Properties

Label 1792.2.m.d.1345.2
Level $1792$
Weight $2$
Character 1792.1345
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1345.2
Root \(0.500000 - 0.0297061i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1345
Dual form 1792.2.m.d.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.236813 - 0.236813i) q^{3} +(2.98593 - 2.98593i) q^{5} -1.00000i q^{7} -2.88784i q^{9} +O(q^{10})\) \(q+(-0.236813 - 0.236813i) q^{3} +(2.98593 - 2.98593i) q^{5} -1.00000i q^{7} -2.88784i q^{9} +(2.55765 - 2.55765i) q^{11} +(-2.17740 - 2.17740i) q^{13} -1.41421 q^{15} -0.473626 q^{17} +(5.12936 + 5.12936i) q^{19} +(-0.236813 + 0.236813i) q^{21} +0.0840215i q^{23} -12.8316i q^{25} +(-1.39432 + 1.39432i) q^{27} +(-1.35480 - 1.35480i) q^{29} +0.196182 q^{31} -1.21137 q^{33} +(-2.98593 - 2.98593i) q^{35} +(1.74441 - 1.74441i) q^{37} +1.03127i q^{39} +5.81324i q^{41} +(-7.30205 + 7.30205i) q^{43} +(-8.62289 - 8.62289i) q^{45} +3.52637 q^{47} -1.00000 q^{49} +(0.112161 + 0.112161i) q^{51} +(-4.88784 + 4.88784i) q^{53} -15.2739i q^{55} -2.42940i q^{57} +(-7.51230 + 7.51230i) q^{59} +(5.77456 + 5.77456i) q^{61} -2.88784 q^{63} -13.0031 q^{65} +(9.77568 + 9.77568i) q^{67} +(0.0198974 - 0.0198974i) q^{69} -8.95668i q^{71} +10.7229i q^{73} +(-3.03868 + 3.03868i) q^{75} +(-2.55765 - 2.55765i) q^{77} +9.51156 q^{79} -8.00313 q^{81} +(-5.85387 - 5.85387i) q^{83} +(-1.41421 + 1.41421i) q^{85} +0.641669i q^{87} -15.3231i q^{89} +(-2.17740 + 2.17740i) q^{91} +(-0.0464585 - 0.0464585i) q^{93} +30.6318 q^{95} +11.2623 q^{97} +(-7.38607 - 7.38607i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{5} - 8 q^{11} - 12 q^{13} + 8 q^{17} + 4 q^{19} + 4 q^{21} - 8 q^{27} + 8 q^{31} - 16 q^{33} - 4 q^{35} + 8 q^{37} - 24 q^{43} - 12 q^{45} + 40 q^{47} - 8 q^{49} + 24 q^{51} - 16 q^{53} - 52 q^{59} + 20 q^{61} - 24 q^{65} + 32 q^{67} - 8 q^{69} - 28 q^{75} + 8 q^{77} + 16 q^{81} - 12 q^{83} - 12 q^{91} + 40 q^{93} + 80 q^{95} + 72 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.236813 0.236813i −0.136724 0.136724i 0.635432 0.772156i \(-0.280822\pi\)
−0.772156 + 0.635432i \(0.780822\pi\)
\(4\) 0 0
\(5\) 2.98593 2.98593i 1.33535 1.33535i 0.434842 0.900507i \(-0.356804\pi\)
0.900507 0.434842i \(-0.143196\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.88784i 0.962613i
\(10\) 0 0
\(11\) 2.55765 2.55765i 0.771160 0.771160i −0.207150 0.978309i \(-0.566419\pi\)
0.978309 + 0.207150i \(0.0664187\pi\)
\(12\) 0 0
\(13\) −2.17740 2.17740i −0.603902 0.603902i 0.337443 0.941346i \(-0.390438\pi\)
−0.941346 + 0.337443i \(0.890438\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) −0.473626 −0.114871 −0.0574356 0.998349i \(-0.518292\pi\)
−0.0574356 + 0.998349i \(0.518292\pi\)
\(18\) 0 0
\(19\) 5.12936 + 5.12936i 1.17676 + 1.17676i 0.980566 + 0.196191i \(0.0628573\pi\)
0.196191 + 0.980566i \(0.437143\pi\)
\(20\) 0 0
\(21\) −0.236813 + 0.236813i −0.0516768 + 0.0516768i
\(22\) 0 0
\(23\) 0.0840215i 0.0175197i 0.999962 + 0.00875985i \(0.00278838\pi\)
−0.999962 + 0.00875985i \(0.997212\pi\)
\(24\) 0 0
\(25\) 12.8316i 2.56631i
\(26\) 0 0
\(27\) −1.39432 + 1.39432i −0.268336 + 0.268336i
\(28\) 0 0
\(29\) −1.35480 1.35480i −0.251580 0.251580i 0.570038 0.821618i \(-0.306928\pi\)
−0.821618 + 0.570038i \(0.806928\pi\)
\(30\) 0 0
\(31\) 0.196182 0.0352354 0.0176177 0.999845i \(-0.494392\pi\)
0.0176177 + 0.999845i \(0.494392\pi\)
\(32\) 0 0
\(33\) −1.21137 −0.210872
\(34\) 0 0
\(35\) −2.98593 2.98593i −0.504714 0.504714i
\(36\) 0 0
\(37\) 1.74441 1.74441i 0.286779 0.286779i −0.549026 0.835805i \(-0.685001\pi\)
0.835805 + 0.549026i \(0.185001\pi\)
\(38\) 0 0
\(39\) 1.03127i 0.165136i
\(40\) 0 0
\(41\) 5.81324i 0.907876i 0.891033 + 0.453938i \(0.149981\pi\)
−0.891033 + 0.453938i \(0.850019\pi\)
\(42\) 0 0
\(43\) −7.30205 + 7.30205i −1.11355 + 1.11355i −0.120886 + 0.992666i \(0.538574\pi\)
−0.992666 + 0.120886i \(0.961426\pi\)
\(44\) 0 0
\(45\) −8.62289 8.62289i −1.28542 1.28542i
\(46\) 0 0
\(47\) 3.52637 0.514375 0.257187 0.966362i \(-0.417204\pi\)
0.257187 + 0.966362i \(0.417204\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0.112161 + 0.112161i 0.0157056 + 0.0157056i
\(52\) 0 0
\(53\) −4.88784 + 4.88784i −0.671396 + 0.671396i −0.958038 0.286642i \(-0.907461\pi\)
0.286642 + 0.958038i \(0.407461\pi\)
\(54\) 0 0
\(55\) 15.2739i 2.05953i
\(56\) 0 0
\(57\) 2.42940i 0.321782i
\(58\) 0 0
\(59\) −7.51230 + 7.51230i −0.978019 + 0.978019i −0.999764 0.0217448i \(-0.993078\pi\)
0.0217448 + 0.999764i \(0.493078\pi\)
\(60\) 0 0
\(61\) 5.77456 + 5.77456i 0.739357 + 0.739357i 0.972454 0.233097i \(-0.0748858\pi\)
−0.233097 + 0.972454i \(0.574886\pi\)
\(62\) 0 0
\(63\) −2.88784 −0.363834
\(64\) 0 0
\(65\) −13.0031 −1.61284
\(66\) 0 0
\(67\) 9.77568 + 9.77568i 1.19429 + 1.19429i 0.975851 + 0.218439i \(0.0700963\pi\)
0.218439 + 0.975851i \(0.429904\pi\)
\(68\) 0 0
\(69\) 0.0198974 0.0198974i 0.00239536 0.00239536i
\(70\) 0 0
\(71\) 8.95668i 1.06296i −0.847070 0.531481i \(-0.821636\pi\)
0.847070 0.531481i \(-0.178364\pi\)
