Properties

Label 48.22.c.b.47.11
Level $48$
Weight $22$
Character 48.47
Analytic conductor $134.149$
Analytic rank $0$
Dimension $12$
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] = [48,22,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.c (of order \(2\), degree \(1\), 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: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{68}\cdot 3^{56}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 47.11
Root \(0.500000 - 5.87639e6i\) of defining polynomial
Character \(\chi\) \(=\) 48.47
Dual form 48.22.c.b.47.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(94418.3 - 39313.4i) q^{3} +2.52523e7i q^{5} -1.86298e8i q^{7} +(7.36926e9 - 7.42381e9i) q^{9} +O(q^{10})\) \(q+(94418.3 - 39313.4i) q^{3} +2.52523e7i q^{5} -1.86298e8i q^{7} +(7.36926e9 - 7.42381e9i) q^{9} +7.28845e10 q^{11} +4.38397e11 q^{13} +(9.92756e11 + 2.38428e12i) q^{15} +5.98100e12i q^{17} +4.42697e12i q^{19} +(-7.32401e12 - 1.75899e13i) q^{21} +4.50087e13 q^{23} -1.60843e14 q^{25} +(4.03937e14 - 9.90654e14i) q^{27} +1.46454e15i q^{29} -7.97978e15i q^{31} +(6.88163e15 - 2.86534e15i) q^{33} +4.70445e15 q^{35} -8.35027e15 q^{37} +(4.13927e16 - 1.72349e16i) q^{39} +2.72853e16i q^{41} -1.09024e16i q^{43} +(1.87469e17 + 1.86091e17i) q^{45} -4.14126e17 q^{47} +5.23839e17 q^{49} +(2.35134e17 + 5.64716e17i) q^{51} +1.78175e18i q^{53} +1.84050e18i q^{55} +(1.74040e17 + 4.17987e17i) q^{57} +4.28502e17 q^{59} +9.27850e18 q^{61} +(-1.38304e18 - 1.37288e18i) q^{63} +1.10705e19i q^{65} -2.56309e19i q^{67} +(4.24964e18 - 1.76945e18i) q^{69} +2.34639e19 q^{71} -2.85702e19 q^{73} +(-1.51865e19 + 6.32328e18i) q^{75} -1.35782e19i q^{77} +8.48441e19i q^{79} +(-8.06992e17 - 1.09416e20i) q^{81} -6.81890e19 q^{83} -1.51034e20 q^{85} +(5.75763e19 + 1.38280e20i) q^{87} +8.17387e19i q^{89} -8.16724e19i q^{91} +(-3.13712e20 - 7.53437e20i) q^{93} -1.11791e20 q^{95} +8.73058e19 q^{97} +(5.37105e20 - 5.41081e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6223178268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6223178268 q^{9} - 1558099630680 q^{13} - 467668605656952 q^{21} - 23\!\cdots\!04 q^{25}+ \cdots - 47\!\cdots\!00 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 94418.3 39313.4i 0.923172 0.384386i
\(4\) 0 0
\(5\) 2.52523e7i 1.15642i 0.815887 + 0.578211i \(0.196249\pi\)
−0.815887 + 0.578211i \(0.803751\pi\)
\(6\) 0 0
\(7\) 1.86298e8i 0.249275i −0.992202 0.124637i \(-0.960223\pi\)
0.992202 0.124637i \(-0.0397768\pi\)
\(8\) 0 0
\(9\) 7.36926e9 7.42381e9i 0.704494 0.709710i
\(10\) 0 0
\(11\) 7.28845e10 0.847251 0.423625 0.905837i \(-0.360757\pi\)
0.423625 + 0.905837i \(0.360757\pi\)
\(12\) 0 0
\(13\) 4.38397e11 0.881987 0.440994 0.897510i \(-0.354626\pi\)
0.440994 + 0.897510i \(0.354626\pi\)
\(14\) 0 0
\(15\) 9.92756e11 + 2.38428e12i 0.444513 + 1.06758i
\(16\) 0 0
\(17\) 5.98100e12i 0.719549i 0.933039 + 0.359774i \(0.117146\pi\)
−0.933039 + 0.359774i \(0.882854\pi\)
\(18\) 0 0
\(19\) 4.42697e12i 0.165651i 0.996564 + 0.0828255i \(0.0263944\pi\)
−0.996564 + 0.0828255i \(0.973606\pi\)
\(20\) 0 0
\(21\) −7.32401e12 1.75899e13i −0.0958178 0.230124i
\(22\) 0 0
\(23\) 4.50087e13 0.226545 0.113272 0.993564i \(-0.463867\pi\)
0.113272 + 0.993564i \(0.463867\pi\)
\(24\) 0 0
\(25\) −1.60843e14 −0.337312
\(26\) 0 0
\(27\) 4.03937e14 9.90654e14i 0.377567 0.925982i
\(28\) 0 0
\(29\) 1.46454e15i 0.646433i 0.946325 + 0.323217i \(0.104764\pi\)
−0.946325 + 0.323217i \(0.895236\pi\)
\(30\) 0 0
\(31\) 7.97978e15i 1.74861i −0.485377 0.874305i \(-0.661318\pi\)
0.485377 0.874305i \(-0.338682\pi\)
\(32\) 0 0
\(33\) 6.88163e15 2.86534e15i 0.782159 0.325672i
\(34\) 0 0
\(35\) 4.70445e15 0.288267
\(36\) 0 0
\(37\) −8.35027e15 −0.285484 −0.142742 0.989760i \(-0.545592\pi\)
−0.142742 + 0.989760i \(0.545592\pi\)
\(38\) 0 0
\(39\) 4.13927e16 1.72349e16i 0.814226 0.339024i
\(40\) 0 0
\(41\) 2.72853e16i 0.317467i 0.987321 + 0.158734i \(0.0507411\pi\)
−0.987321 + 0.158734i \(0.949259\pi\)
\(42\) 0 0
\(43\) 1.09024e16i 0.0769313i −0.999260 0.0384656i \(-0.987753\pi\)
0.999260 0.0384656i \(-0.0122470\pi\)
\(44\) 0 0
\(45\) 1.87469e17 + 1.86091e17i 0.820724 + 0.814693i
\(46\) 0 0
\(47\) −4.14126e17 −1.14843 −0.574215 0.818704i \(-0.694693\pi\)
−0.574215 + 0.818704i \(0.694693\pi\)
\(48\) 0 0
\(49\) 5.23839e17 0.937862
\(50\) 0 0
\(51\) 2.35134e17 + 5.64716e17i 0.276585 + 0.664268i
\(52\) 0 0
\(53\) 1.78175e18i 1.39943i 0.714424 + 0.699713i \(0.246689\pi\)
−0.714424 + 0.699713i \(0.753311\pi\)
\(54\) 0 0
\(55\) 1.84050e18i 0.979780i
\(56\) 0 0
\(57\) 1.74040e17 + 4.17987e17i 0.0636740 + 0.152924i
\(58\) 0 0
\(59\) 4.28502e17 0.109146 0.0545729 0.998510i \(-0.482620\pi\)
0.0545729 + 0.998510i \(0.482620\pi\)
\(60\) 0 0
\(61\) 9.27850e18 1.66538 0.832692 0.553736i \(-0.186798\pi\)
0.832692 + 0.553736i \(0.186798\pi\)
\(62\) 0 0
\(63\) −1.38304e18 1.37288e18i −0.176913 0.175613i
\(64\) 0 0
\(65\) 1.10705e19i 1.01995i
\(66\) 0 0
\(67\) 2.56309e19i 1.71782i −0.512124 0.858912i \(-0.671141\pi\)
