Properties

Label 1300.2.bb.b.1049.2
Level $1300$
Weight $2$
Character 1300.1049
Analytic conductor $10.381$
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] = [1300,2,Mod(549,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1049.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1300.1049
Dual form 1300.2.bb.b.549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(-0.866025 + 0.500000i) q^{7} +(-1.00000 - 1.73205i) q^{9} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{3} +(-0.866025 + 0.500000i) q^{7} +(-1.00000 - 1.73205i) q^{9} +(-1.50000 + 2.59808i) q^{11} +(3.46410 - 1.00000i) q^{13} +(-2.59808 + 1.50000i) q^{17} +(2.50000 + 4.33013i) q^{19} -1.00000 q^{21} +(7.79423 + 4.50000i) q^{23} -5.00000i q^{27} +(-4.50000 + 7.79423i) q^{29} +8.00000 q^{31} +(-2.59808 + 1.50000i) q^{33} +(6.06218 + 3.50000i) q^{37} +(3.50000 + 0.866025i) q^{39} +(-1.50000 + 2.59808i) q^{41} +(0.866025 - 0.500000i) q^{43} +(-3.00000 + 5.19615i) q^{49} -3.00000 q^{51} -6.00000i q^{53} +5.00000i q^{57} +(4.50000 + 7.79423i) q^{59} +(0.500000 + 0.866025i) q^{61} +(1.73205 + 1.00000i) q^{63} +(-4.33013 - 2.50000i) q^{67} +(4.50000 + 7.79423i) q^{69} +(-4.50000 - 7.79423i) q^{71} -2.00000i q^{73} -3.00000i q^{77} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(-7.79423 + 4.50000i) q^{87} +(1.50000 - 2.59808i) q^{89} +(-2.50000 + 2.59808i) q^{91} +(6.92820 + 4.00000i) q^{93} +(14.7224 - 8.50000i) q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 6 q^{11} + 10 q^{19} - 4 q^{21} - 18 q^{29} + 32 q^{31} + 14 q^{39} - 6 q^{41} - 12 q^{49} - 12 q^{51} + 18 q^{59} + 2 q^{61} + 18 q^{69} - 18 q^{71} - 32 q^{79} - 2 q^{81} + 6 q^{89} - 10 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i 0.728714 0.684819i \(-0.240119\pi\)
−0.228714 + 0.973494i \(0.573452\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.327327 + 0.188982i −0.654654 0.755929i \(-0.727186\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) −1.00000 1.73205i −0.333333 0.577350i
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) 3.46410 1.00000i 0.960769 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.59808 + 1.50000i −0.630126 + 0.363803i −0.780801 0.624780i \(-0.785189\pi\)
0.150675 + 0.988583i \(0.451855\pi\)
\(18\) 0 0
\(19\) 2.50000 + 4.33013i 0.573539 + 0.993399i 0.996199 + 0.0871106i \(0.0277634\pi\)
−0.422659 + 0.906289i \(0.638903\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 7.79423 + 4.50000i 1.62521 + 0.938315i 0.985496 + 0.169701i \(0.0542803\pi\)
0.639713 + 0.768613i \(0.279053\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) −4.50000 + 7.79423i −0.835629 + 1.44735i 0.0578882 + 0.998323i \(0.481563\pi\)
−0.893517 + 0.449029i \(0.851770\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −2.59808 + 1.50000i −0.452267 + 0.261116i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.06218 + 3.50000i 0.996616 + 0.575396i 0.907245 0.420602i \(-0.138181\pi\)
0.0893706 + 0.995998i \(0.471514\pi\)
\(38\) 0 0
\(39\) 3.50000 + 0.866025i 0.560449 + 0.138675i
\(40\) 0 0
\(41\) −1.50000 + 2.59808i −0.234261 + 0.405751i −0.959058 0.283211i \(-0.908600\pi\)
0.724797 + 0.688963i \(0.241934\pi\)
\(42\) 0 0
\(43\) 0.866025 0.500000i 0.132068 0.0762493i −0.432511 0.901629i \(-0.642372\pi\)
0.564578 + 0.825380i \(0.309039\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −3.00000 + 5.19615i −0.428571 + 0.742307i
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000i 0.662266i
\(58\) 0 0
\(59\) 4.50000 + 7.79423i 0.585850 + 1.01472i 0.994769 + 0.102151i \(0.0325726\pi\)
−0.408919 + 0.912571i \(0.634094\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 1.73205 + 1.00000i 0.218218 + 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.33013 2.50000i −0.529009 0.305424i 0.211604 0.977356i \(-0.432131\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 4.50000 + 7.79423i 0.541736 + 0.938315i
\(70\) 0 0
\(71\) −4.50000 7.79423i −0.534052 0.925005i −0.999209 0.0397765i \(-0.987335\pi\)
0.465157 0.885228i \(-0.345998\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.79423 + 4.50000i −0.835629 + 0.482451i