\(72\) 0 0
\(73\) 10.7229i 1.25502i 0.778607 + 0.627512i \(0.215927\pi\)
−0.778607 + 0.627512i \(0.784073\pi\)
\(74\) 0 0
\(75\) −3.03868 + 3.03868i −0.350876 + 0.350876i
\(76\) 0 0
\(77\) −2.55765 2.55765i −0.291471 0.291471i
\(78\) 0 0
\(79\) 9.51156 1.07013 0.535067 0.844810i \(-0.320286\pi\)
0.535067 + 0.844810i \(0.320286\pi\)
\(80\) 0 0
\(81\) −8.00313 −0.889237
\(82\) 0 0
\(83\) −5.85387 5.85387i −0.642546 0.642546i 0.308635 0.951181i \(-0.400128\pi\)
−0.951181 + 0.308635i \(0.900128\pi\)
\(84\) 0 0
\(85\) −1.41421 + 1.41421i −0.153393 + 0.153393i
\(86\) 0 0
\(87\) 0.641669i 0.0687941i
\(88\) 0 0
\(89\) 15.3231i 1.62425i −0.583484 0.812124i \(-0.698311\pi\)
0.583484 0.812124i \(-0.301689\pi\)
\(90\) 0 0
\(91\) −2.17740 + 2.17740i −0.228254 + 0.228254i
\(92\) 0 0
\(93\) −0.0464585 0.0464585i −0.00481752 0.00481752i
\(94\) 0 0
\(95\) 30.6318 3.14276
\(96\) 0 0
\(97\) 11.2623 1.14351 0.571755 0.820425i \(-0.306263\pi\)
0.571755 + 0.820425i \(0.306263\pi\)
\(98\) 0 0
\(99\) −7.38607 7.38607i −0.742328 0.742328i
\(100\) 0 0
\(101\) 10.5947 10.5947i 1.05422 1.05422i 0.0557734 0.998443i \(-0.482238\pi\)
0.998443 0.0557734i \(-0.0177624\pi\)
\(102\) 0 0
\(103\) 12.6417i 1.24562i −0.782373 0.622810i \(-0.785991\pi\)
0.782373 0.622810i \(-0.214009\pi\)
\(104\) 0 0
\(105\) 1.41421i 0.138013i
\(106\) 0 0
\(107\) −12.1371 + 12.1371i −1.17334 + 1.17334i −0.191934 + 0.981408i \(0.561476\pi\)
−0.981408 + 0.191934i \(0.938524\pi\)
\(108\) 0 0
\(109\) 7.52951 + 7.52951i 0.721196 + 0.721196i 0.968849 0.247653i \(-0.0796592\pi\)
−0.247653 + 0.968849i \(0.579659\pi\)
\(110\) 0 0
\(111\) −0.826195 −0.0784190
\(112\) 0 0
\(113\) 3.00313 0.282511 0.141256 0.989973i \(-0.454886\pi\)
0.141256 + 0.989973i \(0.454886\pi\)
\(114\) 0 0
\(115\) 0.250882 + 0.250882i 0.0233949 + 0.0233949i
\(116\) 0 0
\(117\) −6.28798 + 6.28798i −0.581324 + 0.581324i
\(118\) 0 0
\(119\) 0.473626i 0.0434172i
\(120\) 0 0
\(121\) 2.08312i 0.189374i
\(122\) 0 0
\(123\) 1.37665 1.37665i 0.124128 0.124128i
\(124\) 0 0
\(125\) −23.3845 23.3845i −2.09157 2.09157i
\(126\) 0 0
\(127\) −18.8562 −1.67321 −0.836607 0.547803i \(-0.815464\pi\)
−0.836607 + 0.547803i \(0.815464\pi\)
\(128\) 0 0
\(129\) 3.45844 0.304499
\(130\) 0 0
\(131\) −1.10280 1.10280i −0.0963524 0.0963524i 0.657288 0.753640i \(-0.271704\pi\)
−0.753640 + 0.657288i \(0.771704\pi\)
\(132\) 0 0
\(133\) 5.12936 5.12936i 0.444772 0.444772i
\(134\) 0 0
\(135\) 8.32666i 0.716645i
\(136\) 0 0
\(137\) 8.04922i 0.687691i −0.939026 0.343846i \(-0.888270\pi\)
0.939026 0.343846i \(-0.111730\pi\)
\(138\) 0 0
\(139\) 4.69553 4.69553i 0.398270 0.398270i −0.479353 0.877622i \(-0.659128\pi\)
0.877622 + 0.479353i \(0.159128\pi\)
\(140\) 0 0
\(141\) −0.835091 0.835091i −0.0703273 0.0703273i
\(142\) 0 0
\(143\) −11.1380 −0.931410
\(144\) 0 0
\(145\) −8.09069 −0.671895
\(146\) 0 0
\(147\) 0.236813 + 0.236813i 0.0195320 + 0.0195320i
\(148\) 0 0
\(149\) −1.00942 + 1.00942i −0.0826952 + 0.0826952i −0.747244 0.664549i \(-0.768624\pi\)
0.664549 + 0.747244i \(0.268624\pi\)
\(150\) 0 0
\(151\) 0.863230i 0.0702487i 0.999383 + 0.0351243i \(0.0111827\pi\)
−0.999383 + 0.0351243i \(0.988817\pi\)
\(152\) 0 0
\(153\) 1.36776i 0.110576i
\(154\) 0 0
\(155\) 0.585786 0.585786i 0.0470515 0.0470515i
\(156\) 0 0
\(157\) −11.6475 11.6475i −0.929571 0.929571i 0.0681066 0.997678i \(-0.478304\pi\)
−0.997678 + 0.0681066i \(0.978304\pi\)
\(158\) 0 0
\(159\) 2.31501 0.183592
\(160\) 0 0
\(161\) 0.0840215 0.00662182
\(162\) 0 0
\(163\) 11.3861 + 11.3861i 0.891826 + 0.891826i 0.994695 0.102869i \(-0.0328022\pi\)
−0.102869 + 0.994695i \(0.532802\pi\)
\(164\) 0 0
\(165\) −3.61706 + 3.61706i −0.281588 + 0.281588i
\(166\) 0 0
\(167\) 4.47363i 0.346180i 0.984906 + 0.173090i \(0.0553751\pi\)
−0.984906 + 0.173090i \(0.944625\pi\)
\(168\) 0 0
\(169\) 3.51785i 0.270604i
\(170\) 0 0
\(171\) 14.8128 14.8128i 1.13276 1.13276i
\(172\) 0 0
\(173\) 2.71896 + 2.71896i 0.206719 + 0.206719i 0.802871 0.596152i \(-0.203305\pi\)
−0.596152 + 0.802871i \(0.703305\pi\)
\(174\) 0 0
\(175\) −12.8316 −0.969975
\(176\) 0 0
\(177\) 3.55802 0.267437
\(178\) 0 0
\(179\) 10.8910 + 10.8910i 0.814030 + 0.814030i 0.985235 0.171206i \(-0.0547663\pi\)
−0.171206 + 0.985235i \(0.554766\pi\)
\(180\) 0 0
\(181\) 2.52563 2.52563i 0.187729 0.187729i −0.606985 0.794713i \(-0.707621\pi\)
0.794713 + 0.606985i \(0.207621\pi\)
\(182\) 0 0
\(183\) 2.73498i 0.202176i
\(184\) 0 0
\(185\) 10.4173i 0.765899i
\(186\) 0 0
\(187\) −1.21137 + 1.21137i −0.0885840 + 0.0885840i
\(188\) 0 0
\(189\) 1.39432 + 1.39432i 0.101422 + 0.101422i
\(190\) 0 0
\(191\) −20.0233 −1.44884 −0.724418 0.689361i \(-0.757891\pi\)
−0.724418 + 0.689361i \(0.757891\pi\)
\(192\) 0 0
\(193\) 5.68943 0.409534 0.204767 0.978811i \(-0.434356\pi\)
0.204767 + 0.978811i \(0.434356\pi\)
\(194\) 0 0
\(195\) 3.07931 + 3.07931i 0.220514 + 0.220514i
\(196\) 0 0
\(197\) 16.7033 16.7033i 1.19006 1.19006i 0.213012 0.977050i \(-0.431673\pi\)
0.977050 0.213012i \(-0.0683274\pi\)
\(198\) 0 0
