Properties

Label 3072.2.d.h.1537.2
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3072,2,Mod(1537,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(2))
 
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("3072.1537");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.2
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.h.1537.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.03528i q^{5} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.03528i q^{5} -2.44949 q^{7} -1.00000 q^{9} +5.46410i q^{11} -4.24264i q^{13} -1.03528 q^{15} +3.46410 q^{17} -0.535898i q^{19} +2.44949i q^{21} +2.82843 q^{23} +3.92820 q^{25} +1.00000i q^{27} +5.93426i q^{29} +7.34847 q^{31} +5.46410 q^{33} +2.53590i q^{35} -9.14162i q^{37} -4.24264 q^{39} -11.4641 q^{41} +3.46410i q^{43} +1.03528i q^{45} +2.82843 q^{47} -1.00000 q^{49} -3.46410i q^{51} -9.52056i q^{53} +5.65685 q^{55} -0.535898 q^{57} -13.8564i q^{59} -9.14162i q^{61} +2.44949 q^{63} -4.39230 q^{65} -1.07180i q^{67} -2.82843i q^{69} +16.2127 q^{71} -4.00000 q^{73} -3.92820i q^{75} -13.3843i q^{77} -2.44949 q^{79} +1.00000 q^{81} +1.46410i q^{83} -3.58630i q^{85} +5.93426 q^{87} -8.92820 q^{89} +10.3923i q^{91} -7.34847i q^{93} -0.554803 q^{95} +14.9282 q^{97} -5.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 24 q^{25} + 16 q^{33} - 64 q^{41} - 8 q^{49} - 32 q^{57} + 48 q^{65} - 32 q^{73} + 8 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) − 1.03528i − 0.462990i −0.972836 0.231495i \(-0.925638\pi\)
0.972836 0.231495i \(-0.0743616\pi\)
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.46410i 1.64749i 0.566961 + 0.823744i \(0.308119\pi\)
−0.566961 + 0.823744i \(0.691881\pi\)
\(12\) 0 0
\(13\) − 4.24264i − 1.17670i −0.808608 0.588348i \(-0.799778\pi\)
0.808608 0.588348i \(-0.200222\pi\)
\(14\) 0 0
\(15\) −1.03528 −0.267307
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) − 0.535898i − 0.122944i −0.998109 0.0614718i \(-0.980421\pi\)
0.998109 0.0614718i \(-0.0195794\pi\)
\(20\) 0 0
\(21\) 2.44949i 0.534522i
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 3.92820 0.785641
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 5.93426i 1.10196i 0.834517 + 0.550982i \(0.185747\pi\)
−0.834517 + 0.550982i \(0.814253\pi\)
\(30\) 0 0
\(31\) 7.34847 1.31982 0.659912 0.751343i \(-0.270594\pi\)
0.659912 + 0.751343i \(0.270594\pi\)
\(32\) 0 0
\(33\) 5.46410 0.951178
\(34\) 0 0
\(35\) 2.53590i 0.428645i
\(36\) 0 0
\(37\) − 9.14162i − 1.50287i −0.659805 0.751437i \(-0.729361\pi\)
0.659805 0.751437i \(-0.270639\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) −11.4641 −1.79039 −0.895196 0.445673i \(-0.852964\pi\)
−0.895196 + 0.445673i \(0.852964\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) 1.03528i 0.154330i
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) − 3.46410i − 0.485071i
\(52\) 0 0
\(53\) − 9.52056i − 1.30775i −0.756603 0.653875i \(-0.773142\pi\)
0.756603 0.653875i \(-0.226858\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) −0.535898 −0.0709815
\(58\) 0 0
\(59\) − 13.8564i − 1.80395i −0.431788 0.901975i \(-0.642117\pi\)
0.431788 0.901975i \(-0.357883\pi\)
\(60\) 0 0
\(61\) − 9.14162i − 1.17046i −0.810866 0.585232i \(-0.801003\pi\)
0.810866 0.585232i \(-0.198997\pi\)
\(62\) 0 0
\(63\) 2.44949 0.308607
\(64\) 0 0
\(65\) −4.39230 −0.544798
\(66\) 0 0
\(67\) − 1.07180i − 0.130941i −0.997855 0.0654704i \(-0.979145\pi\)
0.997855 0.0654704i \(-0.0208548\pi\)
\(68\) 0 0
\(69\) − 2.82843i − 0.340503i
\(70\) 0 0
\(71\) 16.2127 1.92409 0.962046 0.272887i \(-0.0879786\pi\)
0.962046 + 0.272887i \(0.0879786\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) − 3.92820i − 0.453590i
\(76\) 0 0
\(77\) − 13.3843i − 1.52528i
\(78\) 0 0
\(79\) −2.44949 −0.275589 −0.137795 0.990461i \(-0.544001\pi\)
−0.137795 + 0.990461i \(0.544001\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.46410i 0.160706i 0.996766 + 0.0803530i \(0.0256048\pi\)
−0.996766 + 0.0803530i \(0.974395\pi\)
\(84\) 0 0
\(85\) − 3.58630i − 0.388989i
\(86\) 0 0
\(87\) 5.93426 0.636219
\(88\) 0 0
\(89\) −8.92820 −0.946388 −0.473194 0.880958i \(-0.656899\pi\)
−0.473194 + 0.880958i \(0.656899\pi\)
\(90\) 0 0
\(91\) 10.3923i 1.08941i
\(92\) 0 0
\(93\) − 7.34847i − 0.762001i
\(94\) 0 0
\(95\) −0.554803 −0.0569216
