Properties

Label 2736.2.dc.e.449.8
Level $2736$
Weight $2$
Character 2736.449
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.8
Root \(4.76099 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.449
Dual form 2736.2.dc.e.1889.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.80341 - 1.04120i) q^{5} +4.37182 q^{7} +O(q^{10})\) \(q+(1.80341 - 1.04120i) q^{5} +4.37182 q^{7} -5.52720i q^{11} +(4.60625 + 2.65942i) q^{13} +(-0.666338 + 0.384710i) q^{17} +(2.87650 + 3.27502i) q^{19} +(5.78457 + 3.33973i) q^{23} +(-0.331814 + 0.574719i) q^{25} +(-1.50650 + 2.60933i) q^{29} -2.12291i q^{31} +(7.88418 - 4.55193i) q^{35} +8.39061i q^{37} +(-2.76363 - 4.78675i) q^{41} +(-5.71996 - 9.90726i) q^{43} +(3.03169 + 1.75035i) q^{47} +12.1128 q^{49} +(-3.59993 + 6.23526i) q^{53} +(-5.75491 - 9.96779i) q^{55} +(-1.46543 - 2.53820i) q^{59} +(-3.90848 + 6.76968i) q^{61} +11.0759 q^{65} +(-11.5296 - 6.65661i) q^{67} +(1.04929 + 1.81742i) q^{71} +(-0.258159 - 0.447144i) q^{73} -24.1639i q^{77} +(-3.23955 + 1.87035i) q^{79} -12.0996i q^{83} +(-0.801119 + 1.38758i) q^{85} +(-6.22380 + 10.7799i) q^{89} +(20.1377 + 11.6265i) q^{91} +(8.59744 + 2.91119i) q^{95} +(-4.74601 + 2.74011i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.80341 1.04120i 0.806508 0.465638i −0.0392335 0.999230i \(-0.512492\pi\)
0.845742 + 0.533592i \(0.179158\pi\)
\(6\) 0 0
\(7\) 4.37182 1.65239 0.826197 0.563382i \(-0.190500\pi\)
0.826197 + 0.563382i \(0.190500\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.52720i 1.66651i −0.552887 0.833256i \(-0.686474\pi\)
0.552887 0.833256i \(-0.313526\pi\)
\(12\) 0 0
\(13\) 4.60625 + 2.65942i 1.27754 + 0.737591i 0.976396 0.215986i \(-0.0692966\pi\)
0.301149 + 0.953577i \(0.402630\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.666338 + 0.384710i −0.161611 + 0.0933059i −0.578624 0.815594i \(-0.696410\pi\)
0.417013 + 0.908900i \(0.363077\pi\)
\(18\) 0 0
\(19\) 2.87650 + 3.27502i 0.659914 + 0.751341i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.78457 + 3.33973i 1.20617 + 0.696381i 0.961920 0.273332i \(-0.0881258\pi\)
0.244247 + 0.969713i \(0.421459\pi\)
\(24\) 0 0
\(25\) −0.331814 + 0.574719i −0.0663629 + 0.114944i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.50650 + 2.60933i −0.279749 + 0.484540i −0.971322 0.237766i \(-0.923585\pi\)
0.691573 + 0.722307i \(0.256918\pi\)
\(30\) 0 0
\(31\) 2.12291i 0.381286i −0.981659 0.190643i \(-0.938943\pi\)
0.981659 0.190643i \(-0.0610573\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.88418 4.55193i 1.33267 0.769417i
\(36\) 0 0
\(37\) 8.39061i 1.37941i 0.724092 + 0.689704i \(0.242259\pi\)
−0.724092 + 0.689704i \(0.757741\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.76363 4.78675i −0.431606 0.747564i 0.565405 0.824813i \(-0.308720\pi\)
−0.997012 + 0.0772489i \(0.975386\pi\)
\(42\) 0 0
\(43\) −5.71996 9.90726i −0.872285 1.51084i −0.859627 0.510923i \(-0.829304\pi\)
−0.0126586 0.999920i \(-0.504029\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.03169 + 1.75035i 0.442218 + 0.255315i 0.704538 0.709666i \(-0.251154\pi\)
−0.262320 + 0.964981i \(0.584488\pi\)
\(48\) 0 0
\(49\) 12.1128 1.73040
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.59993 + 6.23526i −0.494489 + 0.856479i −0.999980 0.00635234i \(-0.997978\pi\)
0.505491 + 0.862832i \(0.331311\pi\)
\(54\) 0 0
\(55\) −5.75491 9.96779i −0.775991 1.34406i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.46543 2.53820i −0.190783 0.330446i 0.754727 0.656039i \(-0.227769\pi\)
−0.945510 + 0.325593i \(0.894436\pi\)
\(60\) 0 0
\(61\) −3.90848 + 6.76968i −0.500429 + 0.866768i 0.499571 + 0.866273i \(0.333491\pi\)
−1.00000 0.000495417i \(0.999842\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.0759 1.37380
\(66\) 0 0
\(67\) −11.5296 6.65661i −1.40856 0.813234i −0.413313 0.910589i \(-0.635628\pi\)
−0.995250 + 0.0973549i \(0.968962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.04929 + 1.81742i 0.124527 + 0.215688i 0.921548 0.388264i \(-0.126925\pi\)
−0.797021 + 0.603952i \(0.793592\pi\)
\(72\) 0 0
\(73\) −0.258159 0.447144i −0.0302152 0.0523343i 0.850522 0.525939i \(-0.176286\pi\)
−0.880738 + 0.473605i \(0.842953\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 24.1639i 2.75373i
\(78\) 0 0
\(79\) −3.23955 + 1.87035i −0.364478 + 0.210431i −0.671043 0.741418i \(-0.734153\pi\)
0.306565 + 0.951850i \(0.400820\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0996i 1.32811i −0.747686 0.664053i \(-0.768835\pi\)
0.747686 0.664053i \(-0.231165\pi\)
\(84\) 0 0