\(199\) 17.7968i 1.26158i −0.775954 0.630789i \(-0.782731\pi\)
0.775954 0.630789i \(-0.217269\pi\)
\(200\) 0 0
\(201\) 4.63001i 0.326576i
\(202\) 0 0
\(203\) −1.35480 + 1.35480i −0.0950884 + 0.0950884i
\(204\) 0 0
\(205\) 17.3579 + 17.3579i 1.21233 + 1.21233i
\(206\) 0 0
\(207\) 0.242641 0.0168647
\(208\) 0 0
\(209\) 26.2382 1.81493
\(210\) 0 0
\(211\) −0.947252 0.947252i −0.0652115 0.0652115i 0.673749 0.738960i \(-0.264683\pi\)
−0.738960 + 0.673749i \(0.764683\pi\)
\(212\) 0 0
\(213\) −2.12106 + 2.12106i −0.145332 + 0.145332i
\(214\) 0 0
\(215\) 43.6068i 2.97396i
\(216\) 0 0
\(217\) 0.196182i 0.0133177i
\(218\) 0 0
\(219\) 2.53933 2.53933i 0.171592 0.171592i
\(220\) 0 0
\(221\) 1.03127 + 1.03127i 0.0693709 + 0.0693709i
\(222\) 0 0
\(223\) 9.64037 0.645567 0.322783 0.946473i \(-0.395381\pi\)
0.322783 + 0.946473i \(0.395381\pi\)
\(224\) 0 0
\(225\) −37.0555 −2.47037
\(226\) 0 0
\(227\) −9.15750 9.15750i −0.607805 0.607805i 0.334567 0.942372i \(-0.391410\pi\)
−0.942372 + 0.334567i \(0.891410\pi\)
\(228\) 0 0
\(229\) 5.48888 5.48888i 0.362715 0.362715i −0.502096 0.864812i \(-0.667438\pi\)
0.864812 + 0.502096i \(0.167438\pi\)
\(230\) 0 0
\(231\) 1.21137i 0.0797021i
\(232\) 0 0
\(233\) 8.67371i 0.568234i −0.958790 0.284117i \(-0.908300\pi\)
0.958790 0.284117i \(-0.0917004\pi\)
\(234\) 0 0
\(235\) 10.5295 10.5295i 0.686869 0.686869i
\(236\) 0 0
\(237\) −2.25246 2.25246i −0.146313 0.146313i
\(238\) 0 0
\(239\) 6.83286 0.441981 0.220990 0.975276i \(-0.429071\pi\)
0.220990 + 0.975276i \(0.429071\pi\)
\(240\) 0 0
\(241\) 18.2985 1.17871 0.589356 0.807874i \(-0.299382\pi\)
0.589356 + 0.807874i \(0.299382\pi\)
\(242\) 0 0
\(243\) 6.07819 + 6.07819i 0.389916 + 0.389916i
\(244\) 0 0
\(245\) −2.98593 + 2.98593i −0.190764 + 0.190764i
\(246\) 0 0
\(247\) 22.3374i 1.42129i
\(248\) 0 0
\(249\) 2.77254i 0.175703i
\(250\) 0 0
\(251\) 15.2449 15.2449i 0.962252 0.962252i −0.0370610 0.999313i \(-0.511800\pi\)
0.999313 + 0.0370610i \(0.0117996\pi\)
\(252\) 0 0
\(253\) 0.214897 + 0.214897i 0.0135105 + 0.0135105i
\(254\) 0 0
\(255\) 0.669808 0.0419450
\(256\) 0 0
\(257\) −18.4588 −1.15143 −0.575715 0.817651i \(-0.695276\pi\)
−0.575715 + 0.817651i \(0.695276\pi\)
\(258\) 0 0
\(259\) −1.74441 1.74441i −0.108392 0.108392i
\(260\) 0 0
\(261\) −3.91245 + 3.91245i −0.242174 + 0.242174i
\(262\) 0 0
\(263\) 9.99684i 0.616432i −0.951316 0.308216i \(-0.900268\pi\)
0.951316 0.308216i \(-0.0997319\pi\)
\(264\) 0 0
\(265\) 29.1895i 1.79310i
\(266\) 0 0
\(267\) −3.62872 + 3.62872i −0.222074 + 0.222074i
\(268\) 0 0
\(269\) 19.5537 + 19.5537i 1.19221 + 1.19221i 0.976446 + 0.215761i \(0.0692233\pi\)
0.215761 + 0.976446i \(0.430777\pi\)
\(270\) 0 0
\(271\) −12.5947 −0.765072 −0.382536 0.923940i \(-0.624949\pi\)
−0.382536 + 0.923940i \(0.624949\pi\)
\(272\) 0 0
\(273\) 1.03127 0.0624155
\(274\) 0 0
\(275\) −32.8186 32.8186i −1.97904 1.97904i
\(276\) 0 0
\(277\) 1.24707 1.24707i 0.0749293 0.0749293i −0.668649 0.743578i \(-0.733127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(278\) 0 0
\(279\) 0.566543i 0.0339180i
\(280\) 0 0
\(281\) 23.3459i 1.39270i 0.717703 + 0.696349i \(0.245194\pi\)
−0.717703 + 0.696349i \(0.754806\pi\)
\(282\) 0 0
\(283\) −7.67877 + 7.67877i −0.456455 + 0.456455i −0.897490 0.441035i \(-0.854612\pi\)
0.441035 + 0.897490i \(0.354612\pi\)
\(284\) 0 0
\(285\) −7.25402 7.25402i −0.429691 0.429691i
\(286\) 0 0
\(287\) 5.81324 0.343145
\(288\) 0 0
\(289\) −16.7757 −0.986805
\(290\) 0 0
\(291\) −2.66705 2.66705i −0.156345 0.156345i
\(292\) 0 0
\(293\) −14.2002 + 14.2002i −0.829582 + 0.829582i −0.987459 0.157877i \(-0.949535\pi\)
0.157877 + 0.987459i \(0.449535\pi\)
\(294\) 0 0
\(295\) 44.8624i 2.61199i
\(296\) 0 0
\(297\) 7.13234i 0.413860i
\(298\) 0 0
\(299\) 0.182949 0.182949i 0.0105802 0.0105802i
\(300\) 0 0
\(301\) 7.30205 + 7.30205i 0.420883 + 0.420883i
\(302\) 0 0
\(303\) −5.01795 −0.288273
\(304\) 0 0
\(305\) 34.4849 1.97460
\(306\) 0 0
\(307\) 12.4447 + 12.4447i 0.710254 + 0.710254i 0.966588 0.256334i \(-0.0825148\pi\)
−0.256334 + 0.966588i \(0.582515\pi\)
\(308\) 0 0
\(309\) −2.99371 + 2.99371i −0.170306 + 0.170306i
\(310\) 0 0
\(311\) 2.76978i 0.157060i 0.996912 + 0.0785300i \(0.0250227\pi\)
−0.996912 + 0.0785300i \(0.974977\pi\)
\(312\) 0 0
\(313\) 10.8401i 0.612718i 0.951916 + 0.306359i \(0.0991107\pi\)
−0.951916 + 0.306359i \(0.900889\pi\)
\(314\) 0 0
\(315\) −8.62289 + 8.62289i −0.485845 + 0.485845i
\(316\) 0 0
\(317\) 14.1220 + 14.1220i 0.793168 + 0.793168i 0.982008 0.188840i \(-0.0604727\pi\)
−0.188840 + 0.982008i \(0.560473\pi\)
\(318\) 0 0
\(319\) −6.93021 −0.388017
\(320\) 0 0
\(321\) 5.74846 0.320848
\(322\) 0 0
\(323\) −2.42940 2.42940i −0.135175 0.135175i
\(324\) 0 0
\(325\) −27.9394 + 27.9394i −1.54980 + 1.54980i
\(326\) 0 0
\(327\) 3.56617i 0.197210i
\(328\) 0 0
\(329\) 3.52637i 0.194415i
\(330\) 0 0
\(331\) −11.0215 + 11.0215i −0.605794 + 0.605794i −0.941844 0.336050i \(-0.890909\pi\)
0.336050 + 0.941844i \(0.390909\pi\)
\(332\) 0 0
\(333\) −5.03756 5.03756i −0.276057 0.276057i
