Properties

Label 1380.2.t.a.1057.13
Level $1380$
Weight $2$
Character 1380.1057
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
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] = [1380,2,Mod(1057,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.t (of order \(4\), degree \(2\), 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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.13
Character \(\chi\) \(=\) 1380.1057
Dual form 1380.2.t.a.1333.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(-0.951596 + 2.02348i) q^{5} +(3.48236 - 3.48236i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +(-0.951596 + 2.02348i) q^{5} +(3.48236 - 3.48236i) q^{7} -1.00000i q^{9} -2.73277i q^{11} +(-0.367971 + 0.367971i) q^{13} +(0.757935 + 2.10370i) q^{15} +(5.02066 - 5.02066i) q^{17} -5.91117 q^{19} -4.92480i q^{21} +(-2.82390 + 3.87629i) q^{23} +(-3.18893 - 3.85107i) q^{25} +(-0.707107 - 0.707107i) q^{27} +2.00084i q^{29} +2.49808 q^{31} +(-1.93236 - 1.93236i) q^{33} +(3.73268 + 10.3603i) q^{35} +(6.38531 - 6.38531i) q^{37} +0.520390i q^{39} -6.05805 q^{41} +(-2.07971 - 2.07971i) q^{43} +(2.02348 + 0.951596i) q^{45} +(4.01416 + 4.01416i) q^{47} -17.2536i q^{49} -7.10029i q^{51} +(-0.794878 - 0.794878i) q^{53} +(5.52971 + 2.60050i) q^{55} +(-4.17983 + 4.17983i) q^{57} +5.00908i q^{59} -9.72089i q^{61} +(-3.48236 - 3.48236i) q^{63} +(-0.394422 - 1.09474i) q^{65} +(2.09702 - 2.09702i) q^{67} +(0.744154 + 4.73775i) q^{69} +9.35585 q^{71} +(8.18100 - 8.18100i) q^{73} +(-4.97803 - 0.468202i) q^{75} +(-9.51650 - 9.51650i) q^{77} -2.90814 q^{79} -1.00000 q^{81} +(7.48074 + 7.48074i) q^{83} +(5.38156 + 14.9368i) q^{85} +(1.41481 + 1.41481i) q^{87} +11.7859 q^{89} +2.56282i q^{91} +(1.76641 - 1.76641i) q^{93} +(5.62505 - 11.9611i) q^{95} +(0.793720 - 0.793720i) q^{97} -2.73277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 8 q^{13} - 16 q^{23} - 8 q^{25} + 8 q^{31} + 8 q^{35} - 24 q^{41} + 8 q^{47} - 32 q^{55} - 24 q^{71} + 8 q^{73} + 32 q^{75} + 40 q^{77} - 48 q^{81} + 24 q^{85} - 40 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(-1\)

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
\(3\) 0.707107 0.707107i 0.408248 0.408248i
\(4\) 0 0
\(5\) −0.951596 + 2.02348i −0.425567 + 0.904927i
\(6\) 0 0
\(7\) 3.48236 3.48236i 1.31621 1.31621i 0.399454 0.916753i \(-0.369200\pi\)
0.916753 0.399454i \(-0.130800\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 2.73277i 0.823963i −0.911192 0.411981i \(-0.864837\pi\)
0.911192 0.411981i \(-0.135163\pi\)
\(12\) 0 0
\(13\) −0.367971 + 0.367971i −0.102057 + 0.102057i −0.756292 0.654235i \(-0.772991\pi\)
0.654235 + 0.756292i \(0.272991\pi\)
\(14\) 0 0
\(15\) 0.757935 + 2.10370i 0.195698 + 0.543172i
\(16\) 0 0
\(17\) 5.02066 5.02066i 1.21769 1.21769i 0.249251 0.968439i \(-0.419816\pi\)
0.968439 0.249251i \(-0.0801843\pi\)
\(18\) 0 0
\(19\) −5.91117 −1.35612 −0.678058 0.735008i \(-0.737178\pi\)
−0.678058 + 0.735008i \(0.737178\pi\)
\(20\) 0 0
\(21\) 4.92480i 1.07468i
\(22\) 0 0
\(23\) −2.82390 + 3.87629i −0.588823 + 0.808262i
\(24\) 0 0
\(25\) −3.18893 3.85107i −0.637786 0.770214i
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 2.00084i 0.371547i 0.982593 + 0.185773i \(0.0594790\pi\)
−0.982593 + 0.185773i \(0.940521\pi\)
\(30\) 0 0
\(31\) 2.49808 0.448669 0.224334 0.974512i \(-0.427979\pi\)
0.224334 + 0.974512i \(0.427979\pi\)
\(32\) 0 0
\(33\) −1.93236 1.93236i −0.336381 0.336381i
\(34\) 0 0
\(35\) 3.73268 + 10.3603i 0.630938 + 1.75121i
\(36\) 0 0
\(37\) 6.38531 6.38531i 1.04974 1.04974i 0.0510416 0.998697i \(-0.483746\pi\)
0.998697 0.0510416i \(-0.0162541\pi\)
\(38\) 0 0
\(39\) 0.520390i 0.0833291i
\(40\) 0 0
\(41\) −6.05805 −0.946109 −0.473054 0.881033i \(-0.656849\pi\)
−0.473054 + 0.881033i \(0.656849\pi\)
\(42\) 0 0
\(43\) −2.07971 2.07971i −0.317153 0.317153i 0.530520 0.847673i \(-0.321997\pi\)
−0.847673 + 0.530520i \(0.821997\pi\)
\(44\) 0 0
\(45\) 2.02348 + 0.951596i 0.301642 + 0.141856i
\(46\) 0 0
\(47\) 4.01416 + 4.01416i 0.585526 + 0.585526i 0.936416 0.350891i \(-0.114121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(48\) 0 0
\(49\) 17.2536i 2.46480i
\(50\) 0 0
\(51\) 7.10029i 0.994240i
\(52\) 0 0
\(53\) −0.794878 0.794878i −0.109185 0.109185i 0.650404 0.759589i \(-0.274600\pi\)
−0.759589 + 0.650404i \(0.774600\pi\)
\(54\) 0 0
\(55\) 5.52971 + 2.60050i 0.745626 + 0.350651i
\(56\) 0 0
\(57\) −4.17983 + 4.17983i −0.553632 + 0.553632i
\(58\) 0 0
\(59\) 5.00908i 0.652127i 0.945348 + 0.326063i \(0.105722\pi\)
−0.945348 + 0.326063i \(0.894278\pi\)
\(60\) 0 0
\(61\) 9.72089i 1.24463i −0.782766 0.622316i \(-0.786192\pi\)
0.782766 0.622316i \(-0.213808\pi\)
\(62\) 0 0
\(63\) −3.48236 3.48236i −0.438736 0.438736i
\(64\) 0 0
\(65\) −0.394422 1.09474i −0.0489220 0.135786i
\(66\) 0 0
\(67\) 2.09702 2.09702i 0.256191 0.256191i −0.567312 0.823503i \(-0.692017\pi\)
0.823503 + 0.567312i \(0.192017\pi\)
\(68\) 0 0
\(69\) 0.744154 + 4.73775i 0.0895856 + 0.570358i
\(70\) 0 0
\(71\) 9.35585 1.11034 0.555168 0.831738i \(-0.312654\pi\)
0.555168 + 0.831738i \(0.312654\pi\)