\(88\) 0 0
\(89\) 1.50000 2.59808i 0.159000 0.275396i −0.775509 0.631337i \(-0.782506\pi\)
0.934508 + 0.355942i \(0.115840\pi\)
\(90\) 0 0
\(91\) −2.50000 + 2.59808i −0.262071 + 0.272352i
\(92\) 0 0
\(93\) 6.92820 + 4.00000i 0.718421 + 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.7224 8.50000i 1.49484 0.863044i 0.494854 0.868976i \(-0.335222\pi\)
0.999982 + 0.00593185i \(0.00188818\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −7.50000 + 12.9904i −0.746278 + 1.29259i 0.203317 + 0.979113i \(0.434828\pi\)
−0.949595 + 0.313478i \(0.898506\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.59808 + 1.50000i 0.251166 + 0.145010i 0.620298 0.784366i \(-0.287012\pi\)
−0.369132 + 0.929377i \(0.620345\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 3.50000 + 6.06218i 0.332205 + 0.575396i
\(112\) 0 0
\(113\) 12.9904 7.50000i 1.22203 0.705541i 0.256681 0.966496i \(-0.417371\pi\)
0.965351 + 0.260955i \(0.0840376\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.19615 5.00000i −0.480384 0.462250i
\(118\) 0 0
\(119\) 1.50000 2.59808i 0.137505 0.238165i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) −2.59808 + 1.50000i −0.234261 + 0.135250i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.7224 8.50000i −1.30640 0.754253i −0.324910 0.945745i \(-0.605334\pi\)
−0.981494 + 0.191492i \(0.938667\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.33013 2.50000i −0.375470 0.216777i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.59808 + 1.50000i −0.221969 + 0.128154i −0.606861 0.794808i \(-0.707572\pi\)
0.384893 + 0.922961i \(0.374238\pi\)
\(138\) 0 0
\(139\) 2.50000 + 4.33013i 0.212047 + 0.367277i 0.952355 0.304991i \(-0.0986536\pi\)
−0.740308 + 0.672268i \(0.765320\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.59808 + 10.5000i −0.217262 + 0.878054i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.19615 + 3.00000i −0.428571 + 0.247436i
\(148\) 0 0
\(149\) −4.50000 7.79423i −0.368654 0.638528i 0.620701 0.784047i \(-0.286848\pi\)
−0.989355 + 0.145519i \(0.953515\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 5.19615 + 3.00000i 0.420084 + 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) 3.00000 5.19615i 0.237915 0.412082i
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) 0.866025 0.500000i 0.0678323 0.0391630i −0.465700 0.884943i \(-0.654198\pi\)
0.533533 + 0.845780i \(0.320864\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.9904 + 7.50000i 1.00523 + 0.580367i 0.909790 0.415068i \(-0.136242\pi\)
0.0954356 + 0.995436i \(0.469576\pi\)
\(168\) 0 0
\(169\) 11.0000 6.92820i 0.846154 0.532939i
\(170\) 0 0
\(171\) 5.00000 8.66025i 0.382360 0.662266i
\(172\) 0 0
\(173\) −18.1865 + 10.5000i −1.38270 + 0.798300i −0.992478 0.122422i \(-0.960934\pi\)
−0.390218 + 0.920722i \(0.627601\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.00000i 0.676481i
\(178\) 0 0
\(179\) 7.50000 12.9904i 0.560576 0.970947i −0.436870 0.899525i \(-0.643913\pi\)
0.997446 0.0714220i \(-0.0227537\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 1.00000i 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000i 0.658145i
\(188\) 0 0
\(189\) 2.50000 + 4.33013i 0.181848 + 0.314970i
\(190\) 0 0
\(191\) 1.50000 + 2.59808i 0.108536 + 0.187990i 0.915177 0.403051i \(-0.132050\pi\)
−0.806641 + 0.591041i \(0.798717\pi\)
\(192\) 0 0
\(193\) 4.33013 + 2.50000i 0.311689 + 0.179954i 0.647682 0.761911i \(-0.275738\pi\)
−0.335993 + 0.941865i \(0.609072\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.59808 + 1.50000i 0.185105 + 0.106871i 0.589689 0.807630i \(-0.299250\pi\)
−0.404584 + 0.914501i \(0.632584\pi\)
\(198\) 0 0
\(199\) −3.50000 6.06218i −0.248108 0.429736i 0.714893 0.699234i \(-0.246476\pi\)
−0.963001 + 0.269498i \(0.913142\pi\)
\(200\) 0 0
\(201\) −2.50000 4.33013i −0.176336 0.305424i
\(202\) 0 0
\(203\) 9.00000i 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.0000i 1.25109i
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) 12.5000 21.6506i 0.860535 1.49049i −0.0108774 0.999941i \(-0.503462\pi\)
0.871413 0.490550i \(-0.163204\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.92820 + 4.00000i −0.470317 + 0.271538i