\(334\) 0 0
\(335\) 58.3790 3.18959
\(336\) 0 0
\(337\) 0.329422 0.0179448 0.00897239 0.999960i \(-0.497144\pi\)
0.00897239 + 0.999960i \(0.497144\pi\)
\(338\) 0 0
\(339\) −0.711181 0.711181i −0.0386260 0.0386260i
\(340\) 0 0
\(341\) 0.501765 0.501765i 0.0271721 0.0271721i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.118824i 0.00639729i
\(346\) 0 0
\(347\) 14.0117 14.0117i 0.752185 0.752185i −0.222702 0.974887i \(-0.571488\pi\)
0.974887 + 0.222702i \(0.0714876\pi\)
\(348\) 0 0
\(349\) −4.68741 4.68741i −0.250911 0.250911i 0.570433 0.821344i \(-0.306775\pi\)
−0.821344 + 0.570433i \(0.806775\pi\)
\(350\) 0 0
\(351\) 6.07197 0.324098
\(352\) 0 0
\(353\) −1.89060 −0.100626 −0.0503132 0.998733i \(-0.516022\pi\)
−0.0503132 + 0.998733i \(0.516022\pi\)
\(354\) 0 0
\(355\) −26.7440 26.7440i −1.41942 1.41942i
\(356\) 0 0
\(357\) 0.112161 0.112161i 0.00593617 0.00593617i
\(358\) 0 0
\(359\) 32.4334i 1.71177i 0.517165 + 0.855886i \(0.326987\pi\)
−0.517165 + 0.855886i \(0.673013\pi\)
\(360\) 0 0
\(361\) 33.6208i 1.76951i
\(362\) 0 0
\(363\) −0.493309 + 0.493309i −0.0258920 + 0.0258920i
\(364\) 0 0
\(365\) 32.0179 + 32.0179i 1.67589 + 1.67589i
\(366\) 0 0
\(367\) 19.9992 1.04395 0.521975 0.852961i \(-0.325195\pi\)
0.521975 + 0.852961i \(0.325195\pi\)
\(368\) 0 0
\(369\) 16.7877 0.873933
\(370\) 0 0
\(371\) 4.88784 + 4.88784i 0.253764 + 0.253764i
\(372\) 0 0
\(373\) −5.10234 + 5.10234i −0.264189 + 0.264189i −0.826754 0.562564i \(-0.809815\pi\)
0.562564 + 0.826754i \(0.309815\pi\)
\(374\) 0 0
\(375\) 11.0755i 0.571936i
\(376\) 0 0
\(377\) 5.89989i 0.303860i
\(378\) 0 0
\(379\) −9.46067 + 9.46067i −0.485962 + 0.485962i −0.907029 0.421067i \(-0.861656\pi\)
0.421067 + 0.907029i \(0.361656\pi\)
\(380\) 0 0
\(381\) 4.46538 + 4.46538i 0.228769 + 0.228769i
\(382\) 0 0
\(383\) −11.9156 −0.608858 −0.304429 0.952535i \(-0.598466\pi\)
−0.304429 + 0.952535i \(0.598466\pi\)
\(384\) 0 0
\(385\) −15.2739 −0.778431
\(386\) 0 0
\(387\) 21.0872 + 21.0872i 1.07192 + 1.07192i
\(388\) 0 0
\(389\) 13.7905 13.7905i 0.699204 0.699204i −0.265034 0.964239i \(-0.585383\pi\)
0.964239 + 0.265034i \(0.0853833\pi\)
\(390\) 0 0
\(391\) 0.0397948i 0.00201251i
\(392\) 0 0
\(393\) 0.522316i 0.0263474i
\(394\) 0 0
\(395\) 28.4009 28.4009i 1.42900 1.42900i
\(396\) 0 0
\(397\) 15.7768 + 15.7768i 0.791815 + 0.791815i 0.981789 0.189974i \(-0.0608405\pi\)
−0.189974 + 0.981789i \(0.560841\pi\)
\(398\) 0 0
\(399\) −2.42940 −0.121622
\(400\) 0 0
\(401\) 10.0166 0.500208 0.250104 0.968219i \(-0.419535\pi\)
0.250104 + 0.968219i \(0.419535\pi\)
\(402\) 0 0
\(403\) −0.427167 0.427167i −0.0212787 0.0212787i
\(404\) 0 0
\(405\) −23.8968 + 23.8968i −1.18744 + 1.18744i
\(406\) 0 0
\(407\) 8.92315i 0.442304i
\(408\) 0 0
\(409\) 5.54913i 0.274387i 0.990544 + 0.137193i \(0.0438082\pi\)
−0.990544 + 0.137193i \(0.956192\pi\)
\(410\) 0 0
\(411\) −1.90616 + 1.90616i −0.0940239 + 0.0940239i
\(412\) 0 0
\(413\) 7.51230 + 7.51230i 0.369656 + 0.369656i
\(414\) 0 0
\(415\) −34.9585 −1.71605
\(416\) 0 0
\(417\) −2.22393 −0.108906
\(418\) 0 0
\(419\) 5.36376 + 5.36376i 0.262037 + 0.262037i 0.825881 0.563844i \(-0.190678\pi\)
−0.563844 + 0.825881i \(0.690678\pi\)
\(420\) 0 0
\(421\) −19.5439 + 19.5439i −0.952513 + 0.952513i −0.998922 0.0464099i \(-0.985222\pi\)
0.0464099 + 0.998922i \(0.485222\pi\)
\(422\) 0 0
\(423\) 10.1836i 0.495144i
\(424\) 0 0
\(425\) 6.07736i 0.294795i
\(426\) 0 0
\(427\) 5.77456 5.77456i 0.279451 0.279451i
\(428\) 0 0
\(429\) 2.63763 + 2.63763i 0.127346 + 0.127346i
\(430\) 0 0
\(431\) −25.6979 −1.23783 −0.618913 0.785460i \(-0.712427\pi\)
−0.618913 + 0.785460i \(0.712427\pi\)
\(432\) 0 0
\(433\) −4.47599 −0.215102 −0.107551 0.994200i \(-0.534301\pi\)
−0.107551 + 0.994200i \(0.534301\pi\)
\(434\) 0 0
\(435\) 1.91598 + 1.91598i 0.0918641 + 0.0918641i
\(436\) 0 0
\(437\) −0.430977 + 0.430977i −0.0206164 + 0.0206164i
\(438\) 0 0
\(439\) 21.2404i 1.01375i −0.862020 0.506874i \(-0.830801\pi\)
0.862020 0.506874i \(-0.169199\pi\)
\(440\) 0 0
\(441\) 2.88784i 0.137516i
\(442\) 0 0
\(443\) −5.92097 + 5.92097i −0.281314 + 0.281314i −0.833633 0.552319i \(-0.813743\pi\)
0.552319 + 0.833633i \(0.313743\pi\)
\(444\) 0 0
\(445\) −45.7538 45.7538i −2.16894 2.16894i
\(446\) 0 0
\(447\) 0.478089 0.0226128
\(448\) 0 0
\(449\) 7.63095 0.360127 0.180063 0.983655i \(-0.442370\pi\)
0.180063 + 0.983655i \(0.442370\pi\)
\(450\) 0 0
\(451\) 14.8682 + 14.8682i 0.700117 + 0.700117i
\(452\) 0 0
\(453\) 0.204424 0.204424i 0.00960468 0.00960468i
\(454\) 0 0
\(455\) 13.0031i 0.609596i
\(456\) 0 0
\(457\) 29.2078i 1.36628i 0.730286 + 0.683142i \(0.239387\pi\)
−0.730286 + 0.683142i \(0.760613\pi\)
\(458\) 0 0
\(459\) 0.660384 0.660384i 0.0308241 0.0308241i
\(460\) 0 0
\(461\) −0.370169 0.370169i −0.0172405 0.0172405i 0.698434 0.715674i \(-0.253881\pi\)
−0.715674 + 0.698434i \(0.753881\pi\)
\(462\) 0 0
\(463\) −39.2012 −1.82183 −0.910916 0.412592i \(-0.864624\pi\)
−0.910916 + 0.412592i \(0.864624\pi\)
\(464\) 0 0
\(465\) −0.277444 −0.0128661
