Properties

Label 3042.2.b.l.1351.1
Level $3042$
Weight $2$
Character 3042.1351
Analytic conductor $24.290$
Analytic rank $0$
Dimension $4$
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] = [3042,2,Mod(1351,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (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: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3042.1351
Dual form 3042.2.b.l.1351.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.73205i q^{5} +4.73205i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.73205i q^{5} +4.73205i q^{7} +1.00000i q^{8} -1.73205 q^{10} -4.73205i q^{11} +4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} -1.26795i q^{19} +1.73205i q^{20} -4.73205 q^{22} +2.19615 q^{23} +2.00000 q^{25} -4.73205i q^{28} +3.00000 q^{29} -2.53590i q^{31} -1.00000i q^{32} +5.19615i q^{34} +8.19615 q^{35} +3.00000i q^{37} -1.26795 q^{38} +1.73205 q^{40} -0.464102i q^{41} -6.19615 q^{43} +4.73205i q^{44} -2.19615i q^{46} +1.26795i q^{47} -15.3923 q^{49} -2.00000i q^{50} -3.00000 q^{53} -8.19615 q^{55} -4.73205 q^{56} -3.00000i q^{58} -13.8564i q^{59} +4.80385 q^{61} -2.53590 q^{62} -1.00000 q^{64} -10.7321i q^{67} +5.19615 q^{68} -8.19615i q^{70} +8.19615i q^{71} -12.1244i q^{73} +3.00000 q^{74} +1.26795i q^{76} +22.3923 q^{77} -12.3923 q^{79} -1.73205i q^{80} -0.464102 q^{82} -11.6603i q^{83} +9.00000i q^{85} +6.19615i q^{86} +4.73205 q^{88} -2.53590i q^{89} -2.19615 q^{92} +1.26795 q^{94} -2.19615 q^{95} -6.00000i q^{97} +15.3923i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) − 1.73205i − 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 4.73205i 1.78855i 0.447521 + 0.894274i \(0.352307\pi\)
−0.447521 + 0.894274i \(0.647693\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) − 4.73205i − 1.42677i −0.700774 0.713384i \(-0.747162\pi\)
0.700774 0.713384i \(-0.252838\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.73205 1.26469
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.19615 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) − 1.26795i − 0.290887i −0.989367 0.145444i \(-0.953539\pi\)
0.989367 0.145444i \(-0.0464610\pi\)
\(20\) 1.73205i 0.387298i
\(21\) 0 0
\(22\) −4.73205 −1.00888
\(23\) 2.19615 0.457929 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) − 4.73205i − 0.894274i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) − 2.53590i − 0.455461i −0.973724 0.227730i \(-0.926870\pi\)
0.973724 0.227730i \(-0.0731305\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) 8.19615 1.38540
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) − 0.464102i − 0.0724805i −0.999343 0.0362402i \(-0.988462\pi\)
0.999343 0.0362402i \(-0.0115382\pi\)
\(42\) 0 0
\(43\) −6.19615 −0.944904 −0.472452 0.881356i \(-0.656631\pi\)
−0.472452 + 0.881356i \(0.656631\pi\)
\(44\) 4.73205i 0.713384i
\(45\) 0 0
\(46\) − 2.19615i − 0.323805i
\(47\) 1.26795i 0.184949i 0.995715 + 0.0924747i \(0.0294777\pi\)
−0.995715 + 0.0924747i \(0.970522\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) − 2.00000i − 0.282843i
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −8.19615 −1.10517