0.512124 0.858912i \(-0.328859\pi\)
\(68\) 0 0
\(69\) 4.24964e18 1.76945e18i 0.209140 0.0870808i
\(70\) 0 0
\(71\) 2.34639e19 0.855435 0.427718 0.903912i \(-0.359318\pi\)
0.427718 + 0.903912i \(0.359318\pi\)
\(72\) 0 0
\(73\) −2.85702e19 −0.778077 −0.389039 0.921221i \(-0.627193\pi\)
−0.389039 + 0.921221i \(0.627193\pi\)
\(74\) 0 0
\(75\) −1.51865e19 + 6.32328e18i −0.311397 + 0.129658i
\(76\) 0 0
\(77\) 1.35782e19i 0.211198i
\(78\) 0 0
\(79\) 8.48441e19i 1.00818i 0.863652 + 0.504089i \(0.168172\pi\)
−0.863652 + 0.504089i \(0.831828\pi\)
\(80\) 0 0
\(81\) −8.06992e17 1.09416e20i −0.00737525 0.999973i
\(82\) 0 0
\(83\) −6.81890e19 −0.482385 −0.241193 0.970477i \(-0.577539\pi\)
−0.241193 + 0.970477i \(0.577539\pi\)
\(84\) 0 0
\(85\) −1.51034e20 −0.832102
\(86\) 0 0
\(87\) 5.75763e19 + 1.38280e20i 0.248480 + 0.596769i
\(88\) 0 0
\(89\) 8.17387e19i 0.277864i 0.990302 + 0.138932i \(0.0443670\pi\)
−0.990302 + 0.138932i \(0.955633\pi\)
\(90\) 0 0
\(91\) 8.16724e19i 0.219857i
\(92\) 0 0
\(93\) −3.13712e20 7.53437e20i −0.672142 1.61427i
\(94\) 0 0
\(95\) −1.11791e20 −0.191562
\(96\) 0 0
\(97\) 8.73058e19 0.120210 0.0601049 0.998192i \(-0.480856\pi\)
0.0601049 + 0.998192i \(0.480856\pi\)
\(98\) 0 0
\(99\) 5.37105e20 5.41081e20i 0.596884 0.601302i
\(100\) 0 0
\(101\) 1.50948e21i 1.35973i 0.733338 + 0.679864i \(0.237961\pi\)
−0.733338 + 0.679864i \(0.762039\pi\)
\(102\) 0 0
\(103\) 7.56986e19i 0.0555005i −0.999615 0.0277503i \(-0.991166\pi\)
0.999615 0.0277503i \(-0.00883432\pi\)
\(104\) 0 0
\(105\) 4.44186e20 1.84948e20i 0.266120 0.110806i
\(106\) 0 0
\(107\) 2.43960e21 1.19892 0.599459 0.800405i \(-0.295382\pi\)
0.599459 + 0.800405i \(0.295382\pi\)
\(108\) 0 0
\(109\) 2.19375e21 0.887584 0.443792 0.896130i \(-0.353633\pi\)
0.443792 + 0.896130i \(0.353633\pi\)
\(110\) 0 0
\(111\) −7.88418e20 + 3.28278e20i −0.263551 + 0.109736i
\(112\) 0 0
\(113\) 4.05656e21i 1.12418i 0.827077 + 0.562088i \(0.190002\pi\)
−0.827077 + 0.562088i \(0.809998\pi\)
\(114\) 0 0
\(115\) 1.13657e21i 0.261982i
\(116\) 0 0
\(117\) 3.23066e21 3.25458e21i 0.621355 0.625955i
\(118\) 0 0
\(119\) 1.11425e21 0.179365
\(120\) 0 0
\(121\) −2.08810e21 −0.282166
\(122\) 0 0
\(123\) 1.07268e21 + 2.57623e21i 0.122030 + 0.293077i
\(124\) 0 0
\(125\) 7.97959e21i 0.766347i
\(126\) 0 0
\(127\) 2.04934e22i 1.66600i 0.553273 + 0.833000i \(0.313378\pi\)
−0.553273 + 0.833000i \(0.686622\pi\)
\(128\) 0 0
\(129\) −4.28610e20 1.02938e21i −0.0295713 0.0710208i
\(130\) 0 0
\(131\) 2.31046e22 1.35628 0.678140 0.734933i \(-0.262786\pi\)
0.678140 + 0.734933i \(0.262786\pi\)
\(132\) 0 0
\(133\) 8.24736e20 0.0412926
\(134\) 0 0
\(135\) 2.50163e22 + 1.02004e22i 1.07083 + 0.436627i
\(136\) 0 0
\(137\) 4.33031e22i 1.58838i −0.607673 0.794188i \(-0.707897\pi\)
0.607673 0.794188i \(-0.292103\pi\)
\(138\) 0 0
\(139\) 4.70660e22i 1.48269i −0.671121 0.741347i \(-0.734187\pi\)
0.671121 0.741347i \(-0.265813\pi\)
\(140\) 0 0
\(141\) −3.91010e22 + 1.62807e22i −1.06020 + 0.441441i
\(142\) 0 0
\(143\) 3.19523e22 0.747264
\(144\) 0 0
\(145\) −3.69832e22 −0.747550
\(146\) 0 0
\(147\) 4.94600e22 2.05939e22i 0.865808 0.360501i
\(148\) 0 0
\(149\) 6.85592e22i 1.04138i −0.853745 0.520691i \(-0.825674\pi\)
0.853745 0.520691i \(-0.174326\pi\)
\(150\) 0 0
\(151\) 1.04699e23i 1.38257i 0.722583 + 0.691284i \(0.242955\pi\)
−0.722583 + 0.691284i \(0.757045\pi\)
\(152\) 0 0
\(153\) 4.44018e22 + 4.40756e22i 0.510671 + 0.506918i
\(154\) 0 0
\(155\) 2.01508e23 2.02213
\(156\) 0 0
\(157\) 5.06383e22 0.444153 0.222077 0.975029i \(-0.428716\pi\)
0.222077 + 0.975029i \(0.428716\pi\)
\(158\) 0 0
\(159\) 7.00467e22 + 1.68230e23i 0.537920 + 1.29191i
\(160\) 0 0
\(161\) 8.38503e21i 0.0564720i
\(162\) 0 0
\(163\) 2.05777e23i 1.21738i 0.793408 + 0.608690i \(0.208305\pi\)
−0.793408 + 0.608690i \(0.791695\pi\)
\(164\) 0 0
\(165\) 7.23565e22 + 1.73777e23i 0.376614 + 0.904506i
\(166\) 0 0
\(167\) 2.91171e23 1.33544 0.667721 0.744412i \(-0.267270\pi\)
0.667721 + 0.744412i \(0.267270\pi\)
\(168\) 0 0
\(169\) −5.48727e22 −0.222099
\(170\) 0 0
\(171\) 3.28650e22 + 3.26235e22i 0.117564 + 0.116700i
\(172\) 0 0
\(173\) 2.35485e23i 0.745553i 0.927921 + 0.372776i \(0.121594\pi\)
−0.927921 + 0.372776i \(0.878406\pi\)
\(174\) 0 0
\(175\) 2.99647e22i 0.0840834i
\(176\) 0 0
\(177\) 4.04584e22 1.68459e22i 0.100760 0.0419541i
\(178\) 0 0
\(179\) 8.51869e23 1.88545 0.942727 0.333565i \(-0.108252\pi\)
0.942727 + 0.333565i \(0.108252\pi\)
\(180\) 0 0
\(181\) 6.73258e23 1.32604 0.663020 0.748601i \(-0.269274\pi\)
0.663020 + 0.748601i \(0.269274\pi\)
\(182\) 0 0
\(183\) 8.76060e23 3.64770e23i 1.53744 0.640151i
\(184\) 0 0
\(185\) 2.10864e23i 0.330140i
\(186\) 0 0
\(187\) 4.35922e23i 0.609638i
\(188\) 0 0
\(189\) −1.84557e23 7.52526e22i −0.230824 0.0941180i
\(190\) 0 0
\(191\) 6.73954e23 0.754710 0.377355 0.926069i \(-0.376834\pi\)
0.377355 + 0.926069i \(0.376834\pi\)
\(192\) 0 0
\(193\) −1.02993e24 −1.03385 −0.516925 0.856031i \(-0.672923\pi\)
