Properties

Label 136.4.k.b.89.6
Level $136$
Weight $4$
Character 136.89
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
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] = [136,4,Mod(81,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.k (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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 89.6
Root \(3.80701i\) of defining polynomial
Character \(\chi\) \(=\) 136.89
Dual form 136.4.k.b.81.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.80701 - 3.80701i) q^{3} +(2.78003 - 2.78003i) q^{5} +(25.7370 + 25.7370i) q^{7} -1.98658i q^{9} +O(q^{10})\) \(q+(3.80701 - 3.80701i) q^{3} +(2.78003 - 2.78003i) q^{5} +(25.7370 + 25.7370i) q^{7} -1.98658i q^{9} +(-17.1786 - 17.1786i) q^{11} +71.6546 q^{13} -21.1672i q^{15} +(-34.3358 - 61.1069i) q^{17} -72.2853i q^{19} +195.962 q^{21} +(-90.8809 - 90.8809i) q^{23} +109.543i q^{25} +(95.2262 + 95.2262i) q^{27} +(-82.0612 + 82.0612i) q^{29} +(35.2724 - 35.2724i) q^{31} -130.798 q^{33} +143.099 q^{35} +(69.5253 - 69.5253i) q^{37} +(272.789 - 272.789i) q^{39} +(196.536 + 196.536i) q^{41} -344.162i q^{43} +(-5.52277 - 5.52277i) q^{45} -626.172 q^{47} +981.785i q^{49} +(-363.351 - 101.918i) q^{51} +145.329i q^{53} -95.5140 q^{55} +(-275.191 - 275.191i) q^{57} +454.009i q^{59} +(-179.197 - 179.197i) q^{61} +(51.1287 - 51.1287i) q^{63} +(199.202 - 199.202i) q^{65} -874.562 q^{67} -691.968 q^{69} +(58.6994 - 58.6994i) q^{71} +(-275.870 + 275.870i) q^{73} +(417.030 + 417.030i) q^{75} -884.249i q^{77} +(-464.841 - 464.841i) q^{79} +778.691 q^{81} -163.049i q^{83} +(-265.334 - 74.4246i) q^{85} +624.815i q^{87} -418.761 q^{89} +(1844.17 + 1844.17i) q^{91} -268.564i q^{93} +(-200.956 - 200.956i) q^{95} +(162.634 - 162.634i) q^{97} +(-34.1267 + 34.1267i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7} - 66 q^{11} - 124 q^{13} + 130 q^{17} - 148 q^{21} - 162 q^{23} - 204 q^{27} + 158 q^{29} - 350 q^{31} + 116 q^{33} + 236 q^{35} + 582 q^{37} - 320 q^{39} + 878 q^{41} - 26 q^{45} - 448 q^{47} - 590 q^{51} + 1460 q^{55} + 292 q^{57} - 1130 q^{61} + 114 q^{63} + 20 q^{65} - 64 q^{67} + 2076 q^{69} - 3274 q^{71} - 1426 q^{73} + 946 q^{75} + 1718 q^{79} + 166 q^{81} - 2018 q^{85} + 476 q^{89} + 2128 q^{91} - 2444 q^{95} + 1926 q^{97} + 3870 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 3.80701 3.80701i 0.732659 0.732659i −0.238487 0.971146i \(-0.576651\pi\)
0.971146 + 0.238487i \(0.0766515\pi\)
\(4\) 0 0
\(5\) 2.78003 2.78003i 0.248654 0.248654i −0.571764 0.820418i \(-0.693741\pi\)
0.820418 + 0.571764i \(0.193741\pi\)
\(6\) 0 0
\(7\) 25.7370 + 25.7370i 1.38967 + 1.38967i 0.826007 + 0.563660i \(0.190607\pi\)
0.563660 + 0.826007i \(0.309393\pi\)
\(8\) 0 0
\(9\) 1.98658i 0.0735772i
\(10\) 0 0
\(11\) −17.1786 17.1786i −0.470867 0.470867i 0.431328 0.902195i \(-0.358045\pi\)
−0.902195 + 0.431328i \(0.858045\pi\)
\(12\) 0 0
\(13\) 71.6546 1.52872 0.764362 0.644787i \(-0.223054\pi\)
0.764362 + 0.644787i \(0.223054\pi\)
\(14\) 0 0
\(15\) 21.1672i 0.364357i
\(16\) 0 0
\(17\) −34.3358 61.1069i −0.489862 0.871800i
\(18\) 0 0
\(19\) 72.2853i 0.872810i −0.899750 0.436405i \(-0.856251\pi\)
0.899750 0.436405i \(-0.143749\pi\)
\(20\) 0 0
\(21\) 195.962 2.03630
\(22\) 0 0
\(23\) −90.8809 90.8809i −0.823912 0.823912i 0.162755 0.986667i \(-0.447962\pi\)
−0.986667 + 0.162755i \(0.947962\pi\)
\(24\) 0 0
\(25\) 109.543i 0.876343i
\(26\) 0 0
\(27\) 95.2262 + 95.2262i 0.678752 + 0.678752i
\(28\) 0 0
\(29\) −82.0612 + 82.0612i −0.525461 + 0.525461i −0.919216 0.393754i \(-0.871176\pi\)
0.393754 + 0.919216i \(0.371176\pi\)
\(30\) 0 0
\(31\) 35.2724 35.2724i 0.204358 0.204358i −0.597506 0.801864i \(-0.703842\pi\)
0.801864 + 0.597506i \(0.203842\pi\)
\(32\) 0 0
\(33\) −130.798 −0.689969
\(34\) 0 0
\(35\) 143.099 0.691092
\(36\) 0 0
\(37\) 69.5253 69.5253i 0.308916 0.308916i −0.535573 0.844489i \(-0.679904\pi\)
0.844489 + 0.535573i \(0.179904\pi\)
\(38\) 0 0
\(39\) 272.789 272.789i 1.12003 1.12003i
\(40\) 0 0
\(41\) 196.536 + 196.536i 0.748628 + 0.748628i 0.974221 0.225594i \(-0.0724322\pi\)
−0.225594 + 0.974221i \(0.572432\pi\)
\(42\) 0 0
\(43\) 344.162i 1.22056i −0.792185 0.610281i \(-0.791057\pi\)
0.792185 0.610281i \(-0.208943\pi\)
\(44\) 0 0
\(45\) −5.52277 5.52277i −0.0182953 0.0182953i
\(46\) 0 0
\(47\) −626.172 −1.94333 −0.971665 0.236362i \(-0.924045\pi\)
−0.971665 + 0.236362i \(0.924045\pi\)
\(48\) 0 0
\(49\) 981.785i 2.86235i
\(50\) 0 0
\(51\) −363.351 101.918i −0.997633 0.279830i
\(52\) 0 0
\(53\) 145.329i 0.376651i 0.982107 + 0.188326i \(0.0603060\pi\)
−0.982107 + 0.188326i \(0.939694\pi\)
\(54\) 0 0
\(55\) −95.5140 −0.234166
\(56\) 0 0
\(57\) −275.191 275.191i −0.639472 0.639472i
\(58\) 0 0
\(59\) 454.009i 1.00181i 0.865502 + 0.500906i \(0.167000\pi\)
−0.865502 + 0.500906i \(0.833000\pi\)
\(60\) 0 0
\(61\) −179.197 179.197i −0.376129 0.376129i 0.493575 0.869703i \(-0.335690\pi\)
−0.869703 + 0.493575i \(0.835690\pi\)
\(62\) 0 0
\(63\) 51.1287 51.1287i 0.102248 0.102248i