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) − 3.00000i − 0.393919i
\(59\) − 13.8564i − 1.80395i −0.431788 0.901975i \(-0.642117\pi\)
0.431788 0.901975i \(-0.357883\pi\)
\(60\) 0 0
\(61\) 4.80385 0.615070 0.307535 0.951537i \(-0.400496\pi\)
0.307535 + 0.951537i \(0.400496\pi\)
\(62\) −2.53590 −0.322059
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.7321i − 1.31113i −0.755139 0.655564i \(-0.772431\pi\)
0.755139 0.655564i \(-0.227569\pi\)
\(68\) 5.19615 0.630126
\(69\) 0 0
\(70\) − 8.19615i − 0.979628i
\(71\) 8.19615i 0.972704i 0.873763 + 0.486352i \(0.161673\pi\)
−0.873763 + 0.486352i \(0.838327\pi\)
\(72\) 0 0
\(73\) − 12.1244i − 1.41905i −0.704681 0.709524i \(-0.748910\pi\)
0.704681 0.709524i \(-0.251090\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 1.26795i 0.145444i
\(77\) 22.3923 2.55184
\(78\) 0 0
\(79\) −12.3923 −1.39424 −0.697122 0.716953i \(-0.745536\pi\)
−0.697122 + 0.716953i \(0.745536\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) −0.464102 −0.0512514
\(83\) − 11.6603i − 1.27988i −0.768425 0.639940i \(-0.778959\pi\)
0.768425 0.639940i \(-0.221041\pi\)
\(84\) 0 0
\(85\) 9.00000i 0.976187i
\(86\) 6.19615i 0.668148i
\(87\) 0 0
\(88\) 4.73205 0.504438
\(89\) − 2.53590i − 0.268805i −0.990927 0.134402i \(-0.957089\pi\)
0.990927 0.134402i \(-0.0429115\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.19615 −0.228965
\(93\) 0 0
\(94\) 1.26795 0.130779
\(95\) −2.19615 −0.225320
\(96\) 0 0
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 15.3923i 1.55486i
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −1.39230 −0.138540 −0.0692698 0.997598i \(-0.522067\pi\)
−0.0692698 + 0.997598i \(0.522067\pi\)
\(102\) 0 0
\(103\) 4.19615 0.413459 0.206730 0.978398i \(-0.433718\pi\)
0.206730 + 0.978398i \(0.433718\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.00000i 0.291386i
\(107\) −8.19615 −0.792352 −0.396176 0.918175i \(-0.629663\pi\)
−0.396176 + 0.918175i \(0.629663\pi\)
\(108\) 0 0
\(109\) − 16.3923i − 1.57010i −0.619434 0.785049i \(-0.712638\pi\)
0.619434 0.785049i \(-0.287362\pi\)
\(110\) 8.19615i 0.781472i
\(111\) 0 0
\(112\) 4.73205i 0.447137i
\(113\) −11.1962 −1.05325 −0.526623 0.850099i \(-0.676542\pi\)
−0.526623 + 0.850099i \(0.676542\pi\)
\(114\) 0 0
\(115\) − 3.80385i − 0.354711i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −13.8564 −1.27559
\(119\) − 24.5885i − 2.25402i
\(120\) 0 0
\(121\) −11.3923 −1.03566
\(122\) − 4.80385i − 0.434920i
\(123\) 0 0
\(124\) 2.53590i 0.227730i
\(125\) − 12.1244i − 1.08444i
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −16.3923 −1.43220 −0.716101 0.697997i \(-0.754075\pi\)
−0.716101 + 0.697997i \(0.754075\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −10.7321 −0.927108
\(135\) 0 0
\(136\) − 5.19615i − 0.445566i
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −8.19615 −0.692701
\(141\) 0 0
\(142\) 8.19615 0.687806
\(143\) 0 0
\(144\) 0 0
\(145\) − 5.19615i − 0.431517i
\(146\) −12.1244 −1.00342
\(147\) 0 0
\(148\) − 3.00000i − 0.246598i
\(149\) − 18.1244i − 1.48481i −0.669954 0.742403i \(-0.733686\pi\)