\(85\) −0.801119 + 1.38758i −0.0868935 + 0.150504i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.22380 + 10.7799i −0.659721 + 1.14267i 0.320967 + 0.947091i \(0.395992\pi\)
−0.980688 + 0.195580i \(0.937341\pi\)
\(90\) 0 0
\(91\) 20.1377 + 11.6265i 2.11101 + 1.21879i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.59744 + 2.91119i 0.882079 + 0.298682i
\(96\) 0 0
\(97\) −4.74601 + 2.74011i −0.481884 + 0.278216i −0.721201 0.692725i \(-0.756410\pi\)
0.239317 + 0.970941i \(0.423076\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.1547 6.44016i −1.10993 0.640820i −0.171120 0.985250i \(-0.554739\pi\)
−0.938812 + 0.344430i \(0.888072\pi\)
\(102\) 0 0
\(103\) 17.7899i 1.75289i 0.481498 + 0.876447i \(0.340093\pi\)
−0.481498 + 0.876447i \(0.659907\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.6440 −1.51236 −0.756182 0.654361i \(-0.772938\pi\)
−0.756182 + 0.654361i \(0.772938\pi\)
\(108\) 0 0
\(109\) 14.9747 8.64567i 1.43432 0.828105i 0.436874 0.899523i \(-0.356086\pi\)
0.997446 + 0.0714177i \(0.0227523\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7728 1.20156 0.600782 0.799413i \(-0.294856\pi\)
0.600782 + 0.799413i \(0.294856\pi\)
\(114\) 0 0
\(115\) 13.9093 1.29705
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.91311 + 1.68188i −0.267044 + 0.154178i
\(120\) 0 0
\(121\) −19.5499 −1.77726
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.7939i 1.05488i
\(126\) 0 0
\(127\) −2.33607 1.34873i −0.207293 0.119681i 0.392760 0.919641i \(-0.371520\pi\)
−0.600053 + 0.799960i \(0.704854\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.82279 3.93914i 0.596110 0.344164i −0.171400 0.985202i \(-0.554829\pi\)
0.767510 + 0.641037i \(0.221496\pi\)
\(132\) 0 0
\(133\) 12.5755 + 14.3178i 1.09044 + 1.24151i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.8285 9.71596i −1.43776 0.830091i −0.440066 0.897966i \(-0.645045\pi\)
−0.997694 + 0.0678749i \(0.978378\pi\)
\(138\) 0 0
\(139\) 0.723513 1.25316i 0.0613676 0.106292i −0.833709 0.552204i \(-0.813787\pi\)
0.895077 + 0.445912i \(0.147120\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.6992 25.4597i 1.22920 2.12905i
\(144\) 0 0
\(145\) 6.27424i 0.521048i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.54085 3.19901i 0.453924 0.262073i −0.255562 0.966793i \(-0.582261\pi\)
0.709486 + 0.704720i \(0.248927\pi\)
\(150\) 0 0
\(151\) 0.0390540i 0.00317817i 0.999999 + 0.00158908i \(0.000505821\pi\)
−0.999999 + 0.00158908i \(0.999494\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.21037 3.82848i −0.177541 0.307511i
\(156\) 0 0
\(157\) −1.45016 2.51174i −0.115735 0.200459i 0.802338 0.596870i \(-0.203589\pi\)
−0.918073 + 0.396411i \(0.870256\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 25.2891 + 14.6007i 1.99306 + 1.15069i
\(162\) 0 0
\(163\) −9.55244 −0.748205 −0.374102 0.927387i \(-0.622049\pi\)
−0.374102 + 0.927387i \(0.622049\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.16818 5.48745i 0.245161 0.424632i −0.717016 0.697057i \(-0.754492\pi\)
0.962177 + 0.272425i \(0.0878257\pi\)
\(168\) 0 0
\(169\) 7.64505 + 13.2416i 0.588081 + 1.01859i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.2929 17.8278i −0.782555 1.35543i −0.930449 0.366422i \(-0.880583\pi\)
0.147893 0.989003i \(-0.452751\pi\)
\(174\) 0 0
\(175\) −1.45063 + 2.51257i −0.109658 + 0.189932i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.2259 1.13804 0.569018 0.822325i \(-0.307323\pi\)
0.569018 + 0.822325i \(0.307323\pi\)
\(180\) 0 0
\(181\) 7.03559 + 4.06200i 0.522951 + 0.301926i 0.738141 0.674646i \(-0.235704\pi\)
−0.215190 + 0.976572i \(0.569037\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.73628 + 15.1317i 0.642304 + 1.11250i
\(186\) 0 0
\(187\) 2.12637 + 3.68298i 0.155496 + 0.269326i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.6421i 0.842392i 0.906970 + 0.421196i \(0.138390\pi\)
−0.906970 + 0.421196i \(0.861610\pi\)
\(192\) 0 0
\(193\) 7.58184 4.37738i 0.545753 0.315090i −0.201655 0.979457i \(-0.564632\pi\)
0.747407 + 0.664366i \(0.231298\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0975i 1.57438i −0.616711 0.787189i \(-0.711535\pi\)
0.616711 0.787189i \(-0.288465\pi\)
\(198\) 0 0
\(199\) 4.20866 7.28961i 0.298344 0.516747i −0.677413 0.735603i \(-0.736899\pi\)
0.975757 + 0.218856i \(0.0702325\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.58613 + 11.4075i −0.462256 + 0.800651i
\(204\) 0 0
\(205\) −9.96790 5.75497i −0.696188 0.401945i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 18.1017 15.8990i 1.25212 1.09975i
\(210\) 0 0
\(211\) 13.4313 7.75458i 0.924651 0.533847i 0.0395350 0.999218i \(-0.487412\pi\)