−0.516925 + 0.856031i \(0.672923\pi\)
\(194\) 0 0
\(195\) 4.35221e23 + 1.04526e24i 0.392055 + 0.941589i
\(196\) 0 0
\(197\) 5.84549e23i 0.473070i 0.971623 + 0.236535i \(0.0760118\pi\)
−0.971623 + 0.236535i \(0.923988\pi\)
\(198\) 0 0
\(199\) 3.82477e22i 0.0278386i 0.999903 + 0.0139193i \(0.00443080\pi\)
−0.999903 + 0.0139193i \(0.995569\pi\)
\(200\) 0 0
\(201\) −1.00764e24 2.42002e24i −0.660308 1.58585i
\(202\) 0 0
\(203\) 2.72842e23 0.161140
\(204\) 0 0
\(205\) −6.89018e23 −0.367126
\(206\) 0 0
\(207\) 3.31681e23 3.34136e23i 0.159600 0.160781i
\(208\) 0 0
\(209\) 3.22658e23i 0.140348i
\(210\) 0 0
\(211\) 2.45976e24i 0.968116i 0.875036 + 0.484058i \(0.160838\pi\)
−0.875036 + 0.484058i \(0.839162\pi\)
\(212\) 0 0
\(213\) 2.21542e24 9.22446e23i 0.789714 0.328818i
\(214\) 0 0
\(215\) 2.75311e23 0.0889650
\(216\) 0 0
\(217\) −1.48662e24 −0.435885
\(218\) 0 0
\(219\) −2.69755e24 + 1.12319e24i −0.718299 + 0.299082i
\(220\) 0 0
\(221\) 2.62205e24i 0.634633i
\(222\) 0 0
\(223\) 3.13530e24i 0.690365i −0.938536 0.345182i \(-0.887817\pi\)
0.938536 0.345182i \(-0.112183\pi\)
\(224\) 0 0
\(225\) −1.18529e24 + 1.19407e24i −0.237634 + 0.239393i
\(226\) 0 0
\(227\) −5.05206e23 −0.0922991 −0.0461495 0.998935i \(-0.514695\pi\)
−0.0461495 + 0.998935i \(0.514695\pi\)
\(228\) 0 0
\(229\) 4.80507e24 0.800622 0.400311 0.916379i \(-0.368902\pi\)
0.400311 + 0.916379i \(0.368902\pi\)
\(230\) 0 0
\(231\) −5.33807e23 1.28203e24i −0.0811818 0.194972i
\(232\) 0 0
\(233\) 5.01142e24i 0.696184i −0.937460 0.348092i \(-0.886830\pi\)
0.937460 0.348092i \(-0.113170\pi\)
\(234\) 0 0
\(235\) 1.04576e25i 1.32807i
\(236\) 0 0
\(237\) 3.33551e24 + 8.01083e24i 0.387530 + 0.930722i
\(238\) 0 0
\(239\) −9.24013e24 −0.982879 −0.491440 0.870912i \(-0.663529\pi\)
−0.491440 + 0.870912i \(0.663529\pi\)
\(240\) 0 0
\(241\) 5.94660e24 0.579548 0.289774 0.957095i \(-0.406420\pi\)
0.289774 + 0.957095i \(0.406420\pi\)
\(242\) 0 0
\(243\) −4.37771e24 1.02991e25i −0.391184 0.920312i
\(244\) 0 0
\(245\) 1.32282e25i 1.08456i
\(246\) 0 0
\(247\) 1.94077e24i 0.146102i
\(248\) 0 0
\(249\) −6.43828e24 + 2.68074e24i −0.445325 + 0.185422i
\(250\) 0 0
\(251\) −3.06445e25 −1.94885 −0.974424 0.224716i \(-0.927855\pi\)
−0.974424 + 0.224716i \(0.927855\pi\)
\(252\) 0 0
\(253\) 3.28044e24 0.191940
\(254\) 0 0
\(255\) −1.42604e25 + 5.93767e24i −0.768174 + 0.319849i
\(256\) 0 0
\(257\) 3.02169e25i 1.49952i −0.661709 0.749761i \(-0.730169\pi\)
0.661709 0.749761i \(-0.269831\pi\)
\(258\) 0 0
\(259\) 1.55564e24i 0.0711641i
\(260\) 0 0
\(261\) 1.08725e25 + 1.07926e25i 0.458780 + 0.455409i
\(262\) 0 0
\(263\) −3.19177e25 −1.24307 −0.621535 0.783386i \(-0.713491\pi\)
−0.621535 + 0.783386i \(0.713491\pi\)
\(264\) 0 0
\(265\) −4.49933e25 −1.61833
\(266\) 0 0
\(267\) 3.21343e24 + 7.71763e24i 0.106807 + 0.256517i
\(268\) 0 0
\(269\) 4.07028e25i 1.25091i −0.780261 0.625454i \(-0.784914\pi\)
0.780261 0.625454i \(-0.215086\pi\)
\(270\) 0 0
\(271\) 5.29299e24i 0.150496i 0.997165 + 0.0752478i \(0.0239748\pi\)
−0.997165 + 0.0752478i \(0.976025\pi\)
\(272\) 0 0
\(273\) −3.21082e24 7.71137e24i −0.0845101 0.202966i
\(274\) 0 0
\(275\) −1.17230e25 −0.285788
\(276\) 0 0
\(277\) −8.90441e24 −0.201172 −0.100586 0.994928i \(-0.532072\pi\)
−0.100586 + 0.994928i \(0.532072\pi\)
\(278\) 0 0
\(279\) −5.92404e25 5.88051e25i −1.24101 1.23189i
\(280\) 0 0
\(281\) 2.92998e25i 0.569440i 0.958611 + 0.284720i \(0.0919007\pi\)
−0.958611 + 0.284720i \(0.908099\pi\)
\(282\) 0 0
\(283\) 9.56490e25i 1.72553i −0.505603 0.862766i \(-0.668730\pi\)
0.505603 0.862766i \(-0.331270\pi\)
\(284\) 0 0
\(285\) −1.05551e25 + 4.39490e24i −0.176845 + 0.0736340i
\(286\) 0 0
\(287\) 5.08320e24 0.0791366
\(288\) 0 0
\(289\) 3.33196e25 0.482250
\(290\) 0 0
\(291\) 8.24326e24 3.43229e24i 0.110974 0.0462070i
\(292\) 0 0
\(293\) 9.41328e24i 0.117932i 0.998260 + 0.0589659i \(0.0187803\pi\)
−0.998260 + 0.0589659i \(0.981220\pi\)
\(294\) 0 0
\(295\) 1.08207e25i 0.126219i
\(296\) 0 0
\(297\) 2.94408e25 7.22034e25i 0.319894 0.784539i
\(298\) 0 0
\(299\) 1.97317e25 0.199810
\(300\) 0 0
\(301\) −2.03109e24 −0.0191770
\(302\) 0 0
\(303\) 5.93427e25 + 1.42522e26i 0.522661 + 1.25526i
\(304\) 0 0
\(305\) 2.34304e26i 1.92589i
\(306\) 0 0
\(307\) 1.23603e26i 0.948582i 0.880368 + 0.474291i \(0.157296\pi\)
−0.880368 + 0.474291i \(0.842704\pi\)
\(308\) 0 0
\(309\) −2.97597e24 7.14733e24i −0.0213336 0.0512365i
\(310\) 0 0
\(311\) 6.19989e25 0.415336 0.207668 0.978199i \(-0.433413\pi\)
0.207668 + 0.978199i \(0.433413\pi\)
\(312\) 0 0
\(313\) 1.01777e26 0.637433 0.318716 0.947850i \(-0.396748\pi\)
0.318716 + 0.947850i \(0.396748\pi\)
\(314\) 0 0
\(315\) 3.46684e25 3.49250e25i 0.203082 0.204586i
\(316\) 0 0
\(317\) 8.08782e24i 0.0443312i 0.999754 + 0.0221656i \(0.00705611\pi\)
−0.999754 + 0.0221656i \(0.992944\pi\)
\(318\) 0 0
\(319\) 1.06743e26i 0.547691i
\(320\) 0 0
\(321\) 2.30343e26 9.59092e25i 1.10681 0.460848i
\(322\) 0 0
\(323\) −2.64777e25 −0.119194
\(324\) 0 0
