3303i 0.827235i −0.910451 0.413617i \(-0.864265\pi\)
0.910451 0.413617i \(-0.135735\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −15.8745 + 27.4955i −0.713506 + 1.23583i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −27.7804 16.0390i −1.24362 0.718005i −0.273791 0.961789i \(-0.588278\pi\)
−0.969830 + 0.243784i \(0.921611\pi\)
\(500\) 0 0
\(501\) 21.0000 + 36.3731i 0.938211 + 1.62503i
\(502\) 0 0
\(503\) −31.7490 −1.41562 −0.707809 0.706404i \(-0.750316\pi\)
−0.707809 + 0.706404i \(0.750316\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 1.32288 + 2.29129i 0.0587510 + 0.101760i
\(508\) 0 0
\(509\) −22.5000 12.9904i −0.997295 0.575789i −0.0898481 0.995955i \(-0.528638\pi\)
−0.907447 + 0.420167i \(0.861972\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.50000 + 6.06218i −0.154529 + 0.267652i
\(514\) 0 0
\(515\) −3.96863 + 2.29129i −0.174879 + 0.100966i
\(516\) 0 0
\(517\) 36.3731i 1.59969i
\(518\) 0 0
\(519\) 59.5735i 2.61499i
\(520\) 0 0
\(521\) −37.5000 + 21.6506i −1.64290 + 0.948532i −0.663111 + 0.748521i \(0.730764\pi\)
−0.979794 + 0.200011i \(0.935902\pi\)
\(522\) 0 0
\(523\) 6.61438 11.4564i 0.289227 0.500955i −0.684399 0.729108i \(-0.739935\pi\)
0.973625 + 0.228153i \(0.0732686\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9059 + 6.87386i 0.518628 + 0.299430i
\(528\) 0 0
\(529\) −1.00000 1.73205i −0.0434783 0.0753066i
\(530\) 0 0
\(531\) 31.7490 1.37779
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 11.9059 + 20.6216i 0.514736 + 0.891549i
\(536\) 0 0
\(537\) −31.5000 18.1865i −1.35933 0.784807i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.50000 + 11.2583i −0.279457 + 0.484033i −0.971250 0.238062i \(-0.923488\pi\)
0.691793 + 0.722096i \(0.256821\pi\)
\(542\) 0 0
\(543\) −47.6235 + 27.4955i −2.04372 + 1.17994i
\(544\) 0 0
\(545\) 12.1244i 0.519350i
\(546\) 0 0
\(547\) 18.3303i 0.783747i −0.920019 0.391874i \(-0.871827\pi\)
0.920019 0.391874i \(-0.128173\pi\)
\(548\) 0 0
\(549\) 6.00000 3.46410i 0.256074 0.147844i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −27.7804 16.0390i −1.17921 0.680818i
\(556\) 0 0
\(557\) 1.50000 + 2.59808i 0.0635570 + 0.110084i 0.896053 0.443947i \(-0.146422\pi\)
−0.832496 + 0.554031i \(0.813089\pi\)
\(558\) 0 0
\(559\) 31.7490 1.34284
\(560\) 0 0
\(561\) −63.0000 −2.65986
\(562\) 0 0
\(563\) −3.96863 6.87386i −0.167258 0.289699i 0.770197 0.637806i \(-0.220158\pi\)
−0.937455 + 0.348107i \(0.886825\pi\)
\(564\) 0 0
\(565\) 18.0000 + 10.3923i 0.757266 + 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.5000 18.1865i 0.440183 0.762419i −0.557520 0.830164i \(-0.688247\pi\)
0.997703 + 0.0677445i \(0.0215803\pi\)
\(570\) 0 0