\(72\) 0 0
\(73\) 8.18100 8.18100i 0.957513 0.957513i −0.0416204 0.999133i \(-0.513252\pi\)
0.999133 + 0.0416204i \(0.0132520\pi\)
\(74\) 0 0
\(75\) −4.97803 0.468202i −0.574813 0.0540634i
\(76\) 0 0
\(77\) −9.51650 9.51650i −1.08451 1.08451i
\(78\) 0 0
\(79\) −2.90814 −0.327192 −0.163596 0.986527i \(-0.552309\pi\)
−0.163596 + 0.986527i \(0.552309\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 7.48074 + 7.48074i 0.821118 + 0.821118i 0.986268 0.165150i \(-0.0528110\pi\)
−0.165150 + 0.986268i \(0.552811\pi\)
\(84\) 0 0
\(85\) 5.38156 + 14.9368i 0.583712 + 1.62013i
\(86\) 0 0
\(87\) 1.41481 + 1.41481i 0.151683 + 0.151683i
\(88\) 0 0
\(89\) 11.7859 1.24930 0.624650 0.780905i \(-0.285242\pi\)
0.624650 + 0.780905i \(0.285242\pi\)
\(90\) 0 0
\(91\) 2.56282i 0.268656i
\(92\) 0 0
\(93\) 1.76641 1.76641i 0.183168 0.183168i
\(94\) 0 0
\(95\) 5.62505 11.9611i 0.577118 1.22719i
\(96\) 0 0
\(97\) 0.793720 0.793720i 0.0805901 0.0805901i −0.665663 0.746253i \(-0.731851\pi\)
0.746253 + 0.665663i \(0.231851\pi\)
\(98\) 0 0
\(99\) −2.73277 −0.274654
\(100\) 0 0
\(101\) 2.66129 0.264808 0.132404 0.991196i \(-0.457730\pi\)
0.132404 + 0.991196i \(0.457730\pi\)
\(102\) 0 0
\(103\) −3.17934 3.17934i −0.313269 0.313269i 0.532906 0.846175i \(-0.321100\pi\)
−0.846175 + 0.532906i \(0.821100\pi\)
\(104\) 0 0
\(105\) 9.96522 + 4.68642i 0.972506 + 0.457347i
\(106\) 0 0
\(107\) −12.5484 + 12.5484i −1.21310 + 1.21310i −0.243097 + 0.970002i \(0.578163\pi\)
−0.970002 + 0.243097i \(0.921837\pi\)
\(108\) 0 0
\(109\) −10.2093 −0.977876 −0.488938 0.872318i \(-0.662616\pi\)
−0.488938 + 0.872318i \(0.662616\pi\)
\(110\) 0 0
\(111\) 9.03019i 0.857108i
\(112\) 0 0
\(113\) −2.74744 2.74744i −0.258458 0.258458i 0.565969 0.824427i \(-0.308502\pi\)
−0.824427 + 0.565969i \(0.808502\pi\)
\(114\) 0 0
\(115\) −5.15638 9.40275i −0.480835 0.876811i
\(116\) 0 0
\(117\) 0.367971 + 0.367971i 0.0340190 + 0.0340190i
\(118\) 0 0
\(119\) 34.9675i 3.20546i
\(120\) 0 0
\(121\) 3.53194 0.321086
\(122\) 0 0
\(123\) −4.28369 + 4.28369i −0.386247 + 0.386247i
\(124\) 0 0
\(125\) 10.8271 2.78807i 0.968408 0.249373i
\(126\) 0 0
\(127\) 11.1908 + 11.1908i 0.993026 + 0.993026i 0.999976 0.00695004i \(-0.00221228\pi\)
−0.00695004 + 0.999976i \(0.502212\pi\)
\(128\) 0 0
\(129\) −2.94116 −0.258954
\(130\) 0 0
\(131\) −14.2761 −1.24731 −0.623656 0.781699i \(-0.714353\pi\)
−0.623656 + 0.781699i \(0.714353\pi\)
\(132\) 0 0
\(133\) −20.5848 + 20.5848i −1.78493 + 1.78493i
\(134\) 0 0
\(135\) 2.10370 0.757935i 0.181057 0.0652327i
\(136\) 0 0
\(137\) 3.74076 3.74076i 0.319594 0.319594i −0.529017 0.848611i \(-0.677439\pi\)
0.848611 + 0.529017i \(0.177439\pi\)
\(138\) 0 0
\(139\) 3.82831i 0.324713i −0.986732 0.162356i \(-0.948091\pi\)
0.986732 0.162356i \(-0.0519094\pi\)
\(140\) 0 0
\(141\) 5.67688 0.478080
\(142\) 0 0
\(143\) 1.00558 + 1.00558i 0.0840911 + 0.0840911i
\(144\) 0 0
\(145\) −4.04866 1.90399i −0.336223 0.158118i
\(146\) 0 0
\(147\) −12.2002 12.2002i −1.00625 1.00625i
\(148\) 0 0
\(149\) 20.3447 1.66670 0.833351 0.552744i \(-0.186419\pi\)
0.833351 + 0.552744i \(0.186419\pi\)
\(150\) 0 0
\(151\) −17.5132 −1.42521 −0.712604 0.701567i \(-0.752484\pi\)
−0.712604 + 0.701567i \(0.752484\pi\)
\(152\) 0 0
\(153\) −5.02066 5.02066i −0.405897 0.405897i
\(154\) 0 0
\(155\) −2.37716 + 5.05481i −0.190938 + 0.406012i
\(156\) 0 0
\(157\) −0.633264 + 0.633264i −0.0505400 + 0.0505400i −0.731925 0.681385i \(-0.761378\pi\)
0.681385 + 0.731925i \(0.261378\pi\)
\(158\) 0 0
\(159\) −1.12413 −0.0891491
\(160\) 0 0
\(161\) 3.66481 + 23.3324i 0.288827 + 1.83885i
\(162\) 0 0
\(163\) −9.58716 + 9.58716i −0.750925 + 0.750925i −0.974652 0.223727i \(-0.928178\pi\)
0.223727 + 0.974652i \(0.428178\pi\)
\(164\) 0 0
\(165\) 5.74893 2.07127i 0.447553 0.161248i
\(166\) 0 0
\(167\) −9.73651 9.73651i −0.753434 0.753434i 0.221685 0.975118i \(-0.428844\pi\)
−0.975118 + 0.221685i \(0.928844\pi\)
\(168\) 0 0
\(169\) 12.7292i 0.979169i
\(170\) 0 0
\(171\) 5.91117i 0.452039i
\(172\) 0 0
\(173\) −7.66666 + 7.66666i −0.582885 + 0.582885i −0.935695 0.352810i \(-0.885226\pi\)
0.352810 + 0.935695i \(0.385226\pi\)
\(174\) 0 0
\(175\) −24.5158 2.30580i −1.85322 0.174302i
\(176\) 0 0
\(177\) 3.54195 + 3.54195i 0.266230 + 0.266230i
\(178\) 0 0
\(179\) 4.25360i 0.317929i 0.987284 + 0.158964i \(0.0508155\pi\)
−0.987284 + 0.158964i \(0.949185\pi\)
\(180\) 0 0
\(181\) 14.9651i 1.11235i 0.831065 + 0.556175i \(0.187731\pi\)
−0.831065 + 0.556175i \(0.812269\pi\)
\(182\) 0 0
\(183\) −6.87371 6.87371i −0.508119 0.508119i
\(184\) 0 0
\(185\) 6.84430 + 18.9968i 0.503203 + 1.39667i
\(186\) 0 0
\(187\) −13.7203 13.7203i −1.00333 1.00333i
\(188\) 0 0
\(189\) −4.92480 −0.358226
\(190\) 0 0
\(191\) 2.68877i 0.194552i −0.995257 0.0972762i \(-0.968987\pi\)
0.995257 0.0972762i \(-0.0310130\pi\)
\(192\) 0 0
\(193\) −2.19205 + 2.19205i −0.157787 + 0.157787i −0.781585 0.623798i \(-0.785589\pi\)
0.623798 + 0.781585i \(0.285589\pi\)
\(194\) 0 0
\(195\) −1.05300 0.495201i −0.0754068 0.0354621i
\(196\) 0 0