\(96\) 0 0
\(97\) 14.9282 1.51573 0.757865 0.652412i \(-0.226243\pi\)
0.757865 + 0.652412i \(0.226243\pi\)
\(98\) 0 0
\(99\) − 5.46410i − 0.549163i
\(100\) 0 0
\(101\) 0.277401i 0.0276025i 0.999905 + 0.0138012i \(0.00439321\pi\)
−0.999905 + 0.0138012i \(0.995607\pi\)
\(102\) 0 0
\(103\) −8.10634 −0.798742 −0.399371 0.916789i \(-0.630771\pi\)
−0.399371 + 0.916789i \(0.630771\pi\)
\(104\) 0 0
\(105\) 2.53590 0.247478
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) − 1.41421i − 0.135457i −0.997704 0.0677285i \(-0.978425\pi\)
0.997704 0.0677285i \(-0.0215752\pi\)
\(110\) 0 0
\(111\) −9.14162 −0.867684
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) − 2.92820i − 0.273056i
\(116\) 0 0
\(117\) 4.24264i 0.392232i
\(118\) 0 0
\(119\) −8.48528 −0.777844
\(120\) 0 0
\(121\) −18.8564 −1.71422
\(122\) 0 0
\(123\) 11.4641i 1.03368i
\(124\) 0 0
\(125\) − 9.24316i − 0.826733i
\(126\) 0 0
\(127\) −19.4201 −1.72325 −0.861625 0.507545i \(-0.830553\pi\)
−0.861625 + 0.507545i \(0.830553\pi\)
\(128\) 0 0
\(129\) 3.46410 0.304997
\(130\) 0 0
\(131\) 1.07180i 0.0936433i 0.998903 + 0.0468217i \(0.0149092\pi\)
−0.998903 + 0.0468217i \(0.985091\pi\)
\(132\) 0 0
\(133\) 1.31268i 0.113824i
\(134\) 0 0
\(135\) 1.03528 0.0891024
\(136\) 0 0
\(137\) 6.39230 0.546131 0.273066 0.961995i \(-0.411962\pi\)
0.273066 + 0.961995i \(0.411962\pi\)
\(138\) 0 0
\(139\) − 6.92820i − 0.587643i −0.955860 0.293821i \(-0.905073\pi\)
0.955860 0.293821i \(-0.0949270\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) 23.1822 1.93859
\(144\) 0 0
\(145\) 6.14359 0.510198
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) − 20.0764i − 1.64472i −0.568966 0.822361i \(-0.692656\pi\)
0.568966 0.822361i \(-0.307344\pi\)
\(150\) 0 0
\(151\) −17.9043 −1.45703 −0.728516 0.685029i \(-0.759789\pi\)
−0.728516 + 0.685029i \(0.759789\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 0 0
\(155\) − 7.60770i − 0.611065i
\(156\) 0 0
\(157\) − 11.9700i − 0.955314i −0.878546 0.477657i \(-0.841486\pi\)
0.878546 0.477657i \(-0.158514\pi\)
\(158\) 0 0
\(159\) −9.52056 −0.755029
\(160\) 0 0
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) − 11.4641i − 0.897938i −0.893547 0.448969i \(-0.851791\pi\)
0.893547 0.448969i \(-0.148209\pi\)
\(164\) 0 0
\(165\) − 5.65685i − 0.440386i
\(166\) 0 0
\(167\) 2.07055 0.160224 0.0801121 0.996786i \(-0.474472\pi\)
0.0801121 + 0.996786i \(0.474472\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0.535898i 0.0409812i
\(172\) 0 0
\(173\) 2.55103i 0.193951i 0.995287 + 0.0969754i \(0.0309168\pi\)
−0.995287 + 0.0969754i \(0.969083\pi\)
\(174\) 0 0
\(175\) −9.62209 −0.727362
\(176\) 0 0
\(177\) −13.8564 −1.04151
\(178\) 0 0
\(179\) 5.07180i 0.379084i 0.981873 + 0.189542i \(0.0607003\pi\)
−0.981873 + 0.189542i \(0.939300\pi\)
\(180\) 0 0
\(181\) 7.07107i 0.525588i 0.964852 + 0.262794i \(0.0846440\pi\)
−0.964852 + 0.262794i \(0.915356\pi\)
\(182\) 0 0
\(183\) −9.14162 −0.675768
\(184\) 0 0
\(185\) −9.46410 −0.695815
\(186\) 0 0
\(187\) 18.9282i 1.38417i
\(188\) 0 0
\(189\) − 2.44949i − 0.178174i
\(190\) 0 0
\(191\) −2.07055 −0.149820 −0.0749100 0.997190i \(-0.523867\pi\)
−0.0749100 + 0.997190i \(0.523867\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 4.39230i 0.314539i
\(196\) 0 0
\(197\) 5.93426i 0.422798i 0.977400 + 0.211399i \(0.0678020\pi\)
−0.977400 + 0.211399i \(0.932198\pi\)
\(198\) 0 0
\(199\) 3.96524 0.281088 0.140544 0.990074i \(-0.455115\pi\)
0.140544 + 0.990074i \(0.455115\pi\)
\(200\) 0 0
\(201\) −1.07180 −0.0755987
\(202\) 0 0
\(203\) − 14.5359i − 1.02022i
\(204\) 0 0
\(205\) 11.8685i 0.828933i
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 2.92820 0.202548
\(210\) 0 0
\(211\) 6.92820i 0.476957i 0.971148 + 0.238479i \(0.0766487\pi\)
−0.971148 + 0.238479i \(0.923351\pi\)
\(212\) 0 0
\(213\) − 16.2127i − 1.11088i
\(214\) 0 0
\(215\) 3.58630 0.244584
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) 0 0
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) − 14.6969i − 0.988623i
\(222\) 0 0
\(223\) 19.4201 1.30046 0.650231 0.759736i \(-0.274672\pi\)
0.650231 + 0.759736i \(0.274672\pi\)
\(224\) 0 0
\(225\) −3.92820 −0.261880
\(226\) 0 0
\(227\) 23.3205i 1.54784i 0.633286 + 0.773918i \(0.281706\pi\)
−0.633286 + 0.773918i \(0.718294\pi\)