\(571\) −11.9059 + 6.87386i −0.498246 + 0.287662i −0.727989 0.685589i \(-0.759545\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(572\) 0 0
\(573\) 60.6218i 2.53251i
\(574\) 0 0
\(575\) 9.16515i 0.382213i
\(576\) 0 0
\(577\) 16.5000 9.52628i 0.686904 0.396584i −0.115547 0.993302i \(-0.536862\pi\)
0.802451 + 0.596718i \(0.203529\pi\)
\(578\) 0 0
\(579\) −14.5516 + 25.2042i −0.604745 + 1.04745i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.9059 6.87386i −0.493091 0.284686i
\(584\) 0 0
\(585\) −12.0000 20.7846i −0.496139 0.859338i
\(586\) 0 0
\(587\) −31.7490 −1.31042 −0.655211 0.755446i \(-0.727420\pi\)
−0.655211 + 0.755446i \(0.727420\pi\)
\(588\) 0 0
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) −15.8745 27.4955i −0.652990 1.13101i
\(592\) 0 0
\(593\) 19.5000 + 11.2583i 0.800769 + 0.462324i 0.843740 0.536752i \(-0.180349\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.5000 30.3109i 0.716227 1.24054i
\(598\) 0 0
\(599\) −3.96863 + 2.29129i −0.162154 + 0.0936195i −0.578881 0.815412i \(-0.696510\pi\)
0.416727 + 0.909032i \(0.363177\pi\)
\(600\) 0 0
\(601\) 3.46410i 0.141304i −0.997501 0.0706518i \(-0.977492\pi\)
0.997501 0.0706518i \(-0.0225079\pi\)
\(602\) 0 0
\(603\) 18.3303i 0.746468i
\(604\) 0 0
\(605\) 15.0000 8.66025i 0.609837 0.352089i
\(606\) 0 0
\(607\) 1.32288 2.29129i 0.0536939 0.0930005i −0.837929 0.545779i \(-0.816234\pi\)
0.891623 + 0.452778i \(0.149567\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.8118 13.7477i −0.963321 0.556174i
\(612\) 0 0
\(613\) 3.50000 + 6.06218i 0.141364 + 0.244849i 0.928010 0.372554i \(-0.121518\pi\)
−0.786647 + 0.617403i \(0.788185\pi\)
\(614\) 0 0
\(615\) 15.8745 0.640122
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −9.26013 16.0390i −0.372196 0.644662i 0.617707 0.786408i \(-0.288062\pi\)
−0.989903 + 0.141746i \(0.954728\pi\)
\(620\) 0 0
\(621\) −10.5000 6.06218i −0.421350 0.243267i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 27.7804 16.0390i 1.10944 0.640537i
\(628\) 0 0
\(629\) 36.3731i 1.45029i
\(630\) 0 0
\(631\) 45.8258i 1.82429i 0.409863 + 0.912147i \(0.365577\pi\)
−0.409863 + 0.912147i \(0.634423\pi\)
\(632\) 0 0
\(633\) 21.0000 12.1244i 0.834675 0.481900i
\(634\) 0 0
\(635\) −7.93725 + 13.7477i −0.314980 + 0.545562i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 31.7490 + 18.3303i 1.25597 + 0.725136i
\(640\) 0 0
\(641\) 10.5000 + 18.1865i 0.414725 + 0.718325i 0.995400 0.0958109i \(-0.0305444\pi\)
−0.580674 + 0.814136i \(0.697211\pi\)
\(642\) 0 0
\(643\) 21.1660 0.834706 0.417353 0.908744i \(-0.362958\pi\)
0.417353 + 0.908744i \(0.362958\pi\)
\(644\) 0 0
\(645\) 42.0000 1.65375
\(646\) 0 0
\(647\) −19.8431 34.3693i −0.780114 1.35120i −0.931875 0.362780i \(-0.881828\pi\)
0.151761 0.988417i \(-0.451506\pi\)