0.669954 0.742403i \(-0.266314\pi\)
\(150\) 0 0
\(151\) 7.26795i 0.591457i 0.955272 + 0.295729i \(0.0955624\pi\)
−0.955272 + 0.295729i \(0.904438\pi\)
\(152\) 1.26795 0.102844
\(153\) 0 0
\(154\) − 22.3923i − 1.80442i
\(155\) −4.39230 −0.352798
\(156\) 0 0
\(157\) −3.19615 −0.255081 −0.127540 0.991833i \(-0.540708\pi\)
−0.127540 + 0.991833i \(0.540708\pi\)
\(158\) 12.3923i 0.985879i
\(159\) 0 0
\(160\) −1.73205 −0.136931
\(161\) 10.3923i 0.819028i
\(162\) 0 0
\(163\) 9.46410i 0.741286i 0.928775 + 0.370643i \(0.120863\pi\)
−0.928775 + 0.370643i \(0.879137\pi\)
\(164\) 0.464102i 0.0362402i
\(165\) 0 0
\(166\) −11.6603 −0.905011
\(167\) 2.53590i 0.196234i 0.995175 + 0.0981169i \(0.0312819\pi\)
−0.995175 + 0.0981169i \(0.968718\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) 0 0
\(172\) 6.19615 0.472452
\(173\) 16.3923 1.24628 0.623142 0.782109i \(-0.285856\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(174\) 0 0
\(175\) 9.46410i 0.715419i
\(176\) − 4.73205i − 0.356692i
\(177\) 0 0
\(178\) −2.53590 −0.190074
\(179\) −8.19615 −0.612609 −0.306305 0.951934i \(-0.599093\pi\)
−0.306305 + 0.951934i \(0.599093\pi\)
\(180\) 0 0
\(181\) 11.5885 0.861363 0.430682 0.902504i \(-0.358273\pi\)
0.430682 + 0.902504i \(0.358273\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.19615i 0.161903i
\(185\) 5.19615 0.382029
\(186\) 0 0
\(187\) 24.5885i 1.79809i
\(188\) − 1.26795i − 0.0924747i
\(189\) 0 0
\(190\) 2.19615i 0.159326i
\(191\) −20.7846 −1.50392 −0.751961 0.659208i \(-0.770892\pi\)
−0.751961 + 0.659208i \(0.770892\pi\)
\(192\) 0 0
\(193\) − 12.8038i − 0.921641i −0.887493 0.460821i \(-0.847555\pi\)
0.887493 0.460821i \(-0.152445\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 15.3923 1.09945
\(197\) 6.92820i 0.493614i 0.969065 + 0.246807i \(0.0793814\pi\)
−0.969065 + 0.246807i \(0.920619\pi\)
\(198\) 0 0
\(199\) −8.58846 −0.608820 −0.304410 0.952541i \(-0.598459\pi\)
−0.304410 + 0.952541i \(0.598459\pi\)
\(200\) 2.00000i 0.141421i
\(201\) 0 0
\(202\) 1.39230i 0.0979622i
\(203\) 14.1962i 0.996375i
\(204\) 0 0
\(205\) −0.803848 −0.0561432
\(206\) − 4.19615i − 0.292360i
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −3.60770 −0.248364 −0.124182 0.992259i \(-0.539631\pi\)
−0.124182 + 0.992259i \(0.539631\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 8.19615i 0.560277i
\(215\) 10.7321i 0.731920i
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) −16.3923 −1.11023
\(219\) 0 0
\(220\) 8.19615 0.552584
\(221\) 0 0
\(222\) 0 0
\(223\) 18.9282i 1.26753i 0.773527 + 0.633763i \(0.218491\pi\)
−0.773527 + 0.633763i \(0.781509\pi\)
\(224\) 4.73205 0.316173
\(225\) 0 0
\(226\) 11.1962i 0.744757i
\(227\) − 9.80385i − 0.650704i −0.945593 0.325352i \(-0.894517\pi\)
0.945593 0.325352i \(-0.105483\pi\)
\(228\) 0 0
\(229\) 19.8564i 1.31215i 0.754696 + 0.656074i \(0.227784\pi\)
−0.754696 + 0.656074i \(0.772216\pi\)
\(230\) −3.80385 −0.250818
\(231\) 0 0
\(232\) 3.00000i 0.196960i
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 2.19615 0.143261
\(236\) 13.8564i 0.901975i