\(228\) 0 0
\(229\) 2.72689i 0.180198i 0.995933 + 0.0900990i \(0.0287184\pi\)
−0.995933 + 0.0900990i \(0.971282\pi\)
\(230\) 0 0
\(231\) −13.3843 −0.880620
\(232\) 0 0
\(233\) 0.928203 0.0608086 0.0304043 0.999538i \(-0.490321\pi\)
0.0304043 + 0.999538i \(0.490321\pi\)
\(234\) 0 0
\(235\) − 2.92820i − 0.191015i
\(236\) 0 0
\(237\) 2.44949i 0.159111i
\(238\) 0 0
\(239\) 2.07055 0.133933 0.0669664 0.997755i \(-0.478668\pi\)
0.0669664 + 0.997755i \(0.478668\pi\)
\(240\) 0 0
\(241\) 14.7846 0.952360 0.476180 0.879348i \(-0.342021\pi\)
0.476180 + 0.879348i \(0.342021\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 1.03528i 0.0661414i
\(246\) 0 0
\(247\) −2.27362 −0.144667
\(248\) 0 0
\(249\) 1.46410 0.0927837
\(250\) 0 0
\(251\) − 13.4641i − 0.849847i −0.905229 0.424923i \(-0.860301\pi\)
0.905229 0.424923i \(-0.139699\pi\)
\(252\) 0 0
\(253\) 15.4548i 0.971636i
\(254\) 0 0
\(255\) −3.58630 −0.224583
\(256\) 0 0
\(257\) 18.7846 1.17175 0.585876 0.810401i \(-0.300751\pi\)
0.585876 + 0.810401i \(0.300751\pi\)
\(258\) 0 0
\(259\) 22.3923i 1.39139i
\(260\) 0 0
\(261\) − 5.93426i − 0.367321i
\(262\) 0 0
\(263\) 22.6274 1.39527 0.697633 0.716455i \(-0.254237\pi\)
0.697633 + 0.716455i \(0.254237\pi\)
\(264\) 0 0
\(265\) −9.85641 −0.605474
\(266\) 0 0
\(267\) 8.92820i 0.546397i
\(268\) 0 0
\(269\) − 15.1774i − 0.925383i −0.886519 0.462692i \(-0.846884\pi\)
0.886519 0.462692i \(-0.153116\pi\)
\(270\) 0 0
\(271\) 8.10634 0.492425 0.246213 0.969216i \(-0.420814\pi\)
0.246213 + 0.969216i \(0.420814\pi\)
\(272\) 0 0
\(273\) 10.3923 0.628971
\(274\) 0 0
\(275\) 21.4641i 1.29433i
\(276\) 0 0
\(277\) − 26.6670i − 1.60226i −0.598488 0.801132i \(-0.704231\pi\)
0.598488 0.801132i \(-0.295769\pi\)
\(278\) 0 0
\(279\) −7.34847 −0.439941
\(280\) 0 0
\(281\) 12.9282 0.771232 0.385616 0.922659i \(-0.373989\pi\)
0.385616 + 0.922659i \(0.373989\pi\)
\(282\) 0 0
\(283\) − 17.8564i − 1.06145i −0.847543 0.530727i \(-0.821919\pi\)
0.847543 0.530727i \(-0.178081\pi\)
\(284\) 0 0
\(285\) 0.554803i 0.0328637i
\(286\) 0 0
\(287\) 28.0812 1.65758
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) − 14.9282i − 0.875107i
\(292\) 0 0
\(293\) − 13.6617i − 0.798123i −0.916924 0.399061i \(-0.869336\pi\)
0.916924 0.399061i \(-0.130664\pi\)
\(294\) 0 0
\(295\) −14.3452 −0.835210
\(296\) 0 0
\(297\) −5.46410 −0.317059
\(298\) 0 0
\(299\) − 12.0000i − 0.693978i
\(300\) 0 0
\(301\) − 8.48528i − 0.489083i
\(302\) 0 0
\(303\) 0.277401 0.0159363
\(304\) 0 0
\(305\) −9.46410 −0.541913
\(306\) 0 0
\(307\) − 9.07180i − 0.517755i −0.965910 0.258877i \(-0.916647\pi\)
0.965910 0.258877i \(-0.0833526\pi\)
\(308\) 0 0
\(309\) 8.10634i 0.461154i
\(310\) 0 0
\(311\) 12.8295 0.727492 0.363746 0.931498i \(-0.381498\pi\)
0.363746 + 0.931498i \(0.381498\pi\)
\(312\) 0 0
\(313\) 23.8564 1.34844 0.674222 0.738529i \(-0.264479\pi\)
0.674222 + 0.738529i \(0.264479\pi\)
\(314\) 0 0
\(315\) − 2.53590i − 0.142882i
\(316\) 0 0
\(317\) − 1.79315i − 0.100713i −0.998731 0.0503567i \(-0.983964\pi\)
0.998731 0.0503567i \(-0.0160358\pi\)
\(318\) 0 0
\(319\) −32.4254 −1.81547
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 1.85641i − 0.103293i
\(324\) 0 0
\(325\) − 16.6660i − 0.924461i
\(326\) 0 0
\(327\) −1.41421 −0.0782062
\(328\) 0 0
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) 14.9282i 0.820528i 0.911967 + 0.410264i \(0.134563\pi\)
−0.911967 + 0.410264i \(0.865437\pi\)
\(332\) 0 0
\(333\) 9.14162i 0.500958i
\(334\) 0 0
\(335\) −1.10961 −0.0606242
\(336\) 0 0
\(337\) −17.8564 −0.972700 −0.486350 0.873764i \(-0.661672\pi\)
−0.486350 + 0.873764i \(0.661672\pi\)
\(338\) 0 0
\(339\) − 18.0000i − 0.977626i
\(340\) 0 0
\(341\) 40.1528i 2.17440i
\(342\) 0 0
\(343\) 19.5959 1.05808
\(344\) 0 0
\(345\) −2.92820 −0.157649
\(346\) 0 0
\(347\) 9.46410i 0.508060i 0.967196 + 0.254030i \(0.0817561\pi\)
−0.967196 + 0.254030i \(0.918244\pi\)
\(348\) 0 0
\(349\) 1.96902i 0.105399i 0.998610 + 0.0526995i \(0.0167826\pi\)
−0.998610 + 0.0526995i \(0.983217\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) 0 0
\(355\) − 16.7846i − 0.890835i
\(356\) 0 0
\(357\) 8.48528i 0.449089i
\(358\) 0 0
\(359\) −30.1518 −1.59135 −0.795674 0.605725i \(-0.792883\pi\)
−0.795674 + 0.605725i \(0.792883\pi\)
\(360\) 0 0
\(361\) 18.7128 0.984885