\(197\) 9.29614 + 9.29614i 0.662323 + 0.662323i 0.955927 0.293604i \(-0.0948548\pi\)
−0.293604 + 0.955927i \(0.594855\pi\)
\(198\) 0 0
\(199\) 26.6926 1.89219 0.946096 0.323887i \(-0.104990\pi\)
0.946096 + 0.323887i \(0.104990\pi\)
\(200\) 0 0
\(201\) 2.96563i 0.209179i
\(202\) 0 0
\(203\) 6.96764 + 6.96764i 0.489033 + 0.489033i
\(204\) 0 0
\(205\) 5.76482 12.2583i 0.402632 0.856160i
\(206\) 0 0
\(207\) 3.87629 + 2.82390i 0.269421 + 0.196274i
\(208\) 0 0
\(209\) 16.1539i 1.11739i
\(210\) 0 0
\(211\) 22.7587 1.56677 0.783386 0.621536i \(-0.213491\pi\)
0.783386 + 0.621536i \(0.213491\pi\)
\(212\) 0 0
\(213\) 6.61559 6.61559i 0.453293 0.453293i
\(214\) 0 0
\(215\) 6.18730 2.22921i 0.421970 0.152031i
\(216\) 0 0
\(217\) 8.69921 8.69921i 0.590541 0.590541i
\(218\) 0 0
\(219\) 11.5697i 0.781806i
\(220\) 0 0
\(221\) 3.69492i 0.248547i
\(222\) 0 0
\(223\) −12.5272 + 12.5272i −0.838881 + 0.838881i −0.988712 0.149831i \(-0.952127\pi\)
0.149831 + 0.988712i \(0.452127\pi\)
\(224\) 0 0
\(225\) −3.85107 + 3.18893i −0.256738 + 0.212595i
\(226\) 0 0
\(227\) −12.4018 + 12.4018i −0.823138 + 0.823138i −0.986557 0.163419i \(-0.947748\pi\)
0.163419 + 0.986557i \(0.447748\pi\)
\(228\) 0 0
\(229\) −15.2938 −1.01065 −0.505323 0.862930i \(-0.668627\pi\)
−0.505323 + 0.862930i \(0.668627\pi\)
\(230\) 0 0
\(231\) −13.4584 −0.885495
\(232\) 0 0
\(233\) 15.2955 15.2955i 1.00204 1.00204i 0.00204482 0.999998i \(-0.499349\pi\)
0.999998 0.00204482i \(-0.000650886\pi\)
\(234\) 0 0
\(235\) −11.9424 + 4.30271i −0.779038 + 0.280678i
\(236\) 0 0
\(237\) −2.05637 + 2.05637i −0.133575 + 0.133575i
\(238\) 0 0
\(239\) 8.28742i 0.536068i 0.963409 + 0.268034i \(0.0863740\pi\)
−0.963409 + 0.268034i \(0.913626\pi\)
\(240\) 0 0
\(241\) 9.97259i 0.642391i 0.947013 + 0.321196i \(0.104085\pi\)
−0.947013 + 0.321196i \(0.895915\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 34.9123 + 16.4185i 2.23047 + 1.04894i
\(246\) 0 0
\(247\) 2.17514 2.17514i 0.138401 0.138401i
\(248\) 0 0
\(249\) 10.5794 0.670440
\(250\) 0 0
\(251\) 3.73360i 0.235663i 0.993034 + 0.117831i \(0.0375943\pi\)
−0.993034 + 0.117831i \(0.962406\pi\)
\(252\) 0 0
\(253\) 10.5930 + 7.71707i 0.665978 + 0.485168i
\(254\) 0 0
\(255\) 14.3673 + 6.75661i 0.899714 + 0.423115i
\(256\) 0 0
\(257\) −17.9838 17.9838i −1.12180 1.12180i −0.991471 0.130325i \(-0.958398\pi\)
−0.130325 0.991471i \(-0.541602\pi\)
\(258\) 0 0
\(259\) 44.4718i 2.76335i
\(260\) 0 0
\(261\) 2.00084 0.123849
\(262\) 0 0
\(263\) 9.76385 + 9.76385i 0.602065 + 0.602065i 0.940860 0.338795i \(-0.110019\pi\)
−0.338795 + 0.940860i \(0.610019\pi\)
\(264\) 0 0
\(265\) 2.36482 0.852016i 0.145270 0.0523389i
\(266\) 0 0
\(267\) 8.33387 8.33387i 0.510025 0.510025i
\(268\) 0 0
\(269\) 9.86177i 0.601283i 0.953737 + 0.300641i \(0.0972007\pi\)
−0.953737 + 0.300641i \(0.902799\pi\)
\(270\) 0 0
\(271\) 30.9582 1.88058 0.940289 0.340378i \(-0.110555\pi\)
0.940289 + 0.340378i \(0.110555\pi\)
\(272\) 0 0
\(273\) 1.81218 + 1.81218i 0.109678 + 0.109678i
\(274\) 0 0
\(275\) −10.5241 + 8.71463i −0.634627 + 0.525512i
\(276\) 0 0
\(277\) −5.28631 5.28631i −0.317624 0.317624i 0.530230 0.847854i \(-0.322106\pi\)
−0.847854 + 0.530230i \(0.822106\pi\)
\(278\) 0 0
\(279\) 2.49808i 0.149556i
\(280\) 0 0
\(281\) 8.12633i 0.484776i 0.970179 + 0.242388i \(0.0779307\pi\)
−0.970179 + 0.242388i \(0.922069\pi\)
\(282\) 0 0
\(283\) 1.11664 + 1.11664i 0.0663772 + 0.0663772i 0.739516 0.673139i \(-0.235054\pi\)
−0.673139 + 0.739516i \(0.735054\pi\)
\(284\) 0 0
\(285\) −4.48029 12.4353i −0.265389 0.736604i
\(286\) 0 0
\(287\) −21.0963 + 21.0963i −1.24528 + 1.24528i
\(288\) 0 0
\(289\) 33.4141i 1.96554i
\(290\) 0 0
\(291\) 1.12249i 0.0658015i
\(292\) 0 0
\(293\) −11.0252 11.0252i −0.644097 0.644097i 0.307463 0.951560i \(-0.400520\pi\)
−0.951560 + 0.307463i \(0.900520\pi\)
\(294\) 0 0
\(295\) −10.1358 4.76662i −0.590127 0.277523i
\(296\) 0 0
\(297\) −1.93236 + 1.93236i −0.112127 + 0.112127i
\(298\) 0 0
\(299\) −0.387250 2.46548i −0.0223953 0.142582i
\(300\) 0 0
\(301\) −14.4846 −0.834878
\(302\) 0 0
\(303\) 1.88182 1.88182i 0.108108 0.108108i
\(304\) 0 0
\(305\) 19.6700 + 9.25036i 1.12630 + 0.529674i
\(306\) 0 0
\(307\) 22.3268 + 22.3268i 1.27426 + 1.27426i 0.943833 + 0.330424i \(0.107192\pi\)
0.330424 + 0.943833i \(0.392808\pi\)
\(308\) 0 0
\(309\) −4.49626 −0.255783
\(310\) 0 0
\(311\) 3.95458 0.224244 0.112122 0.993694i \(-0.464235\pi\)
0.112122 + 0.993694i \(0.464235\pi\)
\(312\) 0 0
\(313\) 22.9975 + 22.9975i 1.29990 + 1.29990i 0.928457 + 0.371439i \(0.121136\pi\)
0.371439 + 0.928457i \(0.378864\pi\)
\(314\) 0 0
\(315\) 10.3603 3.73268i 0.583735 0.210313i
\(316\) 0 0
\(317\) −13.0622 13.0622i −0.733647 0.733647i 0.237693 0.971340i \(-0.423609\pi\)
−0.971340 + 0.237693i \(0.923609\pi\)
\(318\) 0 0
\(319\) 5.46785 0.306141
\(320\) 0 0
\(321\) 17.7461i 0.990491i
\(322\) 0 0
\(323\) −29.6780 + 29.6780i −1.65133 + 1.65133i
\(324\) 0 0
\(325\) 2.59052 + 0.243648i 0.143696 + 0.0135152i
\(326\) 0 0
\(327\) −7.21908 + 7.21908i −0.399216 + 0.399216i
\(328\) 0 0
\(329\) 27.9575 1.54135
\(330\) 0 0
