Properties

Label 1380.2.t.a.1057.4
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 1057.4
Character \(\chi\) \(=\) 1380.1057
Dual form 1380.2.t.a.1333.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.62167 - 1.53954i) q^{5} +(-2.42328 + 2.42328i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.62167 - 1.53954i) q^{5} +(-2.42328 + 2.42328i) q^{7} -1.00000i q^{9} +4.29399i q^{11} +(-2.97472 + 2.97472i) q^{13} +(-0.0580751 + 2.23531i) q^{15} +(5.55993 - 5.55993i) q^{17} -3.90921 q^{19} -3.42704i q^{21} +(-3.74079 - 3.00108i) q^{23} +(0.259632 - 4.99325i) q^{25} +(0.707107 + 0.707107i) q^{27} +6.44894i q^{29} -7.44378 q^{31} +(-3.03631 - 3.03631i) q^{33} +(-0.199026 + 7.66051i) q^{35} +(1.45878 - 1.45878i) q^{37} -4.20688i q^{39} -4.04416 q^{41} +(-3.37770 - 3.37770i) q^{43} +(-1.53954 - 1.62167i) q^{45} +(-7.41315 - 7.41315i) q^{47} -4.74461i q^{49} +7.86293i q^{51} +(-0.484238 - 0.484238i) q^{53} +(6.61077 + 6.96344i) q^{55} +(2.76423 - 2.76423i) q^{57} +15.1559i q^{59} +12.1499i q^{61} +(2.42328 + 2.42328i) q^{63} +(-0.244315 + 9.40370i) q^{65} +(1.35595 - 1.35595i) q^{67} +(4.76722 - 0.523058i) q^{69} -14.1317 q^{71} +(-8.24680 + 8.24680i) q^{73} +(3.34718 + 3.71435i) q^{75} +(-10.4056 - 10.4056i) q^{77} +7.52299 q^{79} -1.00000 q^{81} +(3.22090 + 3.22090i) q^{83} +(0.456640 - 17.5761i) q^{85} +(-4.56009 - 4.56009i) q^{87} +7.86653 q^{89} -14.4172i q^{91} +(5.26355 - 5.26355i) q^{93} +(-6.33945 + 6.01839i) q^{95} +(4.32463 - 4.32463i) q^{97} +4.29399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) 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
\(5\) 1.62167 1.53954i 0.725233 0.688503i
\(6\) 0 0
\(7\) −2.42328 + 2.42328i −0.915915 + 0.915915i −0.996729 0.0808141i \(-0.974248\pi\)
0.0808141 + 0.996729i \(0.474248\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 4.29399i 1.29469i 0.762198 + 0.647344i \(0.224120\pi\)
−0.762198 + 0.647344i \(0.775880\pi\)
\(12\) 0 0
\(13\) −2.97472 + 2.97472i −0.825038 + 0.825038i −0.986826 0.161788i \(-0.948274\pi\)
0.161788 + 0.986826i \(0.448274\pi\)
\(14\) 0 0
\(15\) −0.0580751 + 2.23531i −0.0149949 + 0.577156i
\(16\) 0 0
\(17\) 5.55993 5.55993i 1.34848 1.34848i 0.461168 0.887313i \(-0.347430\pi\)
0.887313 0.461168i \(-0.152570\pi\)
\(18\) 0 0
\(19\) −3.90921 −0.896834 −0.448417 0.893824i \(-0.648012\pi\)
−0.448417 + 0.893824i \(0.648012\pi\)
\(20\) 0 0
\(21\) 3.42704i 0.747842i
\(22\) 0 0
\(23\) −3.74079 3.00108i −0.780009 0.625768i
\(24\) 0 0
\(25\) 0.259632 4.99325i 0.0519264 0.998651i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 6.44894i 1.19754i 0.800921 + 0.598769i \(0.204343\pi\)
−0.800921 + 0.598769i \(0.795657\pi\)
\(30\) 0 0
\(31\) −7.44378 −1.33694 −0.668471 0.743738i \(-0.733051\pi\)
−0.668471 + 0.743738i \(0.733051\pi\)
\(32\) 0 0
\(33\) −3.03631 3.03631i −0.528554 0.528554i
\(34\) 0 0
\(35\) −0.199026 + 7.66051i −0.0336415 + 1.29486i
\(36\) 0 0
\(37\) 1.45878 1.45878i 0.239822 0.239822i −0.576954 0.816777i \(-0.695759\pi\)
0.816777 + 0.576954i \(0.195759\pi\)
\(38\) 0 0
\(39\) 4.20688i 0.673640i
\(40\) 0 0
\(41\) −4.04416 −0.631592 −0.315796 0.948827i \(-0.602272\pi\)
−0.315796 + 0.948827i \(0.602272\pi\)
\(42\) 0 0
\(43\) −3.37770 3.37770i −0.515094 0.515094i 0.400989 0.916083i \(-0.368667\pi\)
−0.916083 + 0.400989i \(0.868667\pi\)
\(44\) 0 0
\(45\) −1.53954 1.62167i −0.229501 0.241744i
\(46\) 0 0
\(47\) −7.41315 7.41315i −1.08132 1.08132i −0.996387 0.0849330i \(-0.972932\pi\)
−0.0849330 0.996387i \(-0.527068\pi\)
\(48\) 0 0
\(49\) 4.74461i 0.677801i
\(50\) 0 0
\(51\) 7.86293i 1.10103i
\(52\) 0 0
\(53\) −0.484238 0.484238i −0.0665152 0.0665152i 0.673067 0.739582i \(-0.264977\pi\)
−0.739582 + 0.673067i \(0.764977\pi\)
\(54\) 0 0
\(55\) 6.61077 + 6.96344i 0.891397 + 0.938950i
\(56\) 0 0
\(57\) 2.76423 2.76423i 0.366131 0.366131i
\(58\) 0 0
\(59\) 15.1559i 1.97313i 0.163357 + 0.986567i \(0.447768\pi\)
−0.163357 + 0.986567i \(0.552232\pi\)
\(60\) 0 0
\(61\) 12.1499i 1.55563i 0.628494 + 0.777815i \(0.283672\pi\)
−0.628494 + 0.777815i \(0.716328\pi\)
\(62\) 0 0
\(63\) 2.42328 + 2.42328i 0.305305 + 0.305305i
\(64\) 0 0
\(65\) −0.244315 + 9.40370i −0.0303036 + 1.16639i
\(66\) 0 0
\(67\) 1.35595 1.35595i 0.165655 0.165655i −0.619411 0.785067i \(-0.712629\pi\)
0.785067 + 0.619411i \(0.212629\pi\)
\(68\) 0 0
\(69\) 4.76722 0.523058i 0.573906 0.0629688i
\(70\) 0 0
\(71\) −14.1317 −1.67712 −0.838560 0.544809i \(-0.816602\pi\)
−0.838560 + 0.544809i \(0.816602\pi\)
\(72\) 0 0
\(73\) −8.24680 + 8.24680i −0.965214 + 0.965214i −0.999415 0.0342006i \(-0.989111\pi\)
0.0342006 + 0.999415i \(0.489111\pi\)
\(74\) 0 0
\(75\) 3.34718 + 3.71435i 0.386499 + 0.428896i
\(76\) 0 0
\(77\) −10.4056 10.4056i −1.18582 1.18582i
\(78\) 0 0
\(79\) 7.52299 0.846403 0.423201 0.906036i \(-0.360906\pi\)
0.423201 + 0.906036i \(0.360906\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 3.22090 + 3.22090i 0.353540 + 0.353540i 0.861425 0.507885i \(-0.169573\pi\)
−0.507885 + 0.861425i \(0.669573\pi\)
\(84\) 0 0