\(362\) 0 0
\(363\) 18.8564i 0.989705i
\(364\) 0 0
\(365\) 4.14110i 0.216755i
\(366\) 0 0
\(367\) −27.7023 −1.44605 −0.723023 0.690824i \(-0.757248\pi\)
−0.723023 + 0.690824i \(0.757248\pi\)
\(368\) 0 0
\(369\) 11.4641 0.596797
\(370\) 0 0
\(371\) 23.3205i 1.21074i
\(372\) 0 0
\(373\) 16.1112i 0.834204i 0.908860 + 0.417102i \(0.136954\pi\)
−0.908860 + 0.417102i \(0.863046\pi\)
\(374\) 0 0
\(375\) −9.24316 −0.477315
\(376\) 0 0
\(377\) 25.1769 1.29668
\(378\) 0 0
\(379\) − 21.3205i − 1.09516i −0.836753 0.547580i \(-0.815549\pi\)
0.836753 0.547580i \(-0.184451\pi\)
\(380\) 0 0
\(381\) 19.4201i 0.994919i
\(382\) 0 0
\(383\) 0.554803 0.0283491 0.0141746 0.999900i \(-0.495488\pi\)
0.0141746 + 0.999900i \(0.495488\pi\)
\(384\) 0 0
\(385\) −13.8564 −0.706188
\(386\) 0 0
\(387\) − 3.46410i − 0.176090i
\(388\) 0 0
\(389\) 4.62158i 0.234323i 0.993113 + 0.117162i \(0.0373796\pi\)
−0.993113 + 0.117162i \(0.962620\pi\)
\(390\) 0 0
\(391\) 9.79796 0.495504
\(392\) 0 0
\(393\) 1.07180 0.0540650
\(394\) 0 0
\(395\) 2.53590i 0.127595i
\(396\) 0 0
\(397\) − 25.9091i − 1.30034i −0.759788 0.650171i \(-0.774697\pi\)
0.759788 0.650171i \(-0.225303\pi\)
\(398\) 0 0
\(399\) 1.31268 0.0657161
\(400\) 0 0
\(401\) 2.39230 0.119466 0.0597330 0.998214i \(-0.480975\pi\)
0.0597330 + 0.998214i \(0.480975\pi\)
\(402\) 0 0
\(403\) − 31.1769i − 1.55303i
\(404\) 0 0
\(405\) − 1.03528i − 0.0514433i
\(406\) 0 0
\(407\) 49.9507 2.47597
\(408\) 0 0
\(409\) 22.9282 1.13373 0.566863 0.823812i \(-0.308157\pi\)
0.566863 + 0.823812i \(0.308157\pi\)
\(410\) 0 0
\(411\) − 6.39230i − 0.315309i
\(412\) 0 0
\(413\) 33.9411i 1.67013i
\(414\) 0 0
\(415\) 1.51575 0.0744052
\(416\) 0 0
\(417\) −6.92820 −0.339276
\(418\) 0 0
\(419\) 20.3923i 0.996229i 0.867111 + 0.498115i \(0.165974\pi\)
−0.867111 + 0.498115i \(0.834026\pi\)
\(420\) 0 0
\(421\) 0.101536i 0.00494856i 0.999997 + 0.00247428i \(0.000787589\pi\)
−0.999997 + 0.00247428i \(0.999212\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 13.6077 0.660070
\(426\) 0 0
\(427\) 22.3923i 1.08364i
\(428\) 0 0
\(429\) − 23.1822i − 1.11925i
\(430\) 0 0
\(431\) −23.9401 −1.15315 −0.576577 0.817043i \(-0.695612\pi\)
−0.576577 + 0.817043i \(0.695612\pi\)
\(432\) 0 0
\(433\) −2.14359 −0.103015 −0.0515073 0.998673i \(-0.516403\pi\)
−0.0515073 + 0.998673i \(0.516403\pi\)
\(434\) 0 0
\(435\) − 6.14359i − 0.294563i
\(436\) 0 0
\(437\) − 1.51575i − 0.0725081i
\(438\) 0 0
\(439\) −28.4601 −1.35833 −0.679164 0.733986i \(-0.737658\pi\)
−0.679164 + 0.733986i \(0.737658\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) − 0.679492i − 0.0322836i −0.999870 0.0161418i \(-0.994862\pi\)
0.999870 0.0161418i \(-0.00513832\pi\)
\(444\) 0 0
\(445\) 9.24316i 0.438168i
\(446\) 0 0
\(447\) −20.0764 −0.949581
\(448\) 0 0
\(449\) 14.3923 0.679215 0.339607 0.940567i \(-0.389706\pi\)
0.339607 + 0.940567i \(0.389706\pi\)
\(450\) 0 0
\(451\) − 62.6410i − 2.94965i
\(452\) 0 0
\(453\) 17.9043i 0.841218i
\(454\) 0 0
\(455\) 10.7589 0.504385
\(456\) 0 0
\(457\) −20.7846 −0.972263 −0.486132 0.873886i \(-0.661592\pi\)
−0.486132 + 0.873886i \(0.661592\pi\)
\(458\) 0 0
\(459\) 3.46410i 0.161690i
\(460\) 0 0
\(461\) 22.7017i 1.05733i 0.848832 + 0.528663i \(0.177306\pi\)
−0.848832 + 0.528663i \(0.822694\pi\)
\(462\) 0 0
\(463\) 4.72311 0.219502 0.109751 0.993959i \(-0.464995\pi\)
0.109751 + 0.993959i \(0.464995\pi\)
\(464\) 0 0
\(465\) −7.60770 −0.352798
\(466\) 0 0
\(467\) − 18.2487i − 0.844450i −0.906491 0.422225i \(-0.861249\pi\)
0.906491 0.422225i \(-0.138751\pi\)
\(468\) 0 0
\(469\) 2.62536i 0.121228i
\(470\) 0 0
\(471\) −11.9700 −0.551551
\(472\) 0 0
\(473\) −18.9282 −0.870320
\(474\) 0 0
\(475\) − 2.10512i − 0.0965894i
\(476\) 0 0
\(477\) 9.52056i 0.435916i
\(478\) 0 0
\(479\) 38.8401 1.77465 0.887325 0.461145i \(-0.152561\pi\)
0.887325 + 0.461145i \(0.152561\pi\)
\(480\) 0 0
\(481\) −38.7846 −1.76843
\(482\) 0 0
\(483\) 6.92820i 0.315244i
\(484\) 0 0
\(485\) − 15.4548i − 0.701767i
\(486\) 0 0
\(487\) −22.8033 −1.03332 −0.516658 0.856192i \(-0.672824\pi\)
−0.516658 + 0.856192i \(0.672824\pi\)
\(488\) 0 0
\(489\) −11.4641 −0.518425
\(490\) 0 0
\(491\) − 21.8564i − 0.986366i −0.869926 0.493183i \(-0.835833\pi\)
0.869926 0.493183i \(-0.164167\pi\)
\(492\) 0 0
\(493\) 20.5569i 0.925835i