\(218\) 0 0
\(219\) 1.00000 1.73205i 0.0675737 0.117041i
\(220\) 0 0
\(221\) −7.50000 + 7.79423i −0.504505 + 0.524297i
\(222\) 0 0
\(223\) −16.4545 9.50000i −1.10187 0.636167i −0.165161 0.986267i \(-0.552814\pi\)
−0.936713 + 0.350100i \(0.886148\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.9904 7.50000i 0.862202 0.497792i −0.00254715 0.999997i \(-0.500811\pi\)
0.864749 + 0.502204i \(0.167477\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 1.50000 2.59808i 0.0986928 0.170941i
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.92820 4.00000i −0.450035 0.259828i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −11.5000 19.9186i −0.740780 1.28307i −0.952141 0.305661i \(-0.901123\pi\)
0.211360 0.977408i \(-0.432211\pi\)
\(242\) 0 0
\(243\) −13.8564 + 8.00000i −0.888889 + 0.513200i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.9904 + 12.5000i 0.826558 + 0.795356i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.50000 7.79423i −0.284037 0.491967i 0.688338 0.725390i \(-0.258341\pi\)
−0.972375 + 0.233423i \(0.925007\pi\)
\(252\) 0 0
\(253\) −23.3827 + 13.5000i −1.47006 + 0.848738i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.3827 + 13.5000i 1.45857 + 0.842107i 0.998941 0.0460033i \(-0.0146485\pi\)
0.459631 + 0.888110i \(0.347982\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) −2.59808 1.50000i −0.160204 0.0924940i 0.417755 0.908560i \(-0.362817\pi\)
−0.577959 + 0.816066i \(0.696151\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.59808 1.50000i 0.159000 0.0917985i
\(268\) 0 0
\(269\) −10.5000 18.1865i −0.640196 1.10885i −0.985389 0.170321i \(-0.945520\pi\)
0.345192 0.938532i \(-0.387814\pi\)
\(270\) 0 0
\(271\) 6.50000 11.2583i 0.394847 0.683895i −0.598235 0.801321i \(-0.704131\pi\)
0.993082 + 0.117426i \(0.0374643\pi\)
\(272\) 0 0
\(273\) −3.46410 + 1.00000i −0.209657 + 0.0605228i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.4545 + 9.50000i −0.988654 + 0.570800i −0.904872 0.425684i \(-0.860033\pi\)
−0.0837823 + 0.996484i \(0.526700\pi\)
\(278\) 0 0
\(279\) −8.00000 13.8564i −0.478947 0.829561i
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 14.7224 + 8.50000i 0.875158 + 0.505273i 0.869059 0.494709i \(-0.164725\pi\)
0.00609896 + 0.999981i \(0.498059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000i 0.177084i
\(288\) 0 0
\(289\) −4.00000 + 6.92820i −0.235294 + 0.407541i
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) 0 0
\(293\) 2.59808 1.50000i 0.151781 0.0876309i −0.422186 0.906509i \(-0.638737\pi\)
0.573967 + 0.818878i \(0.305404\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 12.9904 + 7.50000i 0.753778 + 0.435194i
\(298\) 0 0
\(299\) 31.5000 + 7.79423i 1.82169 + 0.450752i
\(300\) 0 0
\(301\) −0.500000 + 0.866025i −0.0288195 + 0.0499169i
\(302\) 0 0
\(303\) −12.9904 + 7.50000i −0.746278 + 0.430864i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000i 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) 4.00000 6.92820i 0.227552 0.394132i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) −13.5000 23.3827i −0.755855 1.30918i
\(320\) 0 0
\(321\) 1.50000 + 2.59808i 0.0837218 + 0.145010i
\(322\) 0 0
\(323\) −12.9904 7.50000i −0.722804 0.417311i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.1244 7.00000i −0.670478 0.387101i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.50000 + 16.4545i 0.522167 + 0.904420i 0.999667 + 0.0257885i \(0.00820965\pi\)
−0.477500 + 0.878632i \(0.658457\pi\)
\(332\) 0 0
\(333\) 14.0000i 0.767195i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000i 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) 0 0
\(341\) −12.0000 + 20.7846i −0.649836 + 1.12555i
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.79423 + 4.50000i −0.418416 + 0.241573i −0.694399 0.719590i \(-0.744330\pi\)
0.275983 + 0.961162i \(0.410997\pi\)
\(348\) 0 0
\(349\) 17.5000 30.3109i 0.936754 1.62250i 0.165277 0.986247i \(-0.447148\pi\)
0.771477 0.636257i \(-0.219518\pi\)
\(350\) 0 0
\(351\) −5.00000 17.3205i −0.266880 0.924500i
\(352\) 0 0