\(325\) −7.05130e25 −0.297505
\(326\) 0 0
\(327\) 2.07130e26 8.62439e25i 0.819393 0.341175i
\(328\) 0 0
\(329\) 7.71508e25i 0.286275i
\(330\) 0 0
\(331\) 7.32094e25i 0.254902i 0.991845 + 0.127451i \(0.0406795\pi\)
−0.991845 + 0.127451i \(0.959320\pi\)
\(332\) 0 0
\(333\) −6.15353e25 + 6.19908e25i −0.201122 + 0.202611i
\(334\) 0 0
\(335\) 6.47240e26 1.98653
\(336\) 0 0
\(337\) −5.15782e26 −1.48714 −0.743570 0.668658i \(-0.766869\pi\)
−0.743570 + 0.668658i \(0.766869\pi\)
\(338\) 0 0
\(339\) 1.59477e26 + 3.83013e26i 0.432118 + 1.03781i
\(340\) 0 0
\(341\) 5.81602e26i 1.48151i
\(342\) 0 0
\(343\) 2.01646e26i 0.483060i
\(344\) 0 0
\(345\) 4.46827e25 + 1.07313e26i 0.100702 + 0.241854i
\(346\) 0 0
\(347\) 8.07840e26 1.71343 0.856714 0.515791i \(-0.172502\pi\)
0.856714 + 0.515791i \(0.172502\pi\)
\(348\) 0 0
\(349\) −4.13783e26 −0.826238 −0.413119 0.910677i \(-0.635561\pi\)
−0.413119 + 0.910677i \(0.635561\pi\)
\(350\) 0 0
\(351\) 1.77085e26 4.34300e26i 0.333009 0.816704i
\(352\) 0 0
\(353\) 5.06496e25i 0.0897309i −0.998993 0.0448654i \(-0.985714\pi\)
0.998993 0.0448654i \(-0.0142859\pi\)
\(354\) 0 0
\(355\) 5.92518e26i 0.989244i
\(356\) 0 0
\(357\) 1.05205e26 4.38049e25i 0.165585 0.0689456i
\(358\) 0 0
\(359\) −1.19382e26 −0.177193 −0.0885966 0.996068i \(-0.528238\pi\)
−0.0885966 + 0.996068i \(0.528238\pi\)
\(360\) 0 0
\(361\) 6.94611e26 0.972560
\(362\) 0 0
\(363\) −1.97154e26 + 8.20903e25i −0.260488 + 0.108461i
\(364\) 0 0
\(365\) 7.21463e26i 0.899786i
\(366\) 0 0
\(367\) 2.97480e26i 0.350319i 0.984540 + 0.175159i \(0.0560441\pi\)
−0.984540 + 0.175159i \(0.943956\pi\)
\(368\) 0 0
\(369\) 2.02561e26 + 2.01073e26i 0.225309 + 0.223654i
\(370\) 0 0
\(371\) 3.31936e26 0.348842
\(372\) 0 0
\(373\) −1.26477e27 −1.25623 −0.628113 0.778122i \(-0.716172\pi\)
−0.628113 + 0.778122i \(0.716172\pi\)
\(374\) 0 0
\(375\) 3.13705e26 + 7.53419e26i 0.294573 + 0.707471i
\(376\) 0 0
\(377\) 6.42052e26i 0.570146i
\(378\) 0 0
\(379\) 2.14429e25i 0.0180124i 0.999959 + 0.00900619i \(0.00286680\pi\)
−0.999959 + 0.00900619i \(0.997133\pi\)
\(380\) 0 0
\(381\) 8.05666e26 + 1.93495e27i 0.640387 + 1.53800i
\(382\) 0 0
\(383\) −6.44306e26 −0.484736 −0.242368 0.970184i \(-0.577924\pi\)
−0.242368 + 0.970184i \(0.577924\pi\)
\(384\) 0 0
\(385\) 3.42882e26 0.244234
\(386\) 0 0
\(387\) −8.09373e25 8.03426e25i −0.0545989 0.0541977i
\(388\) 0 0
\(389\) 7.83774e26i 0.500864i −0.968134 0.250432i \(-0.919427\pi\)
0.968134 0.250432i \(-0.0805727\pi\)
\(390\) 0 0
\(391\) 2.69197e26i 0.163010i
\(392\) 0 0
\(393\) 2.18149e27 9.08320e26i 1.25208 0.521336i
\(394\) 0 0
\(395\) −2.14251e27 −1.16588
\(396\) 0 0
\(397\) −4.46679e26 −0.230513 −0.115256 0.993336i \(-0.536769\pi\)
−0.115256 + 0.993336i \(0.536769\pi\)
\(398\) 0 0
\(399\) 7.78701e25 3.24232e25i 0.0381202 0.0158723i
\(400\) 0 0
\(401\) 3.13140e27i 1.45453i 0.686357 + 0.727265i \(0.259209\pi\)
−0.686357 + 0.727265i \(0.740791\pi\)
\(402\) 0 0
\(403\) 3.49831e27i 1.54225i
\(404\) 0 0
\(405\) 2.76301e27 2.03784e25i 1.15639 0.00852890i
\(406\) 0 0
\(407\) −6.08605e26 −0.241877
\(408\) 0 0
\(409\) −3.40894e27 −1.28684 −0.643420 0.765514i \(-0.722485\pi\)
−0.643420 + 0.765514i \(0.722485\pi\)
\(410\) 0 0
\(411\) −1.70239e27 4.08860e27i −0.610550 1.46634i
\(412\) 0 0
\(413\) 7.98290e25i 0.0272073i
\(414\) 0 0
\(415\) 1.72193e27i 0.557841i
\(416\) 0 0
\(417\) −1.85033e27 4.44389e27i −0.569928 1.36878i
\(418\) 0 0
\(419\) −2.37162e27 −0.694700 −0.347350 0.937736i \(-0.612918\pi\)
−0.347350 + 0.937736i \(0.612918\pi\)
\(420\) 0 0
\(421\) −1.84452e27 −0.513949 −0.256975 0.966418i \(-0.582726\pi\)
−0.256975 + 0.966418i \(0.582726\pi\)
\(422\) 0 0
\(423\) −3.05180e27 + 3.07439e27i −0.809063 + 0.815052i
\(424\) 0 0
\(425\) 9.62001e26i 0.242712i
\(426\) 0 0
\(427\) 1.72856e27i 0.415138i
\(428\) 0 0
\(429\) 3.01688e27 1.25616e27i 0.689854 0.287238i
\(430\) 0 0
\(431\) −5.64442e27 −1.22916 −0.614579 0.788855i \(-0.710674\pi\)
−0.614579 + 0.788855i \(0.710674\pi\)
\(432\) 0 0
\(433\) −6.36146e27 −1.31957 −0.659787 0.751452i \(-0.729354\pi\)
−0.659787 + 0.751452i \(0.729354\pi\)
\(434\) 0 0
\(435\) −3.49189e27 + 1.45394e27i −0.690117 + 0.287348i
\(436\) 0 0
\(437\) 1.99252e26i 0.0375274i
\(438\) 0 0
\(439\) 7.32681e27i 1.31534i 0.753307 + 0.657669i \(0.228457\pi\)
−0.753307 + 0.657669i \(0.771543\pi\)
\(440\) 0 0
\(441\) 3.86031e27 3.88888e27i 0.660719 0.665610i
\(442\) 0 0
\(443\) −2.94318e27 −0.480372 −0.240186 0.970727i \(-0.577208\pi\)
−0.240186 + 0.970727i \(0.577208\pi\)
\(444\) 0 0
\(445\) −2.06409e27 −0.321328
\(446\) 0 0
\(447\) −2.69530e27 6.47324e27i −0.400293 0.961375i
\(448\) 0 0
\(449\) 1.05123e27i 0.148974i 0.997222 + 0.0744868i \(0.0237319\pi\)
−0.997222 + 0.0744868i \(0.976268\pi\)
\(450\) 0 0
\(451\) 1.98868e27i 0.268974i
\(452\) 0 0
\(453\) 4.11609e27 + 9.88554e27i 0.531440 + 1.27635i
\(454\) 0 0
\(455\) 2.06242e27 0.254248
\(456\) 0 0
\(457\) −1.20730e28 −1.42132 −0.710662 0.703533i \(-0.751605\pi\)
−0.710662 + 0.703533i \(0.751605\pi\)