\(494\) 0 0
\(495\) −5.65685 −0.254257
\(496\) 0 0
\(497\) −39.7128 −1.78136
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) − 2.07055i − 0.0925055i
\(502\) 0 0
\(503\) 33.1833 1.47957 0.739784 0.672844i \(-0.234928\pi\)
0.739784 + 0.672844i \(0.234928\pi\)
\(504\) 0 0
\(505\) 0.287187 0.0127797
\(506\) 0 0
\(507\) 5.00000i 0.222058i
\(508\) 0 0
\(509\) − 27.0459i − 1.19879i −0.800454 0.599395i \(-0.795408\pi\)
0.800454 0.599395i \(-0.204592\pi\)
\(510\) 0 0
\(511\) 9.79796 0.433436
\(512\) 0 0
\(513\) 0.535898 0.0236605
\(514\) 0 0
\(515\) 8.39230i 0.369809i
\(516\) 0 0
\(517\) 15.4548i 0.679702i
\(518\) 0 0
\(519\) 2.55103 0.111978
\(520\) 0 0
\(521\) −13.3205 −0.583582 −0.291791 0.956482i \(-0.594251\pi\)
−0.291791 + 0.956482i \(0.594251\pi\)
\(522\) 0 0
\(523\) 24.5359i 1.07288i 0.843938 + 0.536440i \(0.180231\pi\)
−0.843938 + 0.536440i \(0.819769\pi\)
\(524\) 0 0
\(525\) 9.62209i 0.419943i
\(526\) 0 0
\(527\) 25.4558 1.10887
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 13.8564i 0.601317i
\(532\) 0 0
\(533\) 48.6381i 2.10675i
\(534\) 0 0
\(535\) −4.14110 −0.179036
\(536\) 0 0
\(537\) 5.07180 0.218864
\(538\) 0 0
\(539\) − 5.46410i − 0.235356i
\(540\) 0 0
\(541\) 8.58682i 0.369176i 0.982816 + 0.184588i \(0.0590951\pi\)
−0.982816 + 0.184588i \(0.940905\pi\)
\(542\) 0 0
\(543\) 7.07107 0.303449
\(544\) 0 0
\(545\) −1.46410 −0.0627152
\(546\) 0 0
\(547\) 39.1769i 1.67508i 0.546373 + 0.837542i \(0.316008\pi\)
−0.546373 + 0.837542i \(0.683992\pi\)
\(548\) 0 0
\(549\) 9.14162i 0.390155i
\(550\) 0 0
\(551\) 3.18016 0.135479
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) 0 0
\(555\) 9.46410i 0.401729i
\(556\) 0 0
\(557\) 42.7038i 1.80942i 0.426030 + 0.904709i \(0.359912\pi\)
−0.426030 + 0.904709i \(0.640088\pi\)
\(558\) 0 0
\(559\) 14.6969 0.621614
\(560\) 0 0
\(561\) 18.9282 0.799149
\(562\) 0 0
\(563\) 5.46410i 0.230284i 0.993349 + 0.115142i \(0.0367324\pi\)
−0.993349 + 0.115142i \(0.963268\pi\)
\(564\) 0 0
\(565\) − 18.6350i − 0.783979i
\(566\) 0 0
\(567\) −2.44949 −0.102869
\(568\) 0 0
\(569\) −3.46410 −0.145223 −0.0726113 0.997360i \(-0.523133\pi\)
−0.0726113 + 0.997360i \(0.523133\pi\)
\(570\) 0 0
\(571\) 23.7128i 0.992350i 0.868222 + 0.496175i \(0.165263\pi\)
−0.868222 + 0.496175i \(0.834737\pi\)
\(572\) 0 0
\(573\) 2.07055i 0.0864986i
\(574\) 0 0
\(575\) 11.1106 0.463346
\(576\) 0 0
\(577\) 5.85641 0.243805 0.121903 0.992542i \(-0.461100\pi\)
0.121903 + 0.992542i \(0.461100\pi\)
\(578\) 0 0
\(579\) − 6.00000i − 0.249351i
\(580\) 0 0
\(581\) − 3.58630i − 0.148785i
\(582\) 0 0
\(583\) 52.0213 2.15450
\(584\) 0 0
\(585\) 4.39230 0.181599
\(586\) 0 0
\(587\) − 29.0718i − 1.19992i −0.800029 0.599961i \(-0.795183\pi\)
0.800029 0.599961i \(-0.204817\pi\)
\(588\) 0 0
\(589\) − 3.93803i − 0.162264i
\(590\) 0 0
\(591\) 5.93426 0.244103
\(592\) 0 0
\(593\) −17.7128 −0.727378 −0.363689 0.931520i \(-0.618483\pi\)
−0.363689 + 0.931520i \(0.618483\pi\)
\(594\) 0 0
\(595\) 8.78461i 0.360134i
\(596\) 0 0
\(597\) − 3.96524i − 0.162286i
\(598\) 0 0
\(599\) −12.6264 −0.515900 −0.257950 0.966158i \(-0.583047\pi\)
−0.257950 + 0.966158i \(0.583047\pi\)
\(600\) 0 0
\(601\) 35.7128 1.45676 0.728378 0.685176i \(-0.240275\pi\)
0.728378 + 0.685176i \(0.240275\pi\)
\(602\) 0 0
\(603\) 1.07180i 0.0436469i
\(604\) 0 0
\(605\) 19.5216i 0.793665i
\(606\) 0 0
\(607\) −31.8434 −1.29248 −0.646241 0.763133i \(-0.723660\pi\)
−0.646241 + 0.763133i \(0.723660\pi\)
\(608\) 0 0
\(609\) −14.5359 −0.589024
\(610\) 0 0
\(611\) − 12.0000i − 0.485468i
\(612\) 0 0
\(613\) 42.8797i 1.73189i 0.500136 + 0.865947i \(0.333283\pi\)
−0.500136 + 0.865947i \(0.666717\pi\)
\(614\) 0 0
\(615\) 11.8685 0.478585
\(616\) 0 0
\(617\) 12.1436 0.488883 0.244441 0.969664i \(-0.421395\pi\)
0.244441 + 0.969664i \(0.421395\pi\)
\(618\) 0 0
\(619\) − 12.0000i − 0.482321i −0.970485 0.241160i \(-0.922472\pi\)
0.970485 0.241160i \(-0.0775280\pi\)
\(620\) 0 0
\(621\) 2.82843i 0.113501i
\(622\) 0 0
\(623\) 21.8695 0.876185
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 0 0
\(627\) − 2.92820i − 0.116941i
\(628\) 0 0
\(629\) − 31.6675i − 1.26267i
\(630\) 0 0
\(631\) 0.933740 0.0371716 0.0185858 0.999827i \(-0.494084\pi\)
0.0185858 + 0.999827i \(0.494084\pi\)
\(632\) 0 0
\(633\) 6.92820 0.275371
\(634\) 0 0