\(331\) 2.29715 0.126263 0.0631314 0.998005i \(-0.479891\pi\)
0.0631314 + 0.998005i \(0.479891\pi\)
\(332\) 0 0
\(333\) −6.38531 6.38531i −0.349913 0.349913i
\(334\) 0 0
\(335\) 2.24776 + 6.23878i 0.122808 + 0.340861i
\(336\) 0 0
\(337\) −23.8232 + 23.8232i −1.29773 + 1.29773i −0.367849 + 0.929885i \(0.619906\pi\)
−0.929885 + 0.367849i \(0.880094\pi\)
\(338\) 0 0
\(339\) −3.88547 −0.211030
\(340\) 0 0
\(341\) 6.82669i 0.369686i
\(342\) 0 0
\(343\) −35.7068 35.7068i −1.92798 1.92798i
\(344\) 0 0
\(345\) −10.2949 3.00264i −0.554257 0.161657i
\(346\) 0 0
\(347\) 15.1228 + 15.1228i 0.811836 + 0.811836i 0.984909 0.173073i \(-0.0553698\pi\)
−0.173073 + 0.984909i \(0.555370\pi\)
\(348\) 0 0
\(349\) 3.85339i 0.206267i 0.994668 + 0.103133i \(0.0328869\pi\)
−0.994668 + 0.103133i \(0.967113\pi\)
\(350\) 0 0
\(351\) 0.520390 0.0277764
\(352\) 0 0
\(353\) −17.0525 + 17.0525i −0.907611 + 0.907611i −0.996079 0.0884682i \(-0.971803\pi\)
0.0884682 + 0.996079i \(0.471803\pi\)
\(354\) 0 0
\(355\) −8.90299 + 18.9314i −0.472522 + 1.00477i
\(356\) 0 0
\(357\) −24.7257 24.7257i −1.30863 1.30863i
\(358\) 0 0
\(359\) 6.39323 0.337422 0.168711 0.985666i \(-0.446040\pi\)
0.168711 + 0.985666i \(0.446040\pi\)
\(360\) 0 0
\(361\) 15.9420 0.839050
\(362\) 0 0
\(363\) 2.49746 2.49746i 0.131083 0.131083i
\(364\) 0 0
\(365\) 8.76906 + 24.3391i 0.458994 + 1.27397i
\(366\) 0 0
\(367\) −1.27685 + 1.27685i −0.0666508 + 0.0666508i −0.739646 0.672996i \(-0.765007\pi\)
0.672996 + 0.739646i \(0.265007\pi\)
\(368\) 0 0
\(369\) 6.05805i 0.315370i
\(370\) 0 0
\(371\) −5.53610 −0.287420
\(372\) 0 0
\(373\) 11.0669 + 11.0669i 0.573021 + 0.573021i 0.932971 0.359951i \(-0.117206\pi\)
−0.359951 + 0.932971i \(0.617206\pi\)
\(374\) 0 0
\(375\) 5.68447 9.62740i 0.293545 0.497157i
\(376\) 0 0
\(377\) −0.736252 0.736252i −0.0379189 0.0379189i
\(378\) 0 0
\(379\) 12.2350 0.628467 0.314234 0.949346i \(-0.398252\pi\)
0.314234 + 0.949346i \(0.398252\pi\)
\(380\) 0 0
\(381\) 15.8262 0.810802
\(382\) 0 0
\(383\) −11.9126 11.9126i −0.608708 0.608708i 0.333901 0.942608i \(-0.391635\pi\)
−0.942608 + 0.333901i \(0.891635\pi\)
\(384\) 0 0
\(385\) 28.3123 10.2006i 1.44293 0.519869i
\(386\) 0 0
\(387\) −2.07971 + 2.07971i −0.105718 + 0.105718i
\(388\) 0 0
\(389\) −6.06586 −0.307551 −0.153776 0.988106i \(-0.549143\pi\)
−0.153776 + 0.988106i \(0.549143\pi\)
\(390\) 0 0
\(391\) 5.28371 + 33.6394i 0.267209 + 1.70122i
\(392\) 0 0
\(393\) −10.0947 + 10.0947i −0.509213 + 0.509213i
\(394\) 0 0
\(395\) 2.76738 5.88456i 0.139242 0.296085i
\(396\) 0 0
\(397\) −5.17259 5.17259i −0.259605 0.259605i 0.565288 0.824893i \(-0.308765\pi\)
−0.824893 + 0.565288i \(0.808765\pi\)
\(398\) 0 0
\(399\) 29.1113i 1.45739i
\(400\) 0 0
\(401\) 30.5270i 1.52445i 0.647314 + 0.762223i \(0.275892\pi\)
−0.647314 + 0.762223i \(0.724108\pi\)
\(402\) 0 0
\(403\) −0.919222 + 0.919222i −0.0457897 + 0.0457897i
\(404\) 0 0
\(405\) 0.951596 2.02348i 0.0472852 0.100547i
\(406\) 0 0
\(407\) −17.4496 17.4496i −0.864945 0.864945i
\(408\) 0 0
\(409\) 14.7162i 0.727668i 0.931464 + 0.363834i \(0.118532\pi\)
−0.931464 + 0.363834i \(0.881468\pi\)
\(410\) 0 0
\(411\) 5.29023i 0.260948i
\(412\) 0 0
\(413\) 17.4434 + 17.4434i 0.858334 + 0.858334i
\(414\) 0 0
\(415\) −22.2558 + 8.01847i −1.09249 + 0.393611i
\(416\) 0 0
\(417\) −2.70702 2.70702i −0.132563 0.132563i
\(418\) 0 0
\(419\) −32.7417 −1.59954 −0.799769 0.600308i \(-0.795045\pi\)
−0.799769 + 0.600308i \(0.795045\pi\)
\(420\) 0 0
\(421\) 17.8474i 0.869828i −0.900472 0.434914i \(-0.856779\pi\)
0.900472 0.434914i \(-0.143221\pi\)
\(422\) 0 0
\(423\) 4.01416 4.01416i 0.195175 0.195175i
\(424\) 0 0
\(425\) −35.3455 3.32437i −1.71451 0.161256i
\(426\) 0 0
\(427\) −33.8516 33.8516i −1.63819 1.63819i
\(428\) 0 0
\(429\) 1.42211 0.0686601
\(430\) 0 0
\(431\) 12.2783i 0.591424i −0.955277 0.295712i \(-0.904443\pi\)
0.955277 0.295712i \(-0.0955569\pi\)
\(432\) 0 0
\(433\) −23.1039 23.1039i −1.11030 1.11030i −0.993109 0.117192i \(-0.962611\pi\)
−0.117192 0.993109i \(-0.537389\pi\)
\(434\) 0 0
\(435\) −4.20916 + 1.51651i −0.201814 + 0.0727110i
\(436\) 0 0
\(437\) 16.6925 22.9134i 0.798512 1.09610i
\(438\) 0 0
\(439\) 16.3865i 0.782084i −0.920373 0.391042i \(-0.872115\pi\)
0.920373 0.391042i \(-0.127885\pi\)
\(440\) 0 0
\(441\) −17.2536 −0.821601
\(442\) 0 0
\(443\) −17.9738 + 17.9738i −0.853959 + 0.853959i −0.990618 0.136659i \(-0.956364\pi\)
0.136659 + 0.990618i \(0.456364\pi\)
\(444\) 0 0
\(445\) −11.2154 + 23.8485i −0.531661 + 1.13053i
\(446\) 0 0
\(447\) 14.3859 14.3859i 0.680429 0.680429i
\(448\) 0 0
\(449\) 2.07546i 0.0979471i −0.998800 0.0489736i \(-0.984405\pi\)
0.998800 0.0489736i \(-0.0155950\pi\)
\(450\) 0 0
\(451\) 16.5553i 0.779558i
\(452\) 0 0
\(453\) −12.3837 + 12.3837i −0.581838 + 0.581838i
\(454\) 0 0
\(455\) −5.18580 2.43876i −0.243114 0.114331i
\(456\) 0 0
\(457\) −17.7693 + 17.7693i −0.831215 + 0.831215i −0.987683 0.156468i \(-0.949989\pi\)
0.156468 + 0.987683i \(0.449989\pi\)
\(458\) 0 0
\(459\) −7.10029 −0.331413
\(460\) 0 0
\(461\) 8.62611 0.401758 0.200879 0.979616i \(-0.435620\pi\)