\(458\) 0 0
\(459\) 5.92511e27 + 2.41595e27i 0.666289 + 0.271678i
\(460\) 0 0
\(461\) 7.61355e26i 0.0817951i −0.999163 0.0408976i \(-0.986978\pi\)
0.999163 0.0408976i \(-0.0130217\pi\)
\(462\) 0 0
\(463\) 1.71097e28i 1.75647i −0.478227 0.878236i \(-0.658720\pi\)
0.478227 0.878236i \(-0.341280\pi\)
\(464\) 0 0
\(465\) 1.90260e28 7.92197e27i 1.86678 0.777279i
\(466\) 0 0
\(467\) −1.44975e28 −1.35977 −0.679885 0.733319i \(-0.737970\pi\)
−0.679885 + 0.733319i \(0.737970\pi\)
\(468\) 0 0
\(469\) −4.77498e27 −0.428210
\(470\) 0 0
\(471\) 4.78118e27 1.99077e27i 0.410030 0.170726i
\(472\) 0 0
\(473\) 7.94615e26i 0.0651801i
\(474\) 0 0
\(475\) 7.12047e26i 0.0558760i
\(476\) 0 0
\(477\) 1.32274e28 + 1.31302e28i 0.993186 + 0.985888i
\(478\) 0 0
\(479\) 2.47684e28 1.77982 0.889910 0.456137i \(-0.150767\pi\)
0.889910 + 0.456137i \(0.150767\pi\)
\(480\) 0 0
\(481\) −3.66073e27 −0.251794
\(482\) 0 0
\(483\) −3.29644e26 7.91700e26i −0.0217070 0.0521333i
\(484\) 0 0
\(485\) 2.20468e27i 0.139013i
\(486\) 0 0
\(487\) 8.13030e27i 0.490967i 0.969401 + 0.245484i \(0.0789468\pi\)
−0.969401 + 0.245484i \(0.921053\pi\)
\(488\) 0 0
\(489\) 8.08979e27 + 1.94291e28i 0.467944 + 1.12385i
\(490\) 0 0
\(491\) 1.17252e28 0.649778 0.324889 0.945752i \(-0.394673\pi\)
0.324889 + 0.945752i \(0.394673\pi\)
\(492\) 0 0
\(493\) −8.75944e27 −0.465140
\(494\) 0 0
\(495\) 1.36636e28 + 1.35632e28i 0.695359 + 0.690249i
\(496\) 0 0
\(497\) 4.37127e27i 0.213239i
\(498\) 0 0
\(499\) 3.49942e28i 1.63659i 0.574798 + 0.818296i \(0.305081\pi\)
−0.574798 + 0.818296i \(0.694919\pi\)
\(500\) 0 0
\(501\) 2.74918e28 1.14469e28i 1.23284 0.513325i
\(502\) 0 0
\(503\) 2.52525e28 1.08603 0.543013 0.839724i \(-0.317283\pi\)
0.543013 + 0.839724i \(0.317283\pi\)
\(504\) 0 0
\(505\) −3.81178e28 −1.57242
\(506\) 0 0
\(507\) −5.18098e27 + 2.15723e27i −0.205035 + 0.0853717i
\(508\) 0 0
\(509\) 2.18459e28i 0.829531i −0.909928 0.414765i \(-0.863864\pi\)
0.909928 0.414765i \(-0.136136\pi\)
\(510\) 0 0
\(511\) 5.32256e27i 0.193955i
\(512\) 0 0
\(513\) 4.38560e27 + 1.78822e27i 0.153390 + 0.0625444i
\(514\) 0 0
\(515\) 1.91157e27 0.0641820
\(516\) 0 0
\(517\) −3.01834e28 −0.973009
\(518\) 0 0
\(519\) 9.25771e27 + 2.22340e28i 0.286580 + 0.688274i
\(520\) 0 0
\(521\) 2.54801e28i 0.757538i −0.925491 0.378769i \(-0.876347\pi\)
0.925491 0.378769i \(-0.123653\pi\)
\(522\) 0 0
\(523\) 3.85601e28i 1.10121i −0.834766 0.550605i \(-0.814397\pi\)
0.834766 0.550605i \(-0.185603\pi\)
\(524\) 0 0
\(525\) 1.17801e27 + 2.82921e27i 0.0323205 + 0.0776234i
\(526\) 0 0
\(527\) 4.77271e28 1.25821
\(528\) 0 0
\(529\) −3.74458e28 −0.948677
\(530\) 0 0
\(531\) 3.15774e27 3.18112e27i 0.0768926 0.0774618i
\(532\) 0 0
\(533\) 1.19618e28i 0.280002i
\(534\) 0 0
\(535\) 6.16057e28i 1.38646i
\(536\) 0 0
\(537\) 8.04320e28 3.34899e28i 1.74060 0.724743i
\(538\) 0 0
\(539\) 3.81798e28 0.794605
\(540\) 0 0
\(541\) −7.99029e28 −1.59953 −0.799763 0.600316i \(-0.795042\pi\)
−0.799763 + 0.600316i \(0.795042\pi\)
\(542\) 0 0
\(543\) 6.35679e28 2.64681e28i 1.22416 0.509712i
\(544\) 0 0
\(545\) 5.53973e28i 1.02642i
\(546\) 0 0
\(547\) 1.74325e28i 0.310809i −0.987851 0.155404i \(-0.950332\pi\)
0.987851 0.155404i \(-0.0496680\pi\)
\(548\) 0 0
\(549\) 6.83757e28 6.88818e28i 1.17325 1.18194i
\(550\) 0 0
\(551\) −6.48350e27 −0.107082
\(552\) 0 0
\(553\) 1.58063e28 0.251313
\(554\) 0 0
\(555\) −8.28978e27 1.99094e28i −0.126901 0.304777i
\(556\) 0 0
\(557\) 2.75381e28i 0.405933i 0.979186 + 0.202966i \(0.0650582\pi\)
−0.979186 + 0.202966i \(0.934942\pi\)
\(558\) 0 0
\(559\) 4.77957e27i 0.0678524i
\(560\) 0 0
\(561\) 1.71376e28 + 4.11590e28i 0.234337 + 0.562801i
\(562\) 0 0
\(563\) −9.77185e28 −1.28718 −0.643589 0.765371i \(-0.722556\pi\)
−0.643589 + 0.765371i \(0.722556\pi\)
\(564\) 0 0
\(565\) −1.02438e29 −1.30002
\(566\) 0 0
\(567\) −2.03840e28 + 1.50341e26i −0.249268 + 0.00183846i
\(568\) 0 0
\(569\) 1.03279e29i 1.21712i 0.793509 + 0.608558i \(0.208252\pi\)
−0.793509 + 0.608558i \(0.791748\pi\)
\(570\) 0 0
\(571\) 9.75848e28i 1.10842i −0.832378 0.554208i \(-0.813021\pi\)
0.832378 0.554208i \(-0.186979\pi\)
\(572\) 0 0
\(573\) 6.36336e28 2.64955e28i 0.696727 0.290100i
\(574\) 0 0
\(575\) −7.23933e27 −0.0764163
\(576\) 0 0
\(577\) 6.63410e28 0.675206 0.337603 0.941289i \(-0.390384\pi\)
0.337603 + 0.941289i \(0.390384\pi\)
\(578\) 0 0
\(579\) −9.72446e28 + 4.04902e28i −0.954422 + 0.397398i
\(580\) 0 0
\(581\) 1.27035e28i 0.120247i
\(582\) 0 0
\(583\) 1.29862e29i 1.18567i
\(584\) 0 0
\(585\) 8.21856e28 + 8.15817e28i 0.723868 + 0.718549i
\(586\) 0 0
\(587\) −1.32148e29 −1.12295 −0.561474 0.827494i \(-0.689766\pi\)
−0.561474 + 0.827494i \(0.689766\pi\)
\(588\) 0 0
\(589\) 3.53263e28 0.289659
\(590\) 0 0
\(591\) 2.29806e28 + 5.51921e28i 0.181842 + 0.436725i
\(592\) 0 0
\(593\) 1.26760e29i 0.968073i −0.875048 0.484037i \(-0.839170\pi\)
0.875048 0.484037i \(-0.160830\pi\)
\(594\) 0 0
\(595\) 2.81374e28i 0.207422i
\(596\) 0 0
\(597\) 1.50365e27 + 3.61128e27i 0.0107008 + 0.0256998i