0.885116 + 0.465371i \(0.154079\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −20.6308 11.9112i −1.40701 0.812338i
\(216\) 0 0
\(217\) 9.28099i 0.630035i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.09243 −0.275286
\(222\) 0 0
\(223\) 2.13784 1.23428i 0.143160 0.0826536i −0.426709 0.904389i \(-0.640327\pi\)
0.569869 + 0.821735i \(0.306994\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.14861 0.408097 0.204049 0.978961i \(-0.434590\pi\)
0.204049 + 0.978961i \(0.434590\pi\)
\(228\) 0 0
\(229\) −0.0945480 −0.00624791 −0.00312395 0.999995i \(-0.500994\pi\)
−0.00312395 + 0.999995i \(0.500994\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.81625 1.04861i 0.118986 0.0686968i −0.439326 0.898328i \(-0.644783\pi\)
0.558312 + 0.829631i \(0.311449\pi\)
\(234\) 0 0
\(235\) 7.28984 0.475537
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.14199i 0.461977i −0.972956 0.230988i \(-0.925804\pi\)
0.972956 0.230988i \(-0.0741960\pi\)
\(240\) 0 0
\(241\) −22.6383 13.0702i −1.45826 0.841928i −0.459336 0.888263i \(-0.651913\pi\)
−0.998926 + 0.0463346i \(0.985246\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.8444 12.6118i 1.39558 0.805741i
\(246\) 0 0
\(247\) 4.54022 + 22.7354i 0.288887 + 1.44662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.1535 + 7.59418i 0.830242 + 0.479340i 0.853936 0.520379i \(-0.174209\pi\)
−0.0236935 + 0.999719i \(0.507543\pi\)
\(252\) 0 0
\(253\) 18.4593 31.9725i 1.16053 2.01009i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.83058 + 3.17065i −0.114188 + 0.197780i −0.917455 0.397840i \(-0.869760\pi\)
0.803267 + 0.595620i \(0.203093\pi\)
\(258\) 0 0
\(259\) 36.6822i 2.27932i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.3315 + 8.27427i −0.883715 + 0.510213i −0.871882 0.489717i \(-0.837100\pi\)
−0.0118336 + 0.999930i \(0.503767\pi\)
\(264\) 0 0
\(265\) 14.9930i 0.921010i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.64612 9.77936i −0.344250 0.596258i 0.640967 0.767568i \(-0.278533\pi\)
−0.985217 + 0.171310i \(0.945200\pi\)
\(270\) 0 0
\(271\) 6.62580 + 11.4762i 0.402489 + 0.697131i 0.994026 0.109147i \(-0.0348119\pi\)
−0.591537 + 0.806278i \(0.701479\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.17659 + 1.83400i 0.191555 + 0.110595i
\(276\) 0 0
\(277\) 13.8936 0.834785 0.417392 0.908726i \(-0.362944\pi\)
0.417392 + 0.908726i \(0.362944\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.94844 6.83889i 0.235544 0.407974i −0.723887 0.689919i \(-0.757646\pi\)
0.959431 + 0.281945i \(0.0909795\pi\)
\(282\) 0 0
\(283\) −8.70512 15.0777i −0.517466 0.896277i −0.999794 0.0202863i \(-0.993542\pi\)
0.482329 0.875990i \(-0.339791\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0821 20.9268i −0.713183 1.23527i
\(288\) 0 0
\(289\) −8.20400 + 14.2097i −0.482588 + 0.835867i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.15383 0.417931 0.208966 0.977923i \(-0.432990\pi\)
0.208966 + 0.977923i \(0.432990\pi\)
\(294\) 0 0
\(295\) −5.28554 3.05161i −0.307736 0.177671i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.7635 + 30.7672i 1.02729 + 1.77932i
\(300\) 0 0
\(301\) −25.0066 43.3128i −1.44136 2.49651i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.2780i 0.932075i
\(306\) 0 0
\(307\) −14.0741 + 8.12569i −0.803252 + 0.463758i −0.844607 0.535387i \(-0.820166\pi\)
0.0413552 + 0.999145i \(0.486832\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.6708i 0.718498i 0.933242 + 0.359249i \(0.116967\pi\)
−0.933242 + 0.359249i \(0.883033\pi\)
\(312\) 0 0
\(313\) 16.3208 28.2684i 0.922506 1.59783i 0.126982 0.991905i \(-0.459471\pi\)
0.795524 0.605922i \(-0.207196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.20924 + 5.55856i −0.180249 + 0.312200i −0.941965 0.335711i \(-0.891024\pi\)
0.761716 + 0.647910i \(0.224357\pi\)
\(318\) 0 0
\(319\) 14.4223 + 8.32670i 0.807492 + 0.466206i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.17665 1.07565i −0.176754 0.0598509i
\(324\) 0 0
\(325\) −3.05684 + 1.76487i −0.169563 + 0.0978973i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.2540 + 7.65221i 0.730718 + 0.421880i
\(330\) 0 0
\(331\) 5.75686i 0.316426i −0.987405 0.158213i \(-0.949427\pi\)
0.987405 0.158213i \(-0.0505732\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.7234 −1.51469
\(336\) 0 0
\(337\) −19.7540 + 11.4050i −1.07607 + 0.621269i −0.929834 0.367980i \(-0.880049\pi\)
−0.146237 + 0.989250i \(0.546716\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11.7338 −0.635418
\(342\) 0 0
\(343\) 22.3523 1.20691