\(648\) 0 0
\(649\) −31.5000 18.1865i −1.23648 0.713884i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.5000 18.1865i 0.410897 0.711694i −0.584091 0.811688i \(-0.698549\pi\)
0.994988 + 0.0999939i \(0.0318823\pi\)
\(654\) 0 0
\(655\) 11.9059 6.87386i 0.465201 0.268584i
\(656\) 0 0
\(657\) 20.7846i 0.810885i
\(658\) 0 0
\(659\) 9.16515i 0.357024i −0.983938 0.178512i \(-0.942872\pi\)
0.983938 0.178512i \(-0.0571283\pi\)
\(660\) 0 0
\(661\) 4.50000 2.59808i 0.175030 0.101053i −0.409926 0.912119i \(-0.634445\pi\)
0.584955 + 0.811065i \(0.301112\pi\)
\(662\) 0 0
\(663\) 23.8118 41.2432i 0.924772 1.60175i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −14.0000 24.2487i −0.541271 0.937509i
\(670\) 0 0
\(671\) −7.93725 −0.306414
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 2.64575 + 4.58258i 0.101835 + 0.176383i
\(676\) 0 0
\(677\) 1.50000 + 0.866025i 0.0576497 + 0.0332841i 0.528548 0.848904i \(-0.322737\pi\)
−0.470898 + 0.882188i \(0.656070\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 31.5000 54.5596i 1.20708 2.09073i
\(682\) 0 0
\(683\) 3.96863 2.29129i 0.151855 0.0876737i −0.422147 0.906527i \(-0.638723\pi\)
0.574002 + 0.818854i \(0.305390\pi\)
\(684\) 0 0
\(685\) 36.3731i 1.38974i
\(686\) 0 0
\(687\) 4.58258i 0.174836i
\(688\) 0 0
\(689\) 9.00000 5.19615i 0.342873 0.197958i
\(690\) 0 0
\(691\) −1.32288 + 2.29129i −0.0503246 + 0.0871647i −0.890090 0.455784i \(-0.849359\pi\)
0.839766 + 0.542949i \(0.182692\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.8745 + 9.16515i 0.602154 + 0.347654i
\(696\) 0 0
\(697\) 9.00000 + 15.5885i 0.340899 + 0.590455i
\(698\) 0 0
\(699\) −55.5608 −2.10150
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 9.26013 + 16.0390i 0.349252 + 0.604923i
\(704\) 0 0
\(705\) −31.5000 18.1865i −1.18636 0.684944i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.5000 + 30.3109i −0.657226 + 1.13835i 0.324104 + 0.946021i \(0.394937\pi\)
−0.981331 + 0.192328i \(0.938396\pi\)
\(710\) 0 0
\(711\) −15.8745 + 9.16515i −0.595341 + 0.343720i
\(712\) 0 0
\(713\) 12.1244i 0.454061i
\(714\) 0 0
\(715\) 27.4955i 1.02827i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.96863 6.87386i 0.148005 0.256352i −0.782485 0.622669i \(-0.786048\pi\)
0.930490 + 0.366317i \(0.119382\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 43.6549 + 25.2042i 1.62354 + 0.937353i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.1660 −0.785004 −0.392502 0.919751i \(-0.628390\pi\)
−0.392502 + 0.919751i \(0.628390\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) 23.8118 + 41.2432i 0.880710 + 1.52543i
\(732\) 0 0
\(733\) −19.5000 11.2583i −0.720249 0.415836i 0.0945954 0.995516i \(-0.469844\pi\)