\(64\) 0 0
\(65\) 199.202 199.202i 0.380123 0.380123i
\(66\) 0 0
\(67\) −874.562 −1.59470 −0.797349 0.603518i \(-0.793765\pi\)
−0.797349 + 0.603518i \(0.793765\pi\)
\(68\) 0 0
\(69\) −691.968 −1.20729
\(70\) 0 0
\(71\) 58.6994 58.6994i 0.0981174 0.0981174i −0.656344 0.754462i \(-0.727898\pi\)
0.754462 + 0.656344i \(0.227898\pi\)
\(72\) 0 0
\(73\) −275.870 + 275.870i −0.442303 + 0.442303i −0.892785 0.450482i \(-0.851252\pi\)
0.450482 + 0.892785i \(0.351252\pi\)
\(74\) 0 0
\(75\) 417.030 + 417.030i 0.642060 + 0.642060i
\(76\) 0 0
\(77\) 884.249i 1.30870i
\(78\) 0 0
\(79\) −464.841 464.841i −0.662009 0.662009i 0.293844 0.955853i \(-0.405065\pi\)
−0.955853 + 0.293844i \(0.905065\pi\)
\(80\) 0 0
\(81\) 778.691 1.06816
\(82\) 0 0
\(83\) 163.049i 0.215626i −0.994171 0.107813i \(-0.965615\pi\)
0.994171 0.107813i \(-0.0343848\pi\)
\(84\) 0 0
\(85\) −265.334 74.4246i −0.338582 0.0949703i
\(86\) 0 0
\(87\) 624.815i 0.769968i
\(88\) 0 0
\(89\) −418.761 −0.498748 −0.249374 0.968407i \(-0.580225\pi\)
−0.249374 + 0.968407i \(0.580225\pi\)
\(90\) 0 0
\(91\) 1844.17 + 1844.17i 2.12442 + 2.12442i
\(92\) 0 0
\(93\) 268.564i 0.299450i
\(94\) 0 0
\(95\) −200.956 200.956i −0.217028 0.217028i
\(96\) 0 0
\(97\) 162.634 162.634i 0.170237 0.170237i −0.616846 0.787084i \(-0.711590\pi\)
0.787084 + 0.616846i \(0.211590\pi\)
\(98\) 0 0
\(99\) −34.1267 + 34.1267i −0.0346451 + 0.0346451i
\(100\) 0 0
\(101\) 1060.55 1.04484 0.522420 0.852689i \(-0.325029\pi\)
0.522420 + 0.852689i \(0.325029\pi\)
\(102\) 0 0
\(103\) 671.076 0.641972 0.320986 0.947084i \(-0.395986\pi\)
0.320986 + 0.947084i \(0.395986\pi\)
\(104\) 0 0
\(105\) 544.780 544.780i 0.506334 0.506334i
\(106\) 0 0
\(107\) −175.136 + 175.136i −0.158234 + 0.158234i −0.781784 0.623550i \(-0.785690\pi\)
0.623550 + 0.781784i \(0.285690\pi\)
\(108\) 0 0
\(109\) 509.552 + 509.552i 0.447764 + 0.447764i 0.894611 0.446847i \(-0.147453\pi\)
−0.446847 + 0.894611i \(0.647453\pi\)
\(110\) 0 0
\(111\) 529.366i 0.452660i
\(112\) 0 0
\(113\) −208.993 208.993i −0.173986 0.173986i 0.614742 0.788728i \(-0.289260\pi\)
−0.788728 + 0.614742i \(0.789260\pi\)
\(114\) 0 0
\(115\) −505.304 −0.409738
\(116\) 0 0
\(117\) 142.348i 0.112479i
\(118\) 0 0
\(119\) 689.008 2456.41i 0.530767 1.89226i
\(120\) 0 0
\(121\) 740.793i 0.556569i
\(122\) 0 0
\(123\) 1496.43 1.09698
\(124\) 0 0
\(125\) 652.037 + 652.037i 0.466560 + 0.466560i
\(126\) 0 0
\(127\) 450.551i 0.314803i 0.987535 + 0.157401i \(0.0503117\pi\)
−0.987535 + 0.157401i \(0.949688\pi\)
\(128\) 0 0
\(129\) −1310.22 1310.22i −0.894255 0.894255i
\(130\) 0 0
\(131\) 562.452 562.452i 0.375127 0.375127i −0.494213 0.869341i \(-0.664544\pi\)
0.869341 + 0.494213i \(0.164544\pi\)
\(132\) 0 0
\(133\) 1860.41 1860.41i 1.21292 1.21292i
\(134\) 0 0
\(135\) 529.464 0.337548
\(136\) 0 0
\(137\) −1480.25 −0.923109 −0.461555 0.887112i \(-0.652708\pi\)
−0.461555 + 0.887112i \(0.652708\pi\)
\(138\) 0 0
\(139\) −2213.66 + 2213.66i −1.35079 + 1.35079i −0.466017 + 0.884776i \(0.654311\pi\)
−0.884776 + 0.466017i \(0.845689\pi\)
\(140\) 0 0
\(141\) −2383.84 + 2383.84i −1.42380 + 1.42380i
\(142\) 0 0
\(143\) −1230.92 1230.92i −0.719825 0.719825i
\(144\) 0 0
\(145\) 456.266i 0.261316i
\(146\) 0 0
\(147\) 3737.66 + 3737.66i 2.09712 + 2.09712i
\(148\) 0 0
\(149\) −2885.80 −1.58667 −0.793336 0.608784i \(-0.791658\pi\)
−0.793336 + 0.608784i \(0.791658\pi\)
\(150\) 0 0
\(151\) 1037.68i 0.559239i −0.960111 0.279619i \(-0.909792\pi\)
0.960111 0.279619i \(-0.0902083\pi\)
\(152\) 0 0
\(153\) −121.394 + 68.2110i −0.0641446 + 0.0360427i
\(154\) 0 0
\(155\) 196.117i 0.101629i
\(156\) 0 0
\(157\) 3420.64 1.73883 0.869415 0.494082i \(-0.164496\pi\)
0.869415 + 0.494082i \(0.164496\pi\)
\(158\) 0 0
\(159\) 553.269 + 553.269i 0.275957 + 0.275957i
\(160\) 0 0
\(161\) 4678.00i 2.28993i
\(162\) 0 0
\(163\) −57.4644 57.4644i −0.0276132 0.0276132i 0.693165 0.720779i \(-0.256216\pi\)
−0.720779 + 0.693165i \(0.756216\pi\)
\(164\) 0 0
\(165\) −363.622 + 363.622i −0.171563 + 0.171563i
\(166\) 0 0
\(167\) 857.266 857.266i 0.397229 0.397229i −0.480025 0.877255i \(-0.659373\pi\)
0.877255 + 0.480025i \(0.159373\pi\)
\(168\) 0 0
\(169\) 2937.38 1.33700
\(170\) 0 0
\(171\) −143.601 −0.0642189
\(172\) 0 0
\(173\) −1010.59 + 1010.59i −0.444127 + 0.444127i −0.893396 0.449269i \(-0.851684\pi\)
0.449269 + 0.893396i \(0.351684\pi\)
\(174\) 0 0
\(175\) −2819.30 + 2819.30i −1.21782 + 1.21782i
\(176\) 0 0
\(177\) 1728.41 + 1728.41i 0.733986 + 0.733986i
\(178\) 0 0
\(179\) 1950.92i 0.814629i −0.913288 0.407315i \(-0.866465\pi\)
0.913288 0.407315i \(-0.133535\pi\)
\(180\) 0 0
\(181\) −1850.30 1850.30i −0.759844 0.759844i 0.216449 0.976294i \(-0.430552\pi\)
−0.976294 + 0.216449i \(0.930552\pi\)
\(182\) 0 0
\(183\) −1364.41 −0.551148
\(184\) 0 0
\(185\) 386.565i 0.153626i
\(186\) 0 0
\(187\) −459.889 + 1639.57i −0.179842 + 0.641161i
\(188\) 0 0
\(189\) 4901.67i 1.88648i
\(190\) 0 0
\(191\) −2192.73 −0.830683 −0.415342 0.909665i \(-0.636338\pi\)
−0.415342 + 0.909665i \(0.636338\pi\)