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.6126 + 11.3233i −1.05286 + 0.607867i −0.923448 0.383724i \(-0.874641\pi\)
−0.129409 + 0.991591i \(0.541308\pi\)
\(348\) 0 0
\(349\) 34.7197 1.85850 0.929251 0.369448i \(-0.120453\pi\)
0.929251 + 0.369448i \(0.120453\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.6750i 1.47299i 0.676441 + 0.736497i \(0.263521\pi\)
−0.676441 + 0.736497i \(0.736479\pi\)
\(354\) 0 0
\(355\) 3.78458 + 2.18503i 0.200865 + 0.115969i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.7075 + 11.3782i −1.04012 + 0.600516i −0.919868 0.392227i \(-0.871705\pi\)
−0.120256 + 0.992743i \(0.538371\pi\)
\(360\) 0 0
\(361\) −2.45152 + 18.8412i −0.129028 + 0.991641i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.931131 0.537589i −0.0487376 0.0281387i
\(366\) 0 0
\(367\) 10.7076 18.5461i 0.558933 0.968101i −0.438653 0.898657i \(-0.644544\pi\)
0.997586 0.0694441i \(-0.0221226\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.7383 + 27.2595i −0.817090 + 1.41524i
\(372\) 0 0
\(373\) 19.1459i 0.991339i 0.868511 + 0.495669i \(0.165077\pi\)
−0.868511 + 0.495669i \(0.834923\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.8786 + 8.01282i −0.714785 + 0.412681i
\(378\) 0 0
\(379\) 3.72739i 0.191463i −0.995407 0.0957315i \(-0.969481\pi\)
0.995407 0.0957315i \(-0.0305190\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0151 20.8108i −0.613945 1.06338i −0.990569 0.137017i \(-0.956248\pi\)
0.376624 0.926366i \(-0.377085\pi\)
\(384\) 0 0
\(385\) −25.1594 43.5774i −1.28224 2.22091i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.67756 + 3.85529i 0.338566 + 0.195471i 0.659638 0.751584i \(-0.270710\pi\)
−0.321072 + 0.947055i \(0.604043\pi\)
\(390\) 0 0
\(391\) −5.13931 −0.259906
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.89482 + 6.74602i −0.195970 + 0.339429i
\(396\) 0 0
\(397\) −2.76925 4.79648i −0.138985 0.240728i 0.788128 0.615511i \(-0.211050\pi\)
−0.927112 + 0.374783i \(0.877717\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.67281 + 4.62944i 0.133474 + 0.231183i 0.925013 0.379935i \(-0.124053\pi\)
−0.791540 + 0.611118i \(0.790720\pi\)
\(402\) 0 0
\(403\) 5.64572 9.77867i 0.281233 0.487110i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 46.3766 2.29880
\(408\) 0 0
\(409\) −9.57997 5.53100i −0.473699 0.273490i 0.244088 0.969753i \(-0.421511\pi\)
−0.717787 + 0.696263i \(0.754845\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.40660 11.0966i −0.315248 0.546026i
\(414\) 0 0
\(415\) −12.5981 21.8205i −0.618416 1.07113i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.60015i 0.127025i −0.997981 0.0635127i \(-0.979770\pi\)
0.997981 0.0635127i \(-0.0202303\pi\)
\(420\) 0 0
\(421\) −6.43902 + 3.71757i −0.313819 + 0.181183i −0.648634 0.761100i \(-0.724659\pi\)
0.334815 + 0.942284i \(0.391326\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.510609i 0.0247682i
\(426\) 0 0
\(427\) −17.0872 + 29.5958i −0.826905 + 1.43224i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.8740 + 22.2984i −0.620118 + 1.07408i 0.369345 + 0.929292i \(0.379582\pi\)
−0.989463 + 0.144784i \(0.953751\pi\)
\(432\) 0 0
\(433\) 11.0457 + 6.37726i 0.530825 + 0.306472i 0.741352 0.671116i \(-0.234185\pi\)
−0.210528 + 0.977588i \(0.567518\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.70164 + 28.5513i 0.272747 + 1.36579i
\(438\) 0 0
\(439\) 20.9241 12.0805i 0.998651 0.576572i 0.0908025 0.995869i \(-0.471057\pi\)
0.907849 + 0.419297i \(0.137723\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.5348 14.7425i −1.21320 0.700439i −0.249741 0.968313i \(-0.580346\pi\)
−0.963454 + 0.267874i \(0.913679\pi\)
\(444\) 0 0
\(445\) 25.9208i 1.22876i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.5059 −1.53405 −0.767023 0.641619i \(-0.778263\pi\)
−0.767023 + 0.641619i \(0.778263\pi\)
\(450\) 0 0
\(451\) −26.4573 + 15.2751i −1.24583 + 0.719278i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 48.4220 2.27006
\(456\) 0 0
\(457\) 10.6325 0.497367 0.248683 0.968585i \(-0.420002\pi\)
0.248683 + 0.968585i \(0.420002\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.778023 + 0.449192i −0.0362362 + 0.0209210i −0.518009 0.855375i \(-0.673327\pi\)
0.481772 + 0.876296i \(0.339993\pi\)
\(462\) 0 0
\(463\) −40.1423 −1.86557 −0.932786 0.360430i \(-0.882630\pi\)
−0.932786 + 0.360430i \(0.882630\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.5765i 1.22981i −0.788600 0.614907i \(-0.789194\pi\)
0.788600 0.614907i \(-0.210806\pi\)
\(468\) 0 0
\(469\) −50.4053 29.1015i −2.32750 1.34378i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −54.7594 + 31.6153i −2.51784 + 1.45367i
\(474\) 0 0