−0.814844 + 0.579680i \(0.803178\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.5000 + 18.1865i −0.386772 + 0.669910i
\(738\) 0 0
\(739\) 19.8431 11.4564i 0.729942 0.421432i −0.0884593 0.996080i \(-0.528194\pi\)
0.818401 + 0.574648i \(0.194861\pi\)
\(740\) 0 0
\(741\) 24.2487i 0.890799i
\(742\) 0 0
\(743\) 45.8258i 1.68118i 0.541669 + 0.840592i \(0.317793\pi\)
−0.541669 + 0.840592i \(0.682207\pi\)
\(744\) 0 0
\(745\) 22.5000 12.9904i 0.824336 0.475931i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11.9059 + 6.87386i 0.434452 + 0.250831i 0.701241 0.712924i \(-0.252630\pi\)
−0.266790 + 0.963755i \(0.585963\pi\)
\(752\) 0 0
\(753\) 21.0000 + 36.3731i 0.765283 + 1.32551i
\(754\) 0 0
\(755\) 39.6863 1.44433
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 0 0
\(759\) 27.7804 + 48.1170i 1.00836 + 1.74654i
\(760\) 0 0
\(761\) −22.5000 12.9904i −0.815624 0.470901i 0.0332809 0.999446i \(-0.489404\pi\)
−0.848905 + 0.528545i \(0.822738\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 18.0000 31.1769i 0.650791 1.12720i
\(766\) 0 0
\(767\) 23.8118 13.7477i 0.859793 0.496402i
\(768\) 0 0
\(769\) 51.9615i 1.87378i 0.349624 + 0.936890i \(0.386309\pi\)
−0.349624 + 0.936890i \(0.613691\pi\)
\(770\) 0 0
\(771\) 4.58258i 0.165037i
\(772\) 0 0
\(773\) 25.5000 14.7224i 0.917171 0.529529i 0.0344397 0.999407i \(-0.489035\pi\)
0.882732 + 0.469878i \(0.155702\pi\)
\(774\) 0 0
\(775\) −2.64575 + 4.58258i −0.0950382 + 0.164611i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.93725 4.58258i −0.284382 0.164188i
\(780\) 0 0
\(781\) −21.0000 36.3731i −0.751439 1.30153i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.00000 0.321224
\(786\) 0 0
\(787\) 6.61438 + 11.4564i 0.235777 + 0.408378i 0.959498 0.281715i \(-0.0909031\pi\)
−0.723721 + 0.690093i \(0.757570\pi\)
\(788\) 0 0
\(789\) −10.5000 6.06218i −0.373810 0.215819i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.00000 5.19615i 0.106533 0.184521i
\(794\) 0 0
\(795\) 11.9059 6.87386i 0.422258 0.243791i
\(796\) 0 0
\(797\) 3.46410i 0.122705i 0.998116 + 0.0613524i \(0.0195413\pi\)
−0.998116 + 0.0613524i \(0.980459\pi\)
\(798\) 0 0
\(799\) 41.2432i 1.45908i
\(800\) 0 0
\(801\) 6.00000 3.46410i 0.212000 0.122398i
\(802\) 0 0
\(803\) 11.9059 20.6216i 0.420149 0.727720i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −67.4667 38.9519i −2.37494 1.37117i
\(808\) 0 0
\(809\) 22.5000 + 38.9711i 0.791058 + 1.37015i 0.925312 + 0.379206i \(0.123803\pi\)
−0.134255 + 0.990947i \(0.542864\pi\)
\(810\) 0 0
\(811\) −21.1660 −0.743239 −0.371620 0.928385i \(-0.621197\pi\)
−0.371620 + 0.928385i \(0.621197\pi\)
\(812\) 0 0
\(813\) 77.0000 2.70051
\(814\) 0 0
\(815\) 3.96863 + 6.87386i 0.139015 + 0.240781i
\(816\) 0 0