\(192\) 0 0
\(193\) 70.8011 + 70.8011i 0.0264061 + 0.0264061i 0.720187 0.693780i \(-0.244056\pi\)
−0.693780 + 0.720187i \(0.744056\pi\)
\(194\) 0 0
\(195\) 1516.73i 0.557001i
\(196\) 0 0
\(197\) −566.472 566.472i −0.204870 0.204870i 0.597213 0.802083i \(-0.296275\pi\)
−0.802083 + 0.597213i \(0.796275\pi\)
\(198\) 0 0
\(199\) −790.293 + 790.293i −0.281519 + 0.281519i −0.833715 0.552195i \(-0.813790\pi\)
0.552195 + 0.833715i \(0.313790\pi\)
\(200\) 0 0
\(201\) −3329.46 + 3329.46i −1.16837 + 1.16837i
\(202\) 0 0
\(203\) −4224.02 −1.46043
\(204\) 0 0
\(205\) 1092.75 0.372298
\(206\) 0 0
\(207\) −180.543 + 180.543i −0.0606211 + 0.0606211i
\(208\) 0 0
\(209\) −1241.76 + 1241.76i −0.410977 + 0.410977i
\(210\) 0 0
\(211\) 399.006 + 399.006i 0.130183 + 0.130183i 0.769196 0.639013i \(-0.220657\pi\)
−0.639013 + 0.769196i \(0.720657\pi\)
\(212\) 0 0
\(213\) 446.938i 0.143773i
\(214\) 0 0
\(215\) −956.781 956.781i −0.303497 0.303497i
\(216\) 0 0
\(217\) 1815.61 0.567980
\(218\) 0 0
\(219\) 2100.48i 0.648114i
\(220\) 0 0
\(221\) −2460.32 4378.59i −0.748864 1.33274i
\(222\) 0 0
\(223\) 5051.79i 1.51701i −0.651667 0.758505i \(-0.725930\pi\)
0.651667 0.758505i \(-0.274070\pi\)
\(224\) 0 0
\(225\) 217.616 0.0644788
\(226\) 0 0
\(227\) 202.315 + 202.315i 0.0591547 + 0.0591547i 0.736065 0.676911i \(-0.236682\pi\)
−0.676911 + 0.736065i \(0.736682\pi\)
\(228\) 0 0
\(229\) 2265.18i 0.653657i 0.945084 + 0.326828i \(0.105980\pi\)
−0.945084 + 0.326828i \(0.894020\pi\)
\(230\) 0 0
\(231\) −3366.34 3366.34i −0.958827 0.958827i
\(232\) 0 0
\(233\) 3776.51 3776.51i 1.06184 1.06184i 0.0638776 0.997958i \(-0.479653\pi\)
0.997958 0.0638776i \(-0.0203467\pi\)
\(234\) 0 0
\(235\) −1740.78 + 1740.78i −0.483216 + 0.483216i
\(236\) 0 0
\(237\) −3539.31 −0.970053
\(238\) 0 0
\(239\) 7017.70 1.89932 0.949660 0.313284i \(-0.101429\pi\)
0.949660 + 0.313284i \(0.101429\pi\)
\(240\) 0 0
\(241\) −3368.64 + 3368.64i −0.900386 + 0.900386i −0.995469 0.0950830i \(-0.969688\pi\)
0.0950830 + 0.995469i \(0.469688\pi\)
\(242\) 0 0
\(243\) 393.374 393.374i 0.103848 0.103848i
\(244\) 0 0
\(245\) 2729.40 + 2729.40i 0.711734 + 0.711734i
\(246\) 0 0
\(247\) 5179.58i 1.33429i
\(248\) 0 0
\(249\) −620.729 620.729i −0.157980 0.157980i
\(250\) 0 0
\(251\) 6592.92 1.65793 0.828967 0.559298i \(-0.188929\pi\)
0.828967 + 0.559298i \(0.188929\pi\)
\(252\) 0 0
\(253\) 3122.41i 0.775905i
\(254\) 0 0
\(255\) −1293.46 + 726.793i −0.317646 + 0.178484i
\(256\) 0 0
\(257\) 6128.40i 1.48747i 0.668476 + 0.743734i \(0.266947\pi\)
−0.668476 + 0.743734i \(0.733053\pi\)
\(258\) 0 0
\(259\) 3578.74 0.858580
\(260\) 0 0
\(261\) 163.022 + 163.022i 0.0386620 + 0.0386620i
\(262\) 0 0
\(263\) 6142.19i 1.44009i −0.693927 0.720045i \(-0.744121\pi\)
0.693927 0.720045i \(-0.255879\pi\)
\(264\) 0 0
\(265\) 404.020 + 404.020i 0.0936557 + 0.0936557i
\(266\) 0 0
\(267\) −1594.22 + 1594.22i −0.365412 + 0.365412i
\(268\) 0 0
\(269\) 831.758 831.758i 0.188525 0.188525i −0.606533 0.795058i \(-0.707440\pi\)
0.795058 + 0.606533i \(0.207440\pi\)
\(270\) 0 0
\(271\) −2498.30 −0.560004 −0.280002 0.959999i \(-0.590335\pi\)
−0.280002 + 0.959999i \(0.590335\pi\)
\(272\) 0 0
\(273\) 14041.6 3.11294
\(274\) 0 0
\(275\) 1881.79 1881.79i 0.412641 0.412641i
\(276\) 0 0
\(277\) 3640.10 3640.10i 0.789576 0.789576i −0.191848 0.981425i \(-0.561448\pi\)
0.981425 + 0.191848i \(0.0614482\pi\)
\(278\) 0 0
\(279\) −70.0716 70.0716i −0.0150361 0.0150361i
\(280\) 0 0
\(281\) 3355.18i 0.712290i 0.934431 + 0.356145i \(0.115909\pi\)
−0.934431 + 0.356145i \(0.884091\pi\)
\(282\) 0 0
\(283\) −1847.55 1847.55i −0.388076 0.388076i 0.485925 0.874001i \(-0.338483\pi\)
−0.874001 + 0.485925i \(0.838483\pi\)
\(284\) 0 0
\(285\) −1530.08 −0.318014
\(286\) 0 0
\(287\) 10116.5i 2.08069i
\(288\) 0 0
\(289\) −2555.11 + 4196.31i −0.520070 + 0.854123i
\(290\) 0 0
\(291\) 1238.30i 0.249452i
\(292\) 0 0
\(293\) −1683.63 −0.335695 −0.167848 0.985813i \(-0.553682\pi\)
−0.167848 + 0.985813i \(0.553682\pi\)
\(294\) 0 0
\(295\) 1262.16 + 1262.16i 0.249104 + 0.249104i
\(296\) 0 0
\(297\) 3271.70i 0.639203i
\(298\) 0 0
\(299\) −6512.03 6512.03i −1.25953 1.25953i
\(300\) 0 0
\(301\) 8857.68 8857.68i 1.69617 1.69617i
\(302\) 0 0
\(303\) 4037.52 4037.52i 0.765511 0.765511i
\(304\) 0 0
\(305\) −996.348 −0.187052
\(306\) 0 0
\(307\) 5864.67 1.09028 0.545138 0.838346i \(-0.316477\pi\)
0.545138 + 0.838346i \(0.316477\pi\)
\(308\) 0 0
\(309\) 2554.79 2554.79i 0.470346 0.470346i
\(310\) 0 0
\(311\) 4507.41 4507.41i 0.821838 0.821838i −0.164534 0.986371i \(-0.552612\pi\)
0.986371 + 0.164534i \(0.0526119\pi\)
\(312\) 0 0
\(313\) −2339.52 2339.52i −0.422484 0.422484i 0.463574 0.886058i \(-0.346567\pi\)
−0.886058 + 0.463574i \(0.846567\pi\)
\(314\) 0 0
\(315\) 284.279i 0.0508486i
\(316\) 0 0
\(317\) 5257.60 + 5257.60i 0.931534 + 0.931534i 0.997802 0.0662682i \(-0.0211093\pi\)
−0.0662682 + 0.997802i \(0.521109\pi\)
\(318\) 0 0
\(319\) 2819.39 0.494845
\(320\) 0 0
\(321\) 1333.49i 0.231863i
\(322\) 0 0