\(475\) −2.83668 + 0.566480i −0.130156 + 0.0259919i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.16489 0.672547i −0.0532250 0.0307295i 0.473151 0.880981i \(-0.343116\pi\)
−0.526376 + 0.850252i \(0.676450\pi\)
\(480\) 0 0
\(481\) −22.3142 + 38.6493i −1.01744 + 1.76226i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.70599 + 9.88307i −0.259096 + 0.448767i
\(486\) 0 0
\(487\) 10.7505i 0.487151i 0.969882 + 0.243576i \(0.0783204\pi\)
−0.969882 + 0.243576i \(0.921680\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −33.6979 + 19.4555i −1.52077 + 0.878015i −0.521066 + 0.853516i \(0.674466\pi\)
−0.999700 + 0.0244988i \(0.992201\pi\)
\(492\) 0 0
\(493\) 2.31826i 0.104409i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.58729 + 7.94542i 0.205768 + 0.356401i
\(498\) 0 0
\(499\) −9.92455 17.1898i −0.444284 0.769523i 0.553718 0.832704i \(-0.313209\pi\)
−0.998002 + 0.0631817i \(0.979875\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.74748 2.16361i −0.167092 0.0964706i 0.414122 0.910221i \(-0.364089\pi\)
−0.581214 + 0.813751i \(0.697422\pi\)
\(504\) 0 0
\(505\) −26.8219 −1.19356
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.31800 12.6752i 0.324365 0.561816i −0.657019 0.753874i \(-0.728183\pi\)
0.981384 + 0.192058i \(0.0615161\pi\)
\(510\) 0 0
\(511\) −1.12862 1.95483i −0.0499274 0.0864768i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.5228 + 32.0825i 0.816214 + 1.41372i
\(516\) 0 0
\(517\) 9.67453 16.7568i 0.425485 0.736962i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.879568 −0.0385346 −0.0192673 0.999814i \(-0.506133\pi\)
−0.0192673 + 0.999814i \(0.506133\pi\)
\(522\) 0 0
\(523\) 4.45456 + 2.57184i 0.194784 + 0.112459i 0.594220 0.804302i \(-0.297461\pi\)
−0.399436 + 0.916761i \(0.630794\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.816706 + 1.41458i 0.0355763 + 0.0616199i
\(528\) 0 0
\(529\) 10.8075 + 18.7192i 0.469893 + 0.813878i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29.3986i 1.27340i
\(534\) 0 0
\(535\) −28.2125 + 16.2885i −1.21973 + 0.704214i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 66.9500i 2.88374i
\(540\) 0 0
\(541\) 10.5854 18.3344i 0.455100 0.788257i −0.543594 0.839348i \(-0.682937\pi\)
0.998694 + 0.0510917i \(0.0162701\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.0037 31.1833i 0.771194 1.33575i
\(546\) 0 0
\(547\) −29.6167 17.0992i −1.26632 0.731110i −0.292030 0.956409i \(-0.594331\pi\)
−0.974290 + 0.225300i \(0.927664\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.8790 + 2.57192i −0.548665 + 0.109567i
\(552\) 0 0
\(553\) −14.1627 + 8.17685i −0.602260 + 0.347715i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.4681 + 16.4360i 1.20623 + 0.696418i 0.961934 0.273283i \(-0.0881095\pi\)
0.244297 + 0.969700i \(0.421443\pi\)
\(558\) 0 0
\(559\) 60.8471i 2.57356i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 35.0701 1.47803 0.739013 0.673691i \(-0.235292\pi\)
0.739013 + 0.673691i \(0.235292\pi\)
\(564\) 0 0
\(565\) 23.0346 13.2990i 0.969072 0.559494i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.24898 0.387737 0.193869 0.981027i \(-0.437896\pi\)
0.193869 + 0.981027i \(0.437896\pi\)
\(570\) 0 0
\(571\) 7.77726 0.325468 0.162734 0.986670i \(-0.447969\pi\)
0.162734 + 0.986670i \(0.447969\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.83881 + 2.21634i −0.160089 + 0.0924276i
\(576\) 0 0
\(577\) −5.35699 −0.223014 −0.111507 0.993764i \(-0.535568\pi\)
−0.111507 + 0.993764i \(0.535568\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 52.8974i 2.19455i
\(582\) 0 0
\(583\) 34.4635 + 19.8975i 1.42733 + 0.824072i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.44952 + 3.14628i −0.224926 + 0.129861i −0.608229 0.793762i \(-0.708120\pi\)
0.383303 + 0.923623i \(0.374786\pi\)
\(588\) 0 0
\(589\) 6.95258 6.10655i 0.286476 0.251616i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.350867 0.202573i −0.0144084 0.00831868i 0.492779 0.870155i \(-0.335981\pi\)
−0.507187 + 0.861836i \(0.669315\pi\)
\(594\) 0 0
\(595\) −3.50235 + 6.06625i −0.143582 + 0.248692i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.23255 + 10.7951i −0.254655 + 0.441075i −0.964802 0.262978i \(-0.915295\pi\)
0.710147 + 0.704054i \(0.248629\pi\)
\(600\) 0 0
\(601\) 33.1841i 1.35361i −0.736164 0.676803i \(-0.763365\pi\)
0.736164 0.676803i \(-0.236635\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −35.2565 + 20.3553i −1.43338 + 0.827562i
\(606\) 0 0
\(607\) 17.0058i 0.690245i −0.938558 0.345123i \(-0.887837\pi\)