\(85\) 0.456640 17.5761i 0.0495296 1.90640i
\(86\) 0 0
\(87\) −4.56009 4.56009i −0.488893 0.488893i
\(88\) 0 0
\(89\) 7.86653 0.833850 0.416925 0.908941i \(-0.363108\pi\)
0.416925 + 0.908941i \(0.363108\pi\)
\(90\) 0 0
\(91\) 14.4172i 1.51133i
\(92\) 0 0
\(93\) 5.26355 5.26355i 0.545804 0.545804i
\(94\) 0 0
\(95\) −6.33945 + 6.01839i −0.650414 + 0.617473i
\(96\) 0 0
\(97\) 4.32463 4.32463i 0.439099 0.439099i −0.452610 0.891709i \(-0.649507\pi\)
0.891709 + 0.452610i \(0.149507\pi\)
\(98\) 0 0
\(99\) 4.29399 0.431563
\(100\) 0 0
\(101\) −8.38025 −0.833866 −0.416933 0.908937i \(-0.636895\pi\)
−0.416933 + 0.908937i \(0.636895\pi\)
\(102\) 0 0
\(103\) −0.329316 0.329316i −0.0324485 0.0324485i 0.690696 0.723145i \(-0.257304\pi\)
−0.723145 + 0.690696i \(0.757304\pi\)
\(104\) 0 0
\(105\) −5.27607 5.55753i −0.514891 0.542360i
\(106\) 0 0
\(107\) 4.62917 4.62917i 0.447519 0.447519i −0.447010 0.894529i \(-0.647511\pi\)
0.894529 + 0.447010i \(0.147511\pi\)
\(108\) 0 0
\(109\) −19.3838 −1.85664 −0.928318 0.371788i \(-0.878745\pi\)
−0.928318 + 0.371788i \(0.878745\pi\)
\(110\) 0 0
\(111\) 2.06303i 0.195814i
\(112\) 0 0
\(113\) −7.60107 7.60107i −0.715048 0.715048i 0.252538 0.967587i \(-0.418735\pi\)
−0.967587 + 0.252538i \(0.918735\pi\)
\(114\) 0 0
\(115\) −10.6866 + 0.892341i −0.996532 + 0.0832113i
\(116\) 0 0
\(117\) 2.97472 + 2.97472i 0.275013 + 0.275013i
\(118\) 0 0
\(119\) 26.9466i 2.47019i
\(120\) 0 0
\(121\) −7.43838 −0.676216
\(122\) 0 0
\(123\) 2.85965 2.85965i 0.257846 0.257846i
\(124\) 0 0
\(125\) −7.26628 8.49713i −0.649916 0.760006i
\(126\) 0 0
\(127\) 4.71025 + 4.71025i 0.417967 + 0.417967i 0.884502 0.466536i \(-0.154498\pi\)
−0.466536 + 0.884502i \(0.654498\pi\)
\(128\) 0 0
\(129\) 4.77678 0.420572
\(130\) 0 0
\(131\) 3.75315 0.327914 0.163957 0.986467i \(-0.447574\pi\)
0.163957 + 0.986467i \(0.447574\pi\)
\(132\) 0 0
\(133\) 9.47312 9.47312i 0.821424 0.821424i
\(134\) 0 0
\(135\) 2.23531 + 0.0580751i 0.192385 + 0.00499831i
\(136\) 0 0
\(137\) −13.5311 + 13.5311i −1.15604 + 1.15604i −0.170724 + 0.985319i \(0.554611\pi\)
−0.985319 + 0.170724i \(0.945389\pi\)
\(138\) 0 0
\(139\) 13.4756i 1.14299i −0.820606 0.571494i \(-0.806364\pi\)
0.820606 0.571494i \(-0.193636\pi\)
\(140\) 0 0
\(141\) 10.4838 0.882894
\(142\) 0 0
\(143\) −12.7734 12.7734i −1.06817 1.06817i
\(144\) 0 0
\(145\) 9.92841 + 10.4581i 0.824509 + 0.868495i
\(146\) 0 0
\(147\) 3.35494 + 3.35494i 0.276711 + 0.276711i
\(148\) 0 0
\(149\) 1.23720 0.101355 0.0506777 0.998715i \(-0.483862\pi\)
0.0506777 + 0.998715i \(0.483862\pi\)
\(150\) 0 0
\(151\) −5.50269 −0.447802 −0.223901 0.974612i \(-0.571879\pi\)
−0.223901 + 0.974612i \(0.571879\pi\)
\(152\) 0 0
\(153\) −5.55993 5.55993i −0.449494 0.449494i
\(154\) 0 0
\(155\) −12.0714 + 11.4600i −0.969595 + 0.920489i
\(156\) 0 0
\(157\) 5.05369 5.05369i 0.403329 0.403329i −0.476076 0.879404i \(-0.657941\pi\)
0.879404 + 0.476076i \(0.157941\pi\)
\(158\) 0 0
\(159\) 0.684816 0.0543095
\(160\) 0 0
\(161\) 16.3375 1.79254i 1.28757 0.141272i
\(162\) 0 0
\(163\) 0.132281 0.132281i 0.0103611 0.0103611i −0.701907 0.712268i \(-0.747668\pi\)
0.712268 + 0.701907i \(0.247668\pi\)
\(164\) 0 0
\(165\) −9.59842 0.249374i −0.747236 0.0194137i
\(166\) 0 0
\(167\) 9.14511 + 9.14511i 0.707670 + 0.707670i 0.966045 0.258375i \(-0.0831870\pi\)
−0.258375 + 0.966045i \(0.583187\pi\)
\(168\) 0 0
\(169\) 4.69787i 0.361374i
\(170\) 0 0
\(171\) 3.90921i 0.298945i
\(172\) 0 0
\(173\) −10.4998 + 10.4998i −0.798283 + 0.798283i −0.982825 0.184541i \(-0.940920\pi\)
0.184541 + 0.982825i \(0.440920\pi\)
\(174\) 0 0
\(175\) 11.4709 + 12.7292i 0.867119 + 0.962240i
\(176\) 0 0
\(177\) −10.7169 10.7169i −0.805529 0.805529i
\(178\) 0 0
\(179\) 18.6034i 1.39048i −0.718777 0.695241i \(-0.755298\pi\)
0.718777 0.695241i \(-0.244702\pi\)
\(180\) 0 0
\(181\) 8.62136i 0.640820i −0.947279 0.320410i \(-0.896179\pi\)
0.947279 0.320410i \(-0.103821\pi\)
\(182\) 0 0
\(183\) −8.59125 8.59125i −0.635083 0.635083i
\(184\) 0 0
\(185\) 0.119811 4.61152i 0.00880866 0.339046i
\(186\) 0 0
\(187\) 23.8743 + 23.8743i 1.74586 + 1.74586i
\(188\) 0 0
\(189\) −3.42704 −0.249281
\(190\) 0 0
\(191\) 7.09143i 0.513118i 0.966529 + 0.256559i \(0.0825888\pi\)
−0.966529 + 0.256559i \(0.917411\pi\)
\(192\) 0 0
\(193\) −2.02699 + 2.02699i −0.145906 + 0.145906i −0.776286 0.630381i \(-0.782899\pi\)
0.630381 + 0.776286i \(0.282899\pi\)
\(194\) 0 0
\(195\) −6.47667 6.82218i −0.463804 0.488546i
\(196\) 0 0
\(197\) 17.1838 + 17.1838i 1.22429 + 1.22429i 0.966091 + 0.258203i \(0.0831302\pi\)
0.258203 + 0.966091i \(0.416870\pi\)
\(198\) 0 0
\(199\) 24.2591 1.71968 0.859839 0.510565i \(-0.170564\pi\)
0.859839 + 0.510565i \(0.170564\pi\)
\(200\) 0 0
\(201\) 1.91760i 0.135257i
\(202\) 0 0
\(203\) −15.6276 15.6276i −1.09684 1.09684i
\(204\) 0 0
\(205\) −6.55830 + 6.22615i −0.458051 + 0.434853i
\(206\) 0 0
\(207\) −3.00108 + 3.74079i −0.208589 + 0.260003i
\(208\) 0 0
\(209\) 16.7861i 1.16112i
\(210\) 0 0
\(211\) 20.0827 1.38255 0.691277 0.722590i \(-0.257049\pi\)