\(323\) −4417.13 + 2481.97i −0.760916 + 0.427556i
\(324\) 0 0
\(325\) 7849.25i 1.33969i
\(326\) 0 0
\(327\) 3879.74 0.656116
\(328\) 0 0
\(329\) −16115.8 16115.8i −2.70058 2.70058i
\(330\) 0 0
\(331\) 953.626i 0.158356i −0.996860 0.0791782i \(-0.974770\pi\)
0.996860 0.0791782i \(-0.0252296\pi\)
\(332\) 0 0
\(333\) −138.118 138.118i −0.0227292 0.0227292i
\(334\) 0 0
\(335\) −2431.31 + 2431.31i −0.396528 + 0.396528i
\(336\) 0 0
\(337\) −6649.51 + 6649.51i −1.07484 + 1.07484i −0.0778798 + 0.996963i \(0.524815\pi\)
−0.996963 + 0.0778798i \(0.975185\pi\)
\(338\) 0 0
\(339\) −1591.27 −0.254944
\(340\) 0 0
\(341\) −1211.86 −0.192451
\(342\) 0 0
\(343\) −16440.4 + 16440.4i −2.58804 + 2.58804i
\(344\) 0 0
\(345\) −1923.69 + 1923.69i −0.300198 + 0.300198i
\(346\) 0 0
\(347\) 7319.13 + 7319.13i 1.13231 + 1.13231i 0.989792 + 0.142518i \(0.0455200\pi\)
0.142518 + 0.989792i \(0.454480\pi\)
\(348\) 0 0
\(349\) 9266.42i 1.42126i −0.703566 0.710630i \(-0.748410\pi\)
0.703566 0.710630i \(-0.251590\pi\)
\(350\) 0 0
\(351\) 6823.40 + 6823.40i 1.03762 + 1.03762i
\(352\) 0 0
\(353\) 3315.92 0.499968 0.249984 0.968250i \(-0.419575\pi\)
0.249984 + 0.968250i \(0.419575\pi\)
\(354\) 0 0
\(355\) 326.373i 0.0487945i
\(356\) 0 0
\(357\) −6728.50 11974.6i −0.997507 1.77525i
\(358\) 0 0
\(359\) 5654.75i 0.831327i −0.909518 0.415664i \(-0.863549\pi\)
0.909518 0.415664i \(-0.136451\pi\)
\(360\) 0 0
\(361\) 1633.83 0.238203
\(362\) 0 0
\(363\) −2820.21 2820.21i −0.407775 0.407775i
\(364\) 0 0
\(365\) 1533.85i 0.219961i
\(366\) 0 0
\(367\) −9.03278 9.03278i −0.00128476 0.00128476i 0.706464 0.707749i \(-0.250289\pi\)
−0.707749 + 0.706464i \(0.750289\pi\)
\(368\) 0 0
\(369\) 390.435 390.435i 0.0550820 0.0550820i
\(370\) 0 0
\(371\) −3740.34 + 3740.34i −0.523419 + 0.523419i
\(372\) 0 0
\(373\) 2955.39 0.410252 0.205126 0.978736i \(-0.434240\pi\)
0.205126 + 0.978736i \(0.434240\pi\)
\(374\) 0 0
\(375\) 4964.62 0.683658
\(376\) 0 0
\(377\) −5880.06 + 5880.06i −0.803286 + 0.803286i
\(378\) 0 0
\(379\) −3054.12 + 3054.12i −0.413931 + 0.413931i −0.883105 0.469175i \(-0.844552\pi\)
0.469175 + 0.883105i \(0.344552\pi\)
\(380\) 0 0
\(381\) 1715.25 + 1715.25i 0.230643 + 0.230643i
\(382\) 0 0
\(383\) 5788.94i 0.772326i 0.922431 + 0.386163i \(0.126200\pi\)
−0.922431 + 0.386163i \(0.873800\pi\)
\(384\) 0 0
\(385\) −2458.24 2458.24i −0.325412 0.325412i
\(386\) 0 0
\(387\) −683.706 −0.0898055
\(388\) 0 0
\(389\) 10329.9i 1.34640i −0.739462 0.673198i \(-0.764920\pi\)
0.739462 0.673198i \(-0.235080\pi\)
\(390\) 0 0
\(391\) −2432.98 + 8673.92i −0.314683 + 1.12189i
\(392\) 0 0
\(393\) 4282.51i 0.549680i
\(394\) 0 0
\(395\) −2584.55 −0.329222
\(396\) 0 0
\(397\) 7727.58 + 7727.58i 0.976917 + 0.976917i 0.999740 0.0228226i \(-0.00726530\pi\)
−0.0228226 + 0.999740i \(0.507265\pi\)
\(398\) 0 0
\(399\) 14165.2i 1.77731i
\(400\) 0 0
\(401\) −4285.93 4285.93i −0.533739 0.533739i 0.387944 0.921683i \(-0.373185\pi\)
−0.921683 + 0.387944i \(0.873185\pi\)
\(402\) 0 0
\(403\) 2527.43 2527.43i 0.312407 0.312407i
\(404\) 0 0
\(405\) 2164.79 2164.79i 0.265603 0.265603i
\(406\) 0 0
\(407\) −2388.69 −0.290916
\(408\) 0 0
\(409\) 12598.8 1.52316 0.761580 0.648071i \(-0.224424\pi\)
0.761580 + 0.648071i \(0.224424\pi\)
\(410\) 0 0
\(411\) −5635.31 + 5635.31i −0.676324 + 0.676324i
\(412\) 0 0
\(413\) −11684.8 + 11684.8i −1.39219 + 1.39219i
\(414\) 0 0
\(415\) −453.282 453.282i −0.0536162 0.0536162i
\(416\) 0 0
\(417\) 16854.8i 1.97934i
\(418\) 0 0
\(419\) 1462.63 + 1462.63i 0.170536 + 0.170536i 0.787215 0.616679i \(-0.211522\pi\)
−0.616679 + 0.787215i \(0.711522\pi\)
\(420\) 0 0
\(421\) −7194.58 −0.832880 −0.416440 0.909163i \(-0.636722\pi\)
−0.416440 + 0.909163i \(0.636722\pi\)
\(422\) 0 0
\(423\) 1243.94i 0.142985i
\(424\) 0 0
\(425\) 6693.82 3761.24i 0.763995 0.429287i
\(426\) 0 0
\(427\) 9223.99i 1.04539i
\(428\) 0 0
\(429\) −9372.27 −1.05477
\(430\) 0 0
\(431\) 9683.60 + 9683.60i 1.08223 + 1.08223i 0.996301 + 0.0859325i \(0.0273869\pi\)
0.0859325 + 0.996301i \(0.472613\pi\)
\(432\) 0 0
\(433\) 12504.4i 1.38782i −0.720063 0.693909i \(-0.755887\pi\)
0.720063 0.693909i \(-0.244113\pi\)
\(434\) 0 0
\(435\) 1737.01 + 1737.01i 0.191455 + 0.191455i
\(436\) 0 0
\(437\) −6569.35 + 6569.35i −0.719119 + 0.719119i
\(438\) 0 0
\(439\) −9878.38 + 9878.38i −1.07396 + 1.07396i −0.0769257 + 0.997037i \(0.524510\pi\)
−0.997037 + 0.0769257i \(0.975490\pi\)
\(440\) 0 0
\(441\) 1950.40 0.210604
\(442\) 0 0
\(443\) 829.809 0.0889964 0.0444982 0.999009i \(-0.485831\pi\)
0.0444982 + 0.999009i \(0.485831\pi\)
\(444\) 0 0
\(445\) −1164.17 + 1164.17i −0.124016 + 0.124016i
\(446\) 0 0
\(447\) −10986.3 + 10986.3i −1.16249 + 1.16249i
\(448\) 0 0
\(449\) 5937.37 + 5937.37i 0.624057 + 0.624057i 0.946566 0.322509i \(-0.104526\pi\)
−0.322509 + 0.946566i \(0.604526\pi\)
\(450\) 0 0
\(451\) 6752.41i 0.705008i
\(452\) 0 0
\(453\) −3950.45 3950.45i −0.409731 0.409731i
\(454\) 0 0
\(455\) 10253.7 1.05649
\(456\) 0 0
\(457\) 4013.92i 0.410861i −0.978672 0.205430i \(-0.934141\pi\)