\(635\) 20.1051i 0.797847i
\(636\) 0 0
\(637\) 4.24264i 0.168100i
\(638\) 0 0
\(639\) −16.2127 −0.641364
\(640\) 0 0
\(641\) −46.3923 −1.83239 −0.916193 0.400737i \(-0.868754\pi\)
−0.916193 + 0.400737i \(0.868754\pi\)
\(642\) 0 0
\(643\) 35.4641i 1.39857i 0.714844 + 0.699284i \(0.246498\pi\)
−0.714844 + 0.699284i \(0.753502\pi\)
\(644\) 0 0
\(645\) − 3.58630i − 0.141210i
\(646\) 0 0
\(647\) 21.3147 0.837969 0.418984 0.907993i \(-0.362386\pi\)
0.418984 + 0.907993i \(0.362386\pi\)
\(648\) 0 0
\(649\) 75.7128 2.97199
\(650\) 0 0
\(651\) 18.0000i 0.705476i
\(652\) 0 0
\(653\) 0.480473i 0.0188024i 0.999956 + 0.00940119i \(0.00299254\pi\)
−0.999956 + 0.00940119i \(0.997007\pi\)
\(654\) 0 0
\(655\) 1.10961 0.0433559
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 24.0000i 0.934907i 0.884018 + 0.467454i \(0.154829\pi\)
−0.884018 + 0.467454i \(0.845171\pi\)
\(660\) 0 0
\(661\) 35.9101i 1.39674i 0.715736 + 0.698371i \(0.246092\pi\)
−0.715736 + 0.698371i \(0.753908\pi\)
\(662\) 0 0
\(663\) −14.6969 −0.570782
\(664\) 0 0
\(665\) 1.35898 0.0526991
\(666\) 0 0
\(667\) 16.7846i 0.649903i
\(668\) 0 0
\(669\) − 19.4201i − 0.750823i
\(670\) 0 0
\(671\) 49.9507 1.92833
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) 3.92820i 0.151197i
\(676\) 0 0
\(677\) 27.8038i 1.06859i 0.845299 + 0.534293i \(0.179422\pi\)
−0.845299 + 0.534293i \(0.820578\pi\)
\(678\) 0 0
\(679\) −36.5665 −1.40329
\(680\) 0 0
\(681\) 23.3205 0.893644
\(682\) 0 0
\(683\) − 37.1769i − 1.42254i −0.702921 0.711268i \(-0.748121\pi\)
0.702921 0.711268i \(-0.251879\pi\)
\(684\) 0 0
\(685\) − 6.61780i − 0.252853i
\(686\) 0 0
\(687\) 2.72689 0.104037
\(688\) 0 0
\(689\) −40.3923 −1.53882
\(690\) 0 0
\(691\) − 5.60770i − 0.213327i −0.994295 0.106663i \(-0.965983\pi\)
0.994295 0.106663i \(-0.0340167\pi\)
\(692\) 0 0
\(693\) 13.3843i 0.508426i
\(694\) 0 0
\(695\) −7.17260 −0.272072
\(696\) 0 0
\(697\) −39.7128 −1.50423
\(698\) 0 0
\(699\) − 0.928203i − 0.0351079i
\(700\) 0 0
\(701\) − 29.1165i − 1.09971i −0.835259 0.549857i \(-0.814682\pi\)
0.835259 0.549857i \(-0.185318\pi\)
\(702\) 0 0
\(703\) −4.89898 −0.184769
\(704\) 0 0
\(705\) −2.92820 −0.110283
\(706\) 0 0
\(707\) − 0.679492i − 0.0255549i
\(708\) 0 0
\(709\) 29.2923i 1.10010i 0.835133 + 0.550048i \(0.185391\pi\)
−0.835133 + 0.550048i \(0.814609\pi\)
\(710\) 0 0
\(711\) 2.44949 0.0918630
\(712\) 0 0
\(713\) 20.7846 0.778390
\(714\) 0 0
\(715\) − 24.0000i − 0.897549i
\(716\) 0 0
\(717\) − 2.07055i − 0.0773262i
\(718\) 0 0
\(719\) 31.6675 1.18100 0.590499 0.807038i \(-0.298931\pi\)
0.590499 + 0.807038i \(0.298931\pi\)
\(720\) 0 0
\(721\) 19.8564 0.739491
\(722\) 0 0
\(723\) − 14.7846i − 0.549846i
\(724\) 0 0
\(725\) 23.3110i 0.865747i
\(726\) 0 0
\(727\) −3.20736 −0.118955 −0.0594773 0.998230i \(-0.518943\pi\)
−0.0594773 + 0.998230i \(0.518943\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) − 7.07107i − 0.261176i −0.991437 0.130588i \(-0.958314\pi\)
0.991437 0.130588i \(-0.0416865\pi\)
\(734\) 0 0
\(735\) 1.03528 0.0381867
\(736\) 0 0
\(737\) 5.85641 0.215724
\(738\) 0 0
\(739\) − 36.0000i − 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) 2.27362i 0.0835237i
\(742\) 0 0
\(743\) −21.6665 −0.794866 −0.397433 0.917631i \(-0.630099\pi\)
−0.397433 + 0.917631i \(0.630099\pi\)
\(744\) 0 0
\(745\) −20.7846 −0.761489
\(746\) 0 0
\(747\) − 1.46410i − 0.0535687i
\(748\) 0 0
\(749\) 9.79796i 0.358010i
\(750\) 0 0
\(751\) −21.6937 −0.791614 −0.395807 0.918334i \(-0.629535\pi\)
−0.395807 + 0.918334i \(0.629535\pi\)
\(752\) 0 0
\(753\) −13.4641 −0.490659
\(754\) 0 0
\(755\) 18.5359i 0.674590i
\(756\) 0 0
\(757\) 25.1512i 0.914137i 0.889431 + 0.457069i \(0.151101\pi\)
−0.889431 + 0.457069i \(0.848899\pi\)
\(758\) 0 0
\(759\) 15.4548 0.560974
\(760\) 0 0
\(761\) 3.75129 0.135984 0.0679921 0.997686i \(-0.478341\pi\)
0.0679921 + 0.997686i \(0.478341\pi\)
\(762\) 0 0
\(763\) 3.46410i 0.125409i
\(764\) 0 0
\(765\) 3.58630i 0.129663i
\(766\) 0 0
\(767\) −58.7878 −2.12270
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) − 18.7846i − 0.676511i
\(772\) 0 0
\(773\) 4.82465i 0.173531i 0.996229 + 0.0867653i \(0.0276530\pi\)
−0.996229 + 0.0867653i \(0.972347\pi\)
\(774\) 0 0
\(775\) 28.8663 1.03691
\(776\) 0 0
\(777\) 22.3923 0.803319
\(778\) 0 0