\(598\) 0 0
\(599\) −2.20473e27 −0.0151486 −0.00757432 0.999971i \(-0.502411\pi\)
−0.00757432 + 0.999971i \(0.502411\pi\)
\(600\) 0 0
\(601\) −9.82280e28 −0.651708 −0.325854 0.945420i \(-0.605652\pi\)
−0.325854 + 0.945420i \(0.605652\pi\)
\(602\) 0 0
\(603\) −1.90279e29 1.88881e29i −1.21916 1.21020i
\(604\) 0 0
\(605\) 5.27293e28i 0.326303i
\(606\) 0 0
\(607\) 1.46388e29i 0.875034i −0.899210 0.437517i \(-0.855858\pi\)
0.899210 0.437517i \(-0.144142\pi\)
\(608\) 0 0
\(609\) 2.57612e28 1.07263e28i 0.148760 0.0619398i
\(610\) 0 0
\(611\) −1.81551e29 −1.01290
\(612\) 0 0
\(613\) −2.64193e29 −1.42425 −0.712125 0.702052i \(-0.752267\pi\)
−0.712125 + 0.702052i \(0.752267\pi\)
\(614\) 0 0
\(615\) −6.50559e28 + 2.70877e28i −0.338920 + 0.141118i
\(616\) 0 0
\(617\) 9.70244e28i 0.488525i −0.969709 0.244263i \(-0.921454\pi\)
0.969709 0.244263i \(-0.0785459\pi\)
\(618\) 0 0
\(619\) 3.48597e29i 1.69657i −0.529539 0.848285i \(-0.677635\pi\)
0.529539 0.848285i \(-0.322365\pi\)
\(620\) 0 0
\(621\) 1.81807e28 4.45881e28i 0.0855359 0.209777i
\(622\) 0 0
\(623\) 1.52277e28 0.0692646
\(624\) 0 0
\(625\) −2.78199e29 −1.22353
\(626\) 0 0
\(627\) 1.26848e28 + 3.04648e28i 0.0539478 + 0.129565i
\(628\) 0 0
\(629\) 4.99430e28i 0.205420i
\(630\) 0 0
\(631\) 1.73495e29i 0.690206i −0.938565 0.345103i \(-0.887844\pi\)
0.938565 0.345103i \(-0.112156\pi\)
\(632\) 0 0
\(633\) 9.67017e28 + 2.32246e29i 0.372130 + 0.893738i
\(634\) 0 0
\(635\) −5.17506e29 −1.92660
\(636\) 0 0
\(637\) 2.29649e29 0.827182
\(638\) 0 0
\(639\) 1.72911e29 1.74191e29i 0.602649 0.607111i
\(640\) 0 0
\(641\) 5.71022e29i 1.92594i −0.269604 0.962971i \(-0.586893\pi\)
0.269604 0.962971i \(-0.413107\pi\)
\(642\) 0 0
\(643\) 1.96431e29i 0.641203i 0.947214 + 0.320601i \(0.103885\pi\)
−0.947214 + 0.320601i \(0.896115\pi\)
\(644\) 0 0
\(645\) 2.59944e28 1.08234e28i 0.0821300 0.0341969i
\(646\) 0 0
\(647\) 5.00829e29 1.53177 0.765887 0.642975i \(-0.222300\pi\)
0.765887 + 0.642975i \(0.222300\pi\)
\(648\) 0 0
\(649\) 3.12312e28 0.0924739
\(650\) 0 0
\(651\) −1.40364e29 + 5.84440e28i −0.402397 + 0.167548i
\(652\) 0 0
\(653\) 5.42218e29i 1.50517i 0.658495 + 0.752585i \(0.271193\pi\)
−0.658495 + 0.752585i \(0.728807\pi\)
\(654\) 0 0
\(655\) 5.83444e29i 1.56843i
\(656\) 0 0
\(657\) −2.10541e29 + 2.12100e29i −0.548151 + 0.552209i
\(658\) 0 0
\(659\) −6.80613e29 −1.71634 −0.858170 0.513366i \(-0.828398\pi\)
−0.858170 + 0.513366i \(0.828398\pi\)
\(660\) 0 0
\(661\) 1.35616e29 0.331280 0.165640 0.986186i \(-0.447031\pi\)
0.165640 + 0.986186i \(0.447031\pi\)
\(662\) 0 0
\(663\) 1.03082e29 + 2.47570e29i 0.243944 + 0.585875i
\(664\) 0 0
\(665\) 2.08265e28i 0.0477517i
\(666\) 0 0
\(667\) 6.59173e28i 0.146446i
\(668\) 0 0
\(669\) −1.23260e29 2.96030e29i −0.265367 0.637325i
\(670\) 0 0
\(671\) 6.76259e29 1.41100
\(672\) 0 0
\(673\) 2.91865e29 0.590233 0.295116 0.955461i \(-0.404642\pi\)
0.295116 + 0.955461i \(0.404642\pi\)
\(674\) 0 0
\(675\) −6.49704e28 + 1.59340e29i −0.127358 + 0.312345i
\(676\) 0 0
\(677\) 2.15331e29i 0.409190i 0.978847 + 0.204595i \(0.0655878\pi\)
−0.978847 + 0.204595i \(0.934412\pi\)
\(678\) 0 0
\(679\) 1.62649e28i 0.0299653i
\(680\) 0 0
\(681\) −4.77007e28 + 1.98614e28i −0.0852080 + 0.0354785i
\(682\) 0 0
\(683\) 4.41610e29 0.764929 0.382465 0.923970i \(-0.375075\pi\)
0.382465 + 0.923970i \(0.375075\pi\)
\(684\) 0 0
\(685\) 1.09350e30 1.83683
\(686\) 0 0
\(687\) 4.53687e29 1.88904e29i 0.739112 0.307748i
\(688\) 0 0
\(689\) 7.81113e29i 1.23428i
\(690\) 0 0
\(691\) 3.29378e29i 0.504865i −0.967615 0.252432i \(-0.918769\pi\)
0.967615 0.252432i \(-0.0812305\pi\)
\(692\) 0 0
\(693\) −1.00802e29 1.00062e29i −0.149889 0.148788i
\(694\) 0 0
\(695\) 1.18853e30 1.71462
\(696\) 0 0
\(697\) −1.63194e29 −0.228433
\(698\) 0 0
\(699\) −1.97016e29 4.73170e29i −0.267604 0.642698i
\(700\) 0 0
\(701\) 4.96564e29i 0.654539i 0.944931 + 0.327270i \(0.106129\pi\)
−0.944931 + 0.327270i \(0.893871\pi\)
\(702\) 0 0
\(703\) 3.69664e28i 0.0472908i
\(704\) 0 0
\(705\) −4.11126e29 9.87392e29i −0.510492 1.22604i
\(706\) 0 0
\(707\) 2.81212e29 0.338946
\(708\) 0 0
\(709\) −7.87065e28 −0.0920926 −0.0460463 0.998939i \(-0.514662\pi\)
−0.0460463 + 0.998939i \(0.514662\pi\)
\(710\) 0 0
\(711\) 6.29866e29 + 6.25238e29i 0.715513 + 0.710255i
\(712\) 0 0
\(713\) 3.59159e29i 0.396139i
\(714\) 0 0
\(715\) 8.06871e29i 0.864153i
\(716\) 0 0
\(717\) −8.72437e29 + 3.63261e29i −0.907367 + 0.377805i
\(718\) 0 0
\(719\) 8.27107e29 0.835427 0.417713 0.908579i \(-0.362832\pi\)
0.417713 + 0.908579i \(0.362832\pi\)
\(720\) 0 0
\(721\) −1.41025e28 −0.0138349
\(722\) 0 0
\(723\) 5.61467e29 2.33781e29i 0.535023 0.222770i
\(724\) 0 0
\(725\) 2.35561e29i 0.218050i
\(726\) 0 0
\(727\) 1.01486e30i 0.912631i 0.889818 + 0.456315i \(0.150831\pi\)
−0.889818 + 0.456315i \(0.849169\pi\)
\(728\) 0 0
\(729\) −8.18231e29 8.00324e29i −0.714886 0.699241i
\(730\) 0 0
\(731\) 6.52072e28 0.0553558
\(732\) 0 0
\(733\) −2.22596e28 −0.0183623 −0.00918114 0.999958i \(-0.502922\pi\)