0.691277 + 0.722590i \(0.257049\pi\)
\(212\) 0 0
\(213\) 9.99260 9.99260i 0.684681 0.684681i
\(214\) 0 0
\(215\) −10.6776 0.277412i −0.728207 0.0189194i
\(216\) 0 0
\(217\) 18.0384 18.0384i 1.22453 1.22453i
\(218\) 0 0
\(219\) 11.6627i 0.788094i
\(220\) 0 0
\(221\) 33.0784i 2.22510i
\(222\) 0 0
\(223\) 9.02181 9.02181i 0.604145 0.604145i −0.337265 0.941410i \(-0.609502\pi\)
0.941410 + 0.337265i \(0.109502\pi\)
\(224\) 0 0
\(225\) −4.99325 0.259632i −0.332884 0.0173088i
\(226\) 0 0
\(227\) 8.48763 8.48763i 0.563343 0.563343i −0.366912 0.930256i \(-0.619585\pi\)
0.930256 + 0.366912i \(0.119585\pi\)
\(228\) 0 0
\(229\) 16.8119 1.11096 0.555480 0.831530i \(-0.312534\pi\)
0.555480 + 0.831530i \(0.312534\pi\)
\(230\) 0 0
\(231\) 14.7157 0.968221
\(232\) 0 0
\(233\) −13.3998 + 13.3998i −0.877849 + 0.877849i −0.993312 0.115463i \(-0.963165\pi\)
0.115463 + 0.993312i \(0.463165\pi\)
\(234\) 0 0
\(235\) −23.4345 0.608847i −1.52870 0.0397168i
\(236\) 0 0
\(237\) −5.31956 + 5.31956i −0.345542 + 0.345542i
\(238\) 0 0
\(239\) 0.795147i 0.0514338i 0.999669 + 0.0257169i \(0.00818685\pi\)
−0.999669 + 0.0257169i \(0.991813\pi\)
\(240\) 0 0
\(241\) 23.1419i 1.49070i 0.666674 + 0.745350i \(0.267718\pi\)
−0.666674 + 0.745350i \(0.732282\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) −7.30451 7.69419i −0.466668 0.491564i
\(246\) 0 0
\(247\) 11.6288 11.6288i 0.739922 0.739922i
\(248\) 0 0
\(249\) −4.55504 −0.288664
\(250\) 0 0
\(251\) 4.99742i 0.315435i −0.987484 0.157717i \(-0.949587\pi\)
0.987484 0.157717i \(-0.0504135\pi\)
\(252\) 0 0
\(253\) 12.8866 16.0629i 0.810174 1.00987i
\(254\) 0 0
\(255\) 12.1053 + 12.7511i 0.758063 + 0.798504i
\(256\) 0 0
\(257\) 9.31312 + 9.31312i 0.580937 + 0.580937i 0.935161 0.354224i \(-0.115255\pi\)
−0.354224 + 0.935161i \(0.615255\pi\)
\(258\) 0 0
\(259\) 7.07009i 0.439314i
\(260\) 0 0
\(261\) 6.44894 0.399180
\(262\) 0 0
\(263\) 2.98627 + 2.98627i 0.184141 + 0.184141i 0.793158 0.609016i \(-0.208436\pi\)
−0.609016 + 0.793158i \(0.708436\pi\)
\(264\) 0 0
\(265\) −1.53078 0.0397708i −0.0940350 0.00244310i
\(266\) 0 0
\(267\) −5.56247 + 5.56247i −0.340418 + 0.340418i
\(268\) 0 0
\(269\) 6.65419i 0.405713i 0.979208 + 0.202857i \(0.0650225\pi\)
−0.979208 + 0.202857i \(0.934978\pi\)
\(270\) 0 0
\(271\) 3.10057 0.188346 0.0941731 0.995556i \(-0.469979\pi\)
0.0941731 + 0.995556i \(0.469979\pi\)
\(272\) 0 0
\(273\) 10.1945 + 10.1945i 0.616997 + 0.616997i
\(274\) 0 0
\(275\) 21.4410 + 1.11486i 1.29294 + 0.0672285i
\(276\) 0 0
\(277\) −14.2733 14.2733i −0.857599 0.857599i 0.133456 0.991055i \(-0.457393\pi\)
−0.991055 + 0.133456i \(0.957393\pi\)
\(278\) 0 0
\(279\) 7.44378i 0.445647i
\(280\) 0 0
\(281\) 5.58516i 0.333183i 0.986026 + 0.166591i \(0.0532760\pi\)
−0.986026 + 0.166591i \(0.946724\pi\)
\(282\) 0 0
\(283\) −15.5902 15.5902i −0.926739 0.926739i 0.0707550 0.997494i \(-0.477459\pi\)
−0.997494 + 0.0707550i \(0.977459\pi\)
\(284\) 0 0
\(285\) 0.227028 8.73831i 0.0134480 0.517613i
\(286\) 0 0
\(287\) 9.80015 9.80015i 0.578485 0.578485i
\(288\) 0 0
\(289\) 44.8256i 2.63680i
\(290\) 0 0
\(291\) 6.11595i 0.358523i
\(292\) 0 0
\(293\) 7.83471 + 7.83471i 0.457708 + 0.457708i 0.897903 0.440194i \(-0.145090\pi\)
−0.440194 + 0.897903i \(0.645090\pi\)
\(294\) 0 0
\(295\) 23.3332 + 24.5779i 1.35851 + 1.43098i
\(296\) 0 0
\(297\) −3.03631 + 3.03631i −0.176185 + 0.176185i
\(298\) 0 0
\(299\) 20.0551 2.20044i 1.15982 0.127255i
\(300\) 0 0
\(301\) 16.3702 0.943564
\(302\) 0 0
\(303\) 5.92573 5.92573i 0.340424 0.340424i
\(304\) 0 0
\(305\) 18.7052 + 19.7031i 1.07106 + 1.12819i
\(306\) 0 0
\(307\) 12.4715 + 12.4715i 0.711788 + 0.711788i 0.966909 0.255121i \(-0.0821154\pi\)
−0.255121 + 0.966909i \(0.582115\pi\)
\(308\) 0 0
\(309\) 0.465723 0.0264941
\(310\) 0 0
\(311\) 6.97357 0.395435 0.197717 0.980259i \(-0.436647\pi\)
0.197717 + 0.980259i \(0.436647\pi\)
\(312\) 0 0
\(313\) 3.65841 + 3.65841i 0.206786 + 0.206786i 0.802900 0.596114i \(-0.203289\pi\)
−0.596114 + 0.802900i \(0.703289\pi\)
\(314\) 0 0
\(315\) 7.66051 + 0.199026i 0.431621 + 0.0112138i
\(316\) 0 0
\(317\) 12.0363 + 12.0363i 0.676024 + 0.676024i 0.959098 0.283074i \(-0.0913542\pi\)
−0.283074 + 0.959098i \(0.591354\pi\)
\(318\) 0 0
\(319\) −27.6917 −1.55044
\(320\) 0 0
\(321\) 6.54664i 0.365398i
\(322\) 0 0
\(323\) −21.7349 + 21.7349i −1.20936 + 1.20936i
\(324\) 0 0
\(325\) 14.0812 + 15.6258i 0.781083 + 0.866766i
\(326\) 0 0
\(327\) 13.7064 13.7064i 0.757968 0.757968i
\(328\) 0 0
\(329\) 35.9283 1.98079
\(330\) 0 0
\(331\) 32.5843 1.79100 0.895499 0.445064i \(-0.146819\pi\)
0.895499 + 0.445064i \(0.146819\pi\)
\(332\) 0 0
\(333\) −1.45878 1.45878i −0.0799408 0.0799408i
\(334\) 0 0
\(335\) 0.111365 4.28643i 0.00608450 0.234193i
\(336\) 0 0
\(337\) −5.03755 + 5.03755i −0.274413 + 0.274413i −0.830874 0.556461i \(-0.812159\pi\)
0.556461 + 0.830874i \(0.312159\pi\)
\(338\) 0 0
\(339\) 10.7495 0.583835
\(340\) 0 0
\(341\) 31.9635i 1.73092i
\(342\) 0 0
\(343\) −5.46546 5.46546i −0.295107 0.295107i