\(779\) 6.14359i 0.220117i
\(780\) 0 0
\(781\) 88.5878i 3.16992i
\(782\) 0 0
\(783\) −5.93426 −0.212073
\(784\) 0 0
\(785\) −12.3923 −0.442300
\(786\) 0 0
\(787\) 16.5359i 0.589441i 0.955583 + 0.294721i \(0.0952266\pi\)
−0.955583 + 0.294721i \(0.904773\pi\)
\(788\) 0 0
\(789\) − 22.6274i − 0.805557i
\(790\) 0 0
\(791\) −44.0908 −1.56769
\(792\) 0 0
\(793\) −38.7846 −1.37728
\(794\) 0 0
\(795\) 9.85641i 0.349571i
\(796\) 0 0
\(797\) 41.7429i 1.47861i 0.673372 + 0.739304i \(0.264845\pi\)
−0.673372 + 0.739304i \(0.735155\pi\)
\(798\) 0 0
\(799\) 9.79796 0.346627
\(800\) 0 0
\(801\) 8.92820 0.315463
\(802\) 0 0
\(803\) − 21.8564i − 0.771296i
\(804\) 0 0
\(805\) 7.17260i 0.252801i
\(806\) 0 0
\(807\) −15.1774 −0.534270
\(808\) 0 0
\(809\) 41.3205 1.45275 0.726376 0.687298i \(-0.241203\pi\)
0.726376 + 0.687298i \(0.241203\pi\)
\(810\) 0 0
\(811\) 33.6077i 1.18013i 0.807357 + 0.590063i \(0.200897\pi\)
−0.807357 + 0.590063i \(0.799103\pi\)
\(812\) 0 0
\(813\) − 8.10634i − 0.284302i
\(814\) 0 0
\(815\) −11.8685 −0.415736
\(816\) 0 0
\(817\) 1.85641 0.0649474
\(818\) 0 0
\(819\) − 10.3923i − 0.363137i
\(820\) 0 0
\(821\) − 39.8754i − 1.39166i −0.718206 0.695830i \(-0.755037\pi\)
0.718206 0.695830i \(-0.244963\pi\)
\(822\) 0 0
\(823\) −43.9149 −1.53078 −0.765389 0.643567i \(-0.777454\pi\)
−0.765389 + 0.643567i \(0.777454\pi\)
\(824\) 0 0
\(825\) 21.4641 0.747284
\(826\) 0 0
\(827\) 3.21539i 0.111810i 0.998436 + 0.0559050i \(0.0178044\pi\)
−0.998436 + 0.0559050i \(0.982196\pi\)
\(828\) 0 0
\(829\) 40.8091i 1.41736i 0.705530 + 0.708680i \(0.250709\pi\)
−0.705530 + 0.708680i \(0.749291\pi\)
\(830\) 0 0
\(831\) −26.6670 −0.925067
\(832\) 0 0
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) − 2.14359i − 0.0741821i
\(836\) 0 0
\(837\) 7.34847i 0.254000i
\(838\) 0 0
\(839\) 3.38323 0.116802 0.0584010 0.998293i \(-0.481400\pi\)
0.0584010 + 0.998293i \(0.481400\pi\)
\(840\) 0 0
\(841\) −6.21539 −0.214324
\(842\) 0 0
\(843\) − 12.9282i − 0.445271i
\(844\) 0 0
\(845\) 5.17638i 0.178073i
\(846\) 0 0
\(847\) 46.1886 1.58706
\(848\) 0 0
\(849\) −17.8564 −0.612830
\(850\) 0 0
\(851\) − 25.8564i − 0.886346i
\(852\) 0 0
\(853\) − 21.5649i − 0.738369i −0.929356 0.369185i \(-0.879637\pi\)
0.929356 0.369185i \(-0.120363\pi\)
\(854\) 0 0
\(855\) 0.554803 0.0189739
\(856\) 0 0
\(857\) 19.1769 0.655071 0.327535 0.944839i \(-0.393782\pi\)
0.327535 + 0.944839i \(0.393782\pi\)
\(858\) 0 0
\(859\) 16.2487i 0.554399i 0.960812 + 0.277199i \(0.0894063\pi\)
−0.960812 + 0.277199i \(0.910594\pi\)
\(860\) 0 0
\(861\) − 28.0812i − 0.957005i
\(862\) 0 0
\(863\) −31.4644 −1.07106 −0.535531 0.844516i \(-0.679888\pi\)
−0.535531 + 0.844516i \(0.679888\pi\)
\(864\) 0 0
\(865\) 2.64102 0.0897972
\(866\) 0 0
\(867\) 5.00000i 0.169809i
\(868\) 0 0
\(869\) − 13.3843i − 0.454030i
\(870\) 0 0
\(871\) −4.54725 −0.154078
\(872\) 0 0
\(873\) −14.9282 −0.505243
\(874\) 0 0
\(875\) 22.6410i 0.765406i
\(876\) 0 0
\(877\) − 32.8786i − 1.11023i −0.831773 0.555116i \(-0.812674\pi\)
0.831773 0.555116i \(-0.187326\pi\)
\(878\) 0 0
\(879\) −13.6617 −0.460796
\(880\) 0 0
\(881\) 32.6410 1.09970 0.549852 0.835262i \(-0.314684\pi\)
0.549852 + 0.835262i \(0.314684\pi\)
\(882\) 0 0
\(883\) 26.3923i 0.888172i 0.895984 + 0.444086i \(0.146471\pi\)
−0.895984 + 0.444086i \(0.853529\pi\)
\(884\) 0 0
\(885\) 14.3452i 0.482209i
\(886\) 0 0
\(887\) 2.62536 0.0881508 0.0440754 0.999028i \(-0.485966\pi\)
0.0440754 + 0.999028i \(0.485966\pi\)
\(888\) 0 0
\(889\) 47.5692 1.59542
\(890\) 0 0
\(891\) 5.46410i 0.183054i
\(892\) 0 0
\(893\) − 1.51575i − 0.0507226i
\(894\) 0 0
\(895\) 5.25071 0.175512
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 43.6077i 1.45440i
\(900\) 0 0
\(901\) − 32.9802i − 1.09873i
\(902\) 0 0
\(903\) −8.48528 −0.282372
\(904\) 0 0
\(905\) 7.32051 0.243342
\(906\) 0 0
\(907\) − 21.6077i − 0.717472i −0.933439 0.358736i \(-0.883208\pi\)
0.933439 0.358736i \(-0.116792\pi\)
\(908\) 0 0
\(909\) − 0.277401i − 0.00920082i
\(910\) 0 0
\(911\) −51.4665 −1.70516 −0.852580 0.522596i \(-0.824964\pi\)
−0.852580 + 0.522596i \(0.824964\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) 9.46410i 0.312874i
\(916\) 0 0
\(917\) − 2.62536i − 0.0866969i
\(918\) 0 0
\(919\) 44.6728 1.47362 0.736810 0.676100i \(-0.236331\pi\)