\(466\) 0 0
\(467\) −10.3291 10.3291i −0.477973 0.477973i 0.426510 0.904483i \(-0.359743\pi\)
−0.904483 + 0.426510i \(0.859743\pi\)
\(468\) 0 0
\(469\) 9.77568 9.77568i 0.451399 0.451399i
\(470\) 0 0
\(471\) 5.51655i 0.254189i
\(472\) 0 0
\(473\) 37.3522i 1.71745i
\(474\) 0 0
\(475\) 65.8177 65.8177i 3.01993 3.01993i
\(476\) 0 0
\(477\) 14.1153 + 14.1153i 0.646295 + 0.646295i
\(478\) 0 0
\(479\) 21.2528 0.971067 0.485533 0.874218i \(-0.338625\pi\)
0.485533 + 0.874218i \(0.338625\pi\)
\(480\) 0 0
\(481\) −7.59654 −0.346372
\(482\) 0 0
\(483\) −0.0198974 0.0198974i −0.000905362 0.000905362i
\(484\) 0 0
\(485\) 33.6283 33.6283i 1.52698 1.52698i
\(486\) 0 0
\(487\) 9.74088i 0.441401i 0.975342 + 0.220701i \(0.0708344\pi\)
−0.975342 + 0.220701i \(0.929166\pi\)
\(488\) 0 0
\(489\) 5.39274i 0.243868i
\(490\) 0 0
\(491\) −14.6931 + 14.6931i −0.663091 + 0.663091i −0.956107 0.293016i \(-0.905341\pi\)
0.293016 + 0.956107i \(0.405341\pi\)
\(492\) 0 0
\(493\) 0.641669 + 0.641669i 0.0288993 + 0.0288993i
\(494\) 0 0
\(495\) −44.1086 −1.98253
\(496\) 0 0
\(497\) −8.95668 −0.401762
\(498\) 0 0
\(499\) −29.0868 29.0868i −1.30210 1.30210i −0.926972 0.375130i \(-0.877598\pi\)
−0.375130 0.926972i \(-0.622402\pi\)
\(500\) 0 0
\(501\) 1.05941 1.05941i 0.0473310 0.0473310i
\(502\) 0 0
\(503\) 3.30038i 0.147157i −0.997289 0.0735784i \(-0.976558\pi\)
0.997289 0.0735784i \(-0.0234419\pi\)
\(504\) 0 0
\(505\) 63.2704i 2.81549i
\(506\) 0 0
\(507\) −0.833073 + 0.833073i −0.0369981 + 0.0369981i
\(508\) 0 0
\(509\) 11.8519 + 11.8519i 0.525327 + 0.525327i 0.919175 0.393848i \(-0.128856\pi\)
−0.393848 + 0.919175i \(0.628856\pi\)
\(510\) 0 0
\(511\) 10.7229 0.474355
\(512\) 0 0
\(513\) −14.3039 −0.631533
\(514\) 0 0
\(515\) −37.7471 37.7471i −1.66334 1.66334i
\(516\) 0 0
\(517\) 9.01922 9.01922i 0.396665 0.396665i
\(518\) 0 0
\(519\) 1.28777i 0.0565268i
\(520\) 0 0
\(521\) 28.9778i 1.26954i −0.772702 0.634769i \(-0.781095\pi\)
0.772702 0.634769i \(-0.218905\pi\)
\(522\) 0 0
\(523\) 17.4747 17.4747i 0.764117 0.764117i −0.212947 0.977064i \(-0.568306\pi\)
0.977064 + 0.212947i \(0.0683061\pi\)
\(524\) 0 0
\(525\) 3.03868 + 3.03868i 0.132619 + 0.132619i
\(526\) 0 0
\(527\) −0.0929169 −0.00404753
\(528\) 0 0
\(529\) 22.9929 0.999693
\(530\) 0 0
\(531\) 21.6943 + 21.6943i 0.941454 + 0.941454i
\(532\) 0 0
\(533\) 12.6578 12.6578i 0.548268 0.548268i
\(534\) 0 0
\(535\) 72.4813i 3.13364i
\(536\) 0 0
\(537\) 5.15825i 0.222595i
\(538\) 0 0
\(539\) −2.55765 + 2.55765i −0.110166 + 0.110166i
\(540\) 0 0
\(541\) 5.22432 + 5.22432i 0.224611 + 0.224611i 0.810437 0.585826i \(-0.199230\pi\)
−0.585826 + 0.810437i \(0.699230\pi\)
\(542\) 0 0
\(543\) −1.19620 −0.0513340
\(544\) 0 0
\(545\) 44.9652 1.92610
\(546\) 0 0
\(547\) 11.2408 + 11.2408i 0.480621 + 0.480621i 0.905330 0.424709i \(-0.139624\pi\)
−0.424709 + 0.905330i \(0.639624\pi\)
\(548\) 0 0
\(549\) 16.6760 16.6760i 0.711715 0.711715i
\(550\) 0 0
\(551\) 13.8985i 0.592098i
\(552\) 0 0
\(553\) 9.51156i 0.404473i
\(554\) 0 0
\(555\) −2.46696 + 2.46696i −0.104717 + 0.104717i
\(556\) 0 0
\(557\) −16.5182 16.5182i −0.699900 0.699900i 0.264489 0.964389i \(-0.414797\pi\)
−0.964389 + 0.264489i \(0.914797\pi\)
\(558\) 0 0
\(559\) 31.7990 1.34495
\(560\) 0 0
\(561\) 0.573735 0.0242231
\(562\) 0 0
\(563\) 18.2728 + 18.2728i 0.770107 + 0.770107i 0.978125 0.208018i \(-0.0667014\pi\)
−0.208018 + 0.978125i \(0.566701\pi\)
\(564\) 0 0
\(565\) 8.96715 8.96715i 0.377251 0.377251i
\(566\) 0 0
\(567\) 8.00313i 0.336100i
\(568\) 0 0
\(569\) 18.7351i 0.785417i −0.919663 0.392708i \(-0.871538\pi\)
0.919663 0.392708i \(-0.128462\pi\)
\(570\) 0 0
\(571\) −12.0415 + 12.0415i −0.503920 + 0.503920i −0.912654 0.408734i \(-0.865970\pi\)
0.408734 + 0.912654i \(0.365970\pi\)
\(572\) 0 0
\(573\) 4.74178 + 4.74178i 0.198091 + 0.198091i
\(574\) 0 0
\(575\) 1.07813 0.0449610
\(576\) 0 0
\(577\) −3.96647 −0.165126 −0.0825632 0.996586i \(-0.526311\pi\)
−0.0825632 + 0.996586i \(0.526311\pi\)
\(578\) 0 0
\(579\) −1.34733 1.34733i −0.0559931 0.0559931i
\(580\) 0 0
\(581\) −5.85387 + 5.85387i −0.242860 + 0.242860i
\(582\) 0 0
\(583\) 25.0027i 1.03551i
\(584\) 0 0
\(585\) 37.5510i 1.55254i
\(586\) 0 0
\(587\) 5.14102 5.14102i 0.212193 0.212193i −0.593006 0.805198i \(-0.702059\pi\)
0.805198 + 0.593006i \(0.202059\pi\)
\(588\) 0 0
\(589\) 1.00629 + 1.00629i 0.0414635 + 0.0414635i
\(590\) 0 0
\(591\) −7.91112 −0.325420
\(592\) 0 0
\(593\) 18.3463 0.753390 0.376695 0.926337i \(-0.377060\pi\)
0.376695 + 0.926337i \(0.377060\pi\)
\(594\) 0 0
\(595\) 1.41421 + 1.41421i 0.0579771 + 0.0579771i
\(596\) 0 0
\(597\) −4.21450 + 4.21450i −0.172488 + 0.172488i
\(598\) 0 0
\(599\) 35.3692i 1.44515i −0.691295 0.722573i \(-0.742959\pi\)
0.691295 0.722573i \(-0.257041\pi\)
\(600\) 0 0
\(601\) 34.1275i 1.39209i −0.717999 0.696044i \(-0.754942\pi\)
0.717999 0.696044i \(-0.245058\pi\)
\(602\) 0 0
\(603\) 28.2306 28.2306i 1.14964 1.14964i
\(604\) 0 0
\(605\) −6.22005 6.22005i −0.252881 0.252881i
\(606\) 0 0