\(344\) 0 0
\(345\) 6.92560 8.18756i 0.372862 0.440803i
\(346\) 0 0
\(347\) −5.38983 5.38983i −0.289341 0.289341i 0.547479 0.836820i \(-0.315588\pi\)
−0.836820 + 0.547479i \(0.815588\pi\)
\(348\) 0 0
\(349\) 29.7785i 1.59400i 0.603976 + 0.797002i \(0.293582\pi\)
−0.603976 + 0.797002i \(0.706418\pi\)
\(350\) 0 0
\(351\) −4.20688 −0.224547
\(352\) 0 0
\(353\) 16.8433 16.8433i 0.896476 0.896476i −0.0986466 0.995123i \(-0.531451\pi\)
0.995123 + 0.0986466i \(0.0314513\pi\)
\(354\) 0 0
\(355\) −22.9169 + 21.7563i −1.21630 + 1.15470i
\(356\) 0 0
\(357\) −19.0541 19.0541i −1.00845 1.00845i
\(358\) 0 0
\(359\) −3.87689 −0.204615 −0.102307 0.994753i \(-0.532622\pi\)
−0.102307 + 0.994753i \(0.532622\pi\)
\(360\) 0 0
\(361\) −3.71808 −0.195688
\(362\) 0 0
\(363\) 5.25973 5.25973i 0.276064 0.276064i
\(364\) 0 0
\(365\) −0.677314 + 26.0699i −0.0354522 + 1.36456i
\(366\) 0 0
\(367\) −10.3782 + 10.3782i −0.541738 + 0.541738i −0.924038 0.382300i \(-0.875132\pi\)
0.382300 + 0.924038i \(0.375132\pi\)
\(368\) 0 0
\(369\) 4.04416i 0.210531i
\(370\) 0 0
\(371\) 2.34689 0.121845
\(372\) 0 0
\(373\) −18.9119 18.9119i −0.979222 0.979222i 0.0205660 0.999788i \(-0.493453\pi\)
−0.999788 + 0.0205660i \(0.993453\pi\)
\(374\) 0 0
\(375\) 11.1464 + 0.870343i 0.575598 + 0.0449443i
\(376\) 0 0
\(377\) −19.1838 19.1838i −0.988015 0.988015i
\(378\) 0 0
\(379\) −23.0884 −1.18597 −0.592985 0.805213i \(-0.702051\pi\)
−0.592985 + 0.805213i \(0.702051\pi\)
\(380\) 0 0
\(381\) −6.66129 −0.341268
\(382\) 0 0
\(383\) 15.2739 + 15.2739i 0.780459 + 0.780459i 0.979908 0.199449i \(-0.0639152\pi\)
−0.199449 + 0.979908i \(0.563915\pi\)
\(384\) 0 0
\(385\) −32.8942 0.854615i −1.67644 0.0435552i
\(386\) 0 0
\(387\) −3.37770 + 3.37770i −0.171698 + 0.171698i
\(388\) 0 0
\(389\) 7.72655 0.391751 0.195876 0.980629i \(-0.437245\pi\)
0.195876 + 0.980629i \(0.437245\pi\)
\(390\) 0 0
\(391\) −37.4843 + 4.11277i −1.89566 + 0.207991i
\(392\) 0 0
\(393\) −2.65388 + 2.65388i −0.133870 + 0.133870i
\(394\) 0 0
\(395\) 12.1998 11.5819i 0.613839 0.582751i
\(396\) 0 0
\(397\) −25.7465 25.7465i −1.29218 1.29218i −0.933436 0.358745i \(-0.883205\pi\)
−0.358745 0.933436i \(-0.616795\pi\)
\(398\) 0 0
\(399\) 13.3970i 0.670690i
\(400\) 0 0
\(401\) 4.30417i 0.214940i 0.994208 + 0.107470i \(0.0342750\pi\)
−0.994208 + 0.107470i \(0.965725\pi\)
\(402\) 0 0
\(403\) 22.1431 22.1431i 1.10303 1.10303i
\(404\) 0 0
\(405\) −1.62167 + 1.53954i −0.0805815 + 0.0765004i
\(406\) 0 0
\(407\) 6.26400 + 6.26400i 0.310495 + 0.310495i
\(408\) 0 0
\(409\) 10.1730i 0.503022i −0.967854 0.251511i \(-0.919073\pi\)
0.967854 0.251511i \(-0.0809274\pi\)
\(410\) 0 0
\(411\) 19.1359i 0.943905i
\(412\) 0 0
\(413\) −36.7271 36.7271i −1.80722 1.80722i
\(414\) 0 0
\(415\) 10.1819 + 0.264534i 0.499812 + 0.0129855i
\(416\) 0 0
\(417\) 9.52871 + 9.52871i 0.466623 + 0.466623i
\(418\) 0 0
\(419\) 0.763920 0.0373199 0.0186600 0.999826i \(-0.494060\pi\)
0.0186600 + 0.999826i \(0.494060\pi\)
\(420\) 0 0
\(421\) 2.69690i 0.131439i −0.997838 0.0657194i \(-0.979066\pi\)
0.997838 0.0657194i \(-0.0209342\pi\)
\(422\) 0 0
\(423\) −7.41315 + 7.41315i −0.360440 + 0.360440i
\(424\) 0 0
\(425\) −26.3186 29.2057i −1.27664 1.41668i
\(426\) 0 0
\(427\) −29.4425 29.4425i −1.42482 1.42482i
\(428\) 0 0
\(429\) 18.0643 0.872154
\(430\) 0 0
\(431\) 12.1006i 0.582868i 0.956591 + 0.291434i \(0.0941323\pi\)
−0.956591 + 0.291434i \(0.905868\pi\)
\(432\) 0 0
\(433\) −16.5487 16.5487i −0.795281 0.795281i 0.187066 0.982347i \(-0.440102\pi\)
−0.982347 + 0.187066i \(0.940102\pi\)
\(434\) 0 0
\(435\) −14.4154 0.374523i −0.691166 0.0179570i
\(436\) 0 0
\(437\) 14.6235 + 11.7318i 0.699539 + 0.561210i
\(438\) 0 0
\(439\) 26.7012i 1.27438i 0.770706 + 0.637190i \(0.219904\pi\)
−0.770706 + 0.637190i \(0.780096\pi\)
\(440\) 0 0
\(441\) −4.74461 −0.225934
\(442\) 0 0
\(443\) 8.58880 8.58880i 0.408066 0.408066i −0.472998 0.881064i \(-0.656828\pi\)
0.881064 + 0.472998i \(0.156828\pi\)
\(444\) 0 0
\(445\) 12.7569 12.1108i 0.604736 0.574109i
\(446\) 0 0
\(447\) −0.874832 + 0.874832i −0.0413781 + 0.0413781i
\(448\) 0 0
\(449\) 5.02794i 0.237283i −0.992937 0.118642i \(-0.962146\pi\)
0.992937 0.118642i \(-0.0378540\pi\)
\(450\) 0 0
\(451\) 17.3656i 0.817714i
\(452\) 0 0
\(453\) 3.89099 3.89099i 0.182815 0.182815i
\(454\) 0 0
\(455\) −22.1958 23.3799i −1.04055 1.09607i
\(456\) 0 0
\(457\) 22.2129 22.2129i 1.03907 1.03907i 0.0398684 0.999205i \(-0.487306\pi\)
0.999205 0.0398684i \(-0.0126939\pi\)
\(458\) 0 0
\(459\) 7.86293 0.367010
\(460\) 0 0
\(461\) −3.08115 −0.143503 −0.0717517 0.997423i \(-0.522859\pi\)
−0.0717517 + 0.997423i \(0.522859\pi\)
\(462\) 0 0
\(463\) −12.8994 + 12.8994i −0.599486 + 0.599486i −0.940176 0.340690i \(-0.889339\pi\)
0.340690 + 0.940176i \(0.389339\pi\)
\(464\) 0 0
\(465\) 0.432298 16.6392i 0.0200473 0.771623i
\(466\) 0 0
\(467\) −21.6873 + 21.6873i −1.00357 + 1.00357i −0.00357715 + 0.999994i \(0.501139\pi\)
−0.999994 + 0.00357715i \(0.998861\pi\)
\(468\) 0 0