0.200879 + 0.979616i \(0.435620\pi\)
\(462\) 0 0
\(463\) −8.70757 + 8.70757i −0.404675 + 0.404675i −0.879877 0.475202i \(-0.842375\pi\)
0.475202 + 0.879877i \(0.342375\pi\)
\(464\) 0 0
\(465\) 1.89338 + 5.25520i 0.0878036 + 0.243704i
\(466\) 0 0
\(467\) −2.96286 + 2.96286i −0.137105 + 0.137105i −0.772328 0.635224i \(-0.780908\pi\)
0.635224 + 0.772328i \(0.280908\pi\)
\(468\) 0 0
\(469\) 14.6051i 0.674402i
\(470\) 0 0
\(471\) 0.895570i 0.0412657i
\(472\) 0 0
\(473\) −5.68338 + 5.68338i −0.261322 + 0.261322i
\(474\) 0 0
\(475\) 18.8503 + 22.7643i 0.864912 + 1.04450i
\(476\) 0 0
\(477\) −0.794878 + 0.794878i −0.0363950 + 0.0363950i
\(478\) 0 0
\(479\) −23.4955 −1.07354 −0.536768 0.843730i \(-0.680355\pi\)
−0.536768 + 0.843730i \(0.680355\pi\)
\(480\) 0 0
\(481\) 4.69922i 0.214266i
\(482\) 0 0
\(483\) 19.0899 + 13.9071i 0.868622 + 0.632796i
\(484\) 0 0
\(485\) 0.850775 + 2.36138i 0.0386317 + 0.107225i
\(486\) 0 0
\(487\) 16.5628 + 16.5628i 0.750532 + 0.750532i 0.974578 0.224047i \(-0.0719269\pi\)
−0.224047 + 0.974578i \(0.571927\pi\)
\(488\) 0 0
\(489\) 13.5583i 0.613127i
\(490\) 0 0
\(491\) 2.39587 0.108124 0.0540620 0.998538i \(-0.482783\pi\)
0.0540620 + 0.998538i \(0.482783\pi\)
\(492\) 0 0
\(493\) 10.0455 + 10.0455i 0.452429 + 0.452429i
\(494\) 0 0
\(495\) 2.60050 5.52971i 0.116884 0.248542i
\(496\) 0 0
\(497\) 32.5804 32.5804i 1.46143 1.46143i
\(498\) 0 0
\(499\) 2.30108i 0.103010i −0.998673 0.0515052i \(-0.983598\pi\)
0.998673 0.0515052i \(-0.0164019\pi\)
\(500\) 0 0
\(501\) −13.7695 −0.615176
\(502\) 0 0
\(503\) 5.59225 + 5.59225i 0.249346 + 0.249346i 0.820702 0.571356i \(-0.193582\pi\)
−0.571356 + 0.820702i \(0.693582\pi\)
\(504\) 0 0
\(505\) −2.53247 + 5.38506i −0.112694 + 0.239632i
\(506\) 0 0
\(507\) 9.00090 + 9.00090i 0.399744 + 0.399744i
\(508\) 0 0
\(509\) 3.88491i 0.172195i −0.996287 0.0860977i \(-0.972560\pi\)
0.996287 0.0860977i \(-0.0274397\pi\)
\(510\) 0 0
\(511\) 56.9783i 2.52057i
\(512\) 0 0
\(513\) 4.17983 + 4.17983i 0.184544 + 0.184544i
\(514\) 0 0
\(515\) 9.45876 3.40787i 0.416803 0.150169i
\(516\) 0 0
\(517\) 10.9698 10.9698i 0.482451 0.482451i
\(518\) 0 0
\(519\) 10.8423i 0.475924i
\(520\) 0 0
\(521\) 15.4956i 0.678876i −0.940628 0.339438i \(-0.889763\pi\)
0.940628 0.339438i \(-0.110237\pi\)
\(522\) 0 0
\(523\) −20.3365 20.3365i −0.889254 0.889254i 0.105197 0.994451i \(-0.466453\pi\)
−0.994451 + 0.105197i \(0.966453\pi\)
\(524\) 0 0
\(525\) −18.9657 + 15.7048i −0.827732 + 0.685415i
\(526\) 0 0
\(527\) 12.5420 12.5420i 0.546339 0.546339i
\(528\) 0 0
\(529\) −7.05123 21.8925i −0.306575 0.951846i
\(530\) 0 0
\(531\) 5.00908 0.217376
\(532\) 0 0
\(533\) 2.22919 2.22919i 0.0965569 0.0965569i
\(534\) 0 0
\(535\) −13.4504 37.3324i −0.581512 1.61402i
\(536\) 0 0
\(537\) 3.00775 + 3.00775i 0.129794 + 0.129794i
\(538\) 0 0
\(539\) −47.1503 −2.03091
\(540\) 0 0
\(541\) 33.7381 1.45051 0.725257 0.688478i \(-0.241721\pi\)
0.725257 + 0.688478i \(0.241721\pi\)
\(542\) 0 0
\(543\) 10.5819 + 10.5819i 0.454115 + 0.454115i
\(544\) 0 0
\(545\) 9.71515 20.6584i 0.416152 0.884907i
\(546\) 0 0
\(547\) 31.0432 + 31.0432i 1.32731 + 1.32731i 0.907706 + 0.419606i \(0.137832\pi\)
0.419606 + 0.907706i \(0.362168\pi\)
\(548\) 0 0
\(549\) −9.72089 −0.414877
\(550\) 0 0
\(551\) 11.8273i 0.503861i
\(552\) 0 0
\(553\) −10.1272 + 10.1272i −0.430652 + 0.430652i
\(554\) 0 0
\(555\) 18.2724 + 8.59309i 0.775620 + 0.364756i
\(556\) 0 0
\(557\) 16.3749 16.3749i 0.693827 0.693827i −0.269245 0.963072i \(-0.586774\pi\)
0.963072 + 0.269245i \(0.0867741\pi\)
\(558\) 0 0
\(559\) 1.53055 0.0647353
\(560\) 0 0
\(561\) −19.4035 −0.819216
\(562\) 0 0
\(563\) 23.8732 + 23.8732i 1.00613 + 1.00613i 0.999981 + 0.00615292i \(0.00195855\pi\)
0.00615292 + 0.999981i \(0.498041\pi\)
\(564\) 0 0
\(565\) 8.17384 2.94493i 0.343876 0.123894i
\(566\) 0 0
\(567\) −3.48236 + 3.48236i −0.146245 + 0.146245i
\(568\) 0 0
\(569\) −24.7218 −1.03639 −0.518195 0.855262i \(-0.673396\pi\)
−0.518195 + 0.855262i \(0.673396\pi\)
\(570\) 0 0
\(571\) 18.4534i 0.772249i −0.922447 0.386124i \(-0.873814\pi\)
0.922447 0.386124i \(-0.126186\pi\)
\(572\) 0 0
\(573\) −1.90125 1.90125i −0.0794257 0.0794257i
\(574\) 0 0
\(575\) 23.9331 1.48620i 0.998077 0.0619787i
\(576\) 0 0
\(577\) −27.1580 27.1580i −1.13060 1.13060i −0.990077 0.140525i \(-0.955121\pi\)
−0.140525 0.990077i \(-0.544879\pi\)
\(578\) 0 0
\(579\) 3.10002i 0.128833i
\(580\) 0 0
\(581\) 52.1012 2.16152
\(582\) 0 0
\(583\) −2.17222 + 2.17222i −0.0899643 + 0.0899643i
\(584\) 0 0
\(585\) −1.09474 + 0.394422i −0.0452620 + 0.0163073i
\(586\) 0 0
\(587\) −7.91964 7.91964i −0.326878 0.326878i 0.524520 0.851398i \(-0.324245\pi\)
−0.851398 + 0.524520i \(0.824245\pi\)
\(588\) 0 0
\(589\) −14.7666 −0.608447
\(590\) 0 0
\(591\) 13.1467 0.540784
\(592\) 0 0
\(593\) 21.1679 21.1679i 0.869263 0.869263i −0.123128 0.992391i \(-0.539292\pi\)
0.992391 + 0.123128i \(0.0392925\pi\)
\(594\) 0 0
\(595\) 70.7560 + 33.2749i 2.90071 + 1.36414i
\(596\) 0 0
\(597\) 18.8745 18.8745i 0.772484 0.772484i
\(598\) 0 0
\(599\) 11.8908i 0.485843i −0.970046 0.242922i \(-0.921894\pi\)