\(607\) −12.3626 −0.501781 −0.250890 0.968016i \(-0.580723\pi\)
−0.250890 + 0.968016i \(0.580723\pi\)
\(608\) 0 0
\(609\) 0.641669 0.0260017
\(610\) 0 0
\(611\) −7.67833 7.67833i −0.310632 0.310632i
\(612\) 0 0
\(613\) −19.6385 + 19.6385i −0.793192 + 0.793192i −0.982012 0.188820i \(-0.939534\pi\)
0.188820 + 0.982012i \(0.439534\pi\)
\(614\) 0 0
\(615\) 8.22117i 0.331509i
\(616\) 0 0
\(617\) 1.01019i 0.0406689i 0.999793 + 0.0203344i \(0.00647310\pi\)
−0.999793 + 0.0203344i \(0.993527\pi\)
\(618\) 0 0
\(619\) 0.818167 0.818167i 0.0328849 0.0328849i −0.690473 0.723358i \(-0.742598\pi\)
0.723358 + 0.690473i \(0.242598\pi\)
\(620\) 0 0
\(621\) −0.117153 0.117153i −0.00470117 0.00470117i
\(622\) 0 0
\(623\) −15.3231 −0.613908
\(624\) 0 0
\(625\) −75.4912 −3.01965
\(626\) 0 0
\(627\) −6.21355 6.21355i −0.248145 0.248145i
\(628\) 0 0
\(629\) −0.826195 + 0.826195i −0.0329426 + 0.0329426i
\(630\) 0 0
\(631\) 31.6898i 1.26155i 0.775966 + 0.630775i \(0.217263\pi\)
−0.775966 + 0.630775i \(0.782737\pi\)
\(632\) 0 0
\(633\) 0.448643i 0.0178319i
\(634\) 0 0
\(635\) −56.3032 + 56.3032i −2.23432 + 2.23432i
\(636\) 0 0
\(637\) 2.17740 + 2.17740i 0.0862718 + 0.0862718i
\(638\) 0 0
\(639\) −25.8654 −1.02322
\(640\) 0 0
\(641\) 15.2445 0.602121 0.301061 0.953605i \(-0.402659\pi\)
0.301061 + 0.953605i \(0.402659\pi\)
\(642\) 0 0
\(643\) 21.6030 + 21.6030i 0.851939 + 0.851939i 0.990372 0.138433i \(-0.0442066\pi\)
−0.138433 + 0.990372i \(0.544207\pi\)
\(644\) 0 0
\(645\) 10.3267 10.3267i 0.406612 0.406612i
\(646\) 0 0
\(647\) 22.8048i 0.896547i −0.893896 0.448274i \(-0.852039\pi\)
0.893896 0.448274i \(-0.147961\pi\)
\(648\) 0 0
\(649\) 38.4277i 1.50842i
\(650\) 0 0
\(651\) −0.0464585 + 0.0464585i −0.00182085 + 0.00182085i
\(652\) 0 0
\(653\) −13.3213 13.3213i −0.521302 0.521302i 0.396663 0.917964i \(-0.370168\pi\)
−0.917964 + 0.396663i \(0.870168\pi\)
\(654\) 0 0
\(655\) −6.58579 −0.257328
\(656\) 0 0
\(657\) 30.9661 1.20810
\(658\) 0 0
\(659\) −9.62688 9.62688i −0.375010 0.375010i 0.494288 0.869298i \(-0.335429\pi\)
−0.869298 + 0.494288i \(0.835429\pi\)
\(660\) 0 0
\(661\) −18.1305 + 18.1305i −0.705197 + 0.705197i −0.965521 0.260325i \(-0.916170\pi\)
0.260325 + 0.965521i \(0.416170\pi\)
\(662\) 0 0
\(663\) 0.488437i 0.0189693i
\(664\) 0 0
\(665\) 30.6318i 1.18785i
\(666\) 0 0
\(667\) 0.113832 0.113832i 0.00440761 0.00440761i
\(668\) 0 0
\(669\) −2.28296 2.28296i −0.0882645 0.0882645i
\(670\) 0 0
\(671\) 29.5386 1.14032
\(672\) 0 0
\(673\) −4.57745 −0.176448 −0.0882239 0.996101i \(-0.528119\pi\)
−0.0882239 + 0.996101i \(0.528119\pi\)
\(674\) 0 0
\(675\) 17.8913 + 17.8913i 0.688635 + 0.688635i
\(676\) 0 0
\(677\) 18.1035 18.1035i 0.695772 0.695772i −0.267723 0.963496i \(-0.586271\pi\)
0.963496 + 0.267723i \(0.0862714\pi\)
\(678\) 0 0
\(679\) 11.2623i 0.432206i
\(680\) 0 0
\(681\) 4.33723i 0.166203i
\(682\) 0 0
\(683\) −7.94815 + 7.94815i −0.304128 + 0.304128i −0.842626 0.538499i \(-0.818992\pi\)
0.538499 + 0.842626i \(0.318992\pi\)
\(684\) 0 0
\(685\) −24.0344 24.0344i −0.918307 0.918307i
\(686\) 0 0
\(687\) −2.59967 −0.0991837
\(688\) 0 0
\(689\) 21.2856 0.810916
\(690\) 0 0
\(691\) 6.34398 + 6.34398i 0.241336 + 0.241336i 0.817403 0.576066i \(-0.195413\pi\)
−0.576066 + 0.817403i \(0.695413\pi\)
\(692\) 0 0
\(693\) −7.38607 + 7.38607i −0.280574 + 0.280574i
\(694\) 0 0
\(695\) 28.0411i 1.06366i
\(696\) 0 0
\(697\) 2.75330i 0.104289i
\(698\) 0 0
\(699\) −2.05405 + 2.05405i −0.0776912 + 0.0776912i
\(700\) 0 0
\(701\) 19.5173 + 19.5173i 0.737159 + 0.737159i 0.972027 0.234868i \(-0.0754658\pi\)
−0.234868 + 0.972027i \(0.575466\pi\)
\(702\) 0 0
\(703\) 17.8954 0.674937
\(704\) 0 0
\(705\) −4.98705 −0.187823
\(706\) 0 0
\(707\) −10.5947 10.5947i −0.398457 0.398457i
\(708\) 0 0
\(709\) 4.30481 4.30481i 0.161671 0.161671i −0.621636 0.783306i \(-0.713532\pi\)
0.783306 + 0.621636i \(0.213532\pi\)
\(710\) 0 0
\(711\) 27.4679i 1.03013i
\(712\) 0 0
\(713\) 0.0164835i 0.000617313i
\(714\) 0 0
\(715\) −33.2574 + 33.2574i −1.24376 + 1.24376i
\(716\) 0 0
\(717\) −1.61811 1.61811i −0.0604294 0.0604294i
\(718\) 0 0
\(719\) 26.6322 0.993215 0.496608 0.867975i \(-0.334579\pi\)
0.496608 + 0.867975i \(0.334579\pi\)
\(720\) 0 0
\(721\) −12.6417 −0.470800
\(722\) 0 0
\(723\) −4.33333 4.33333i −0.161158 0.161158i
\(724\) 0 0
\(725\) −17.3842 + 17.3842i −0.645634 + 0.645634i
\(726\) 0 0
\(727\) 38.4103i 1.42456i −0.701896 0.712279i \(-0.747663\pi\)
0.701896 0.712279i \(-0.252337\pi\)
\(728\) 0 0
\(729\) 21.1306i 0.782615i
\(730\) 0 0
\(731\) 3.45844 3.45844i 0.127915 0.127915i
\(732\) 0 0
\(733\) −9.64855 9.64855i −0.356377 0.356377i 0.506099 0.862476i \(-0.331087\pi\)
−0.862476 + 0.506099i \(0.831087\pi\)
\(734\) 0 0
\(735\) 1.41421 0.0521641
\(736\) 0 0
\(737\) 50.0055 1.84198
\(738\) 0 0
\(739\) −26.3825 26.3825i −0.970498 0.970498i 0.0290795 0.999577i \(-0.490742\pi\)
−0.999577 + 0.0290795i \(0.990742\pi\)
\(740\) 0 0
\(741\) −5.28978 + 5.28978i −0.194325 + 0.194325i
\(742\) 0 0
\(743\) 1.53540i 0.0563284i 0.999603 + 0.0281642i \(0.00896614\pi\)