\(469\) 6.57169i 0.303452i
\(470\) 0 0
\(471\) 7.14700i 0.329316i
\(472\) 0 0
\(473\) 14.5038 14.5038i 0.666885 0.666885i
\(474\) 0 0
\(475\) −1.01496 + 19.5197i −0.0465694 + 0.895624i
\(476\) 0 0
\(477\) −0.484238 + 0.484238i −0.0221717 + 0.0221717i
\(478\) 0 0
\(479\) −18.8616 −0.861807 −0.430903 0.902398i \(-0.641805\pi\)
−0.430903 + 0.902398i \(0.641805\pi\)
\(480\) 0 0
\(481\) 8.67893i 0.395725i
\(482\) 0 0
\(483\) −10.2848 + 12.8198i −0.467975 + 0.583323i
\(484\) 0 0
\(485\) 0.355184 13.6711i 0.0161281 0.620771i
\(486\) 0 0
\(487\) 10.9118 + 10.9118i 0.494460 + 0.494460i 0.909708 0.415248i \(-0.136305\pi\)
−0.415248 + 0.909708i \(0.636305\pi\)
\(488\) 0 0
\(489\) 0.187074i 0.00845976i
\(490\) 0 0
\(491\) −12.8654 −0.580605 −0.290303 0.956935i \(-0.593756\pi\)
−0.290303 + 0.956935i \(0.593756\pi\)
\(492\) 0 0
\(493\) 35.8557 + 35.8557i 1.61486 + 1.61486i
\(494\) 0 0
\(495\) 6.96344 6.61077i 0.312983 0.297132i
\(496\) 0 0
\(497\) 34.2450 34.2450i 1.53610 1.53610i
\(498\) 0 0
\(499\) 16.6623i 0.745908i 0.927850 + 0.372954i \(0.121655\pi\)
−0.927850 + 0.372954i \(0.878345\pi\)
\(500\) 0 0
\(501\) −12.9331 −0.577810
\(502\) 0 0
\(503\) −2.18468 2.18468i −0.0974102 0.0974102i 0.656722 0.754133i \(-0.271942\pi\)
−0.754133 + 0.656722i \(0.771942\pi\)
\(504\) 0 0
\(505\) −13.5900 + 12.9017i −0.604747 + 0.574119i
\(506\) 0 0
\(507\) 3.32189 + 3.32189i 0.147530 + 0.147530i
\(508\) 0 0
\(509\) 23.5788i 1.04511i 0.852605 + 0.522557i \(0.175022\pi\)
−0.852605 + 0.522557i \(0.824978\pi\)
\(510\) 0 0
\(511\) 39.9686i 1.76811i
\(512\) 0 0
\(513\) −2.76423 2.76423i −0.122044 0.122044i
\(514\) 0 0
\(515\) −1.04104 0.0270469i −0.0458736 0.00119183i
\(516\) 0 0
\(517\) 31.8320 31.8320i 1.39997 1.39997i
\(518\) 0 0
\(519\) 14.8489i 0.651796i
\(520\) 0 0
\(521\) 40.5459i 1.77635i 0.459509 + 0.888173i \(0.348025\pi\)
−0.459509 + 0.888173i \(0.651975\pi\)
\(522\) 0 0
\(523\) 16.7767 + 16.7767i 0.733592 + 0.733592i 0.971329 0.237737i \(-0.0764057\pi\)
−0.237737 + 0.971329i \(0.576406\pi\)
\(524\) 0 0
\(525\) −17.1121 0.889770i −0.746833 0.0388327i
\(526\) 0 0
\(527\) −41.3869 + 41.3869i −1.80284 + 1.80284i
\(528\) 0 0
\(529\) 4.98707 + 22.4528i 0.216829 + 0.976210i
\(530\) 0 0
\(531\) 15.1559 0.657711
\(532\) 0 0
\(533\) 12.0302 12.0302i 0.521087 0.521087i
\(534\) 0 0
\(535\) 0.380197 14.6338i 0.0164373 0.632674i
\(536\) 0 0
\(537\) 13.1546 + 13.1546i 0.567662 + 0.567662i
\(538\) 0 0
\(539\) 20.3733 0.877540
\(540\) 0 0
\(541\) −5.07553 −0.218214 −0.109107 0.994030i \(-0.534799\pi\)
−0.109107 + 0.994030i \(0.534799\pi\)
\(542\) 0 0
\(543\) 6.09622 + 6.09622i 0.261614 + 0.261614i
\(544\) 0 0
\(545\) −31.4342 + 29.8422i −1.34649 + 1.27830i
\(546\) 0 0
\(547\) 10.0257 + 10.0257i 0.428667 + 0.428667i 0.888174 0.459507i \(-0.151974\pi\)
−0.459507 + 0.888174i \(0.651974\pi\)
\(548\) 0 0
\(549\) 12.1499 0.518543
\(550\) 0 0
\(551\) 25.2103i 1.07399i
\(552\) 0 0
\(553\) −18.2303 + 18.2303i −0.775233 + 0.775233i
\(554\) 0 0
\(555\) 3.17612 + 3.34556i 0.134819 + 0.142011i
\(556\) 0 0
\(557\) 14.2312 14.2312i 0.602996 0.602996i −0.338110 0.941106i \(-0.609788\pi\)
0.941106 + 0.338110i \(0.109788\pi\)
\(558\) 0 0
\(559\) 20.0954 0.849943
\(560\) 0 0
\(561\) −33.7634 −1.42549
\(562\) 0 0
\(563\) 4.04643 + 4.04643i 0.170537 + 0.170537i 0.787215 0.616678i \(-0.211522\pi\)
−0.616678 + 0.787215i \(0.711522\pi\)
\(564\) 0 0
\(565\) −24.0286 0.624280i −1.01089 0.0262637i
\(566\) 0 0
\(567\) 2.42328 2.42328i 0.101768 0.101768i
\(568\) 0 0
\(569\) −0.676811 −0.0283734 −0.0141867 0.999899i \(-0.504516\pi\)
−0.0141867 + 0.999899i \(0.504516\pi\)
\(570\) 0 0
\(571\) 23.5218i 0.984357i −0.870494 0.492178i \(-0.836201\pi\)
0.870494 0.492178i \(-0.163799\pi\)
\(572\) 0 0
\(573\) −5.01440 5.01440i −0.209480 0.209480i
\(574\) 0 0
\(575\) −15.9564 + 17.8996i −0.665427 + 0.746463i
\(576\) 0 0
\(577\) −11.6048 11.6048i −0.483113 0.483113i 0.423011 0.906124i \(-0.360973\pi\)
−0.906124 + 0.423011i \(0.860973\pi\)
\(578\) 0 0
\(579\) 2.86659i 0.119132i
\(580\) 0 0
\(581\) −15.6103 −0.647624
\(582\) 0 0
\(583\) 2.07932 2.07932i 0.0861164 0.0861164i
\(584\) 0 0
\(585\) 9.40370 + 0.244315i 0.388795 + 0.0101012i
\(586\) 0 0
\(587\) 9.63593 + 9.63593i 0.397717 + 0.397717i 0.877427 0.479710i \(-0.159258\pi\)
−0.479710 + 0.877427i \(0.659258\pi\)
\(588\) 0 0
\(589\) 29.0993 1.19902
\(590\) 0 0
\(591\) −24.3015 −0.999631
\(592\) 0 0
\(593\) 16.4629 16.4629i 0.676049 0.676049i −0.283055 0.959104i \(-0.591348\pi\)
0.959104 + 0.283055i \(0.0913478\pi\)
\(594\) 0 0
\(595\) 41.4853 + 43.6985i 1.70073 + 1.79146i
\(596\) 0 0
\(597\) −17.1537 + 17.1537i −0.702056 + 0.702056i
\(598\) 0 0
\(599\) 23.9384i 0.978097i −0.872257 0.489048i \(-0.837344\pi\)
0.872257 0.489048i \(-0.162656\pi\)
\(600\) 0 0
\(601\) −3.29617 −0.134454 −0.0672269 0.997738i \(-0.521415\pi\)
−0.0672269 + 0.997738i \(0.521415\pi\)
\(602\) 0 0
\(603\) −1.35595 1.35595i −0.0552184 0.0552184i
\(604\) 0 0
\(605\) −12.0626 + 11.4517i −0.490414 + 0.465577i
\(606\) 0 0