\(237\) 0 0
\(238\) −24.5885 −1.59383
\(239\) − 24.5885i − 1.59050i −0.606285 0.795248i \(-0.707341\pi\)
0.606285 0.795248i \(-0.292659\pi\)
\(240\) 0 0
\(241\) 0.803848i 0.0517804i 0.999665 + 0.0258902i \(0.00824202\pi\)
−0.999665 + 0.0258902i \(0.991758\pi\)
\(242\) 11.3923i 0.732325i
\(243\) 0 0
\(244\) −4.80385 −0.307535
\(245\) 26.6603i 1.70326i
\(246\) 0 0
\(247\) 0 0
\(248\) 2.53590 0.161030
\(249\) 0 0
\(250\) −12.1244 −0.766812
\(251\) −4.39230 −0.277240 −0.138620 0.990346i \(-0.544267\pi\)
−0.138620 + 0.990346i \(0.544267\pi\)
\(252\) 0 0
\(253\) − 10.3923i − 0.653359i
\(254\) − 4.00000i − 0.250982i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.8038 0.798682 0.399341 0.916802i \(-0.369239\pi\)
0.399341 + 0.916802i \(0.369239\pi\)
\(258\) 0 0
\(259\) −14.1962 −0.882106
\(260\) 0 0
\(261\) 0 0
\(262\) 16.3923i 1.01272i
\(263\) 2.19615 0.135421 0.0677103 0.997705i \(-0.478431\pi\)
0.0677103 + 0.997705i \(0.478431\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) − 6.00000i − 0.367884i
\(267\) 0 0
\(268\) 10.7321i 0.655564i
\(269\) 28.3923 1.73111 0.865555 0.500814i \(-0.166966\pi\)
0.865555 + 0.500814i \(0.166966\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −5.19615 −0.315063
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) − 9.46410i − 0.570707i
\(276\) 0 0
\(277\) −15.1962 −0.913048 −0.456524 0.889711i \(-0.650906\pi\)
−0.456524 + 0.889711i \(0.650906\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 8.19615i 0.489814i
\(281\) − 24.4641i − 1.45941i −0.683764 0.729703i \(-0.739658\pi\)
0.683764 0.729703i \(-0.260342\pi\)
\(282\) 0 0
\(283\) 30.1962 1.79497 0.897487 0.441040i \(-0.145390\pi\)
0.897487 + 0.441040i \(0.145390\pi\)
\(284\) − 8.19615i − 0.486352i
\(285\) 0 0
\(286\) 0 0
\(287\) 2.19615 0.129635
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) −5.19615 −0.305129
\(291\) 0 0
\(292\) 12.1244i 0.709524i
\(293\) 14.6603i 0.856461i 0.903669 + 0.428231i \(0.140863\pi\)
−0.903669 + 0.428231i \(0.859137\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −3.00000 −0.174371
\(297\) 0 0
\(298\) −18.1244 −1.04992
\(299\) 0 0
\(300\) 0 0
\(301\) − 29.3205i − 1.69001i
\(302\) 7.26795 0.418223
\(303\) 0 0
\(304\) − 1.26795i − 0.0727219i
\(305\) − 8.32051i − 0.476431i
\(306\) 0 0
\(307\) 10.7321i 0.612510i 0.951949 + 0.306255i \(0.0990761\pi\)
−0.951949 + 0.306255i \(0.900924\pi\)
\(308\) −22.3923 −1.27592
\(309\) 0 0
\(310\) 4.39230i 0.249466i
\(311\) 2.19615 0.124532 0.0622662 0.998060i \(-0.480167\pi\)
0.0622662 + 0.998060i \(0.480167\pi\)
\(312\) 0 0
\(313\) −24.3923 −1.37873 −0.689367 0.724412i \(-0.742111\pi\)
−0.689367 + 0.724412i \(0.742111\pi\)
\(314\) 3.19615i 0.180369i
\(315\) 0 0
\(316\) 12.3923 0.697122
\(317\) 6.12436i 0.343978i 0.985099 + 0.171989i \(0.0550194\pi\)
−0.985099 + 0.171989i \(0.944981\pi\)
\(318\) 0 0
\(319\) − 14.1962i − 0.794832i
\(320\) 1.73205i 0.0968246i
\(321\) 0 0
\(322\) 10.3923 0.579141
\(323\) 6.58846i 0.366592i
\(324\) 0 0
\(325\) 0 0
\(326\) 9.46410 0.524168
\(327\) 0 0
\(328\) 0.464102 0.0256257