−0.999603 + 0.0281642i \(0.991034\pi\)
\(744\) 0 0
\(745\) 6.02814i 0.220854i
\(746\) 0 0
\(747\) −16.9050 + 16.9050i −0.618523 + 0.618523i
\(748\) 0 0
\(749\) 12.1371 + 12.1371i 0.443482 + 0.443482i
\(750\) 0 0
\(751\) 20.7562 0.757404 0.378702 0.925519i \(-0.376370\pi\)
0.378702 + 0.925519i \(0.376370\pi\)
\(752\) 0 0
\(753\) −7.22040 −0.263126
\(754\) 0 0
\(755\) 2.57754 + 2.57754i 0.0938065 + 0.0938065i
\(756\) 0 0
\(757\) −20.8405 + 20.8405i −0.757459 + 0.757459i −0.975859 0.218400i \(-0.929916\pi\)
0.218400 + 0.975859i \(0.429916\pi\)
\(758\) 0 0
\(759\) 0.101781i 0.00369442i
\(760\) 0 0
\(761\) 42.3705i 1.53593i 0.640492 + 0.767965i \(0.278730\pi\)
−0.640492 + 0.767965i \(0.721270\pi\)
\(762\) 0 0
\(763\) 7.52951 7.52951i 0.272586 0.272586i
\(764\) 0 0
\(765\) 4.08402 + 4.08402i 0.147658 + 0.147658i
\(766\) 0 0
\(767\) 32.7146 1.18126
\(768\) 0 0
\(769\) −13.4764 −0.485970 −0.242985 0.970030i \(-0.578127\pi\)
−0.242985 + 0.970030i \(0.578127\pi\)
\(770\) 0 0
\(771\) 4.37128 + 4.37128i 0.157428 + 0.157428i
\(772\) 0 0
\(773\) 13.7392 13.7392i 0.494166 0.494166i −0.415450 0.909616i \(-0.636376\pi\)
0.909616 + 0.415450i \(0.136376\pi\)
\(774\) 0 0
\(775\) 2.51732i 0.0904249i
\(776\) 0 0
\(777\) 0.826195i 0.0296396i
\(778\) 0 0
\(779\) −29.8182 + 29.8182i −1.06835 + 1.06835i
\(780\) 0 0
\(781\) −22.9080 22.9080i −0.819713 0.819713i
\(782\) 0 0
\(783\) 3.77804 0.135016
\(784\) 0 0
\(785\) −69.5572 −2.48260
\(786\) 0 0
\(787\) −14.7111 14.7111i −0.524394 0.524394i 0.394501 0.918895i \(-0.370917\pi\)
−0.918895 + 0.394501i \(0.870917\pi\)
\(788\) 0 0
\(789\) −2.36738 + 2.36738i −0.0842810 + 0.0842810i
\(790\) 0 0
\(791\) 3.00313i 0.106779i
\(792\) 0 0
\(793\) 25.1471i 0.892999i
\(794\) 0 0
\(795\) 6.91245 6.91245i 0.245159 0.245159i
\(796\) 0 0
\(797\) −3.24496 3.24496i −0.114942 0.114942i 0.647296 0.762239i \(-0.275900\pi\)
−0.762239 + 0.647296i \(0.775900\pi\)
\(798\) 0 0
\(799\) −1.67018 −0.0590868
\(800\) 0 0
\(801\) −44.2507 −1.56352
\(802\) 0 0
\(803\) 27.4255 + 27.4255i 0.967824 + 0.967824i
\(804\) 0 0
\(805\) 0.250882 0.250882i 0.00884244 0.00884244i
\(806\) 0 0
\(807\) 9.26111i 0.326007i
\(808\) 0 0
\(809\) 43.9611i 1.54559i −0.634655 0.772795i \(-0.718858\pi\)
0.634655 0.772795i \(-0.281142\pi\)
\(810\) 0 0
\(811\) 12.7261 12.7261i 0.446875 0.446875i −0.447439 0.894314i \(-0.647664\pi\)
0.894314 + 0.447439i \(0.147664\pi\)
\(812\) 0 0
\(813\) 2.98258 + 2.98258i 0.104604 + 0.104604i
\(814\) 0 0
\(815\) 67.9961 2.38180
\(816\) 0 0
\(817\) −74.9098 −2.62076
\(818\) 0 0
\(819\) 6.28798 + 6.28798i 0.219720 + 0.219720i
\(820\) 0 0
\(821\) −15.5549 + 15.5549i −0.542869 + 0.542869i −0.924369 0.381500i \(-0.875408\pi\)
0.381500 + 0.924369i \(0.375408\pi\)
\(822\) 0 0
\(823\) 26.7572i 0.932698i −0.884601 0.466349i \(-0.845569\pi\)
0.884601 0.466349i \(-0.154431\pi\)
\(824\) 0 0
\(825\) 15.5437i 0.541163i
\(826\) 0 0
\(827\) −9.47992 + 9.47992i −0.329649 + 0.329649i −0.852453 0.522804i \(-0.824886\pi\)
0.522804 + 0.852453i \(0.324886\pi\)
\(828\) 0 0
\(829\) −11.2500 11.2500i −0.390730 0.390730i 0.484217 0.874948i \(-0.339104\pi\)
−0.874948 + 0.484217i \(0.839104\pi\)
\(830\) 0 0
\(831\) −0.590646 −0.0204893
\(832\) 0 0
\(833\) 0.473626 0.0164102
\(834\) 0 0
\(835\) 13.3579 + 13.3579i 0.462270 + 0.462270i
\(836\) 0 0
\(837\) −0.273540 + 0.273540i −0.00945493 + 0.00945493i
\(838\) 0 0
\(839\) 30.4767i 1.05217i 0.850431 + 0.526087i \(0.176341\pi\)
−0.850431 + 0.526087i \(0.823659\pi\)
\(840\) 0 0
\(841\) 25.3290i 0.873415i
\(842\) 0 0
\(843\) 5.52861 5.52861i 0.190415 0.190415i
\(844\) 0 0
\(845\) −10.5041 10.5041i −0.361351 0.361351i
\(846\) 0 0
\(847\) −2.08312 −0.0715768
\(848\) 0 0
\(849\) 3.63686 0.124817
\(850\) 0 0
\(851\) 0.146568 + 0.146568i 0.00502427 + 0.00502427i
\(852\) 0 0
\(853\) −29.9235 + 29.9235i −1.02456 + 1.02456i −0.0248689 + 0.999691i \(0.507917\pi\)
−0.999691 + 0.0248689i \(0.992083\pi\)
\(854\) 0 0
\(855\) 88.4599i 3.02526i
\(856\) 0 0
\(857\) 17.5515i 0.599547i 0.954010 + 0.299774i \(0.0969112\pi\)
−0.954010 + 0.299774i \(0.903089\pi\)
\(858\) 0 0
\(859\) 6.71986 6.71986i 0.229279 0.229279i −0.583113 0.812391i \(-0.698165\pi\)
0.812391 + 0.583113i \(0.198165\pi\)
\(860\) 0 0
\(861\) −1.37665 1.37665i −0.0469161 0.0469161i
\(862\) 0 0
\(863\) −52.7588 −1.79593 −0.897965 0.440067i \(-0.854955\pi\)
−0.897965 + 0.440067i \(0.854955\pi\)
\(864\) 0 0
\(865\) 16.2373 0.552083
\(866\) 0 0
\(867\) 3.97270 + 3.97270i 0.134920 + 0.134920i
\(868\) 0 0
\(869\) 24.3272 24.3272i 0.825244 0.825244i
\(870\) 0 0
\(871\) 42.5711i 1.44247i
\(872\) 0 0
\(873\) 32.5236i 1.10076i
\(874\) 0 0
\(875\) −23.3845 + 23.3845i −0.790540 + 0.790540i
\(876\) 0 0
\(877\) 11.1764 + 11.1764i 0.377399 + 0.377399i 0.870163 0.492764i \(-0.164013\pi\)
−0.492764 + 0.870163i \(0.664013\pi\)
\(878\) 0 0
\(879\) 6.72556 0.226847
\(880\) 0 0
\(881\) 23.6304 0.796128 0.398064 0.917358i \(-0.369682\pi\)
0.398064 + 0.917358i \(0.369682\pi\)
\(882\) 0 0