0.736810 + 0.676100i \(0.236331\pi\)
\(920\) 0 0
\(921\) −9.07180 −0.298926
\(922\) 0 0
\(923\) − 68.7846i − 2.26407i
\(924\) 0 0
\(925\) − 35.9101i − 1.18072i
\(926\) 0 0
\(927\) 8.10634 0.266247
\(928\) 0 0
\(929\) −10.6795 −0.350383 −0.175191 0.984534i \(-0.556054\pi\)
−0.175191 + 0.984534i \(0.556054\pi\)
\(930\) 0 0
\(931\) 0.535898i 0.0175634i
\(932\) 0 0
\(933\) − 12.8295i − 0.420018i
\(934\) 0 0
\(935\) 19.5959 0.640855
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) − 23.8564i − 0.778524i
\(940\) 0 0
\(941\) 11.7942i 0.384479i 0.981348 + 0.192240i \(0.0615751\pi\)
−0.981348 + 0.192240i \(0.938425\pi\)
\(942\) 0 0
\(943\) −32.4254 −1.05592
\(944\) 0 0
\(945\) −2.53590 −0.0824928
\(946\) 0 0
\(947\) − 42.9282i − 1.39498i −0.716595 0.697490i \(-0.754300\pi\)
0.716595 0.697490i \(-0.245700\pi\)
\(948\) 0 0
\(949\) 16.9706i 0.550888i
\(950\) 0 0
\(951\) −1.79315 −0.0581469
\(952\) 0 0
\(953\) −25.6077 −0.829515 −0.414757 0.909932i \(-0.636134\pi\)
−0.414757 + 0.909932i \(0.636134\pi\)
\(954\) 0 0
\(955\) 2.14359i 0.0693651i
\(956\) 0 0
\(957\) 32.4254i 1.04816i
\(958\) 0 0
\(959\) −15.6579 −0.505619
\(960\) 0 0
\(961\) 23.0000 0.741935
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) − 6.21166i − 0.199960i
\(966\) 0 0
\(967\) −34.8749 −1.12150 −0.560750 0.827985i \(-0.689487\pi\)
−0.560750 + 0.827985i \(0.689487\pi\)
\(968\) 0 0
\(969\) −1.85641 −0.0596364
\(970\) 0 0
\(971\) − 26.2487i − 0.842361i −0.906977 0.421181i \(-0.861616\pi\)
0.906977 0.421181i \(-0.138384\pi\)
\(972\) 0 0
\(973\) 16.9706i 0.544051i
\(974\) 0 0
\(975\) −16.6660 −0.533738
\(976\) 0 0
\(977\) 27.1769 0.869467 0.434733 0.900559i \(-0.356843\pi\)
0.434733 + 0.900559i \(0.356843\pi\)
\(978\) 0 0
\(979\) − 48.7846i − 1.55916i
\(980\) 0 0
\(981\) 1.41421i 0.0451524i
\(982\) 0 0
\(983\) 35.0507 1.11794 0.558972 0.829186i \(-0.311196\pi\)
0.558972 + 0.829186i \(0.311196\pi\)
\(984\) 0 0
\(985\) 6.14359 0.195751
\(986\) 0 0
\(987\) 6.92820i 0.220527i
\(988\) 0 0
\(989\) 9.79796i 0.311557i
\(990\) 0 0
\(991\) 12.2474 0.389053 0.194527 0.980897i \(-0.437683\pi\)
0.194527 + 0.980897i \(0.437683\pi\)
\(992\) 0 0
\(993\) 14.9282 0.473732
\(994\) 0 0
\(995\) − 4.10512i − 0.130141i
\(996\) 0 0
\(997\) 7.82894i 0.247945i 0.992286 + 0.123973i \(0.0395635\pi\)
−0.992286 + 0.123973i \(0.960437\pi\)
\(998\) 0 0
\(999\) 9.14162 0.289228
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 3072.2.d.h.1537.2 8
4.3 odd 2 inner 3072.2.d.h.1537.6 8
8.3 odd 2 inner 3072.2.d.h.1537.3 8
8.5 even 2 inner 3072.2.d.h.1537.7 8
16.3 odd 4 3072.2.a.l.1.2 4
16.5 even 4 3072.2.a.l.1.3 4
16.11 odd 4 3072.2.a.r.1.3 4
16.13 even 4 3072.2.a.r.1.2 4
32.3 odd 8 768.2.j.e.193.2 8
32.5 even 8 768.2.j.e.577.4 yes 8
32.11 odd 8 768.2.j.f.577.3 yes 8
32.13 even 8 768.2.j.f.193.1 yes 8
32.19 odd 8 768.2.j.f.193.3 yes 8
32.21 even 8 768.2.j.f.577.1 yes 8
32.27 odd 8 768.2.j.e.577.2 yes 8
32.29 even 8 768.2.j.e.193.4 yes 8
48.5 odd 4 9216.2.a.bi.1.2 4
48.11 even 4 9216.2.a.bc.1.2 4
48.29 odd 4 9216.2.a.bc.1.3 4
48.35 even 4 9216.2.a.bi.1.3 4
96.5 odd 8 2304.2.k.l.577.1 8
96.11 even 8 2304.2.k.e.577.4 8
96.29 odd 8 2304.2.k.l.1729.2 8
96.35 even 8 2304.2.k.l.1729.1 8
96.53 odd 8 2304.2.k.e.577.3 8
96.59 even 8 2304.2.k.l.577.2 8
96.77 odd 8 2304.2.k.e.1729.4 8
96.83 even 8 2304.2.k.e.1729.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.e.193.2 8 32.3 odd 8
768.2.j.e.193.4 yes 8 32.29 even 8
768.2.j.e.577.2 yes 8 32.27 odd 8
768.2.j.e.577.4 yes 8 32.5 even 8
768.2.j.f.193.1 yes 8 32.13 even 8
768.2.j.f.193.3 yes 8 32.19 odd 8
768.2.j.f.577.1 yes 8 32.21 even 8
768.2.j.f.577.3 yes 8 32.11 odd 8
2304.2.k.e.577.3 8 96.53 odd 8
2304.2.k.e.577.4 8 96.11 even 8
2304.2.k.e.1729.3 8 96.83 even 8
2304.2.k.e.1729.4 8 96.77 odd 8
2304.2.k.l.577.1 8 96.5 odd 8
2304.2.k.l.577.2 8 96.59 even 8
2304.2.k.l.1729.1 8 96.35 even 8
2304.2.k.l.1729.2 8 96.29 odd 8
3072.2.a.l.1.2 4 16.3 odd 4
3072.2.a.l.1.3 4 16.5 even 4
3072.2.a.r.1.2 4 16.13 even 4
3072.2.a.r.1.3 4 16.11 odd 4
3072.2.d.h.1537.2 8 1.1 even 1 trivial
3072.2.d.h.1537.3 8 8.3 odd 2 inner
3072.2.d.h.1537.6 8 4.3 odd 2 inner
3072.2.d.h.1537.7 8 8.5 even 2 inner
9216.2.a.bc.1.2 4 48.11 even 4
9216.2.a.bc.1.3 4 48.29 odd 4
9216.2.a.bi.1.2 4 48.5 odd 4
9216.2.a.bi.1.3 4 48.35 even 4