\(607\) −2.10815 2.10815i −0.0855670 0.0855670i 0.663028 0.748595i \(-0.269271\pi\)
−0.748595 + 0.663028i \(0.769271\pi\)
\(608\) 0 0
\(609\) 22.1008 0.895569
\(610\) 0 0
\(611\) 44.1041 1.78426
\(612\) 0 0
\(613\) 18.3502 + 18.3502i 0.741157 + 0.741157i 0.972801 0.231644i \(-0.0744104\pi\)
−0.231644 + 0.972801i \(0.574410\pi\)
\(614\) 0 0
\(615\) 0.234865 9.03997i 0.00947068 0.364527i
\(616\) 0 0
\(617\) −31.6621 + 31.6621i −1.27467 + 1.27467i −0.331061 + 0.943610i \(0.607406\pi\)
−0.943610 + 0.331061i \(0.892594\pi\)
\(618\) 0 0
\(619\) −2.85168 −0.114619 −0.0573093 0.998356i \(-0.518252\pi\)
−0.0573093 + 0.998356i \(0.518252\pi\)
\(620\) 0 0
\(621\) −0.523058 4.76722i −0.0209896 0.191302i
\(622\) 0 0
\(623\) −19.0628 + 19.0628i −0.763736 + 0.763736i
\(624\) 0 0
\(625\) −24.8652 2.59282i −0.994607 0.103713i
\(626\) 0 0
\(627\) 11.8696 + 11.8696i 0.474025 + 0.474025i
\(628\) 0 0
\(629\) 16.2215i 0.646792i
\(630\) 0 0
\(631\) 8.75077i 0.348362i 0.984714 + 0.174181i \(0.0557279\pi\)
−0.984714 + 0.174181i \(0.944272\pi\)
\(632\) 0 0
\(633\) −14.2006 + 14.2006i −0.564425 + 0.564425i
\(634\) 0 0
\(635\) 14.8901 + 0.386855i 0.590895 + 0.0153519i
\(636\) 0 0
\(637\) 14.1139 + 14.1139i 0.559211 + 0.559211i
\(638\) 0 0
\(639\) 14.1317i 0.559040i
\(640\) 0 0
\(641\) 24.3110i 0.960228i −0.877206 0.480114i \(-0.840595\pi\)
0.877206 0.480114i \(-0.159405\pi\)
\(642\) 0 0
\(643\) 1.56365 + 1.56365i 0.0616643 + 0.0616643i 0.737266 0.675602i \(-0.236116\pi\)
−0.675602 + 0.737266i \(0.736116\pi\)
\(644\) 0 0
\(645\) 7.74637 7.35405i 0.305013 0.289565i
\(646\) 0 0
\(647\) −31.7988 31.7988i −1.25014 1.25014i −0.955657 0.294483i \(-0.904853\pi\)
−0.294483 0.955657i \(-0.595147\pi\)
\(648\) 0 0
\(649\) −65.0795 −2.55459
\(650\) 0 0
\(651\) 25.5101i 0.999821i
\(652\) 0 0
\(653\) 10.8488 10.8488i 0.424546 0.424546i −0.462220 0.886765i \(-0.652947\pi\)
0.886765 + 0.462220i \(0.152947\pi\)
\(654\) 0 0
\(655\) 6.08637 5.77813i 0.237814 0.225770i
\(656\) 0 0
\(657\) 8.24680 + 8.24680i 0.321738 + 0.321738i
\(658\) 0 0
\(659\) 38.4656 1.49841 0.749203 0.662341i \(-0.230437\pi\)
0.749203 + 0.662341i \(0.230437\pi\)
\(660\) 0 0
\(661\) 25.7565i 1.00181i 0.865502 + 0.500905i \(0.166999\pi\)
−0.865502 + 0.500905i \(0.833001\pi\)
\(662\) 0 0
\(663\) −23.3900 23.3900i −0.908391 0.908391i
\(664\) 0 0
\(665\) 0.778033 29.9465i 0.0301708 1.16128i
\(666\) 0 0
\(667\) 19.3538 24.1242i 0.749381 0.934091i
\(668\) 0 0
\(669\) 12.7588i 0.493282i
\(670\) 0 0
\(671\) −52.1714 −2.01405
\(672\) 0 0
\(673\) −30.2159 + 30.2159i −1.16474 + 1.16474i −0.181312 + 0.983426i \(0.558034\pi\)
−0.983426 + 0.181312i \(0.941966\pi\)
\(674\) 0 0
\(675\) 3.71435 3.34718i 0.142965 0.128833i
\(676\) 0 0
\(677\) 0.437662 0.437662i 0.0168207 0.0168207i −0.698646 0.715467i \(-0.746214\pi\)
0.715467 + 0.698646i \(0.246214\pi\)
\(678\) 0 0
\(679\) 20.9596i 0.804355i
\(680\) 0 0
\(681\) 12.0033i 0.459968i
\(682\) 0 0
\(683\) −4.39542 + 4.39542i −0.168186 + 0.168186i −0.786182 0.617996i \(-0.787945\pi\)
0.617996 + 0.786182i \(0.287945\pi\)
\(684\) 0 0
\(685\) −1.11132 + 42.7748i −0.0424614 + 1.63434i
\(686\) 0 0
\(687\) −11.8878 + 11.8878i −0.453548 + 0.453548i
\(688\) 0 0
\(689\) 2.88094 0.109755
\(690\) 0 0
\(691\) 6.66838 0.253677 0.126839 0.991923i \(-0.459517\pi\)
0.126839 + 0.991923i \(0.459517\pi\)
\(692\) 0 0
\(693\) −10.4056 + 10.4056i −0.395275 + 0.395275i
\(694\) 0 0
\(695\) −20.7463 21.8530i −0.786951 0.828933i
\(696\) 0 0
\(697\) −22.4853 + 22.4853i −0.851690 + 0.851690i
\(698\) 0 0
\(699\) 18.9502i 0.716761i
\(700\) 0 0
\(701\) 14.9673i 0.565309i 0.959222 + 0.282654i \(0.0912150\pi\)
−0.959222 + 0.282654i \(0.908785\pi\)
\(702\) 0 0
\(703\) −5.70269 + 5.70269i −0.215081 + 0.215081i
\(704\) 0 0
\(705\) 17.0012 16.1402i 0.640304 0.607875i
\(706\) 0 0
\(707\) 20.3077 20.3077i 0.763750 0.763750i
\(708\) 0 0
\(709\) −16.4512 −0.617838 −0.308919 0.951088i \(-0.599967\pi\)
−0.308919 + 0.951088i \(0.599967\pi\)
\(710\) 0 0
\(711\) 7.52299i 0.282134i
\(712\) 0 0
\(713\) 27.8456 + 22.3394i 1.04283 + 0.836615i
\(714\) 0 0
\(715\) −40.3794 1.04909i −1.51011 0.0392337i
\(716\) 0 0
\(717\) −0.562254 0.562254i −0.0209978 0.0209978i
\(718\) 0 0
\(719\) 13.5756i 0.506286i 0.967429 + 0.253143i \(0.0814642\pi\)
−0.967429 + 0.253143i \(0.918536\pi\)
\(720\) 0 0
\(721\) 1.59605 0.0594401
\(722\) 0 0
\(723\) −16.3638 16.3638i −0.608576 0.608576i
\(724\) 0 0
\(725\) 32.2012 + 1.67435i 1.19592 + 0.0621839i
\(726\) 0 0
\(727\) −1.54345 + 1.54345i −0.0572435 + 0.0572435i −0.735149 0.677906i \(-0.762888\pi\)
0.677906 + 0.735149i \(0.262888\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −37.5595 −1.38919
\(732\) 0 0
\(733\) 9.44932 + 9.44932i 0.349019 + 0.349019i 0.859744 0.510725i \(-0.170623\pi\)
−0.510725 + 0.859744i \(0.670623\pi\)
\(734\) 0 0
\(735\) 10.6057 + 0.275544i 0.391197 + 0.0101636i
\(736\) 0 0
\(737\) 5.82242 + 5.82242i 0.214472 + 0.214472i
\(738\) 0 0
\(739\) 39.5398i 1.45450i −0.686374 0.727248i \(-0.740799\pi\)