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) − 12.0000i − 0.659580i −0.944054 0.329790i \(-0.893022\pi\)
0.944054 0.329790i \(-0.106978\pi\)
\(332\) 11.6603i 0.639940i
\(333\) 0 0
\(334\) 2.53590 0.138758
\(335\) −18.5885 −1.01560
\(336\) 0 0
\(337\) 31.0000 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 9.00000i − 0.488094i
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) − 39.7128i − 2.14429i
\(344\) − 6.19615i − 0.334074i
\(345\) 0 0
\(346\) − 16.3923i − 0.881256i
\(347\) 12.5885 0.675784 0.337892 0.941185i \(-0.390286\pi\)
0.337892 + 0.941185i \(0.390286\pi\)
\(348\) 0 0
\(349\) 2.53590i 0.135744i 0.997694 + 0.0678718i \(0.0216209\pi\)
−0.997694 + 0.0678718i \(0.978379\pi\)
\(350\) 9.46410 0.505878
\(351\) 0 0
\(352\) −4.73205 −0.252219
\(353\) 5.78461i 0.307884i 0.988080 + 0.153942i \(0.0491968\pi\)
−0.988080 + 0.153942i \(0.950803\pi\)
\(354\) 0 0
\(355\) 14.1962 0.753454
\(356\) 2.53590i 0.134402i
\(357\) 0 0
\(358\) 8.19615i 0.433180i
\(359\) − 22.0526i − 1.16389i −0.813228 0.581945i \(-0.802292\pi\)
0.813228 0.581945i \(-0.197708\pi\)
\(360\) 0 0
\(361\) 17.3923 0.915384
\(362\) − 11.5885i − 0.609076i
\(363\) 0 0
\(364\) 0 0
\(365\) −21.0000 −1.09919
\(366\) 0 0
\(367\) −24.1962 −1.26303 −0.631514 0.775364i \(-0.717566\pi\)
−0.631514 + 0.775364i \(0.717566\pi\)
\(368\) 2.19615 0.114482
\(369\) 0 0
\(370\) − 5.19615i − 0.270135i
\(371\) − 14.1962i − 0.737028i
\(372\) 0 0
\(373\) 23.9808 1.24168 0.620838 0.783939i \(-0.286792\pi\)
0.620838 + 0.783939i \(0.286792\pi\)
\(374\) 24.5885 1.27144
\(375\) 0 0
\(376\) −1.26795 −0.0653895
\(377\) 0 0
\(378\) 0 0
\(379\) 18.2487i 0.937373i 0.883364 + 0.468687i \(0.155273\pi\)
−0.883364 + 0.468687i \(0.844727\pi\)
\(380\) 2.19615 0.112660
\(381\) 0 0
\(382\) 20.7846i 1.06343i
\(383\) 11.3205i 0.578451i 0.957261 + 0.289225i \(0.0933977\pi\)
−0.957261 + 0.289225i \(0.906602\pi\)
\(384\) 0 0
\(385\) − 38.7846i − 1.97665i
\(386\) −12.8038 −0.651699
\(387\) 0 0
\(388\) 6.00000i 0.304604i
\(389\) −13.3923 −0.679017 −0.339508 0.940603i \(-0.610261\pi\)
−0.339508 + 0.940603i \(0.610261\pi\)
\(390\) 0 0
\(391\) −11.4115 −0.577107
\(392\) − 15.3923i − 0.777429i
\(393\) 0 0
\(394\) 6.92820 0.349038
\(395\) 21.4641i 1.07998i
\(396\) 0 0
\(397\) − 16.3923i − 0.822706i −0.911476 0.411353i \(-0.865056\pi\)
0.911476 0.411353i \(-0.134944\pi\)
\(398\) 8.58846i 0.430500i
\(399\) 0 0
\(400\) 2.00000 0.100000
\(401\) − 21.0000i − 1.04869i −0.851506 0.524345i \(-0.824310\pi\)
0.851506 0.524345i \(-0.175690\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.39230 0.0692698
\(405\) 0 0
\(406\) 14.1962 0.704543
\(407\) 14.1962 0.703677
\(408\) 0 0
\(409\) 3.33975i 0.165140i 0.996585 + 0.0825699i \(0.0263128\pi\)
−0.996585 + 0.0825699i \(0.973687\pi\)
\(410\) 0.803848i 0.0396992i
\(411\) 0 0
\(412\) −4.19615 −0.206730
\(413\) 65.5692 3.22645
\(414\) 0 0
\(415\) −20.1962 −0.991390
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000i 0.293470i
\(419\) −16.3923 −0.800816 −0.400408 0.916337i \(-0.631132\pi\)
−0.400408 + 0.916337i \(0.631132\pi\)
\(420\) 0 0
\(421\) − 0.464102i − 0.0226189i −0.999936 0.0113095i \(-0.996400\pi\)