0.970046 0.242922i \(-0.0781058\pi\)
\(600\) 0 0
\(601\) 32.3702 1.32041 0.660203 0.751087i \(-0.270470\pi\)
0.660203 + 0.751087i \(0.270470\pi\)
\(602\) 0 0
\(603\) −2.09702 2.09702i −0.0853971 0.0853971i
\(604\) 0 0
\(605\) −3.36098 + 7.14681i −0.136643 + 0.290559i
\(606\) 0 0
\(607\) 28.1596 + 28.1596i 1.14296 + 1.14296i 0.987906 + 0.155057i \(0.0495561\pi\)
0.155057 + 0.987906i \(0.450444\pi\)
\(608\) 0 0
\(609\) 9.85373 0.399293
\(610\) 0 0
\(611\) −2.95419 −0.119514
\(612\) 0 0
\(613\) 13.4800 + 13.4800i 0.544450 + 0.544450i 0.924830 0.380380i \(-0.124207\pi\)
−0.380380 + 0.924830i \(0.624207\pi\)
\(614\) 0 0
\(615\) −4.59161 12.7443i −0.185152 0.513900i
\(616\) 0 0
\(617\) 14.1443 14.1443i 0.569430 0.569430i −0.362539 0.931969i \(-0.618090\pi\)
0.931969 + 0.362539i \(0.118090\pi\)
\(618\) 0 0
\(619\) 5.83694 0.234607 0.117303 0.993096i \(-0.462575\pi\)
0.117303 + 0.993096i \(0.462575\pi\)
\(620\) 0 0
\(621\) 4.73775 0.744154i 0.190119 0.0298619i
\(622\) 0 0
\(623\) 41.0426 41.0426i 1.64434 1.64434i
\(624\) 0 0
\(625\) −4.66145 + 24.5616i −0.186458 + 0.982463i
\(626\) 0 0
\(627\) 11.4225 + 11.4225i 0.456172 + 0.456172i
\(628\) 0 0
\(629\) 64.1170i 2.55651i
\(630\) 0 0
\(631\) 8.84121i 0.351963i 0.984393 + 0.175982i \(0.0563099\pi\)
−0.984393 + 0.175982i \(0.943690\pi\)
\(632\) 0 0
\(633\) 16.0928 16.0928i 0.639632 0.639632i
\(634\) 0 0
\(635\) −33.2936 + 11.9953i −1.32121 + 0.476017i
\(636\) 0 0
\(637\) 6.34884 + 6.34884i 0.251550 + 0.251550i
\(638\) 0 0
\(639\) 9.35585i 0.370112i
\(640\) 0 0
\(641\) 31.0727i 1.22730i −0.789579 0.613649i \(-0.789701\pi\)
0.789579 0.613649i \(-0.210299\pi\)
\(642\) 0 0
\(643\) −25.4613 25.4613i −1.00410 1.00410i −0.999992 0.00410535i \(-0.998693\pi\)
−0.00410535 0.999992i \(-0.501307\pi\)
\(644\) 0 0
\(645\) 2.79879 5.95137i 0.110202 0.234335i
\(646\) 0 0
\(647\) −10.6871 10.6871i −0.420155 0.420155i 0.465102 0.885257i \(-0.346018\pi\)
−0.885257 + 0.465102i \(0.846018\pi\)
\(648\) 0 0
\(649\) 13.6887 0.537328
\(650\) 0 0
\(651\) 12.3025i 0.482175i
\(652\) 0 0
\(653\) 0.982888 0.982888i 0.0384634 0.0384634i −0.687614 0.726077i \(-0.741342\pi\)
0.726077 + 0.687614i \(0.241342\pi\)
\(654\) 0 0
\(655\) 13.5851 28.8874i 0.530814 1.12873i
\(656\) 0 0
\(657\) −8.18100 8.18100i −0.319171 0.319171i
\(658\) 0 0
\(659\) 14.2495 0.555084 0.277542 0.960714i \(-0.410480\pi\)
0.277542 + 0.960714i \(0.410480\pi\)
\(660\) 0 0
\(661\) 27.1286i 1.05518i −0.849499 0.527591i \(-0.823095\pi\)
0.849499 0.527591i \(-0.176905\pi\)
\(662\) 0 0
\(663\) 2.61270 + 2.61270i 0.101469 + 0.101469i
\(664\) 0 0
\(665\) −22.0645 61.2413i −0.855625 2.37484i
\(666\) 0 0
\(667\) −7.75584 5.65017i −0.300307 0.218775i
\(668\) 0 0
\(669\) 17.7161i 0.684943i
\(670\) 0 0
\(671\) −26.5650 −1.02553
\(672\) 0 0
\(673\) −10.3225 + 10.3225i −0.397902 + 0.397902i −0.877493 0.479590i \(-0.840785\pi\)
0.479590 + 0.877493i \(0.340785\pi\)
\(674\) 0 0
\(675\) −0.468202 + 4.97803i −0.0180211 + 0.191604i
\(676\) 0 0
\(677\) 27.1421 27.1421i 1.04316 1.04316i 0.0441292 0.999026i \(-0.485949\pi\)
0.999026 0.0441292i \(-0.0140513\pi\)
\(678\) 0 0
\(679\) 5.52804i 0.212147i
\(680\) 0 0
\(681\) 17.5388i 0.672089i
\(682\) 0 0
\(683\) −6.74055 + 6.74055i −0.257920 + 0.257920i −0.824208 0.566288i \(-0.808379\pi\)
0.566288 + 0.824208i \(0.308379\pi\)
\(684\) 0 0
\(685\) 4.00965 + 11.1290i 0.153201 + 0.425218i
\(686\) 0 0
\(687\) −10.8144 + 10.8144i −0.412594 + 0.412594i
\(688\) 0 0
\(689\) 0.584985 0.0222861
\(690\) 0 0
\(691\) 34.7074 1.32033 0.660166 0.751120i \(-0.270486\pi\)
0.660166 + 0.751120i \(0.270486\pi\)
\(692\) 0 0
\(693\) −9.51650 + 9.51650i −0.361502 + 0.361502i
\(694\) 0 0
\(695\) 7.74650 + 3.64300i 0.293841 + 0.138187i
\(696\) 0 0
\(697\) −30.4154 + 30.4154i −1.15207 + 1.15207i
\(698\) 0 0
\(699\) 21.6311i 0.818164i
\(700\) 0 0
\(701\) 38.3373i 1.44798i −0.689811 0.723990i \(-0.742306\pi\)
0.689811 0.723990i \(-0.257694\pi\)
\(702\) 0 0
\(703\) −37.7447 + 37.7447i −1.42357 + 1.42357i
\(704\) 0 0
\(705\) −5.40210 + 11.4871i −0.203455 + 0.432627i
\(706\) 0 0
\(707\) 9.26756 9.26756i 0.348543 0.348543i
\(708\) 0 0
\(709\) −16.7671 −0.629700 −0.314850 0.949141i \(-0.601954\pi\)
−0.314850 + 0.949141i \(0.601954\pi\)
\(710\) 0 0
\(711\) 2.90814i 0.109064i
\(712\) 0 0
\(713\) −7.05432 + 9.68328i −0.264186 + 0.362642i
\(714\) 0 0
\(715\) −2.99168 + 1.07787i −0.111883 + 0.0403099i
\(716\) 0 0
\(717\) 5.86009 + 5.86009i 0.218849 + 0.218849i
\(718\) 0 0
\(719\) 19.0730i 0.711302i 0.934619 + 0.355651i \(0.115741\pi\)
−0.934619 + 0.355651i \(0.884259\pi\)
\(720\) 0 0
\(721\) −22.1432 −0.824655
\(722\) 0 0
\(723\) 7.05169 + 7.05169i 0.262255 + 0.262255i
\(724\) 0 0
\(725\) 7.70537 6.38054i 0.286170 0.236967i
\(726\) 0 0
\(727\) 33.0514 33.0514i 1.22581 1.22581i 0.260273 0.965535i \(-0.416187\pi\)
0.965535 0.260273i \(-0.0838126\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −20.8831 −0.772388
\(732\) 0 0
\(733\) −9.13550 9.13550i −0.337427 0.337427i 0.517971 0.855398i \(-0.326688\pi\)
−0.855398 + 0.517971i \(0.826688\pi\)