0.978672 0.205430i \(-0.0658594\pi\)
\(458\) 0 0
\(459\) 2549.31 9088.65i 0.259241 0.924230i
\(460\) 0 0
\(461\) 9576.45i 0.967505i 0.875205 + 0.483753i \(0.160727\pi\)
−0.875205 + 0.483753i \(0.839273\pi\)
\(462\) 0 0
\(463\) 8366.78 0.839822 0.419911 0.907565i \(-0.362061\pi\)
0.419911 + 0.907565i \(0.362061\pi\)
\(464\) 0 0
\(465\) −746.618 746.618i −0.0744593 0.0744593i
\(466\) 0 0
\(467\) 8324.94i 0.824909i −0.910978 0.412454i \(-0.864672\pi\)
0.910978 0.412454i \(-0.135328\pi\)
\(468\) 0 0
\(469\) −22508.6 22508.6i −2.21610 2.21610i
\(470\) 0 0
\(471\) 13022.4 13022.4i 1.27397 1.27397i
\(472\) 0 0
\(473\) −5912.20 + 5912.20i −0.574722 + 0.574722i
\(474\) 0 0
\(475\) 7918.34 0.764881
\(476\) 0 0
\(477\) 288.709 0.0277129
\(478\) 0 0
\(479\) −619.058 + 619.058i −0.0590511 + 0.0590511i −0.736016 0.676965i \(-0.763295\pi\)
0.676965 + 0.736016i \(0.263295\pi\)
\(480\) 0 0
\(481\) 4981.81 4981.81i 0.472247 0.472247i
\(482\) 0 0
\(483\) −17809.2 17809.2i −1.67773 1.67773i
\(484\) 0 0
\(485\) 904.258i 0.0846603i
\(486\) 0 0
\(487\) 2801.50 + 2801.50i 0.260674 + 0.260674i 0.825328 0.564654i \(-0.190990\pi\)
−0.564654 + 0.825328i \(0.690990\pi\)
\(488\) 0 0
\(489\) −437.534 −0.0404621
\(490\) 0 0
\(491\) 5348.65i 0.491611i 0.969319 + 0.245806i \(0.0790525\pi\)
−0.969319 + 0.245806i \(0.920947\pi\)
\(492\) 0 0
\(493\) 7832.14 + 2196.87i 0.715501 + 0.200694i
\(494\) 0 0
\(495\) 189.747i 0.0172293i
\(496\) 0 0
\(497\) 3021.49 0.272701
\(498\) 0 0
\(499\) −2291.68 2291.68i −0.205591 0.205591i 0.596800 0.802390i \(-0.296439\pi\)
−0.802390 + 0.596800i \(0.796439\pi\)
\(500\) 0 0
\(501\) 6527.23i 0.582067i
\(502\) 0 0
\(503\) 12161.4 + 12161.4i 1.07803 + 1.07803i 0.996686 + 0.0813446i \(0.0259214\pi\)
0.0813446 + 0.996686i \(0.474079\pi\)
\(504\) 0 0
\(505\) 2948.37 2948.37i 0.259803 0.259803i
\(506\) 0 0
\(507\) 11182.6 11182.6i 0.979562 0.979562i
\(508\) 0 0
\(509\) 19542.7 1.70179 0.850897 0.525333i \(-0.176059\pi\)
0.850897 + 0.525333i \(0.176059\pi\)
\(510\) 0 0
\(511\) −14200.1 −1.22931
\(512\) 0 0
\(513\) 6883.46 6883.46i 0.592421 0.592421i
\(514\) 0 0
\(515\) 1865.61 1865.61i 0.159629 0.159629i
\(516\) 0 0
\(517\) 10756.7 + 10756.7i 0.915049 + 0.915049i
\(518\) 0 0
\(519\) 7694.67i 0.650787i
\(520\) 0 0
\(521\) 13819.8 + 13819.8i 1.16211 + 1.16211i 0.984014 + 0.178093i \(0.0569926\pi\)
0.178093 + 0.984014i \(0.443007\pi\)
\(522\) 0 0
\(523\) −13396.9 −1.12009 −0.560044 0.828463i \(-0.689216\pi\)
−0.560044 + 0.828463i \(0.689216\pi\)
\(524\) 0 0
\(525\) 21466.2i 1.78450i
\(526\) 0 0
\(527\) −3366.49 944.281i −0.278267 0.0780522i
\(528\) 0 0
\(529\) 4351.67i 0.357662i
\(530\) 0 0
\(531\) 901.927 0.0737106
\(532\) 0 0
\(533\) 14082.7 + 14082.7i 1.14445 + 1.14445i
\(534\) 0 0
\(535\) 973.770i 0.0786911i
\(536\) 0 0
\(537\) −7427.16 7427.16i −0.596845 0.596845i
\(538\) 0 0
\(539\) 16865.7 16865.7i 1.34778 1.34778i
\(540\) 0 0
\(541\) 7739.88 7739.88i 0.615090 0.615090i −0.329178 0.944268i \(-0.606772\pi\)
0.944268 + 0.329178i \(0.106772\pi\)
\(542\) 0 0
\(543\) −14088.2 −1.11341
\(544\) 0 0
\(545\) 2833.15 0.222676
\(546\) 0 0
\(547\) −8933.89 + 8933.89i −0.698328 + 0.698328i −0.964050 0.265722i \(-0.914390\pi\)
0.265722 + 0.964050i \(0.414390\pi\)
\(548\) 0 0
\(549\) −355.990 + 355.990i −0.0276745 + 0.0276745i
\(550\) 0 0
\(551\) 5931.82 + 5931.82i 0.458628 + 0.458628i
\(552\) 0 0
\(553\) 23927.2i 1.83994i
\(554\) 0 0
\(555\) −1471.66 1471.66i −0.112556 0.112556i
\(556\) 0 0
\(557\) 3440.59 0.261728 0.130864 0.991400i \(-0.458225\pi\)
0.130864 + 0.991400i \(0.458225\pi\)
\(558\) 0 0
\(559\) 24660.8i 1.86590i
\(560\) 0 0
\(561\) 4491.05 + 7992.65i 0.337990 + 0.601515i
\(562\) 0 0
\(563\) 11973.1i 0.896278i 0.893964 + 0.448139i \(0.147913\pi\)
−0.893964 + 0.448139i \(0.852087\pi\)
\(564\) 0 0
\(565\) −1162.01 −0.0865243
\(566\) 0 0
\(567\) 20041.2 + 20041.2i 1.48439 + 1.48439i
\(568\) 0 0
\(569\) 12536.9i 0.923680i −0.886963 0.461840i \(-0.847189\pi\)
0.886963 0.461840i \(-0.152811\pi\)
\(570\) 0 0
\(571\) 2485.78 + 2485.78i 0.182183 + 0.182183i 0.792306 0.610123i \(-0.208880\pi\)
−0.610123 + 0.792306i \(0.708880\pi\)
\(572\) 0 0
\(573\) −8347.74 + 8347.74i −0.608607 + 0.608607i
\(574\) 0 0
\(575\) 9955.35 9955.35i 0.722029 0.722029i
\(576\) 0 0
\(577\) 15196.0 1.09639 0.548195 0.836350i \(-0.315315\pi\)
0.548195 + 0.836350i \(0.315315\pi\)
\(578\) 0 0
\(579\) 539.080 0.0386933
\(580\) 0 0
\(581\) 4196.39 4196.39i 0.299648 0.299648i
\(582\) 0 0
\(583\) 2496.55 2496.55i 0.177352 0.177352i
\(584\) 0 0
\(585\) −395.732 395.732i −0.0279684 0.0279684i
\(586\) 0 0
\(587\) 11409.2i 0.802230i −0.916028 0.401115i \(-0.868623\pi\)
0.916028 0.401115i \(-0.131377\pi\)
\(588\) 0 0
\(589\) −2549.68 2549.68i −0.178366 0.178366i
\(590\) 0 0
\(591\) −4313.12 −0.300200
\(592\) 0 0
\(593\) 3295.58i 0.228218i 0.993468 + 0.114109i \(0.0364013\pi\)
−0.993468 + 0.114109i \(0.963599\pi\)
\(594\) 0 0
\(595\) −4913.43 8744.36i −0.338540 0.602494i
\(596\) 0 0
\(597\) 6017.30i 0.412515i
\(598\) 0 0