0.938558 0.345123i \(-0.112163\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.30983 + 16.1251i 0.376636 + 0.652352i
\(612\) 0 0
\(613\) 0.650804 + 1.12723i 0.0262857 + 0.0455282i 0.878869 0.477063i \(-0.158299\pi\)
−0.852583 + 0.522591i \(0.824965\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.5182 + 7.80476i 0.544224 + 0.314208i 0.746789 0.665061i \(-0.231594\pi\)
−0.202565 + 0.979269i \(0.564928\pi\)
\(618\) 0 0
\(619\) 1.29498 0.0520496 0.0260248 0.999661i \(-0.491715\pi\)
0.0260248 + 0.999661i \(0.491715\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −27.2093 + 47.1279i −1.09012 + 1.88814i
\(624\) 0 0
\(625\) 10.6207 + 18.3956i 0.424829 + 0.735826i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.22795 5.59098i −0.128707 0.222927i
\(630\) 0 0
\(631\) 0.340598 0.589933i 0.0135590 0.0234849i −0.859166 0.511696i \(-0.829017\pi\)
0.872725 + 0.488212i \(0.162351\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.61719 −0.222911
\(636\) 0 0
\(637\) 55.7947 + 32.2131i 2.21067 + 1.27633i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.01662 + 10.4211i 0.237642 + 0.411608i 0.960037 0.279872i \(-0.0902921\pi\)
−0.722395 + 0.691481i \(0.756959\pi\)
\(642\) 0 0
\(643\) −5.80977 10.0628i −0.229115 0.396839i 0.728431 0.685119i \(-0.240250\pi\)
−0.957546 + 0.288280i \(0.906917\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.9986i 1.10074i −0.834921 0.550370i \(-0.814487\pi\)
0.834921 0.550370i \(-0.185513\pi\)
\(648\) 0 0
\(649\) −14.0291 + 8.09973i −0.550692 + 0.317942i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.7203i 1.08478i −0.840127 0.542389i \(-0.817520\pi\)
0.840127 0.542389i \(-0.182480\pi\)
\(654\) 0 0
\(655\) 8.20284 14.2077i 0.320512 0.555142i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.98534 15.5631i 0.350019 0.606251i −0.636233 0.771497i \(-0.719508\pi\)
0.986252 + 0.165246i \(0.0528417\pi\)
\(660\) 0 0
\(661\) −2.53986 1.46639i −0.0987892 0.0570360i 0.449791 0.893134i \(-0.351498\pi\)
−0.548581 + 0.836098i \(0.684832\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 37.5865 + 12.7272i 1.45754 + 0.493540i
\(666\) 0 0
\(667\) −17.4289 + 10.0626i −0.674849 + 0.389624i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 37.4173 + 21.6029i 1.44448 + 0.833971i
\(672\) 0 0
\(673\) 41.1188i 1.58501i 0.609863 + 0.792507i \(0.291224\pi\)
−0.609863 + 0.792507i \(0.708776\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.1236 −0.504380 −0.252190 0.967678i \(-0.581151\pi\)
−0.252190 + 0.967678i \(0.581151\pi\)
\(678\) 0 0
\(679\) −20.7487 + 11.9793i −0.796262 + 0.459722i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.3325 0.433626 0.216813 0.976213i \(-0.430434\pi\)
0.216813 + 0.976213i \(0.430434\pi\)
\(684\) 0 0
\(685\) −40.4650 −1.54609
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −33.1644 + 19.1475i −1.26346 + 0.729461i
\(690\) 0 0
\(691\) 6.02314 0.229131 0.114566 0.993416i \(-0.463452\pi\)
0.114566 + 0.993416i \(0.463452\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.01328i 0.114300i
\(696\) 0 0
\(697\) 3.68302 + 2.12639i 0.139504 + 0.0805429i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −38.0784 + 21.9845i −1.43820 + 0.830345i −0.997725 0.0674200i \(-0.978523\pi\)
−0.440475 + 0.897765i \(0.645190\pi\)
\(702\) 0 0
\(703\) −27.4794 + 24.1356i −1.03641 + 0.910290i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −48.7663 28.1552i −1.83404 1.05889i
\(708\) 0 0
\(709\) 14.2700 24.7164i 0.535921 0.928243i −0.463197 0.886255i \(-0.653298\pi\)
0.999118 0.0419874i \(-0.0133689\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.08994 12.2801i 0.265520 0.459895i
\(714\) 0 0
\(715\) 61.2189i 2.28946i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.27534 + 4.77777i −0.308618 + 0.178181i −0.646308 0.763077i \(-0.723688\pi\)
0.337690 + 0.941257i \(0.390355\pi\)
\(720\) 0 0
\(721\) 77.7744i 2.89647i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.999754 1.73162i −0.0371299 0.0643109i
\(726\) 0 0
\(727\) 10.2380 + 17.7327i 0.379705 + 0.657669i 0.991019 0.133719i \(-0.0426920\pi\)
−0.611314 + 0.791388i \(0.709359\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.62285 + 4.40105i 0.281941 + 0.162779i
\(732\) 0 0
\(733\) −14.6915 −0.542641 −0.271321 0.962489i \(-0.587460\pi\)
−0.271321 + 0.962489i \(0.587460\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.7924 + 63.7263i −1.35527 + 2.34739i
\(738\) 0 0
\(739\) −13.4390 23.2770i −0.494361 0.856259i 0.505617 0.862758i \(-0.331265\pi\)
−0.999979 + 0.00649866i \(0.997931\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.9369 + 34.5318i 0.731415 + 1.26685i 0.956279 + 0.292457i \(0.0944729\pi\)