\(734\) 0 0
\(735\) 36.2964 13.0771i 1.33881 0.482357i
\(736\) 0 0
\(737\) −5.73067 5.73067i −0.211092 0.211092i
\(738\) 0 0
\(739\) 28.8701i 1.06200i 0.847370 + 0.531002i \(0.178184\pi\)
−0.847370 + 0.531002i \(0.821816\pi\)
\(740\) 0 0
\(741\) 3.07612i 0.113004i
\(742\) 0 0
\(743\) −35.7107 35.7107i −1.31010 1.31010i −0.921341 0.388756i \(-0.872905\pi\)
−0.388756 0.921341i \(-0.627095\pi\)
\(744\) 0 0
\(745\) −19.3599 + 41.1671i −0.709293 + 1.50824i
\(746\) 0 0
\(747\) 7.48074 7.48074i 0.273706 0.273706i
\(748\) 0 0
\(749\) 87.3960i 3.19338i
\(750\) 0 0
\(751\) 18.1412i 0.661982i −0.943634 0.330991i \(-0.892617\pi\)
0.943634 0.330991i \(-0.107383\pi\)
\(752\) 0 0
\(753\) 2.64006 + 2.64006i 0.0962090 + 0.0962090i
\(754\) 0 0
\(755\) 16.6655 35.4377i 0.606521 1.28971i
\(756\) 0 0
\(757\) −18.5361 + 18.5361i −0.673707 + 0.673707i −0.958569 0.284861i \(-0.908052\pi\)
0.284861 + 0.958569i \(0.408052\pi\)
\(758\) 0 0
\(759\) 12.9472 2.03361i 0.469953 0.0738152i
\(760\) 0 0
\(761\) −44.5294 −1.61419 −0.807094 0.590422i \(-0.798961\pi\)
−0.807094 + 0.590422i \(0.798961\pi\)
\(762\) 0 0
\(763\) −35.5525 + 35.5525i −1.28709 + 1.28709i
\(764\) 0 0
\(765\) 14.9368 5.38156i 0.540043 0.194571i
\(766\) 0 0
\(767\) −1.84320 1.84320i −0.0665540 0.0665540i
\(768\) 0 0
\(769\) 22.4202 0.808494 0.404247 0.914650i \(-0.367534\pi\)
0.404247 + 0.914650i \(0.367534\pi\)
\(770\) 0 0
\(771\) −25.4329 −0.915943
\(772\) 0 0
\(773\) 36.2172 + 36.2172i 1.30264 + 1.30264i 0.926603 + 0.376040i \(0.122715\pi\)
0.376040 + 0.926603i \(0.377285\pi\)
\(774\) 0 0
\(775\) −7.96621 9.62028i −0.286155 0.345571i
\(776\) 0 0
\(777\) −31.4463 31.4463i −1.12813 1.12813i
\(778\) 0 0
\(779\) 35.8102 1.28303
\(780\) 0 0
\(781\) 25.5674i 0.914875i
\(782\) 0 0
\(783\) 1.41481 1.41481i 0.0505611 0.0505611i
\(784\) 0 0
\(785\) −0.678784 1.88401i −0.0242269 0.0672431i
\(786\) 0 0
\(787\) −35.9021 + 35.9021i −1.27977 + 1.27977i −0.338979 + 0.940794i \(0.610082\pi\)
−0.940794 + 0.338979i \(0.889918\pi\)
\(788\) 0 0
\(789\) 13.8082 0.491584
\(790\) 0 0
\(791\) −19.1351 −0.680367
\(792\) 0 0
\(793\) 3.57701 + 3.57701i 0.127023 + 0.127023i
\(794\) 0 0
\(795\) 1.06971 2.27465i 0.0379389 0.0806734i
\(796\) 0 0
\(797\) −20.4445 + 20.4445i −0.724181 + 0.724181i −0.969454 0.245273i \(-0.921122\pi\)
0.245273 + 0.969454i \(0.421122\pi\)
\(798\) 0 0
\(799\) 40.3075 1.42598
\(800\) 0 0
\(801\) 11.7859i 0.416433i
\(802\) 0 0
\(803\) −22.3568 22.3568i −0.788955 0.788955i
\(804\) 0 0
\(805\) −50.7001 14.7874i −1.78694 0.521187i
\(806\) 0 0
\(807\) 6.97332 + 6.97332i 0.245473 + 0.245473i
\(808\) 0 0
\(809\) 45.7899i 1.60989i 0.593352 + 0.804943i \(0.297804\pi\)
−0.593352 + 0.804943i \(0.702196\pi\)
\(810\) 0 0
\(811\) −40.9944 −1.43951 −0.719755 0.694228i \(-0.755746\pi\)
−0.719755 + 0.694228i \(0.755746\pi\)
\(812\) 0 0
\(813\) 21.8908 21.8908i 0.767742 0.767742i
\(814\) 0 0
\(815\) −10.2763 28.5225i −0.359964 0.999101i
\(816\) 0 0
\(817\) 12.2935 + 12.2935i 0.430096 + 0.430096i
\(818\) 0 0
\(819\) 2.56282 0.0895520
\(820\) 0 0
\(821\) 38.5043 1.34381 0.671905 0.740637i \(-0.265476\pi\)
0.671905 + 0.740637i \(0.265476\pi\)
\(822\) 0 0
\(823\) 14.6433 14.6433i 0.510433 0.510433i −0.404226 0.914659i \(-0.632459\pi\)
0.914659 + 0.404226i \(0.132459\pi\)
\(824\) 0 0
\(825\) −1.27949 + 13.6038i −0.0445462 + 0.473625i
\(826\) 0 0
\(827\) −9.37209 + 9.37209i −0.325900 + 0.325900i −0.851025 0.525125i \(-0.824018\pi\)
0.525125 + 0.851025i \(0.324018\pi\)
\(828\) 0 0
\(829\) 32.6312i 1.13333i −0.823949 0.566664i \(-0.808234\pi\)
0.823949 0.566664i \(-0.191766\pi\)
\(830\) 0 0
\(831\) −7.47597 −0.259339
\(832\) 0 0
\(833\) −86.6246 86.6246i −3.00137 3.00137i
\(834\) 0 0
\(835\) 28.9668 10.4364i 1.00244 0.361166i
\(836\) 0 0
\(837\) −1.76641 1.76641i −0.0610561 0.0610561i
\(838\) 0 0
\(839\) −24.2817 −0.838298 −0.419149 0.907918i \(-0.637671\pi\)
−0.419149 + 0.907918i \(0.637671\pi\)
\(840\) 0 0
\(841\) 24.9966 0.861953
\(842\) 0 0
\(843\) 5.74618 + 5.74618i 0.197909 + 0.197909i
\(844\) 0 0
\(845\) −25.7572 12.1131i −0.886076 0.416702i
\(846\) 0 0
\(847\) 12.2995 12.2995i 0.422615 0.422615i
\(848\) 0 0
\(849\) 1.57916 0.0541968
\(850\) 0 0
\(851\) 6.71985 + 42.7827i 0.230354 + 1.46657i
\(852\) 0 0
\(853\) 39.7179 39.7179i 1.35992 1.35992i 0.485903 0.874013i \(-0.338491\pi\)
0.874013 0.485903i \(-0.161509\pi\)
\(854\) 0 0
\(855\) −11.9611 5.62505i −0.409062 0.192373i
\(856\) 0 0
\(857\) 22.7395 + 22.7395i 0.776767 + 0.776767i 0.979280 0.202512i \(-0.0649106\pi\)
−0.202512 + 0.979280i \(0.564911\pi\)
\(858\) 0 0
\(859\) 49.0394i 1.67320i −0.547813 0.836601i \(-0.684540\pi\)
0.547813 0.836601i \(-0.315460\pi\)
\(860\) 0 0
\(861\) 29.8347i 1.01676i
\(862\) 0 0
\(863\) −19.6158 + 19.6158i −0.667730 + 0.667730i −0.957190 0.289460i \(-0.906524\pi\)
0.289460 + 0.957190i \(0.406524\pi\)
\(864\) 0 0
\(865\) −8.21775 22.8089i −0.279412 0.775525i
\(866\) 0 0
\(867\) −23.6274 23.6274i −0.802427 0.802427i
\(868\) 0 0
\(869\) 7.94730i 0.269594i
\(870\) 0 0
\(871\) 1.54328i 0.0522922i
\(872\) 0 0