\(599\) −1426.99 −0.0973378 −0.0486689 0.998815i \(-0.515498\pi\)
−0.0486689 + 0.998815i \(0.515498\pi\)
\(600\) 0 0
\(601\) −20369.5 20369.5i −1.38251 1.38251i −0.840136 0.542376i \(-0.817525\pi\)
−0.542376 0.840136i \(-0.682475\pi\)
\(602\) 0 0
\(603\) 1737.39i 0.117333i
\(604\) 0 0
\(605\) −2059.43 2059.43i −0.138393 0.138393i
\(606\) 0 0
\(607\) −2322.12 + 2322.12i −0.155275 + 0.155275i −0.780469 0.625194i \(-0.785020\pi\)
0.625194 + 0.780469i \(0.285020\pi\)
\(608\) 0 0
\(609\) −16080.9 + 16080.9i −1.07000 + 1.07000i
\(610\) 0 0
\(611\) −44868.1 −2.97082
\(612\) 0 0
\(613\) 15397.0 1.01449 0.507244 0.861803i \(-0.330664\pi\)
0.507244 + 0.861803i \(0.330664\pi\)
\(614\) 0 0
\(615\) 4160.12 4160.12i 0.272768 0.272768i
\(616\) 0 0
\(617\) −20796.5 + 20796.5i −1.35695 + 1.35695i −0.479293 + 0.877655i \(0.659107\pi\)
−0.877655 + 0.479293i \(0.840893\pi\)
\(618\) 0 0
\(619\) −1841.95 1841.95i −0.119603 0.119603i 0.644772 0.764375i \(-0.276952\pi\)
−0.764375 + 0.644772i \(0.776952\pi\)
\(620\) 0 0
\(621\) 17308.5i 1.11846i
\(622\) 0 0
\(623\) −10777.6 10777.6i −0.693093 0.693093i
\(624\) 0 0
\(625\) −10067.5 −0.644319
\(626\) 0 0
\(627\) 9454.76i 0.602212i
\(628\) 0 0
\(629\) −6635.68 1861.27i −0.420639 0.117987i
\(630\) 0 0
\(631\) 6384.47i 0.402792i 0.979510 + 0.201396i \(0.0645478\pi\)
−0.979510 + 0.201396i \(0.935452\pi\)
\(632\) 0 0
\(633\) 3038.03 0.190760
\(634\) 0 0
\(635\) 1252.55 + 1252.55i 0.0782769 + 0.0782769i
\(636\) 0 0
\(637\) 70349.4i 4.37574i
\(638\) 0 0
\(639\) −116.611 116.611i −0.00721920 0.00721920i
\(640\) 0 0
\(641\) −21042.7 + 21042.7i −1.29663 + 1.29663i −0.366021 + 0.930607i \(0.619280\pi\)
−0.930607 + 0.366021i \(0.880720\pi\)
\(642\) 0 0
\(643\) 11224.9 11224.9i 0.688442 0.688442i −0.273445 0.961888i \(-0.588163\pi\)
0.961888 + 0.273445i \(0.0881632\pi\)
\(644\) 0 0
\(645\) −7284.94 −0.444720
\(646\) 0 0
\(647\) −19644.7 −1.19368 −0.596842 0.802359i \(-0.703578\pi\)
−0.596842 + 0.802359i \(0.703578\pi\)
\(648\) 0 0
\(649\) 7799.22 7799.22i 0.471720 0.471720i
\(650\) 0 0
\(651\) 6912.04 6912.04i 0.416135 0.416135i
\(652\) 0 0
\(653\) −3999.32 3999.32i −0.239672 0.239672i 0.577043 0.816714i \(-0.304207\pi\)
−0.816714 + 0.577043i \(0.804207\pi\)
\(654\) 0 0
\(655\) 3127.27i 0.186554i
\(656\) 0 0
\(657\) 548.039 + 548.039i 0.0325434 + 0.0325434i
\(658\) 0 0
\(659\) 22818.1 1.34881 0.674407 0.738360i \(-0.264400\pi\)
0.674407 + 0.738360i \(0.264400\pi\)
\(660\) 0 0
\(661\) 5851.00i 0.344293i −0.985071 0.172146i \(-0.944930\pi\)
0.985071 0.172146i \(-0.0550702\pi\)
\(662\) 0 0
\(663\) −26035.8 7302.88i −1.52511 0.427783i
\(664\) 0 0
\(665\) 10344.0i 0.603192i
\(666\) 0 0
\(667\) 14915.6 0.865868
\(668\) 0 0
\(669\) −19232.2 19232.2i −1.11145 1.11145i
\(670\) 0 0
\(671\) 6156.70i 0.354213i
\(672\) 0 0
\(673\) 8905.21 + 8905.21i 0.510060 + 0.510060i 0.914545 0.404484i \(-0.132549\pi\)
−0.404484 + 0.914545i \(0.632549\pi\)
\(674\) 0 0
\(675\) −10431.3 + 10431.3i −0.594819 + 0.594819i
\(676\) 0 0
\(677\) 17965.5 17965.5i 1.01990 1.01990i 0.0200980 0.999798i \(-0.493602\pi\)
0.999798 0.0200980i \(-0.00639784\pi\)
\(678\) 0 0
\(679\) 8371.43 0.473146
\(680\) 0 0
\(681\) 1540.43 0.0866804
\(682\) 0 0
\(683\) −14089.4 + 14089.4i −0.789338 + 0.789338i −0.981386 0.192048i \(-0.938487\pi\)
0.192048 + 0.981386i \(0.438487\pi\)
\(684\) 0 0
\(685\) −4115.13 + 4115.13i −0.229535 + 0.229535i
\(686\) 0 0
\(687\) 8623.56 + 8623.56i 0.478907 + 0.478907i
\(688\) 0 0
\(689\) 10413.5i 0.575795i
\(690\) 0 0
\(691\) −18417.0 18417.0i −1.01391 1.01391i −0.999902 0.0140112i \(-0.995540\pi\)
−0.0140112 0.999902i \(-0.504460\pi\)
\(692\) 0 0
\(693\) −1756.64 −0.0962902
\(694\) 0 0
\(695\) 12308.1i 0.671759i
\(696\) 0 0
\(697\) 5261.48 18757.9i 0.285929 1.01938i
\(698\) 0 0
\(699\) 28754.4i 1.55593i
\(700\) 0 0
\(701\) 8521.04 0.459109 0.229554 0.973296i \(-0.426273\pi\)
0.229554 + 0.973296i \(0.426273\pi\)
\(702\) 0 0
\(703\) −5025.66 5025.66i −0.269625 0.269625i
\(704\) 0 0
\(705\) 13254.3i 0.708065i
\(706\) 0 0
\(707\) 27295.4 + 27295.4i 1.45198 + 1.45198i
\(708\) 0 0
\(709\) −19634.5 + 19634.5i −1.04004 + 1.04004i −0.0408774 + 0.999164i \(0.513015\pi\)
−0.999164 + 0.0408774i \(0.986985\pi\)
\(710\) 0 0
\(711\) −923.446 + 923.446i −0.0487088 + 0.0487088i
\(712\) 0 0
\(713\) −6411.17 −0.336746
\(714\) 0 0
\(715\) −6844.02 −0.357975
\(716\) 0 0
\(717\) 26716.4 26716.4i 1.39155 1.39155i
\(718\) 0 0
\(719\) 20219.0 20219.0i 1.04874 1.04874i 0.0499880 0.998750i \(-0.484082\pi\)
0.998750 0.0499880i \(-0.0159183\pi\)
\(720\) 0 0
\(721\) 17271.5 + 17271.5i 0.892127 + 0.892127i
\(722\) 0 0
\(723\) 25648.9i 1.31935i
\(724\) 0 0
\(725\) −8989.22 8989.22i −0.460484 0.460484i
\(726\) 0 0
\(727\) −22672.4 −1.15663 −0.578317 0.815812i \(-0.696290\pi\)
−0.578317 + 0.815812i \(0.696290\pi\)
\(728\) 0 0
\(729\) 18029.5i 0.915994i
\(730\) 0 0
\(731\) −21030.6 + 11817.1i −1.06409 + 0.597907i
\(732\) 0 0
\(733\) 3241.83i 0.163356i 0.996659 + 0.0816779i \(0.0260279\pi\)
−0.996659 + 0.0816779i \(0.973972\pi\)