−0.224864 + 0.974390i \(0.572194\pi\)
\(744\) 0 0
\(745\) 6.66160 11.5382i 0.244062 0.422728i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −68.3928 −2.49902
\(750\) 0 0
\(751\) 7.26488 + 4.19438i 0.265099 + 0.153055i 0.626658 0.779294i \(-0.284422\pi\)
−0.361559 + 0.932349i \(0.617755\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.0406629 + 0.0704302i 0.00147987 + 0.00256322i
\(756\) 0 0
\(757\) 12.9281 + 22.3922i 0.469881 + 0.813857i 0.999407 0.0344364i \(-0.0109636\pi\)
−0.529526 + 0.848294i \(0.677630\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 37.3997i 1.35574i 0.735184 + 0.677868i \(0.237096\pi\)
−0.735184 + 0.677868i \(0.762904\pi\)
\(762\) 0 0
\(763\) 65.4669 37.7973i 2.37006 1.36836i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.5888i 0.562879i
\(768\) 0 0
\(769\) 25.3762 43.9529i 0.915089 1.58498i 0.108319 0.994116i \(-0.465453\pi\)
0.806771 0.590865i \(-0.201213\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.5326 + 32.0994i −0.666571 + 1.15454i 0.312285 + 0.949988i \(0.398905\pi\)
−0.978857 + 0.204547i \(0.934428\pi\)
\(774\) 0 0
\(775\) 1.22008 + 0.704413i 0.0438265 + 0.0253032i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.72712 22.8200i 0.276853 0.817612i
\(780\) 0 0
\(781\) 10.0452 5.79961i 0.359446 0.207526i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.23044 3.01980i −0.186683 0.107781i
\(786\) 0 0
\(787\) 49.5627i 1.76672i −0.468696 0.883360i \(-0.655276\pi\)
0.468696 0.883360i \(-0.344724\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 55.8404 1.98546
\(792\) 0 0
\(793\) −36.0069 + 20.7886i −1.27864 + 0.738224i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.3979 −1.28928 −0.644639 0.764487i \(-0.722992\pi\)
−0.644639 + 0.764487i \(0.722992\pi\)
\(798\) 0 0
\(799\) −2.69351 −0.0952895
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.47145 + 1.42690i −0.0872157 + 0.0503540i
\(804\) 0 0
\(805\) 60.8088 2.14323
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.4361i 1.10524i 0.833435 + 0.552618i \(0.186371\pi\)
−0.833435 + 0.552618i \(0.813629\pi\)
\(810\) 0 0
\(811\) 15.0476 + 8.68771i 0.528391 + 0.305067i 0.740361 0.672209i \(-0.234655\pi\)
−0.211970 + 0.977276i \(0.567988\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −17.2269 + 9.94598i −0.603433 + 0.348392i
\(816\) 0 0
\(817\) 15.9930 47.2312i 0.559525 1.65241i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.8876 + 9.17269i 0.554480 + 0.320129i 0.750927 0.660385i \(-0.229607\pi\)
−0.196447 + 0.980514i \(0.562940\pi\)
\(822\) 0 0
\(823\) −6.79392 + 11.7674i −0.236821 + 0.410186i −0.959800 0.280683i \(-0.909439\pi\)
0.722979 + 0.690870i \(0.242772\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.2548 21.2259i 0.426140 0.738096i −0.570386 0.821376i \(-0.693207\pi\)
0.996526 + 0.0832809i \(0.0265399\pi\)
\(828\) 0 0
\(829\) 48.7193i 1.69209i 0.533111 + 0.846045i \(0.321023\pi\)
−0.533111 + 0.846045i \(0.678977\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.07123 + 4.65993i −0.279652 + 0.161457i
\(834\) 0 0
\(835\) 13.1948i 0.456626i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.76438 + 9.98419i 0.199008 + 0.344693i 0.948207 0.317653i \(-0.102895\pi\)
−0.749199 + 0.662345i \(0.769561\pi\)
\(840\) 0 0
\(841\) 9.96094 + 17.2528i 0.343481 + 0.594926i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 27.5743 + 15.9200i 0.948584 + 0.547665i
\(846\) 0 0
\(847\) −85.4687 −2.93674
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −28.0223 + 48.5361i −0.960593 + 1.66380i
\(852\) 0 0
\(853\) −22.9540 39.7574i −0.785929 1.36127i −0.928443 0.371476i \(-0.878852\pi\)
0.142514 0.989793i \(-0.454482\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.77668 + 13.4696i 0.265646 + 0.460113i 0.967733 0.251979i \(-0.0810813\pi\)
−0.702086 + 0.712092i \(0.747748\pi\)
\(858\) 0 0
\(859\) −2.93347 + 5.08091i −0.100089 + 0.173359i −0.911721 0.410810i \(-0.865246\pi\)
0.811632 + 0.584169i \(0.198579\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.34756 −0.318195 −0.159097 0.987263i \(-0.550858\pi\)
−0.159097 + 0.987263i \(0.550858\pi\)
\(864\) 0 0
\(865\) −37.1246 21.4339i −1.26227 0.728775i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.3378 + 17.9056i 0.350686 + 0.607407i
\(870\) 0 0
\(871\) −35.4055 61.3241i −1.19967 2.07789i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 51.5609i 1.74308i
\(876\) 0 0
\(877\) −29.6636 + 17.1263i −1.00167 + 0.578313i −0.908741 0.417360i \(-0.862955\pi\)
−0.0929261 + 0.995673i \(0.529622\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.43700i 0.317941i 0.987283 + 0.158970i \(0.0508174\pi\)