0.999936 0.0113095i \(-0.00359999\pi\)
\(422\) 3.60770i 0.175620i
\(423\) 0 0
\(424\) − 3.00000i − 0.145693i
\(425\) −10.3923 −0.504101
\(426\) 0 0
\(427\) 22.7321i 1.10008i
\(428\) 8.19615 0.396176
\(429\) 0 0
\(430\) 10.7321 0.517545
\(431\) 27.8038i 1.33926i 0.742693 + 0.669632i \(0.233548\pi\)
−0.742693 + 0.669632i \(0.766452\pi\)
\(432\) 0 0
\(433\) 33.7846 1.62358 0.811792 0.583946i \(-0.198492\pi\)
0.811792 + 0.583946i \(0.198492\pi\)
\(434\) − 12.0000i − 0.576018i
\(435\) 0 0
\(436\) 16.3923i 0.785049i
\(437\) − 2.78461i − 0.133206i
\(438\) 0 0
\(439\) −16.5885 −0.791724 −0.395862 0.918310i \(-0.629554\pi\)
−0.395862 + 0.918310i \(0.629554\pi\)
\(440\) − 8.19615i − 0.390736i
\(441\) 0 0
\(442\) 0 0
\(443\) 4.39230 0.208685 0.104342 0.994541i \(-0.466726\pi\)
0.104342 + 0.994541i \(0.466726\pi\)
\(444\) 0 0
\(445\) −4.39230 −0.208215
\(446\) 18.9282 0.896276
\(447\) 0 0
\(448\) − 4.73205i − 0.223568i
\(449\) − 33.4641i − 1.57927i −0.613578 0.789634i \(-0.710270\pi\)
0.613578 0.789634i \(-0.289730\pi\)
\(450\) 0 0
\(451\) −2.19615 −0.103413
\(452\) 11.1962 0.526623
\(453\) 0 0
\(454\) −9.80385 −0.460117
\(455\) 0 0
\(456\) 0 0
\(457\) − 19.9808i − 0.934661i −0.884083 0.467330i \(-0.845216\pi\)
0.884083 0.467330i \(-0.154784\pi\)
\(458\) 19.8564 0.927829
\(459\) 0 0
\(460\) 3.80385i 0.177355i
\(461\) − 19.9808i − 0.930597i −0.885154 0.465298i \(-0.845947\pi\)
0.885154 0.465298i \(-0.154053\pi\)
\(462\) 0 0
\(463\) − 26.1962i − 1.21744i −0.793386 0.608719i \(-0.791684\pi\)
0.793386 0.608719i \(-0.208316\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 18.0000i 0.833834i
\(467\) 36.5885 1.69311 0.846556 0.532300i \(-0.178672\pi\)
0.846556 + 0.532300i \(0.178672\pi\)
\(468\) 0 0
\(469\) 50.7846 2.34502
\(470\) − 2.19615i − 0.101301i
\(471\) 0 0
\(472\) 13.8564 0.637793
\(473\) 29.3205i 1.34816i
\(474\) 0 0
\(475\) − 2.53590i − 0.116355i
\(476\) 24.5885i 1.12701i
\(477\) 0 0
\(478\) −24.5885 −1.12465
\(479\) 35.3205i 1.61384i 0.590664 + 0.806918i \(0.298866\pi\)
−0.590664 + 0.806918i \(0.701134\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.803848 0.0366143
\(483\) 0 0
\(484\) 11.3923 0.517832
\(485\) −10.3923 −0.471890
\(486\) 0 0
\(487\) − 9.12436i − 0.413464i −0.978398 0.206732i \(-0.933717\pi\)
0.978398 0.206732i \(-0.0662828\pi\)
\(488\) 4.80385i 0.217460i
\(489\) 0 0
\(490\) 26.6603 1.20439
\(491\) 0.588457 0.0265567 0.0132784 0.999912i \(-0.495773\pi\)
0.0132784 + 0.999912i \(0.495773\pi\)
\(492\) 0 0
\(493\) −15.5885 −0.702069
\(494\) 0 0
\(495\) 0 0
\(496\) − 2.53590i − 0.113865i
\(497\) −38.7846 −1.73973
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 12.1244i 0.542218i
\(501\) 0 0
\(502\) 4.39230i 0.196038i
\(503\) 18.5885 0.828818 0.414409 0.910091i \(-0.363988\pi\)
0.414409 + 0.910091i \(0.363988\pi\)
\(504\) 0 0
\(505\) 2.41154i 0.107312i
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 9.33975i 0.413977i 0.978343 + 0.206988i \(0.0663663\pi\)
−0.978343 + 0.206988i \(0.933634\pi\)
\(510\) 0 0
\(511\) 57.3731 2.53804