\(883\) 18.7140 + 18.7140i 0.629777 + 0.629777i 0.948012 0.318235i \(-0.103090\pi\)
−0.318235 + 0.948012i \(0.603090\pi\)
\(884\) 0 0
\(885\) 10.6240 10.6240i 0.357122 0.357122i
\(886\) 0 0
\(887\) 46.0412i 1.54591i 0.634459 + 0.772957i \(0.281223\pi\)
−0.634459 + 0.772957i \(0.718777\pi\)
\(888\) 0 0
\(889\) 18.8562i 0.632416i
\(890\) 0 0
\(891\) −20.4692 + 20.4692i −0.685744 + 0.685744i
\(892\) 0 0
\(893\) 18.0881 + 18.0881i 0.605294 + 0.605294i
\(894\) 0 0
\(895\) 65.0394 2.17403
\(896\) 0 0
\(897\) −0.0866491 −0.00289313
\(898\) 0 0
\(899\) −0.265788 0.265788i −0.00886452 0.00886452i
\(900\) 0 0
\(901\) 2.31501 2.31501i 0.0771241 0.0771241i
\(902\) 0 0
\(903\) 3.45844i 0.115090i
\(904\) 0 0
\(905\) 15.0827i 0.501367i
\(906\) 0 0
\(907\) 19.9531 19.9531i 0.662533 0.662533i −0.293443 0.955977i \(-0.594801\pi\)
0.955977 + 0.293443i \(0.0948012\pi\)
\(908\) 0 0
\(909\) −30.5959 30.5959i −1.01480 1.01480i
\(910\) 0 0
\(911\) 7.94815 0.263334 0.131667 0.991294i \(-0.457967\pi\)
0.131667 + 0.991294i \(0.457967\pi\)
\(912\) 0 0
\(913\) −29.9443 −0.991011
\(914\) 0 0
\(915\) −8.16647 8.16647i −0.269975 0.269975i
\(916\) 0 0
\(917\) −1.10280 + 1.10280i −0.0364178 + 0.0364178i
\(918\) 0 0
\(919\) 30.2767i 0.998735i 0.866390 + 0.499367i \(0.166434\pi\)
−0.866390 + 0.499367i \(0.833566\pi\)
\(920\) 0 0
\(921\) 5.89411i 0.194217i
\(922\) 0 0
\(923\) −19.5023 + 19.5023i −0.641925 + 0.641925i
\(924\) 0 0
\(925\) −22.3834 22.3834i −0.735963 0.735963i
\(926\) 0 0
\(927\) −36.5071 −1.19905
\(928\) 0 0
\(929\) 50.9720 1.67234 0.836169 0.548473i \(-0.184790\pi\)
0.836169 + 0.548473i \(0.184790\pi\)
\(930\) 0 0
\(931\) −5.12936 5.12936i −0.168108 0.168108i
\(932\) 0 0
\(933\) 0.655921 0.655921i 0.0214739 0.0214739i
\(934\) 0 0
\(935\) 7.23412i 0.236581i
\(936\) 0 0
\(937\) 6.62686i 0.216490i −0.994124 0.108245i \(-0.965477\pi\)
0.994124 0.108245i \(-0.0345231\pi\)
\(938\) 0 0
\(939\) 2.56707 2.56707i 0.0837732 0.0837732i
\(940\) 0 0
\(941\) −12.5666 12.5666i −0.409660 0.409660i 0.471960 0.881620i \(-0.343547\pi\)
−0.881620 + 0.471960i \(0.843547\pi\)
\(942\) 0 0
\(943\) −0.488437 −0.0159057
\(944\) 0 0
\(945\) 8.32666 0.270866
\(946\) 0 0
\(947\) 28.2069 + 28.2069i 0.916601 + 0.916601i 0.996780 0.0801799i \(-0.0255495\pi\)
−0.0801799 + 0.996780i \(0.525549\pi\)
\(948\) 0 0
\(949\) 23.3481 23.3481i 0.757912 0.757912i
\(950\) 0 0
\(951\) 6.68852i 0.216890i
\(952\) 0 0
\(953\) 38.6729i 1.25274i −0.779526 0.626369i \(-0.784540\pi\)
0.779526 0.626369i \(-0.215460\pi\)
\(954\) 0 0
\(955\) −59.7882 + 59.7882i −1.93470 + 1.93470i
\(956\) 0 0
\(957\) 1.64116 + 1.64116i 0.0530513 + 0.0530513i
\(958\) 0 0
\(959\) −8.04922 −0.259923
\(960\) 0 0
\(961\) −30.9615 −0.998758
\(962\) 0 0
\(963\) 35.0501 + 35.0501i 1.12947 + 1.12947i
\(964\) 0 0
\(965\) 16.9882 16.9882i 0.546870 0.546870i
\(966\) 0 0
\(967\) 16.4567i 0.529213i 0.964356 + 0.264607i \(0.0852421\pi\)
−0.964356 + 0.264607i \(0.914758\pi\)
\(968\) 0 0
\(969\) 1.15063i 0.0369634i
\(970\) 0 0
\(971\) 3.05753 3.05753i 0.0981207 0.0981207i −0.656342 0.754463i \(-0.727897\pi\)
0.754463 + 0.656342i \(0.227897\pi\)
\(972\) 0 0
\(973\) −4.69553 4.69553i −0.150532 0.150532i
\(974\) 0 0
\(975\) 13.2328 0.423790
\(976\) 0 0
\(977\) −3.21193 −0.102759 −0.0513793 0.998679i \(-0.516362\pi\)
−0.0513793 + 0.998679i \(0.516362\pi\)
\(978\) 0 0
\(979\) −39.1912 39.1912i −1.25256 1.25256i
\(980\) 0 0
\(981\) 21.7440 21.7440i 0.694233 0.694233i
\(982\) 0 0
\(983\) 8.33539i 0.265858i −0.991126 0.132929i \(-0.957562\pi\)
0.991126 0.132929i \(-0.0424382\pi\)
\(984\) 0 0
\(985\) 99.7499i 3.17829i
\(986\) 0 0
\(987\) −0.835091 + 0.835091i −0.0265812 + 0.0265812i
\(988\) 0 0
\(989\) −0.613530 0.613530i −0.0195091 0.0195091i
\(990\) 0 0
\(991\) −4.05237 −0.128728 −0.0643640 0.997926i \(-0.520502\pi\)
−0.0643640 + 0.997926i \(0.520502\pi\)
\(992\) 0 0
\(993\) 5.22004 0.165653
\(994\) 0 0
\(995\) −53.1399 53.1399i −1.68465 1.68465i
\(996\) 0 0
\(997\) −11.0203 + 11.0203i −0.349018 + 0.349018i −0.859744 0.510726i \(-0.829377\pi\)
0.510726 + 0.859744i \(0.329377\pi\)
\(998\) 0 0
\(999\) 4.86451i 0.153906i
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 1792.2.m.d.1345.2 yes 8
4.3 odd 2 1792.2.m.b.1345.3 yes 8
8.3 odd 2 1792.2.m.c.1345.2 yes 8
8.5 even 2 1792.2.m.a.1345.3 yes 8
16.3 odd 4 1792.2.m.c.449.2 yes 8
16.5 even 4 inner 1792.2.m.d.449.2 yes 8
16.11 odd 4 1792.2.m.b.449.3 yes 8
16.13 even 4 1792.2.m.a.449.3 8
32.5 even 8 7168.2.a.x.1.3 4
32.11 odd 8 7168.2.a.s.1.3 4
32.21 even 8 7168.2.a.t.1.2 4
32.27 odd 8 7168.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.a.449.3 8 16.13 even 4
1792.2.m.a.1345.3 yes 8 8.5 even 2
1792.2.m.b.449.3 yes 8 16.11 odd 4
1792.2.m.b.1345.3 yes 8 4.3 odd 2
1792.2.m.c.449.2 yes 8 16.3 odd 4
1792.2.m.c.1345.2 yes 8 8.3 odd 2
1792.2.m.d.449.2 yes 8 16.5 even 4 inner
1792.2.m.d.1345.2 yes 8 1.1 even 1 trivial
7168.2.a.s.1.3 4 32.11 odd 8
7168.2.a.t.1.2 4 32.21 even 8
7168.2.a.w.1.2 4 32.27 odd 8
7168.2.a.x.1.3 4 32.5 even 8