\(734\) 0 0
\(735\) 20781.7 1.04292
\(736\) 0 0
\(737\) 15023.7 + 15023.7i 0.750890 + 0.750890i
\(738\) 0 0
\(739\) 8977.81i 0.446893i −0.974716 0.223447i \(-0.928269\pi\)
0.974716 0.223447i \(-0.0717309\pi\)
\(740\) 0 0
\(741\) −19718.7 19718.7i −0.977576 0.977576i
\(742\) 0 0
\(743\) −3759.91 + 3759.91i −0.185650 + 0.185650i −0.793812 0.608163i \(-0.791907\pi\)
0.608163 + 0.793812i \(0.291907\pi\)
\(744\) 0 0
\(745\) −8022.63 + 8022.63i −0.394532 + 0.394532i
\(746\) 0 0
\(747\) −323.911 −0.0158652
\(748\) 0 0
\(749\) −9014.96 −0.439786
\(750\) 0 0
\(751\) 13408.0 13408.0i 0.651485 0.651485i −0.301865 0.953351i \(-0.597609\pi\)
0.953351 + 0.301865i \(0.0976093\pi\)
\(752\) 0 0
\(753\) 25099.3 25099.3i 1.21470 1.21470i
\(754\) 0 0
\(755\) −2884.78 2884.78i −0.139057 0.139057i
\(756\) 0 0
\(757\) 28601.5i 1.37324i −0.727018 0.686619i \(-0.759094\pi\)
0.727018 0.686619i \(-0.240906\pi\)
\(758\) 0 0
\(759\) 11887.0 + 11887.0i 0.568474 + 0.568474i
\(760\) 0 0
\(761\) −23710.9 −1.12946 −0.564730 0.825276i \(-0.691020\pi\)
−0.564730 + 0.825276i \(0.691020\pi\)
\(762\) 0 0
\(763\) 26228.7i 1.24449i
\(764\) 0 0
\(765\) −147.851 + 527.108i −0.00698765 + 0.0249120i
\(766\) 0 0
\(767\) 32531.8i 1.53149i
\(768\) 0 0
\(769\) −3392.20 −0.159071 −0.0795357 0.996832i \(-0.525344\pi\)
−0.0795357 + 0.996832i \(0.525344\pi\)
\(770\) 0 0
\(771\) 23330.9 + 23330.9i 1.08981 + 1.08981i
\(772\) 0 0
\(773\) 10444.4i 0.485977i −0.970029 0.242988i \(-0.921872\pi\)
0.970029 0.242988i \(-0.0781277\pi\)
\(774\) 0 0
\(775\) 3863.84 + 3863.84i 0.179088 + 0.179088i
\(776\) 0 0
\(777\) 13624.3 13624.3i 0.629046 0.629046i
\(778\) 0 0
\(779\) 14206.7 14206.7i 0.653410 0.653410i
\(780\) 0 0
\(781\) −2016.74 −0.0924004
\(782\) 0 0
\(783\) −15628.8 −0.713316
\(784\) 0 0
\(785\) 9509.48 9509.48i 0.432367 0.432367i
\(786\) 0 0
\(787\) −8973.36 + 8973.36i −0.406437 + 0.406437i −0.880494 0.474057i \(-0.842789\pi\)
0.474057 + 0.880494i \(0.342789\pi\)
\(788\) 0 0
\(789\) −23383.4 23383.4i −1.05509 1.05509i
\(790\) 0 0
\(791\) 10757.7i 0.483564i
\(792\) 0 0
\(793\) −12840.3 12840.3i −0.574997 0.574997i
\(794\) 0 0
\(795\) 3076.21 0.137235
\(796\) 0 0
\(797\) 31949.2i 1.41995i −0.704228 0.709974i \(-0.748707\pi\)
0.704228 0.709974i \(-0.251293\pi\)
\(798\) 0 0
\(799\) 21500.1 + 38263.4i 0.951964 + 1.69420i
\(800\) 0 0
\(801\) 831.904i 0.0366965i
\(802\) 0 0
\(803\) 9478.10 0.416531
\(804\) 0 0
\(805\) −13005.0 13005.0i −0.569399 0.569399i
\(806\) 0 0
\(807\) 6333.02i 0.276249i
\(808\) 0 0
\(809\) 13072.2 + 13072.2i 0.568101 + 0.568101i 0.931596 0.363495i \(-0.118417\pi\)
−0.363495 + 0.931596i \(0.618417\pi\)
\(810\) 0 0
\(811\) 15766.6 15766.6i 0.682665 0.682665i −0.277935 0.960600i \(-0.589650\pi\)
0.960600 + 0.277935i \(0.0896500\pi\)
\(812\) 0 0
\(813\) −9511.05 + 9511.05i −0.410292 + 0.410292i
\(814\) 0 0
\(815\) −319.506 −0.0137323
\(816\) 0 0
\(817\) −24877.8 −1.06532
\(818\) 0 0
\(819\) 3663.61 3663.61i 0.156309 0.156309i
\(820\) 0 0
\(821\) 13357.5 13357.5i 0.567818 0.567818i −0.363699 0.931517i \(-0.618486\pi\)
0.931517 + 0.363699i \(0.118486\pi\)
\(822\) 0 0
\(823\) 5642.25 + 5642.25i 0.238975 + 0.238975i 0.816426 0.577451i \(-0.195952\pi\)
−0.577451 + 0.816426i \(0.695952\pi\)
\(824\) 0 0
\(825\) 14328.0i 0.604649i
\(826\) 0 0
\(827\) −3153.39 3153.39i −0.132593 0.132593i 0.637696 0.770288i \(-0.279888\pi\)
−0.770288 + 0.637696i \(0.779888\pi\)
\(828\) 0 0
\(829\) 19686.3 0.824769 0.412384 0.911010i \(-0.364696\pi\)
0.412384 + 0.911010i \(0.364696\pi\)
\(830\) 0 0
\(831\) 27715.8i 1.15698i
\(832\) 0 0
\(833\) 59993.8 33710.4i 2.49539 1.40216i
\(834\) 0 0
\(835\) 4766.46i 0.197545i
\(836\) 0 0
\(837\) 6717.71 0.277417
\(838\) 0 0
\(839\) −20299.7 20299.7i −0.835307 0.835307i 0.152930 0.988237i \(-0.451129\pi\)
−0.988237 + 0.152930i \(0.951129\pi\)
\(840\) 0 0
\(841\) 10920.9i 0.447780i
\(842\) 0 0
\(843\) 12773.2 + 12773.2i 0.521865 + 0.521865i
\(844\) 0 0
\(845\) 8166.02 8166.02i 0.332449 0.332449i
\(846\) 0 0
\(847\) 19065.8 19065.8i 0.773446 0.773446i
\(848\) 0 0
\(849\) −14067.3 −0.568654
\(850\) 0 0
\(851\) −12637.0 −0.509039
\(852\) 0 0
\(853\) 5307.45 5307.45i 0.213040 0.213040i −0.592517 0.805558i \(-0.701866\pi\)
0.805558 + 0.592517i \(0.201866\pi\)
\(854\) 0 0
\(855\) −399.215 + 399.215i −0.0159683 + 0.0159683i
\(856\) 0 0
\(857\) −55.7082 55.7082i −0.00222048 0.00222048i 0.705996 0.708216i \(-0.250500\pi\)
−0.708216 + 0.705996i \(0.750500\pi\)
\(858\) 0 0
\(859\) 5441.53i 0.216138i 0.994143 + 0.108069i \(0.0344668\pi\)
−0.994143 + 0.108069i \(0.965533\pi\)
\(860\) 0 0
\(861\) 38513.5 + 38513.5i 1.52443 + 1.52443i
\(862\) 0 0
\(863\) 7156.86 0.282297 0.141148 0.989988i \(-0.454921\pi\)
0.141148 + 0.989988i \(0.454921\pi\)
\(864\) 0 0
\(865\) 5618.97i 0.220868i
\(866\) 0 0
\(867\) 6248.06 + 25702.7i 0.244747 + 1.00681i
\(868\) 0 0
\(869\) 15970.6i 0.623436i
\(870\) 0 0
\(871\) −62666.4 −2.43785
\(872\) 0 0
\(873\) −323.087 323.087i −0.0125256 0.0125256i
\(874\) 0 0