−0.00918114 + 0.999958i \(0.502922\pi\)
\(734\) 0 0
\(735\) 5.20044e29 + 1.24898e30i 0.416892 + 1.00124i
\(736\) 0 0
\(737\) 1.86810e30i 1.45543i
\(738\) 0 0
\(739\) 1.57388e30i 1.19180i −0.803057 0.595902i \(-0.796795\pi\)
0.803057 0.595902i \(-0.203205\pi\)
\(740\) 0 0
\(741\) 7.62984e28 + 1.83244e29i 0.0561596 + 0.134877i
\(742\) 0 0
\(743\) 6.70225e29 0.479554 0.239777 0.970828i \(-0.422926\pi\)
0.239777 + 0.970828i \(0.422926\pi\)
\(744\) 0 0
\(745\) 1.73128e30 1.20428
\(746\) 0 0
\(747\) −5.02502e29 + 5.06222e29i −0.339838 + 0.342353i
\(748\) 0 0
\(749\) 4.54493e29i 0.298860i
\(750\) 0 0
\(751\) 1.47240e30i 0.941469i 0.882275 + 0.470734i \(0.156011\pi\)
−0.882275 + 0.470734i \(0.843989\pi\)
\(752\) 0 0
\(753\) −2.89340e30 + 1.20474e30i −1.79912 + 0.749111i
\(754\) 0 0
\(755\) −2.64390e30 −1.59883
\(756\) 0 0
\(757\) 1.20620e30 0.709432 0.354716 0.934974i \(-0.384578\pi\)
0.354716 + 0.934974i \(0.384578\pi\)
\(758\) 0 0
\(759\) 3.09733e29 1.28965e29i 0.177194 0.0737793i
\(760\) 0 0
\(761\) 2.79298e30i 1.55428i 0.629330 + 0.777139i \(0.283330\pi\)
−0.629330 + 0.777139i \(0.716670\pi\)
\(762\) 0 0
\(763\) 4.08691e29i 0.221252i
\(764\) 0 0
\(765\) −1.11301e30 + 1.12125e30i −0.586211 + 0.590551i
\(766\) 0 0
\(767\) 1.87854e29 0.0962652
\(768\) 0 0
\(769\) 2.21748e30 1.10569 0.552844 0.833285i \(-0.313542\pi\)
0.552844 + 0.833285i \(0.313542\pi\)
\(770\) 0 0
\(771\) −1.18793e30 2.85303e30i −0.576395 1.38432i
\(772\) 0 0
\(773\) 3.67008e30i 1.73297i −0.499205 0.866484i \(-0.666375\pi\)
0.499205 0.866484i \(-0.333625\pi\)
\(774\) 0 0
\(775\) 1.28349e30i 0.589827i
\(776\) 0 0
\(777\) 6.11575e28 + 1.46881e29i 0.0273545 + 0.0656967i
\(778\) 0 0
\(779\) −1.20791e29 −0.0525887
\(780\) 0 0
\(781\) 1.71015e30 0.724769
\(782\) 0 0
\(783\) 1.45086e30 + 5.91584e29i 0.598586 + 0.244072i
\(784\) 0 0
\(785\) 1.27874e30i 0.513629i
\(786\) 0 0
\(787\) 4.02180e30i 1.57284i −0.617690 0.786422i \(-0.711931\pi\)
0.617690 0.786422i \(-0.288069\pi\)
\(788\) 0 0
\(789\) −3.01361e30 + 1.25479e30i −1.14757 + 0.477819i
\(790\) 0 0
\(791\) 7.55729e29 0.280229
\(792\) 0 0
\(793\) 4.06767e30 1.46885
\(794\) 0 0
\(795\) −4.24819e30 + 1.76884e30i −1.49400 + 0.622063i
\(796\) 0 0
\(797\) 2.15532e30i 0.738242i 0.929381 + 0.369121i \(0.120341\pi\)
−0.929381 + 0.369121i \(0.879659\pi\)
\(798\) 0 0
\(799\) 2.47689e30i 0.826352i
\(800\) 0 0
\(801\) 6.06813e29 + 6.02354e29i 0.197203 + 0.195754i
\(802\) 0 0
\(803\) −2.08232e30 −0.659227
\(804\) 0 0
\(805\) 2.11741e29 0.0653054
\(806\) 0 0
\(807\) −1.60017e30 3.84309e30i −0.480832 1.15480i
\(808\) 0 0
\(809\) 2.35179e29i 0.0688556i 0.999407 + 0.0344278i \(0.0109609\pi\)
−0.999407 + 0.0344278i \(0.989039\pi\)
\(810\) 0 0
\(811\) 1.43433e30i 0.409195i 0.978846 + 0.204598i \(0.0655886\pi\)
−0.978846 + 0.204598i \(0.934411\pi\)
\(812\) 0 0
\(813\) 2.08086e29 + 4.99755e29i 0.0578484 + 0.138933i
\(814\) 0 0
\(815\) −5.19634e30 −1.40780
\(816\) 0 0
\(817\) 4.82646e28 0.0127437
\(818\) 0 0
\(819\) −6.06321e29 6.01865e29i −0.156035 0.154888i
\(820\) 0 0
\(821\) 5.72474e30i 1.43600i −0.696046 0.717998i \(-0.745059\pi\)
0.696046 0.717998i \(-0.254941\pi\)
\(822\) 0 0
\(823\) 3.72061e30i 0.909738i −0.890558 0.454869i \(-0.849686\pi\)
0.890558 0.454869i \(-0.150314\pi\)
\(824\) 0 0
\(825\) −1.10686e30 + 4.60869e29i −0.263831 + 0.109853i
\(826\) 0 0
\(827\) −7.34712e30 −1.70730 −0.853649 0.520849i \(-0.825616\pi\)
−0.853649 + 0.520849i \(0.825616\pi\)
\(828\) 0 0
\(829\) −7.50473e28 −0.0170025 −0.00850125 0.999964i \(-0.502706\pi\)
−0.00850125 + 0.999964i \(0.502706\pi\)
\(830\) 0 0
\(831\) −8.40739e29 + 3.50063e29i −0.185716 + 0.0773277i
\(832\) 0 0
\(833\) 3.13308e30i 0.674838i
\(834\) 0 0
\(835\) 7.35274e30i 1.54433i
\(836\) 0 0
\(837\) −7.90520e30 3.22333e30i −1.61918 0.660218i
\(838\) 0 0
\(839\) 1.85066e30 0.369681 0.184840 0.982769i \(-0.440823\pi\)
0.184840 + 0.982769i \(0.440823\pi\)
\(840\) 0 0
\(841\) 2.98795e30 0.582124
\(842\) 0 0
\(843\) 1.15188e30 + 2.76644e30i 0.218885 + 0.525692i
\(844\) 0 0
\(845\) 1.38566e30i 0.256840i
\(846\) 0 0
\(847\) 3.89008e29i 0.0703368i
\(848\) 0 0
\(849\) −3.76029e30 9.03102e30i −0.663271 1.59296i
\(850\) 0 0
\(851\) −3.75835e29 −0.0646750
\(852\) 0 0
\(853\) −8.34530e29 −0.140113 −0.0700563 0.997543i \(-0.522318\pi\)
−0.0700563 + 0.997543i \(0.522318\pi\)
\(854\) 0 0
\(855\) −8.23820e29 + 8.29918e29i −0.134955 + 0.135954i
\(856\) 0 0
\(857\) 6.07917e30i 0.971730i 0.874034 + 0.485865i \(0.161495\pi\)
−0.874034 + 0.485865i \(0.838505\pi\)
\(858\) 0 0
\(859\) 1.75485e29i 0.0273724i −0.999906 0.0136862i \(-0.995643\pi\)
0.999906 0.0136862i \(-0.00435658\pi\)
\(860\) 0 0
\(861\) 4.79947e29 1.99838e29i 0.0730567 0.0304190i
\(862\) 0 0
\(863\) −2.73201e30 −0.405853 −0.202926 0.979194i \(-0.565045\pi\)
−0.202926 + 0.979194i \(0.565045\pi\)
\(864\) 0 0
\(865\) −5.94653e30 −0.862174
\(866\) 0 0
\(867\) 3.14597e30 1.30991e30i 0.445199 0.185370i
\(868\) 0 0
\(869\) 6.18382e30i 0.854179i