\(817\) −21.0000 12.1244i −0.734697 0.424178i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.5000 23.3827i 0.471153 0.816061i −0.528302 0.849056i \(-0.677171\pi\)
0.999456 + 0.0329950i \(0.0105045\pi\)
\(822\) 0 0
\(823\) 27.7804 16.0390i 0.968363 0.559085i 0.0696265 0.997573i \(-0.477819\pi\)
0.898737 + 0.438488i \(0.144486\pi\)
\(824\) 0 0
\(825\) 24.2487i 0.844232i
\(826\) 0 0
\(827\) 9.16515i 0.318704i 0.987222 + 0.159352i \(0.0509404\pi\)
−0.987222 + 0.159352i \(0.949060\pi\)
\(828\) 0 0
\(829\) −4.50000 + 2.59808i −0.156291 + 0.0902349i −0.576106 0.817375i \(-0.695428\pi\)
0.419815 + 0.907610i \(0.362095\pi\)
\(830\) 0 0
\(831\) −22.4889 + 38.9519i −0.780131 + 1.35123i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 23.8118 + 13.7477i 0.824040 + 0.475760i
\(836\) 0 0
\(837\) 3.50000 + 6.06218i 0.120978 + 0.209540i
\(838\) 0 0
\(839\) 31.7490 1.09610 0.548049 0.836446i \(-0.315371\pi\)
0.548049 + 0.836446i \(0.315371\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.50000 + 0.866025i 0.0516016 + 0.0297922i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 17.5000 30.3109i 0.600598 1.04027i
\(850\) 0 0
\(851\) −27.7804 + 16.0390i −0.952299 + 0.549810i
\(852\) 0 0
\(853\) 51.9615i 1.77913i −0.456810 0.889564i \(-0.651008\pi\)
0.456810 0.889564i \(-0.348992\pi\)
\(854\) 0 0
\(855\) 18.3303i 0.626883i
\(856\) 0 0
\(857\) −25.5000 + 14.7224i −0.871063 + 0.502909i −0.867701 0.497086i \(-0.834403\pi\)
−0.00336193 + 0.999994i \(0.501070\pi\)
\(858\) 0 0
\(859\) 1.32288 2.29129i 0.0451359 0.0781777i −0.842575 0.538579i \(-0.818961\pi\)
0.887711 + 0.460402i \(0.152295\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.9059 6.87386i −0.405281 0.233989i 0.283479 0.958978i \(-0.408511\pi\)
−0.688760 + 0.724989i \(0.741845\pi\)
\(864\) 0 0
\(865\) −19.5000 33.7750i −0.663020 1.14838i
\(866\) 0 0
\(867\) 26.4575 0.898544
\(868\) 0 0
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) −7.93725 13.7477i −0.268944 0.465824i
\(872\) 0 0
\(873\) 12.0000 + 6.92820i 0.406138 + 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.50000 6.06218i 0.118187 0.204705i −0.800862 0.598848i \(-0.795625\pi\)
0.919049 + 0.394143i \(0.128959\pi\)
\(878\) 0 0
\(879\) 47.6235 27.4955i 1.60630 0.927399i
\(880\) 0 0
\(881\) 27.7128i 0.933668i 0.884345 + 0.466834i \(0.154606\pi\)
−0.884345 + 0.466834i \(0.845394\pi\)
\(882\) 0 0
\(883\) 27.4955i 0.925296i −0.886542 0.462648i \(-0.846899\pi\)
0.886542 0.462648i \(-0.153101\pi\)
\(884\) 0 0
\(885\) 31.5000 18.1865i 1.05886 0.611334i
\(886\) 0 0
\(887\) 11.9059 20.6216i 0.399760 0.692405i −0.593936 0.804512i \(-0.702427\pi\)
0.993696 + 0.112107i \(0.0357600\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 19.8431 + 11.4564i 0.664770 + 0.383805i