0.686374 0.727248i \(-0.259201\pi\)
\(740\) 0 0
\(741\) 16.4456i 0.604144i
\(742\) 0 0
\(743\) −13.1890 13.1890i −0.483858 0.483858i 0.422503 0.906361i \(-0.361152\pi\)
−0.906361 + 0.422503i \(0.861152\pi\)
\(744\) 0 0
\(745\) 2.00633 1.90472i 0.0735062 0.0697835i
\(746\) 0 0
\(747\) 3.22090 3.22090i 0.117847 0.117847i
\(748\) 0 0
\(749\) 22.4356i 0.819779i
\(750\) 0 0
\(751\) 37.2467i 1.35915i −0.733606 0.679576i \(-0.762164\pi\)
0.733606 0.679576i \(-0.237836\pi\)
\(752\) 0 0
\(753\) 3.53371 + 3.53371i 0.128776 + 0.128776i
\(754\) 0 0
\(755\) −8.92355 + 8.47161i −0.324761 + 0.308313i
\(756\) 0 0
\(757\) 10.5183 10.5183i 0.382296 0.382296i −0.489633 0.871929i \(-0.662869\pi\)
0.871929 + 0.489633i \(0.162869\pi\)
\(758\) 0 0
\(759\) 2.24601 + 20.4704i 0.0815249 + 0.743029i
\(760\) 0 0
\(761\) 52.6037 1.90688 0.953441 0.301579i \(-0.0975136\pi\)
0.953441 + 0.301579i \(0.0975136\pi\)
\(762\) 0 0
\(763\) 46.9725 46.9725i 1.70052 1.70052i
\(764\) 0 0
\(765\) −17.5761 0.456640i −0.635466 0.0165099i
\(766\) 0 0
\(767\) −45.0846 45.0846i −1.62791 1.62791i
\(768\) 0 0
\(769\) −24.2044 −0.872833 −0.436417 0.899745i \(-0.643753\pi\)
−0.436417 + 0.899745i \(0.643753\pi\)
\(770\) 0 0
\(771\) −13.1707 −0.474333
\(772\) 0 0
\(773\) 3.19000 + 3.19000i 0.114736 + 0.114736i 0.762144 0.647408i \(-0.224147\pi\)
−0.647408 + 0.762144i \(0.724147\pi\)
\(774\) 0 0
\(775\) −1.93264 + 37.1687i −0.0694226 + 1.33514i
\(776\) 0 0
\(777\) −4.99931 4.99931i −0.179349 0.179349i
\(778\) 0 0
\(779\) 15.8095 0.566433
\(780\) 0 0
\(781\) 60.6813i 2.17135i
\(782\) 0 0
\(783\) −4.56009 + 4.56009i −0.162964 + 0.162964i
\(784\) 0 0
\(785\) 0.415063 15.9758i 0.0148142 0.570200i
\(786\) 0 0
\(787\) −22.6154 + 22.6154i −0.806151 + 0.806151i −0.984049 0.177898i \(-0.943070\pi\)
0.177898 + 0.984049i \(0.443070\pi\)
\(788\) 0 0
\(789\) −4.22323 −0.150351
\(790\) 0 0
\(791\) 36.8391 1.30985
\(792\) 0 0
\(793\) −36.1424 36.1424i −1.28345 1.28345i
\(794\) 0 0
\(795\) 1.11055 1.05430i 0.0393870 0.0373922i
\(796\) 0 0
\(797\) −3.37906 + 3.37906i −0.119693 + 0.119693i −0.764416 0.644723i \(-0.776973\pi\)
0.644723 + 0.764416i \(0.276973\pi\)
\(798\) 0 0
\(799\) −82.4332 −2.91628
\(800\) 0 0
\(801\) 7.86653i 0.277950i
\(802\) 0 0
\(803\) −35.4117 35.4117i −1.24965 1.24965i
\(804\) 0 0
\(805\) 23.7343 28.0591i 0.836524 0.988953i
\(806\) 0 0
\(807\) −4.70522 4.70522i −0.165632 0.165632i
\(808\) 0 0
\(809\) 47.1962i 1.65933i −0.558261 0.829665i \(-0.688531\pi\)
0.558261 0.829665i \(-0.311469\pi\)
\(810\) 0 0
\(811\) −3.16242 −0.111048 −0.0555238 0.998457i \(-0.517683\pi\)
−0.0555238 + 0.998457i \(0.517683\pi\)
\(812\) 0 0
\(813\) −2.19243 + 2.19243i −0.0768920 + 0.0768920i
\(814\) 0 0
\(815\) 0.0108643 0.418168i 0.000380561 0.0146478i
\(816\) 0 0
\(817\) 13.2041 + 13.2041i 0.461954 + 0.461954i
\(818\) 0 0
\(819\) −14.4172 −0.503776
\(820\) 0 0
\(821\) −22.8676 −0.798085 −0.399043 0.916932i \(-0.630657\pi\)
−0.399043 + 0.916932i \(0.630657\pi\)
\(822\) 0 0
\(823\) −23.2903 + 23.2903i −0.811847 + 0.811847i −0.984911 0.173064i \(-0.944633\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(824\) 0 0
\(825\) −15.9494 + 14.3728i −0.555287 + 0.500395i
\(826\) 0 0
\(827\) −3.50826 + 3.50826i −0.121994 + 0.121994i −0.765468 0.643474i \(-0.777493\pi\)
0.643474 + 0.765468i \(0.277493\pi\)
\(828\) 0 0
\(829\) 4.58447i 0.159225i −0.996826 0.0796126i \(-0.974632\pi\)
0.996826 0.0796126i \(-0.0253683\pi\)
\(830\) 0 0
\(831\) 20.1855 0.700227
\(832\) 0 0
\(833\) −26.3797 26.3797i −0.914002 0.914002i
\(834\) 0 0
\(835\) 28.9096 + 0.751093i 1.00046 + 0.0259926i
\(836\) 0 0
\(837\) −5.26355 5.26355i −0.181935 0.181935i
\(838\) 0 0
\(839\) −0.796335 −0.0274925 −0.0137463 0.999906i \(-0.504376\pi\)
−0.0137463 + 0.999906i \(0.504376\pi\)
\(840\) 0 0
\(841\) −12.5889 −0.434099
\(842\) 0 0
\(843\) −3.94930 3.94930i −0.136021 0.136021i
\(844\) 0 0
\(845\) −7.23255 7.61839i −0.248807 0.262081i
\(846\) 0 0
\(847\) 18.0253 18.0253i 0.619356 0.619356i
\(848\) 0 0
\(849\) 22.0478 0.756679
\(850\) 0 0
\(851\) −9.83493 + 1.07908i −0.337137 + 0.0369905i
\(852\) 0 0
\(853\) 1.75847 1.75847i 0.0602089 0.0602089i −0.676361 0.736570i \(-0.736444\pi\)
0.736570 + 0.676361i \(0.236444\pi\)
\(854\) 0 0
\(855\) 6.01839 + 6.33945i 0.205824 + 0.216805i
\(856\) 0 0
\(857\) 0.0724985 + 0.0724985i 0.00247650 + 0.00247650i 0.708344 0.705867i \(-0.249443\pi\)
−0.705867 + 0.708344i \(0.749443\pi\)
\(858\) 0 0
\(859\) 45.5671i 1.55473i 0.629050 + 0.777365i \(0.283444\pi\)
−0.629050 + 0.777365i \(0.716556\pi\)
\(860\) 0 0
\(861\) 13.8595i 0.472331i
\(862\) 0 0
\(863\) −33.9486 + 33.9486i −1.15562 + 1.15562i −0.170217 + 0.985407i \(0.554447\pi\)
−0.985407 + 0.170217i \(0.945553\pi\)
\(864\) 0 0
\(865\) −0.862353 + 33.1920i −0.0293209 + 1.12856i
\(866\) 0 0
\(867\) 31.6965 + 31.6965i 1.07647 + 1.07647i
\(868\) 0 0
\(869\) 32.3037i 1.09583i
\(870\) 0 0
\(871\) 8.06711i 0.273344i
\(872\) 0 0
\(873\) −4.32463 4.32463i −0.146366 0.146366i
\(874\) 0 0
\(875\) 38.1992 + 2.98270i 1.29137 + 0.100834i