\(353\) −23.3827 13.5000i −1.24453 0.718532i −0.274521 0.961581i \(-0.588519\pi\)
−0.970014 + 0.243049i \(0.921853\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.59808 1.50000i 0.137505 0.0793884i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) 2.00000i 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −14.7224 8.50000i −0.768505 0.443696i 0.0638362 0.997960i \(-0.479666\pi\)
−0.832341 + 0.554264i \(0.813000\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 3.00000 + 5.19615i 0.155752 + 0.269771i
\(372\) 0 0
\(373\) −4.33013 + 2.50000i −0.224205 + 0.129445i −0.607896 0.794017i \(-0.707986\pi\)
0.383691 + 0.923462i \(0.374653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.79423 + 31.5000i −0.401423 + 1.62233i
\(378\) 0 0
\(379\) 5.50000 9.52628i 0.282516 0.489332i −0.689488 0.724297i \(-0.742164\pi\)
0.972004 + 0.234965i \(0.0754976\pi\)
\(380\) 0 0
\(381\) −8.50000 14.7224i −0.435468 0.754253i
\(382\) 0 0
\(383\) 18.1865 10.5000i 0.929288 0.536525i 0.0427020 0.999088i \(-0.486403\pi\)
0.886586 + 0.462563i \(0.153070\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.73205 1.00000i −0.0880451 0.0508329i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −27.0000 −1.36545
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.1147 14.5000i 1.26047 0.727734i 0.287307 0.957839i \(-0.407240\pi\)
0.973166 + 0.230105i \(0.0739068\pi\)
\(398\) 0 0
\(399\) −2.50000 4.33013i −0.125157 0.216777i
\(400\) 0 0
\(401\) −7.50000 + 12.9904i −0.374532 + 0.648709i −0.990257 0.139253i \(-0.955530\pi\)
0.615725 + 0.787961i \(0.288863\pi\)
\(402\) 0 0
\(403\) 27.7128 8.00000i 1.38047 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.1865 + 10.5000i −0.901473 + 0.520466i
\(408\) 0 0
\(409\) −12.5000 21.6506i −0.618085 1.07056i −0.989835 0.142222i \(-0.954575\pi\)
0.371750 0.928333i \(-0.378758\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) −7.79423 4.50000i −0.383529 0.221431i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000i 0.244851i
\(418\) 0 0
\(419\) −10.5000 + 18.1865i −0.512959 + 0.888470i 0.486928 + 0.873442i \(0.338117\pi\)
−0.999887 + 0.0150285i \(0.995216\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.866025 0.500000i −0.0419099 0.0241967i
\(428\) 0 0
\(429\) −7.50000 + 7.79423i −0.362103 + 0.376309i
\(430\) 0 0
\(431\) −1.50000 + 2.59808i −0.0722525 + 0.125145i −0.899888 0.436121i \(-0.856352\pi\)
0.827636 + 0.561266i \(0.189685\pi\)
\(432\) 0 0
\(433\) −4.33013 + 2.50000i −0.208093 + 0.120142i −0.600425 0.799681i \(-0.705002\pi\)
0.392332 + 0.919824i \(0.371668\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 45.0000i 2.15264i
\(438\) 0 0
\(439\) −0.500000 + 0.866025i −0.0238637 + 0.0413331i −0.877711 0.479191i \(-0.840930\pi\)
0.853847 + 0.520524i \(0.174263\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 12.0000i 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.00000i 0.425685i
\(448\) 0 0
\(449\) 7.50000 + 12.9904i 0.353947 + 0.613054i 0.986937 0.161106i \(-0.0515060\pi\)
−0.632990 + 0.774160i \(0.718173\pi\)
\(450\) 0 0
\(451\) −4.50000 7.79423i −0.211897 0.367016i
\(452\) 0 0
\(453\) 17.3205 + 10.0000i 0.813788 + 0.469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.5070 20.5000i −1.66095 0.958950i −0.972263 0.233890i \(-0.924854\pi\)
−0.688686 0.725059i \(-0.741812\pi\)
\(458\) 0 0
\(459\) 7.50000 + 12.9904i 0.350070 + 0.606339i
\(460\) 0 0
\(461\) −7.50000 12.9904i −0.349310 0.605022i 0.636817 0.771015i \(-0.280251\pi\)
−0.986127 + 0.165992i \(0.946917\pi\)
\(462\) 0 0
\(463\) 40.0000i 1.85896i 0.368875 + 0.929479i \(0.379743\pi\)
−0.368875 + 0.929479i \(0.620257\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) −7.00000 + 12.1244i −0.322543 + 0.558661i
\(472\) 0 0
\(473\) 3.00000i 0.137940i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.3923 + 6.00000i −0.475831 + 0.274721i
\(478\) 0 0
\(479\) 7.50000 12.9904i 0.342684 0.593546i −0.642246 0.766498i \(-0.721997\pi\)
0.984930 + 0.172953i \(0.0553307\pi\)
\(480\) 0 0
\(481\) 24.5000 + 6.06218i 1.11710 + 0.276412i
\(482\) 0 0