−0.987283 + 0.158970i \(0.949183\pi\)
\(882\) 0 0
\(883\) −15.3774 + 26.6345i −0.517492 + 0.896322i 0.482302 + 0.876005i \(0.339801\pi\)
−0.999794 + 0.0203171i \(0.993532\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.6841 + 46.2181i −0.895963 + 1.55185i −0.0633545 + 0.997991i \(0.520180\pi\)
−0.832608 + 0.553862i \(0.813153\pi\)
\(888\) 0 0
\(889\) −10.2129 5.89642i −0.342529 0.197759i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.98823 + 14.9637i 0.0999973 + 0.500742i
\(894\) 0 0
\(895\) 27.4585 15.8532i 0.917836 0.529913i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.53938 + 3.19816i 0.184749 + 0.106665i
\(900\) 0 0
\(901\) 5.53972i 0.184555i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.9174 0.562353
\(906\) 0 0
\(907\) −19.8692 + 11.4715i −0.659746 + 0.380905i −0.792180 0.610287i \(-0.791054\pi\)
0.132434 + 0.991192i \(0.457721\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.5125 0.712740 0.356370 0.934345i \(-0.384014\pi\)
0.356370 + 0.934345i \(0.384014\pi\)
\(912\) 0 0
\(913\) −66.8770 −2.21331
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 29.8280 17.2212i 0.985008 0.568694i
\(918\) 0 0
\(919\) 7.06126 0.232929 0.116465 0.993195i \(-0.462844\pi\)
0.116465 + 0.993195i \(0.462844\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11.1620i 0.367401i
\(924\) 0 0
\(925\) −4.82224 2.78412i −0.158554 0.0915414i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.29476 0.747529i 0.0424796 0.0245256i −0.478610 0.878028i \(-0.658859\pi\)
0.521089 + 0.853502i \(0.325526\pi\)
\(930\) 0 0
\(931\) 34.8425 + 39.6697i 1.14192 + 1.30012i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.66942 + 4.42794i 0.250817 + 0.144809i
\(936\) 0 0
\(937\) 0.844255 1.46229i 0.0275806 0.0477710i −0.851906 0.523695i \(-0.824553\pi\)
0.879486 + 0.475924i \(0.157886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11.7064 20.2762i 0.381619 0.660984i −0.609675 0.792652i \(-0.708700\pi\)
0.991294 + 0.131668i \(0.0420333\pi\)
\(942\) 0 0
\(943\) 36.9191i 1.20225i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.2879 21.5282i 1.21170 0.699573i 0.248567 0.968615i \(-0.420040\pi\)
0.963128 + 0.269042i \(0.0867071\pi\)
\(948\) 0 0
\(949\) 2.74621i 0.0891459i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.51936 + 14.7560i 0.275969 + 0.477993i 0.970379 0.241587i \(-0.0776679\pi\)
−0.694410 + 0.719580i \(0.744335\pi\)
\(954\) 0 0
\(955\) 12.1217 + 20.9954i 0.392249 + 0.679396i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −73.5714 42.4765i −2.37574 1.37164i
\(960\) 0 0
\(961\) 26.4932 0.854621
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.11543 15.7884i 0.293436 0.508246i
\(966\) 0 0
\(967\) −1.47237 2.55021i −0.0473481 0.0820094i 0.841380 0.540444i \(-0.181744\pi\)
−0.888728 + 0.458435i \(0.848410\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.47923 12.9544i −0.240020 0.415727i 0.720700 0.693247i \(-0.243821\pi\)
−0.960720 + 0.277521i \(0.910487\pi\)
\(972\) 0 0
\(973\) 3.16307 5.47860i 0.101403 0.175636i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.7678 0.984350 0.492175 0.870496i \(-0.336202\pi\)
0.492175 + 0.870496i \(0.336202\pi\)
\(978\) 0 0
\(979\) 59.5828 + 34.4002i 1.90428 + 1.09943i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.6453 39.2228i −0.722273 1.25101i −0.960087 0.279702i \(-0.909764\pi\)
0.237814 0.971311i \(-0.423569\pi\)
\(984\) 0 0
\(985\) −23.0078 39.8507i −0.733090 1.26975i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 76.4123i 2.42977i
\(990\) 0 0
\(991\) −3.19630 + 1.84539i −0.101534 + 0.0586206i −0.549907 0.835226i \(-0.685337\pi\)
0.448373 + 0.893846i \(0.352004\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.5282i 0.555681i
\(996\) 0 0
\(997\) 3.38079 5.85570i 0.107071 0.185452i −0.807512 0.589852i \(-0.799186\pi\)
0.914582 + 0.404400i \(0.132520\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2736.2.dc.e.449.8 20
3.2 odd 2 2736.2.dc.f.449.3 20
4.3 odd 2 1368.2.cu.a.449.8 20
12.11 even 2 1368.2.cu.b.449.3 yes 20
19.8 odd 6 2736.2.dc.f.1889.3 20
57.8 even 6 inner 2736.2.dc.e.1889.8 20
76.27 even 6 1368.2.cu.b.521.3 yes 20
228.179 odd 6 1368.2.cu.a.521.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.cu.a.449.8 20 4.3 odd 2
1368.2.cu.a.521.8 yes 20 228.179 odd 6
1368.2.cu.b.449.3 yes 20 12.11 even 2
1368.2.cu.b.521.3 yes 20 76.27 even 6
2736.2.dc.e.449.8 20 1.1 even 1 trivial
2736.2.dc.e.1889.8 20 57.8 even 6 inner
2736.2.dc.f.449.3 20 3.2 odd 2
2736.2.dc.f.1889.3 20 19.8 odd 6