\(870\) 0 0
\(871\) 1.12365e31i 1.51510i
\(872\) 0 0
\(873\) 6.43379e29 6.48142e29i 0.0846872 0.0853141i
\(874\) 0 0
\(875\) 1.48658e30 0.191031
\(876\) 0 0
\(877\) 1.32126e31 1.65765 0.828824 0.559509i \(-0.189010\pi\)
0.828824 + 0.559509i \(0.189010\pi\)
\(878\) 0 0
\(879\) 3.70068e29 + 8.88785e29i 0.0453313 + 0.108871i
\(880\) 0 0
\(881\) 8.38546e30i 1.00295i −0.865172 0.501476i \(-0.832791\pi\)
0.865172 0.501476i \(-0.167209\pi\)
\(882\) 0 0
\(883\) 1.14860e31i 1.34147i −0.741698 0.670734i \(-0.765979\pi\)
0.741698 0.670734i \(-0.234021\pi\)
\(884\) 0 0
\(885\) 4.25398e29 + 1.02167e30i 0.0485167 + 0.116521i
\(886\) 0 0
\(887\) 1.19825e31 1.33460 0.667298 0.744791i \(-0.267451\pi\)
0.667298 + 0.744791i \(0.267451\pi\)
\(888\) 0 0
\(889\) 3.81788e30 0.415292
\(890\) 0 0
\(891\) −5.88173e28 7.97473e30i −0.00624869 0.847228i
\(892\) 0 0
\(893\) 1.83332e30i 0.190239i
\(894\) 0 0
\(895\) 2.15117e31i 2.18038i
\(896\) 0 0
\(897\) 1.86303e30 7.75720e29i 0.184459 0.0768041i
\(898\) 0 0
\(899\) 1.16867e31 1.13036
\(900\) 0 0
\(901\) −1.06566e31 −1.00696
\(902\) 0 0
\(903\) −1.91772e29 + 7.98492e28i −0.0177037 + 0.00737139i
\(904\) 0 0
\(905\) 1.70013e31i 1.53346i
\(906\) 0 0
\(907\) 2.08168e30i 0.183458i 0.995784 + 0.0917291i \(0.0292394\pi\)
−0.995784 + 0.0917291i \(0.970761\pi\)
\(908\) 0 0
\(909\) 1.12061e31 + 1.11237e31i 0.965012 + 0.957921i
\(910\) 0 0
\(911\) 8.52106e30 0.717052 0.358526 0.933520i \(-0.383279\pi\)
0.358526 + 0.933520i \(0.383279\pi\)
\(912\) 0 0
\(913\) −4.96992e30 −0.408701
\(914\) 0 0
\(915\) 9.21128e30 + 2.21225e31i 0.740284 + 1.77793i
\(916\) 0 0
\(917\) 4.30433e30i 0.338087i
\(918\) 0 0
\(919\) 1.50335e31i 1.15411i 0.816705 + 0.577056i \(0.195798\pi\)
−0.816705 + 0.577056i \(0.804202\pi\)
\(920\) 0 0
\(921\) 4.85925e30 + 1.16703e31i 0.364622 + 0.875705i
\(922\) 0 0
\(923\) 1.02865e31 0.754483
\(924\) 0 0
\(925\) 1.34308e30 0.0962973
\(926\) 0 0
\(927\) −5.61972e29 5.57843e29i −0.0393892 0.0390998i
\(928\) 0 0
\(929\) 2.18121e31i 1.49463i −0.664472 0.747314i \(-0.731343\pi\)
0.664472 0.747314i \(-0.268657\pi\)
\(930\) 0 0
\(931\) 2.31902e30i 0.155358i
\(932\) 0 0
\(933\) 5.85383e30 2.43739e30i 0.383427 0.159650i
\(934\) 0 0
\(935\) −1.10081e31 −0.704999
\(936\) 0 0
\(937\) 9.27481e30 0.580817 0.290409 0.956903i \(-0.406209\pi\)
0.290409 + 0.956903i \(0.406209\pi\)
\(938\) 0 0
\(939\) 9.60961e30 4.00121e30i 0.588460 0.245020i
\(940\) 0 0
\(941\) 2.62274e31i 1.57059i 0.619120 + 0.785296i \(0.287489\pi\)
−0.619120 + 0.785296i \(0.712511\pi\)
\(942\) 0 0
\(943\) 1.22808e30i 0.0719205i
\(944\) 0 0
\(945\) 1.90030e30 4.66049e30i 0.108840 0.266930i
\(946\) 0 0
\(947\) 1.86298e31 1.04360 0.521800 0.853068i \(-0.325261\pi\)
0.521800 + 0.853068i \(0.325261\pi\)
\(948\) 0 0
\(949\) −1.25251e31 −0.686254
\(950\) 0 0
\(951\) 3.17960e29 + 7.63638e29i 0.0170403 + 0.0409253i
\(952\) 0 0
\(953\) 3.44161e31i 1.80420i −0.431523 0.902102i \(-0.642024\pi\)
0.431523 0.902102i \(-0.357976\pi\)
\(954\) 0 0
\(955\) 1.70189e31i 0.872763i
\(956\) 0 0
\(957\) 4.19642e30 + 1.00785e31i 0.210525 + 0.505613i
\(958\) 0 0
\(959\) −8.06728e30 −0.395942
\(960\) 0 0
\(961\) −4.28513e31 −2.05764
\(962\) 0 0
\(963\) 1.79781e31 1.81112e31i 0.844632 0.850884i
\(964\) 0 0
\(965\) 2.60082e31i 1.19557i
\(966\) 0 0
\(967\) 3.06414e31i 1.37826i −0.724639 0.689129i \(-0.757993\pi\)
0.724639 0.689129i \(-0.242007\pi\)
\(968\) 0 0
\(969\) −2.49998e30 + 1.04093e30i −0.110037 + 0.0458165i
\(970\) 0 0
\(971\) 3.62652e31 1.56203 0.781014 0.624514i \(-0.214703\pi\)
0.781014 + 0.624514i \(0.214703\pi\)
\(972\) 0 0
\(973\) −8.76829e30 −0.369599
\(974\) 0 0
\(975\) −6.65771e30 + 2.77211e30i −0.274648 + 0.114357i
\(976\) 0 0
\(977\) 1.32636e31i 0.535512i 0.963487 + 0.267756i \(0.0862820\pi\)
−0.963487 + 0.267756i \(0.913718\pi\)
\(978\) 0 0
\(979\) 5.95749e30i 0.235421i
\(980\) 0 0
\(981\) 1.61663e31 1.62860e31i 0.625298 0.629927i
\(982\) 0 0
\(983\) −3.95223e31 −1.49634 −0.748171 0.663506i \(-0.769068\pi\)
−0.748171 + 0.663506i \(0.769068\pi\)
\(984\) 0 0
\(985\) −1.47612e31 −0.547069
\(986\) 0 0
\(987\) 3.03306e30 + 7.28444e30i 0.110040 + 0.264281i
\(988\) 0 0
\(989\) 4.90703e29i 0.0174284i
\(990\) 0 0
\(991\) 2.70998e31i 0.942306i −0.882051 0.471153i \(-0.843838\pi\)
0.882051 0.471153i \(-0.156162\pi\)
\(992\) 0 0
\(993\) 2.87811e30 + 6.91230e30i 0.0979807 + 0.235318i
\(994\) 0 0
\(995\) −9.65842e29 −0.0321932
\(996\) 0 0
\(997\) 1.09599e30 0.0357690 0.0178845 0.999840i \(-0.494307\pi\)
0.0178845 + 0.999840i \(0.494307\pi\)
\(998\) 0 0
\(999\) −3.37298e30 + 8.27223e30i −0.107790 + 0.264353i
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 48.22.c.b.47.11 yes 12
3.2 odd 2 inner 48.22.c.b.47.1 12
4.3 odd 2 inner 48.22.c.b.47.2 yes 12
12.11 even 2 inner 48.22.c.b.47.12 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.22.c.b.47.1 12 3.2 odd 2 inner
48.22.c.b.47.2 yes 12 4.3 odd 2 inner
48.22.c.b.47.11 yes 12 1.1 even 1 trivial
48.22.c.b.47.12 yes 12 12.11 even 2 inner