\(483\) −7.79423 4.50000i −0.354650 0.204757i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 30.3109 17.5000i 1.37352 0.793001i 0.382148 0.924101i \(-0.375184\pi\)
0.991369 + 0.131100i \(0.0418510\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −7.50000 + 12.9904i −0.338470 + 0.586248i −0.984145 0.177365i \(-0.943243\pi\)
0.645675 + 0.763612i \(0.276576\pi\)
\(492\) 0 0
\(493\) 27.0000i 1.21602i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.79423 + 4.50000i 0.349619 + 0.201853i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 7.50000 + 12.9904i 0.335075 + 0.580367i
\(502\) 0 0
\(503\) −33.7750 + 19.5000i −1.50595 + 0.869462i −0.505976 + 0.862547i \(0.668868\pi\)
−0.999976 + 0.00691465i \(0.997799\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.9904 0.500000i 0.576923 0.0222058i
\(508\) 0 0
\(509\) −16.5000 + 28.5788i −0.731350 + 1.26673i 0.224957 + 0.974369i \(0.427776\pi\)
−0.956306 + 0.292366i \(0.905557\pi\)
\(510\) 0 0
\(511\) 1.00000 + 1.73205i 0.0442374 + 0.0766214i
\(512\) 0 0
\(513\) 21.6506 12.5000i 0.955899 0.551888i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 4.33013 + 2.50000i 0.189343 + 0.109317i 0.591675 0.806177i \(-0.298467\pi\)
−0.402332 + 0.915494i \(0.631800\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.7846 + 12.0000i −0.905392 + 0.522728i
\(528\) 0 0
\(529\) 29.0000 + 50.2295i 1.26087 + 2.18389i
\(530\) 0 0
\(531\) 9.00000 15.5885i 0.390567 0.676481i
\(532\) 0 0
\(533\) −2.59808 + 10.5000i −0.112535 + 0.454805i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.9904 7.50000i 0.560576 0.323649i
\(538\) 0 0
\(539\) −9.00000 15.5885i −0.387657 0.671442i
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) −8.66025 5.00000i −0.371647 0.214571i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) 1.00000 1.73205i 0.0426790 0.0739221i
\(550\) 0 0
\(551\) −45.0000 −1.91706
\(552\) 0 0
\(553\) 6.92820 4.00000i 0.294617 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.7750 + 19.5000i 1.43109 + 0.826242i 0.997204 0.0747252i \(-0.0238080\pi\)
0.433888 + 0.900967i \(0.357141\pi\)
\(558\) 0 0
\(559\) 2.50000 2.59808i 0.105739 0.109887i
\(560\) 0 0
\(561\) 4.50000 7.79423i 0.189990 0.329073i
\(562\) 0 0
\(563\) 28.5788 16.5000i 1.20445 0.695392i 0.242912 0.970048i \(-0.421897\pi\)
0.961542 + 0.274656i \(0.0885641\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −16.5000 + 28.5788i −0.691716 + 1.19809i 0.279559 + 0.960128i \(0.409812\pi\)
−0.971275 + 0.237959i \(0.923522\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 3.00000i 0.125327i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000i 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 0 0
\(579\) 2.50000 + 4.33013i 0.103896 + 0.179954i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15.5885 + 9.00000i 0.645608 + 0.372742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.3827 + 13.5000i 0.965107 + 0.557205i 0.897741 0.440524i \(-0.145207\pi\)
0.0673658 + 0.997728i \(0.478541\pi\)
\(588\) 0 0
\(589\) 20.0000 + 34.6410i 0.824086 + 1.42736i
\(590\) 0 0
\(591\) 1.50000 + 2.59808i 0.0617018 + 0.106871i
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000i 0.286491i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0.500000 0.866025i 0.0203954 0.0353259i −0.855648 0.517559i \(-0.826841\pi\)
0.876043 + 0.482233i \(0.160174\pi\)
\(602\) 0 0
\(603\) 10.0000i 0.407231i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.9186 11.5000i 0.808470 0.466771i −0.0379540 0.999279i \(-0.512084\pi\)
0.846424 + 0.532509i \(0.178751\pi\)
\(608\) 0 0
\(609\) 4.50000 7.79423i 0.182349 0.315838i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 25.1147 + 14.5000i 1.01437 + 0.585649i 0.912470 0.409145i \(-0.134173\pi\)
0.101905 + 0.994794i \(0.467506\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.3827 + 13.5000i −0.941351 + 0.543490i −0.890384 0.455211i \(-0.849564\pi\)
−0.0509678 + 0.998700i \(0.516231\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 22.5000 38.9711i 0.902894 1.56386i
\(622\) 0 0
\(623\) 3.00000i 0.120192i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −12.9904 7.50000i −0.518786 0.299521i