\(876\) 0 0
\(877\) −32.7483 32.7483i −1.10583 1.10583i −0.993692 0.112140i \(-0.964229\pi\)
−0.112140 0.993692i \(-0.535771\pi\)
\(878\) 0 0
\(879\) −11.0799 −0.373717
\(880\) 0 0
\(881\) 1.99317i 0.0671517i 0.999436 + 0.0335759i \(0.0106895\pi\)
−0.999436 + 0.0335759i \(0.989310\pi\)
\(882\) 0 0
\(883\) −34.5670 + 34.5670i −1.16327 + 1.16327i −0.179519 + 0.983754i \(0.557454\pi\)
−0.983754 + 0.179519i \(0.942546\pi\)
\(884\) 0 0
\(885\) −33.8783 0.880182i −1.13881 0.0295870i
\(886\) 0 0
\(887\) 5.83938 + 5.83938i 0.196067 + 0.196067i 0.798312 0.602244i \(-0.205727\pi\)
−0.602244 + 0.798312i \(0.705727\pi\)
\(888\) 0 0
\(889\) −22.8285 −0.765644
\(890\) 0 0
\(891\) 4.29399i 0.143854i
\(892\) 0 0
\(893\) 28.9796 + 28.9796i 0.969764 + 0.969764i
\(894\) 0 0
\(895\) −28.6406 30.1685i −0.957351 1.00842i
\(896\) 0 0
\(897\) −12.6252 + 15.7371i −0.421543 + 0.525446i
\(898\) 0 0
\(899\) 48.0045i 1.60104i
\(900\) 0 0
\(901\) −5.38466 −0.179389
\(902\) 0 0
\(903\) −11.5755 + 11.5755i −0.385208 + 0.385208i
\(904\) 0 0
\(905\) −13.2729 13.9810i −0.441207 0.464744i
\(906\) 0 0
\(907\) 19.9819 19.9819i 0.663489 0.663489i −0.292712 0.956201i \(-0.594558\pi\)
0.956201 + 0.292712i \(0.0945577\pi\)
\(908\) 0 0
\(909\) 8.38025i 0.277955i
\(910\) 0 0
\(911\) 46.9352i 1.55503i 0.628862 + 0.777517i \(0.283521\pi\)
−0.628862 + 0.777517i \(0.716479\pi\)
\(912\) 0 0
\(913\) −13.8305 + 13.8305i −0.457723 + 0.457723i
\(914\) 0 0
\(915\) −27.1587 0.705604i −0.897840 0.0233266i
\(916\) 0 0
\(917\) −9.09495 + 9.09495i −0.300342 + 0.300342i
\(918\) 0 0
\(919\) 30.7173 1.01327 0.506634 0.862161i \(-0.330889\pi\)
0.506634 + 0.862161i \(0.330889\pi\)
\(920\) 0 0
\(921\) −17.6374 −0.581172
\(922\) 0 0
\(923\) 42.0377 42.0377i 1.38369 1.38369i
\(924\) 0 0
\(925\) −6.90533 7.66282i −0.227046 0.251952i
\(926\) 0 0
\(927\) −0.329316 + 0.329316i −0.0108162 + 0.0108162i
\(928\) 0 0
\(929\) 13.8148i 0.453250i 0.973982 + 0.226625i \(0.0727692\pi\)
−0.973982 + 0.226625i \(0.927231\pi\)
\(930\) 0 0
\(931\) 18.5477i 0.607875i
\(932\) 0 0
\(933\) −4.93106 + 4.93106i −0.161436 + 0.161436i
\(934\) 0 0
\(935\) 75.4717 + 1.96081i 2.46819 + 0.0641254i
\(936\) 0 0
\(937\) 17.6846 17.6846i 0.577730 0.577730i −0.356548 0.934277i \(-0.616046\pi\)
0.934277 + 0.356548i \(0.116046\pi\)
\(938\) 0 0
\(939\) −5.17378 −0.168840
\(940\) 0 0
\(941\) 28.1623i 0.918064i 0.888420 + 0.459032i \(0.151804\pi\)
−0.888420 + 0.459032i \(0.848196\pi\)
\(942\) 0 0
\(943\) 15.1284 + 12.1368i 0.492648 + 0.395230i
\(944\) 0 0
\(945\) −5.55753 + 5.27607i −0.180787 + 0.171630i
\(946\) 0 0
\(947\) −43.3171 43.3171i −1.40762 1.40762i −0.772033 0.635583i \(-0.780760\pi\)
−0.635583 0.772033i \(-0.719240\pi\)
\(948\) 0 0
\(949\) 49.0637i 1.59268i
\(950\) 0 0
\(951\) −17.0219 −0.551971
\(952\) 0 0
\(953\) 36.1667 + 36.1667i 1.17155 + 1.17155i 0.981839 + 0.189716i \(0.0607566\pi\)
0.189716 + 0.981839i \(0.439243\pi\)
\(954\) 0 0
\(955\) 10.9175 + 11.5000i 0.353283 + 0.372130i
\(956\) 0 0
\(957\) 19.5810 19.5810i 0.632964 0.632964i
\(958\) 0 0
\(959\) 65.5795i 2.11767i
\(960\) 0 0
\(961\) 24.4098 0.787414
\(962\) 0 0
\(963\) −4.62917 4.62917i −0.149173 0.149173i
\(964\) 0 0
\(965\) −0.166478 + 6.40774i −0.00535911 + 0.206272i
\(966\) 0 0
\(967\) −27.8045 27.8045i −0.894133 0.894133i 0.100776 0.994909i \(-0.467868\pi\)
−0.994909 + 0.100776i \(0.967868\pi\)
\(968\) 0 0
\(969\) 30.7378i 0.987441i
\(970\) 0 0
\(971\) 20.9889i 0.673567i 0.941582 + 0.336784i \(0.109339\pi\)
−0.941582 + 0.336784i \(0.890661\pi\)
\(972\) 0 0
\(973\) 32.6553 + 32.6553i 1.04688 + 1.04688i
\(974\) 0 0
\(975\) −21.0060 1.09224i −0.672732 0.0349797i
\(976\) 0 0
\(977\) 12.7848 12.7848i 0.409022 0.409022i −0.472376 0.881397i \(-0.656603\pi\)
0.881397 + 0.472376i \(0.156603\pi\)
\(978\) 0 0
\(979\) 33.7788i 1.07958i
\(980\) 0 0
\(981\) 19.3838i 0.618878i
\(982\) 0 0
\(983\) −22.2420 22.2420i −0.709411 0.709411i 0.257000 0.966411i \(-0.417266\pi\)
−0.966411 + 0.257000i \(0.917266\pi\)
\(984\) 0 0
\(985\) 54.3215 + 1.41131i 1.73083 + 0.0449682i
\(986\) 0 0
\(987\) −25.4052 + 25.4052i −0.808656 + 0.808656i
\(988\) 0 0
\(989\) 2.49853 + 22.7720i 0.0794487 + 0.724107i
\(990\) 0 0
\(991\) −37.8734 −1.20309 −0.601543 0.798840i \(-0.705447\pi\)
−0.601543 + 0.798840i \(0.705447\pi\)
\(992\) 0 0
\(993\) −23.0406 + 23.0406i −0.731172 + 0.731172i
\(994\) 0 0
\(995\) 39.3402 37.3478i 1.24717 1.18400i
\(996\) 0 0
\(997\) 24.5475 + 24.5475i 0.777428 + 0.777428i 0.979393 0.201965i \(-0.0647328\pi\)
−0.201965 + 0.979393i \(0.564733\pi\)
\(998\) 0 0
\(999\) 2.06303 0.0652714
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.1057.4 yes 48
5.3 odd 4 inner 1380.2.t.a.1333.3 yes 48
23.22 odd 2 inner 1380.2.t.a.1057.3 48
115.68 even 4 inner 1380.2.t.a.1333.4 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.t.a.1057.3 48 23.22 odd 2 inner
1380.2.t.a.1057.4 yes 48 1.1 even 1 trivial
1380.2.t.a.1333.3 yes 48 5.3 odd 4 inner
1380.2.t.a.1333.4 yes 48 115.68 even 4 inner