\(892\) 0 0
\(893\) 10.5000 + 18.1865i 0.351369 + 0.608589i
\(894\) 0 0
\(895\) −23.8118 −0.795939
\(896\) 0 0
\(897\) −42.0000 −1.40234
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 13.5000 + 7.79423i 0.449750 + 0.259663i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 + 31.1769i −0.598340 + 1.03636i
\(906\) 0 0
\(907\) −19.8431 + 11.4564i −0.658880 + 0.380405i −0.791850 0.610715i \(-0.790882\pi\)
0.132970 + 0.991120i \(0.457549\pi\)
\(908\) 0 0
\(909\) 20.7846i 0.689382i
\(910\) 0 0
\(911\) 27.4955i 0.910965i −0.890245 0.455483i \(-0.849467\pi\)
0.890245 0.455483i \(-0.150533\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.96863 6.87386i 0.131199 0.227243i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −19.8431 11.4564i −0.654565 0.377913i 0.135638 0.990758i \(-0.456692\pi\)
−0.790203 + 0.612845i \(0.790025\pi\)
\(920\) 0 0
\(921\) −14.0000 24.2487i −0.461316 0.799022i
\(922\) 0 0
\(923\) 31.7490 1.04503
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) 0 0
\(927\) −5.29150 9.16515i −0.173796 0.301023i
\(928\) 0 0
\(929\) 40.5000 + 23.3827i 1.32876 + 0.767161i 0.985108 0.171935i \(-0.0550020\pi\)
0.343654 + 0.939096i \(0.388335\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −31.5000 + 54.5596i −1.03126 + 1.78620i
\(934\) 0 0
\(935\) −35.7176 + 20.6216i −1.16809 + 0.674398i
\(936\) 0 0
\(937\) 27.7128i 0.905338i −0.891679 0.452669i \(-0.850472\pi\)
0.891679 0.452669i \(-0.149528\pi\)
\(938\) 0 0
\(939\) 68.7386i 2.24320i
\(940\) 0 0
\(941\) 16.5000 9.52628i 0.537885 0.310548i −0.206337 0.978481i \(-0.566154\pi\)
0.744221 + 0.667933i \(0.232821\pi\)
\(942\) 0 0
\(943\) 7.93725 13.7477i 0.258473 0.447688i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.96863 + 2.29129i 0.128963 + 0.0744569i 0.563094 0.826393i \(-0.309611\pi\)
−0.434131 + 0.900850i \(0.642944\pi\)
\(948\) 0 0
\(949\) 9.00000 + 15.5885i 0.292152 + 0.506023i
\(950\) 0 0
\(951\) −39.6863 −1.28692
\(952\) 0 0
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) 0 0
\(955\) −19.8431 34.3693i −0.642109 1.11217i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 12.0000 20.7846i 0.387097 0.670471i
\(962\) 0 0
\(963\) −47.6235 + 27.4955i −1.53465 + 0.886029i
\(964\) 0 0
\(965\) 19.0526i 0.613324i
\(966\) 0 0
\(967\) 45.8258i 1.47366i −0.676080 0.736828i \(-0.736323\pi\)
0.676080 0.736828i \(-0.263677\pi\)
\(968\) 0 0
\(969\) −31.5000 + 18.1865i −1.01193 + 0.584236i
\(970\) 0 0
\(971\) −27.7804 + 48.1170i −0.891515 + 1.54415i −0.0534559 + 0.998570i \(0.517024\pi\)
−0.838059 + 0.545579i \(0.816310\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 15.8745 + 9.16515i 0.508391 + 0.293520i
\(976\) 0 0