\(873\) −0.793720 0.793720i −0.0268634 0.0268634i
\(874\) 0 0
\(875\) 27.9949 47.4130i 0.946399 1.60285i
\(876\) 0 0
\(877\) 29.7427 + 29.7427i 1.00434 + 1.00434i 0.999991 + 0.00434836i \(0.00138413\pi\)
0.00434836 + 0.999991i \(0.498616\pi\)
\(878\) 0 0
\(879\) −15.5919 −0.525903
\(880\) 0 0
\(881\) 6.36193i 0.214339i 0.994241 + 0.107169i \(0.0341787\pi\)
−0.994241 + 0.107169i \(0.965821\pi\)
\(882\) 0 0
\(883\) 4.69497 4.69497i 0.157998 0.157998i −0.623681 0.781679i \(-0.714363\pi\)
0.781679 + 0.623681i \(0.214363\pi\)
\(884\) 0 0
\(885\) −10.5376 + 3.79656i −0.354217 + 0.127620i
\(886\) 0 0
\(887\) 31.0726 + 31.0726i 1.04331 + 1.04331i 0.999018 + 0.0442963i \(0.0141046\pi\)
0.0442963 + 0.999018i \(0.485895\pi\)
\(888\) 0 0
\(889\) 77.9409 2.61406
\(890\) 0 0
\(891\) 2.73277i 0.0915514i
\(892\) 0 0
\(893\) −23.7284 23.7284i −0.794041 0.794041i
\(894\) 0 0
\(895\) −8.60706 4.04770i −0.287702 0.135300i
\(896\) 0 0
\(897\) −2.01718 1.46953i −0.0673517 0.0490661i
\(898\) 0 0
\(899\) 4.99826i 0.166701i
\(900\) 0 0
\(901\) −7.98163 −0.265907
\(902\) 0 0
\(903\) −10.2422 + 10.2422i −0.340838 + 0.340838i
\(904\) 0 0
\(905\) −30.2816 14.2408i −1.00660 0.473379i
\(906\) 0 0
\(907\) 6.22151 6.22151i 0.206582 0.206582i −0.596231 0.802813i \(-0.703336\pi\)
0.802813 + 0.596231i \(0.203336\pi\)
\(908\) 0 0
\(909\) 2.66129i 0.0882694i
\(910\) 0 0
\(911\) 6.14528i 0.203602i 0.994805 + 0.101801i \(0.0324605\pi\)
−0.994805 + 0.101801i \(0.967539\pi\)
\(912\) 0 0
\(913\) 20.4432 20.4432i 0.676570 0.676570i
\(914\) 0 0
\(915\) 20.4498 7.36780i 0.676049 0.243572i
\(916\) 0 0
\(917\) −49.7146 + 49.7146i −1.64172 + 1.64172i
\(918\) 0 0
\(919\) 8.86802 0.292529 0.146265 0.989246i \(-0.453275\pi\)
0.146265 + 0.989246i \(0.453275\pi\)
\(920\) 0 0
\(921\) 31.5748 1.04043
\(922\) 0 0
\(923\) −3.44269 + 3.44269i −0.113317 + 0.113317i
\(924\) 0 0
\(925\) −44.9526 4.22796i −1.47803 0.139014i
\(926\) 0 0
\(927\) −3.17934 + 3.17934i −0.104423 + 0.104423i
\(928\) 0 0
\(929\) 38.3528i 1.25831i 0.777278 + 0.629157i \(0.216600\pi\)
−0.777278 + 0.629157i \(0.783400\pi\)
\(930\) 0 0
\(931\) 101.989i 3.34256i
\(932\) 0 0
\(933\) 2.79631 2.79631i 0.0915470 0.0915470i
\(934\) 0 0
\(935\) 40.8190 14.7066i 1.33493 0.480957i
\(936\) 0 0
\(937\) 15.6795 15.6795i 0.512226 0.512226i −0.402982 0.915208i \(-0.632026\pi\)
0.915208 + 0.402982i \(0.132026\pi\)
\(938\) 0 0
\(939\) 32.5234 1.06136
\(940\) 0 0
\(941\) 47.2322i 1.53973i −0.638209 0.769863i \(-0.720324\pi\)
0.638209 0.769863i \(-0.279676\pi\)
\(942\) 0 0
\(943\) 17.1073 23.4828i 0.557091 0.764704i
\(944\) 0 0
\(945\) 4.68642 9.96522i 0.152449 0.324169i
\(946\) 0 0
\(947\) 7.60447 + 7.60447i 0.247112 + 0.247112i 0.819784 0.572672i \(-0.194093\pi\)
−0.572672 + 0.819784i \(0.694093\pi\)
\(948\) 0 0
\(949\) 6.02074i 0.195442i
\(950\) 0 0
\(951\) −18.4728 −0.599020
\(952\) 0 0
\(953\) 27.6487 + 27.6487i 0.895628 + 0.895628i 0.995046 0.0994175i \(-0.0316979\pi\)
−0.0994175 + 0.995046i \(0.531698\pi\)
\(954\) 0 0
\(955\) 5.44066 + 2.55862i 0.176056 + 0.0827950i
\(956\) 0 0
\(957\) 3.86635 3.86635i 0.124981 0.124981i
\(958\) 0 0
\(959\) 26.0533i 0.841305i
\(960\) 0 0
\(961\) −24.7596 −0.798696
\(962\) 0 0
\(963\) 12.5484 + 12.5484i 0.404366 + 0.404366i
\(964\) 0 0
\(965\) −2.34962 6.52151i −0.0756369 0.209935i
\(966\) 0 0
\(967\) −9.16686 9.16686i −0.294786 0.294786i 0.544181 0.838968i \(-0.316840\pi\)
−0.838968 + 0.544181i \(0.816840\pi\)
\(968\) 0 0
\(969\) 41.9710i 1.34830i
\(970\) 0 0
\(971\) 14.3130i 0.459327i 0.973270 + 0.229663i \(0.0737626\pi\)
−0.973270 + 0.229663i \(0.926237\pi\)
\(972\) 0 0
\(973\) −13.3315 13.3315i −0.427389 0.427389i
\(974\) 0 0
\(975\) 2.00406 1.65949i 0.0641812 0.0531461i
\(976\) 0 0
\(977\) 3.39164 3.39164i 0.108508 0.108508i −0.650768 0.759276i \(-0.725553\pi\)
0.759276 + 0.650768i \(0.225553\pi\)
\(978\) 0 0
\(979\) 32.2081i 1.02938i
\(980\) 0 0
\(981\) 10.2093i 0.325959i
\(982\) 0 0
\(983\) −28.3572 28.3572i −0.904454 0.904454i 0.0913638 0.995818i \(-0.470877\pi\)
−0.995818 + 0.0913638i \(0.970877\pi\)
\(984\) 0 0
\(985\) −27.6567 + 9.96437i −0.881216 + 0.317491i
\(986\) 0 0
\(987\) 19.7689 19.7689i 0.629252 0.629252i
\(988\) 0 0
\(989\) 13.9345 2.18867i 0.443090 0.0695958i
\(990\) 0 0
\(991\) −5.98666 −0.190172 −0.0950862 0.995469i \(-0.530313\pi\)
−0.0950862 + 0.995469i \(0.530313\pi\)
\(992\) 0 0
\(993\) 1.62433 1.62433i 0.0515466 0.0515466i
\(994\) 0 0
\(995\) −25.4006 + 54.0120i −0.805254 + 1.71230i
\(996\) 0 0
\(997\) 0.598100 + 0.598100i 0.0189420 + 0.0189420i 0.716514 0.697572i \(-0.245736\pi\)
−0.697572 + 0.716514i \(0.745736\pi\)
\(998\) 0 0
\(999\) −9.03019 −0.285703
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 1380.2.t.a.1057.13 48
5.3 odd 4 inner 1380.2.t.a.1333.14 yes 48
23.22 odd 2 inner 1380.2.t.a.1057.14 yes 48
115.68 even 4 inner 1380.2.t.a.1333.13 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.t.a.1057.13 48 1.1 even 1 trivial
1380.2.t.a.1057.14 yes 48 23.22 odd 2 inner
1380.2.t.a.1333.13 yes 48 115.68 even 4 inner
1380.2.t.a.1333.14 yes 48 5.3 odd 4 inner