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 3.50000 + 6.06218i 0.139333 + 0.241331i 0.927244 0.374457i \(-0.122171\pi\)
−0.787911 + 0.615789i \(0.788838\pi\)
\(632\) 0 0
\(633\) 21.6506 12.5000i 0.860535 0.496830i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.19615 + 21.0000i −0.205879 + 0.832050i
\(638\) 0 0
\(639\) −9.00000 + 15.5885i −0.356034 + 0.616670i
\(640\) 0 0
\(641\) −1.50000 2.59808i −0.0592464 0.102618i 0.834881 0.550431i \(-0.185536\pi\)
−0.894127 + 0.447813i \(0.852203\pi\)
\(642\) 0 0
\(643\) 21.6506 12.5000i 0.853818 0.492952i −0.00811944 0.999967i \(-0.502585\pi\)
0.861937 + 0.507015i \(0.169251\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.3827 + 13.5000i 0.919268 + 0.530740i 0.883402 0.468617i \(-0.155247\pi\)
0.0358667 + 0.999357i \(0.488581\pi\)
\(648\) 0 0
\(649\) −27.0000 −1.05984
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) −23.3827 13.5000i −0.915035 0.528296i −0.0329874 0.999456i \(-0.510502\pi\)
−0.882048 + 0.471160i \(0.843835\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.46410 + 2.00000i −0.135147 + 0.0780274i
\(658\) 0 0
\(659\) 22.5000 + 38.9711i 0.876476 + 1.51810i 0.855183 + 0.518327i \(0.173445\pi\)
0.0212930 + 0.999773i \(0.493222\pi\)
\(660\) 0 0
\(661\) −17.5000 + 30.3109i −0.680671 + 1.17896i 0.294105 + 0.955773i \(0.404978\pi\)
−0.974776 + 0.223184i \(0.928355\pi\)
\(662\) 0 0
\(663\) −10.3923 + 3.00000i −0.403604 + 0.116510i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −70.1481 + 40.5000i −2.71614 + 1.56817i
\(668\) 0 0
\(669\) −9.50000 16.4545i −0.367291 0.636167i
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 0 0
\(673\) −16.4545 9.50000i −0.634274 0.366198i 0.148132 0.988968i \(-0.452674\pi\)
−0.782405 + 0.622770i \(0.786007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) −8.50000 + 14.7224i −0.326200 + 0.564995i
\(680\) 0 0
\(681\) 15.0000 0.574801
\(682\) 0 0
\(683\) 18.1865 10.5000i 0.695888 0.401771i −0.109926 0.993940i \(-0.535061\pi\)
0.805814 + 0.592168i \(0.201728\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 19.0526 + 11.0000i 0.726900 + 0.419676i
\(688\) 0 0
\(689\) −6.00000 20.7846i −0.228582 0.791831i
\(690\) 0 0
\(691\) −11.5000 + 19.9186i −0.437481 + 0.757739i −0.997494 0.0707446i \(-0.977462\pi\)
0.560014 + 0.828483i \(0.310796\pi\)
\(692\) 0 0
\(693\) −5.19615 + 3.00000i −0.197386 + 0.113961i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9.00000i 0.340899i
\(698\) 0 0
\(699\) −3.00000 + 5.19615i −0.113470 + 0.196537i
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 35.0000i 1.32005i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.0000i 0.564133i
\(708\) 0 0
\(709\) 17.5000 + 30.3109i 0.657226 + 1.13835i 0.981331 + 0.192328i \(0.0616038\pi\)
−0.324104 + 0.946021i \(0.605063\pi\)
\(710\) 0 0
\(711\) 8.00000 + 13.8564i 0.300023 + 0.519656i
\(712\) 0 0
\(713\) 62.3538 + 36.0000i 2.33517 + 1.34821i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.7846 12.0000i −0.776215 0.448148i
\(718\) 0 0
\(719\) 22.5000 + 38.9711i 0.839108 + 1.45338i 0.890641 + 0.454707i \(0.150256\pi\)
−0.0515326 + 0.998671i \(0.516411\pi\)
\(720\) 0 0
\(721\) 4.00000 + 6.92820i 0.148968 + 0.258020i
\(722\) 0 0
\(723\) 23.0000i 0.855379i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −1.50000 + 2.59808i −0.0554795 + 0.0960933i
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.9904 7.50000i 0.478507 0.276266i
\(738\) 0 0
\(739\) 5.50000 9.52628i 0.202321 0.350430i −0.746955 0.664875i \(-0.768485\pi\)
0.949276 + 0.314445i \(0.101818\pi\)
\(740\) 0 0
\(741\) 5.00000 + 17.3205i 0.183680 + 0.636285i
\(742\) 0 0
\(743\) 28.5788 + 16.5000i 1.04846 + 0.605326i 0.922217 0.386674i \(-0.126376\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 6.50000 11.2583i 0.237188 0.410822i −0.722718 0.691143i \(-0.757107\pi\)
0.959906 + 0.280321i \(0.0904408\pi\)
\(752\) 0 0
\(753\) 9.00000i 0.327978i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.33013 2.50000i −0.157381 0.0908640i 0.419241 0.907875i \(-0.362296\pi\)
−0.576622 + 0.817011i \(0.695630\pi\)
\(758\) 0 0
\(759\) −27.0000 −0.980038