\(875\) 33562.9i 1.29673i
\(876\) 0 0
\(877\) 10284.2 + 10284.2i 0.395979 + 0.395979i 0.876812 0.480833i \(-0.159666\pi\)
−0.480833 + 0.876812i \(0.659666\pi\)
\(878\) 0 0
\(879\) −6409.59 + 6409.59i −0.245950 + 0.245950i
\(880\) 0 0
\(881\) −15872.5 + 15872.5i −0.606990 + 0.606990i −0.942158 0.335168i \(-0.891207\pi\)
0.335168 + 0.942158i \(0.391207\pi\)
\(882\) 0 0
\(883\) 34760.2 1.32477 0.662385 0.749163i \(-0.269544\pi\)
0.662385 + 0.749163i \(0.269544\pi\)
\(884\) 0 0
\(885\) 9610.10 0.365017
\(886\) 0 0
\(887\) 4228.29 4228.29i 0.160059 0.160059i −0.622534 0.782593i \(-0.713897\pi\)
0.782593 + 0.622534i \(0.213897\pi\)
\(888\) 0 0
\(889\) −11595.8 + 11595.8i −0.437471 + 0.437471i
\(890\) 0 0
\(891\) −13376.8 13376.8i −0.502963 0.502963i
\(892\) 0 0
\(893\) 45263.0i 1.69616i
\(894\) 0 0
\(895\) −5423.62 5423.62i −0.202561 0.202561i
\(896\) 0 0
\(897\) −49582.7 −1.84562
\(898\) 0 0
\(899\) 5788.99i 0.214765i
\(900\) 0 0
\(901\) 8880.62 4989.99i 0.328364 0.184507i
\(902\) 0 0
\(903\) 67442.5i 2.48543i
\(904\) 0 0
\(905\) −10287.8 −0.377876
\(906\) 0 0
\(907\) 26461.5 + 26461.5i 0.968731 + 0.968731i 0.999526 0.0307949i \(-0.00980387\pi\)
−0.0307949 + 0.999526i \(0.509804\pi\)
\(908\) 0 0
\(909\) 2106.87i 0.0768764i
\(910\) 0 0
\(911\) −18.0094 18.0094i −0.000654970 0.000654970i 0.706779 0.707434i \(-0.250147\pi\)
−0.707434 + 0.706779i \(0.750147\pi\)
\(912\) 0 0
\(913\) −2800.95 + 2800.95i −0.101531 + 0.101531i
\(914\) 0 0
\(915\) −3793.10 + 3793.10i −0.137045 + 0.137045i
\(916\) 0 0
\(917\) 28951.6 1.04260
\(918\) 0 0
\(919\) −11502.6 −0.412878 −0.206439 0.978459i \(-0.566188\pi\)
−0.206439 + 0.978459i \(0.566188\pi\)
\(920\) 0 0
\(921\) 22326.8 22326.8i 0.798800 0.798800i
\(922\) 0 0
\(923\) 4206.08 4206.08i 0.149994 0.149994i
\(924\) 0 0
\(925\) 7616.00 + 7616.00i 0.270716 + 0.270716i
\(926\) 0 0
\(927\) 1333.15i 0.0472345i
\(928\) 0 0
\(929\) −26325.7 26325.7i −0.929729 0.929729i 0.0679592 0.997688i \(-0.478351\pi\)
−0.997688 + 0.0679592i \(0.978351\pi\)
\(930\) 0 0
\(931\) 70968.7 2.49829
\(932\) 0 0
\(933\) 34319.4i 1.20425i
\(934\) 0 0
\(935\) 3279.55 + 5836.57i 0.114709 + 0.204146i
\(936\) 0 0
\(937\) 72.1695i 0.00251619i −0.999999 0.00125810i \(-0.999600\pi\)
0.999999 0.00125810i \(-0.000400465\pi\)
\(938\) 0 0
\(939\) −17813.1 −0.619072
\(940\) 0 0
\(941\) 6951.94 + 6951.94i 0.240836 + 0.240836i 0.817196 0.576360i \(-0.195527\pi\)
−0.576360 + 0.817196i \(0.695527\pi\)
\(942\) 0 0
\(943\) 35722.7i 1.23361i
\(944\) 0 0
\(945\) 13626.8 + 13626.8i 0.469080 + 0.469080i
\(946\) 0 0
\(947\) −14532.5 + 14532.5i −0.498671 + 0.498671i −0.911024 0.412353i \(-0.864707\pi\)
0.412353 + 0.911024i \(0.364707\pi\)
\(948\) 0 0
\(949\) −19767.3 + 19767.3i −0.676159 + 0.676159i
\(950\) 0 0
\(951\) 40031.4 1.36499
\(952\) 0 0
\(953\) −2360.48 −0.0802345 −0.0401172 0.999195i \(-0.512773\pi\)
−0.0401172 + 0.999195i \(0.512773\pi\)
\(954\) 0 0
\(955\) −6095.87 + 6095.87i −0.206553 + 0.206553i
\(956\) 0 0
\(957\) 10733.4 10733.4i 0.362552 0.362552i
\(958\) 0 0
\(959\) −38097.1 38097.1i −1.28281 1.28281i
\(960\) 0 0
\(961\) 27302.7i 0.916475i
\(962\) 0 0
\(963\) 347.923 + 347.923i 0.0116424 + 0.0116424i
\(964\) 0 0
\(965\) 393.659 0.0131319
\(966\) 0 0
\(967\) 9395.70i 0.312456i −0.987721 0.156228i \(-0.950066\pi\)
0.987721 0.156228i \(-0.0499335\pi\)
\(968\) 0 0
\(969\) −7367.16 + 26264.9i −0.244239 + 0.870744i
\(970\) 0 0
\(971\) 24081.5i 0.795893i −0.917409 0.397947i \(-0.869723\pi\)
0.917409 0.397947i \(-0.130277\pi\)
\(972\) 0 0
\(973\) −113946. −3.75430
\(974\) 0 0
\(975\) 29882.1 + 29882.1i 0.981532 + 0.981532i
\(976\) 0 0
\(977\) 7503.29i 0.245703i 0.992425 + 0.122851i \(0.0392038\pi\)
−0.992425 + 0.122851i \(0.960796\pi\)
\(978\) 0 0
\(979\) 7193.71 + 7193.71i 0.234844 + 0.234844i
\(980\) 0 0
\(981\) 1012.27 1012.27i 0.0329452 0.0329452i
\(982\) 0 0
\(983\) −9334.52 + 9334.52i −0.302874 + 0.302874i −0.842137 0.539263i \(-0.818703\pi\)
0.539263 + 0.842137i \(0.318703\pi\)
\(984\) 0 0
\(985\) −3149.62 −0.101884
\(986\) 0 0
\(987\) −122706. −3.95721
\(988\) 0 0
\(989\) −31277.7 + 31277.7i −1.00564 + 1.00564i
\(990\) 0 0
\(991\) −27451.4 + 27451.4i −0.879940 + 0.879940i −0.993528 0.113588i \(-0.963766\pi\)
0.113588 + 0.993528i \(0.463766\pi\)
\(992\) 0 0
\(993\) −3630.46 3630.46i −0.116021 0.116021i
\(994\) 0 0
\(995\) 4394.08i 0.140002i
\(996\) 0 0
\(997\) 14656.5 + 14656.5i 0.465573 + 0.465573i 0.900477 0.434904i \(-0.143218\pi\)
−0.434904 + 0.900477i \(0.643218\pi\)
\(998\) 0 0
\(999\) 13241.3 0.419354
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 136.4.k.b.89.6 yes 14
4.3 odd 2 272.4.o.f.225.2 14
17.8 even 8 2312.4.a.l.1.4 14
17.9 even 8 2312.4.a.l.1.11 14
17.13 even 4 inner 136.4.k.b.81.6 14
68.47 odd 4 272.4.o.f.81.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.k.b.81.6 14 17.13 even 4 inner
136.4.k.b.89.6 yes 14 1.1 even 1 trivial
272.4.o.f.81.2 14 68.47 odd 4
272.4.o.f.225.2 14 4.3 odd 2
2312.4.a.l.1.4 14 17.8 even 8
2312.4.a.l.1.11 14 17.9 even 8