\(977\) −10.5000 18.1865i −0.335925 0.581839i 0.647737 0.761864i \(-0.275715\pi\)
−0.983662 + 0.180025i \(0.942382\pi\)
\(978\) 0 0
\(979\) −7.93725 −0.253676
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) 0 0
\(983\) 3.96863 + 6.87386i 0.126580 + 0.219242i 0.922349 0.386357i \(-0.126267\pi\)
−0.795770 + 0.605599i \(0.792933\pi\)
\(984\) 0 0
\(985\) −18.0000 10.3923i −0.573528 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.0000 36.3731i 0.667761 1.15660i
\(990\) 0 0
\(991\) −3.96863 + 2.29129i −0.126068 + 0.0727852i −0.561708 0.827336i \(-0.689855\pi\)
0.435640 + 0.900121i \(0.356522\pi\)
\(992\) 0 0
\(993\) 36.3731i 1.15426i
\(994\) 0 0
\(995\) 22.9129i 0.726387i
\(996\) 0 0
\(997\) −37.5000 + 21.6506i −1.18764 + 0.685682i −0.957769 0.287539i \(-0.907163\pi\)
−0.229868 + 0.973222i \(0.573829\pi\)
\(998\) 0 0
\(999\) 9.26013 16.0390i 0.292978 0.507452i
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 784.2.p.g.607.2 4
4.3 odd 2 inner 784.2.p.g.607.1 4
7.2 even 3 784.2.f.d.783.1 4
7.3 odd 6 inner 784.2.p.g.31.1 4
7.4 even 3 112.2.p.c.31.2 yes 4
7.5 odd 6 784.2.f.d.783.4 4
7.6 odd 2 112.2.p.c.47.1 yes 4
21.2 odd 6 7056.2.b.s.1567.3 4
21.5 even 6 7056.2.b.s.1567.1 4
21.11 odd 6 1008.2.cs.q.703.2 4
21.20 even 2 1008.2.cs.q.271.1 4
28.3 even 6 inner 784.2.p.g.31.2 4
28.11 odd 6 112.2.p.c.31.1 4
28.19 even 6 784.2.f.d.783.2 4
28.23 odd 6 784.2.f.d.783.3 4
28.27 even 2 112.2.p.c.47.2 yes 4
56.5 odd 6 3136.2.f.f.3135.1 4
56.11 odd 6 448.2.p.c.255.2 4
56.13 odd 2 448.2.p.c.383.2 4
56.19 even 6 3136.2.f.f.3135.3 4
56.27 even 2 448.2.p.c.383.1 4
56.37 even 6 3136.2.f.f.3135.4 4
56.51 odd 6 3136.2.f.f.3135.2 4
56.53 even 6 448.2.p.c.255.1 4
84.11 even 6 1008.2.cs.q.703.1 4
84.23 even 6 7056.2.b.s.1567.4 4
84.47 odd 6 7056.2.b.s.1567.2 4
84.83 odd 2 1008.2.cs.q.271.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.p.c.31.1 4 28.11 odd 6
112.2.p.c.31.2 yes 4 7.4 even 3
112.2.p.c.47.1 yes 4 7.6 odd 2
112.2.p.c.47.2 yes 4 28.27 even 2
448.2.p.c.255.1 4 56.53 even 6
448.2.p.c.255.2 4 56.11 odd 6
448.2.p.c.383.1 4 56.27 even 2
448.2.p.c.383.2 4 56.13 odd 2
784.2.f.d.783.1 4 7.2 even 3
784.2.f.d.783.2 4 28.19 even 6
784.2.f.d.783.3 4 28.23 odd 6
784.2.f.d.783.4 4 7.5 odd 6
784.2.p.g.31.1 4 7.3 odd 6 inner
784.2.p.g.31.2 4 28.3 even 6 inner
784.2.p.g.607.1 4 4.3 odd 2 inner
784.2.p.g.607.2 4 1.1 even 1 trivial
1008.2.cs.q.271.1 4 21.20 even 2
1008.2.cs.q.271.2 4 84.83 odd 2
1008.2.cs.q.703.1 4 84.11 even 6
1008.2.cs.q.703.2 4 21.11 odd 6
3136.2.f.f.3135.1 4 56.5 odd 6
3136.2.f.f.3135.2 4 56.51 odd 6
3136.2.f.f.3135.3 4 56.19 even 6
3136.2.f.f.3135.4 4 56.37 even 6
7056.2.b.s.1567.1 4 21.5 even 6
7056.2.b.s.1567.2 4 84.47 odd 6
7056.2.b.s.1567.3 4 21.2 odd 6
7056.2.b.s.1567.4 4 84.23 even 6