\(760\) 0 0
\(761\) 4.50000 + 7.79423i 0.163125 + 0.282541i 0.935988 0.352032i \(-0.114509\pi\)
−0.772863 + 0.634573i \(0.781176\pi\)
\(762\) 0 0
\(763\) 12.1244 7.00000i 0.438931 0.253417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.3827 + 22.5000i 0.844300 + 0.812428i
\(768\) 0 0
\(769\) 5.50000 9.52628i 0.198335 0.343526i −0.749654 0.661830i \(-0.769780\pi\)
0.947989 + 0.318304i \(0.103113\pi\)
\(770\) 0 0
\(771\) 13.5000 + 23.3827i 0.486191 + 0.842107i
\(772\) 0 0
\(773\) 12.9904 7.50000i 0.467232 0.269756i −0.247849 0.968799i \(-0.579724\pi\)
0.715080 + 0.699043i \(0.246390\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.06218 3.50000i −0.217479 0.125562i
\(778\) 0 0
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) 27.0000 0.966136
\(782\) 0 0
\(783\) 38.9711 + 22.5000i 1.39272 + 0.804084i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −42.4352 + 24.5000i −1.51265 + 0.873331i −0.512763 + 0.858530i \(0.671378\pi\)
−0.999890 + 0.0148003i \(0.995289\pi\)
\(788\) 0 0
\(789\) −1.50000 2.59808i −0.0534014 0.0924940i
\(790\) 0 0
\(791\) −7.50000 + 12.9904i −0.266669 + 0.461885i
\(792\) 0 0
\(793\) 2.59808 + 2.50000i 0.0922604 + 0.0887776i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.3827 + 13.5000i −0.828257 + 0.478195i −0.853256 0.521493i \(-0.825375\pi\)
0.0249984 + 0.999687i \(0.492042\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 5.19615 + 3.00000i 0.183368 + 0.105868i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 21.0000i 0.739235i
\(808\) 0 0
\(809\) 19.5000 33.7750i 0.685583 1.18747i −0.287670 0.957730i \(-0.592880\pi\)
0.973253 0.229736i \(-0.0737862\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 11.2583 6.50000i 0.394847 0.227965i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.33013 + 2.50000i 0.151492 + 0.0874639i
\(818\) 0 0
\(819\) 7.00000 + 1.73205i 0.244600 + 0.0605228i
\(820\) 0 0
\(821\) 10.5000 18.1865i 0.366453 0.634714i −0.622556 0.782576i \(-0.713906\pi\)
0.989008 + 0.147861i \(0.0472389\pi\)
\(822\) 0 0
\(823\) −40.7032 + 23.5000i −1.41882 + 0.819159i −0.996196 0.0871445i \(-0.972226\pi\)
−0.422628 + 0.906303i \(0.638892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 5.50000 9.52628i 0.191023 0.330861i −0.754567 0.656223i \(-0.772153\pi\)
0.945589 + 0.325362i \(0.105486\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000i 1.38260i
\(838\) 0 0
\(839\) −7.50000 12.9904i −0.258929 0.448478i 0.707026 0.707187i \(-0.250036\pi\)
−0.965955 + 0.258709i \(0.916703\pi\)
\(840\) 0 0
\(841\) −26.0000 45.0333i −0.896552 1.55287i
\(842\) 0 0
\(843\) −15.5885 9.00000i −0.536895 0.309976i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.73205 1.00000i −0.0595140 0.0343604i
\(848\) 0 0
\(849\) 8.50000 + 14.7224i 0.291719 + 0.505273i
\(850\) 0 0
\(851\) 31.5000 + 54.5596i 1.07981 + 1.87028i
\(852\) 0 0
\(853\) 14.0000i 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000i 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 1.50000 2.59808i 0.0511199 0.0885422i
\(862\) 0 0
\(863\) 48.0000i 1.63394i −0.576681 0.816970i \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.92820 + 4.00000i −0.235294 + 0.135847i
\(868\) 0 0
\(869\) 12.0000 20.7846i 0.407072 0.705070i
\(870\) 0 0
\(871\) −17.5000 4.33013i −0.592965 0.146721i
\(872\) 0 0
\(873\) −29.4449 17.0000i −0.996558 0.575363i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.7224 8.50000i 0.497141 0.287025i −0.230391 0.973098i \(-0.574001\pi\)
0.727532 + 0.686074i \(0.240667\pi\)
\(878\) 0 0
\(879\) 3.00000 0.101187
\(880\) 0 0
\(881\) −1.50000 + 2.59808i −0.0505363 + 0.0875314i −0.890187 0.455595i \(-0.849426\pi\)
0.839651 + 0.543127i \(0.182760\pi\)
\(882\) 0 0
\(883\) 56.0000i 1.88455i −0.334840 0.942275i \(-0.608682\pi\)
0.334840 0.942275i \(-0.391318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.5788 16.5000i −0.959583 0.554016i −0.0635387 0.997979i \(-0.520239\pi\)
−0.896045 + 0.443964i \(0.853572\pi\)
\(888\) 0 0
\(889\) 17.0000 0.570162
\(890\) 0 0
\(891\) −1.50000 2.59808i −0.0502519 0.0870388i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 23.3827 + 22.5000i 0.780725 + 0.751253i
\(898\) 0 0
\(899\) −36.0000 + 62.3538i −1.20067 + 2.07962i
\(900\) 0 0
\(901\) 9.00000 + 15.5885i 0.299833 + 0.519327i
\(902\) 0 0
\(903\) −0.866025 + 0.500000i −0.0288195 + 0.0166390i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 47.6314 + 27.5000i 1.58157 + 0.913123i 0.994630 + 0.103495i \(0.0330027\pi\)
0.586945 + 0.809627i \(0.300331\pi\)
\(908\) 0 0
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3.50000 6.06218i −0.115454 0.199973i 0.802507 0.596643i \(-0.203499\pi\)
−0.917961 + 0.396670i \(0.870166\pi\)
\(920\) 0 0
\(921\) 2.00000 3.46410i 0.0659022 0.114146i
\(922\) 0 0
\(923\) −23.3827 22.5000i −0.769650 0.740597i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −13.8564 + 8.00000i −0.455104 + 0.262754i
\(928\) 0 0
\(929\) 1.50000 + 2.59808i 0.0492134 + 0.0852401i 0.889583 0.456774i \(-0.150995\pi\)
−0.840369 + 0.542014i \(0.817662\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 0 0
\(933\) 10.3923 + 6.00000i 0.340229 + 0.196431i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000i 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) −5.00000 + 8.66025i −0.163169 + 0.282617i
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −23.3827 + 13.5000i −0.761445 + 0.439620i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44.1673 + 25.5000i 1.43524 + 0.828639i 0.997514 0.0704677i \(-0.0224492\pi\)
0.437730 + 0.899106i \(0.355783\pi\)
\(948\) 0 0
\(949\) −2.00000 6.92820i −0.0649227 0.224899i
\(950\) 0 0
\(951\) 9.00000 15.5885i 0.291845 0.505490i
\(952\) 0 0
\(953\) −18.1865 + 10.5000i −0.589120 + 0.340128i −0.764749 0.644328i \(-0.777137\pi\)
0.175630 + 0.984456i \(0.443804\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 27.0000i 0.872786i
\(958\) 0 0
\(959\) 1.50000 2.59808i 0.0484375 0.0838963i
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000i 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) 0 0
\(969\) −7.50000 12.9904i −0.240935 0.417311i
\(970\) 0 0
\(971\) 7.50000 + 12.9904i 0.240686 + 0.416881i 0.960910 0.276861i \(-0.0892941\pi\)
−0.720224 + 0.693742i \(0.755961\pi\)
\(972\) 0 0
\(973\) −4.33013 2.50000i −0.138817 0.0801463i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.59808 + 1.50000i 0.0831198 + 0.0479893i 0.540984 0.841033i \(-0.318052\pi\)
−0.457864 + 0.889022i \(0.651385\pi\)
\(978\) 0 0
\(979\) 4.50000 + 7.79423i 0.143821 + 0.249105i
\(980\) 0 0
\(981\) 14.0000 + 24.2487i 0.446986 + 0.774202i
\(982\) 0 0
\(983\) 48.0000i 1.53096i 0.643458 + 0.765481i \(0.277499\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 24.5000 42.4352i 0.778268 1.34800i −0.154671 0.987966i \(-0.549432\pi\)
0.932939 0.360034i \(-0.117235\pi\)
\(992\) 0 0
\(993\) 19.0000i 0.602947i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.33013 2.50000i 0.137136 0.0791758i −0.429862 0.902895i \(-0.641438\pi\)
0.566999 + 0.823719i \(0.308104\pi\)
\(998\) 0 0
\(999\) 17.5000 30.3109i 0.553675 0.958994i
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 1300.2.bb.b.1049.2 4
5.2 odd 4 1300.2.i.d.1101.1 2
5.3 odd 4 260.2.i.a.61.1 2
5.4 even 2 inner 1300.2.bb.b.1049.1 4
13.3 even 3 inner 1300.2.bb.b.549.1 4
15.8 even 4 2340.2.q.f.1621.1 2
20.3 even 4 1040.2.q.i.321.1 2
65.3 odd 12 260.2.i.a.81.1 yes 2
65.29 even 6 inner 1300.2.bb.b.549.2 4
65.33 even 12 3380.2.f.d.3041.1 2
65.42 odd 12 1300.2.i.d.601.1 2
65.43 odd 12 3380.2.a.i.1.1 1
65.48 odd 12 3380.2.a.f.1.1 1
65.58 even 12 3380.2.f.d.3041.2 2
195.68 even 12 2340.2.q.f.2161.1 2
260.3 even 12 1040.2.q.i.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.a.61.1 2 5.3 odd 4
260.2.i.a.81.1 yes 2 65.3 odd 12
1040.2.q.i.81.1 2 260.3 even 12
1040.2.q.i.321.1 2 20.3 even 4
1300.2.i.d.601.1 2 65.42 odd 12
1300.2.i.d.1101.1 2 5.2 odd 4
1300.2.bb.b.549.1 4 13.3 even 3 inner
1300.2.bb.b.549.2 4 65.29 even 6 inner
1300.2.bb.b.1049.1 4 5.4 even 2 inner
1300.2.bb.b.1049.2 4 1.1 even 1 trivial
2340.2.q.f.1621.1 2 15.8 even 4
2340.2.q.f.2161.1 2 195.68 even 12
3380.2.a.f.1.1 1 65.48 odd 12
3380.2.a.i.1.1 1 65.43 odd 12
3380.2.f.d.3041.1 2 65.33 even 12
3380.2.f.d.3041.2 2 65.58 even 12