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) − 12.8038i − 0.564754i
\(515\) − 7.26795i − 0.320264i
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 14.1962i 0.623743i
\(519\) 0 0
\(520\) 0 0
\(521\) 18.8038 0.823812 0.411906 0.911226i \(-0.364863\pi\)
0.411906 + 0.911226i \(0.364863\pi\)
\(522\) 0 0
\(523\) 1.41154 0.0617225 0.0308612 0.999524i \(-0.490175\pi\)
0.0308612 + 0.999524i \(0.490175\pi\)
\(524\) 16.3923 0.716101
\(525\) 0 0
\(526\) − 2.19615i − 0.0957568i
\(527\) 13.1769i 0.573995i
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) 5.19615 0.225706
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) 14.1962i 0.613753i
\(536\) 10.7321 0.463554
\(537\) 0 0
\(538\) − 28.3923i − 1.22408i
\(539\) 72.8372i 3.13732i
\(540\) 0 0
\(541\) 16.8564i 0.724714i 0.932039 + 0.362357i \(0.118028\pi\)
−0.932039 + 0.362357i \(0.881972\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 5.19615i 0.222783i
\(545\) −28.3923 −1.21619
\(546\) 0 0
\(547\) −6.19615 −0.264928 −0.132464 0.991188i \(-0.542289\pi\)
−0.132464 + 0.991188i \(0.542289\pi\)
\(548\) − 9.00000i − 0.384461i
\(549\) 0 0
\(550\) −9.46410 −0.403551
\(551\) − 3.80385i − 0.162049i
\(552\) 0 0
\(553\) − 58.6410i − 2.49367i
\(554\) 15.1962i 0.645623i
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) − 22.2679i − 0.943523i −0.881726 0.471762i \(-0.843618\pi\)
0.881726 0.471762i \(-0.156382\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 8.19615 0.346351
\(561\) 0 0
\(562\) −24.4641 −1.03196
\(563\) 8.78461 0.370227 0.185114 0.982717i \(-0.440735\pi\)
0.185114 + 0.982717i \(0.440735\pi\)
\(564\) 0 0
\(565\) 19.3923i 0.815840i
\(566\) − 30.1962i − 1.26924i
\(567\) 0 0
\(568\) −8.19615 −0.343903
\(569\) −32.7846 −1.37440 −0.687201 0.726467i \(-0.741161\pi\)
−0.687201 + 0.726467i \(0.741161\pi\)
\(570\) 0 0
\(571\) 13.8038 0.577673 0.288837 0.957378i \(-0.406732\pi\)
0.288837 + 0.957378i \(0.406732\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 2.19615i − 0.0916656i
\(575\) 4.39230 0.183172
\(576\) 0 0
\(577\) 16.2679i 0.677244i 0.940923 + 0.338622i \(0.109961\pi\)
−0.940923 + 0.338622i \(0.890039\pi\)
\(578\) − 10.0000i − 0.415945i
\(579\) 0 0
\(580\) 5.19615i 0.215758i
\(581\) 55.1769 2.28912
\(582\) 0 0
\(583\) 14.1962i 0.587945i
\(584\) 12.1244 0.501709
\(585\) 0 0
\(586\) 14.6603 0.605610
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −3.21539 −0.132488
\(590\) 24.0000i 0.988064i
\(591\) 0 0
\(592\) 3.00000i 0.123299i
\(593\) 46.8564i 1.92416i 0.272764 + 0.962081i \(0.412062\pi\)
−0.272764 + 0.962081i \(0.587938\pi\)
\(594\) 0 0
\(595\) −42.5885 −1.74596
\(596\) 18.1244i 0.742403i
\(597\) 0 0
\(598\) 0 0
\(599\) 4.39230 0.179465 0.0897324 0.995966i \(-0.471399\pi\)
0.0897324 + 0.995966i \(0.471399\pi\)
\(600\) 0 0
\(601\) 21.7846 0.888613 0.444306 0.895875i \(-0.353450\pi\)
0.444306 + 0.895875i \(0.353450\pi\)
\(602\) −29.3205 −1.19501
\(603\) 0 0
\(604\) − 7.26795i − 0.295729i
\(605\) 19.7321i 0.802222i
\(606\) 0 0
\(607\) −48.7846 −1.98011 −0.990053 0.140694i \(-0.955067\pi\)
−0.990053 + 0.140694i \(0.955067\pi\)
\(608\) −1.26795 −0.0514221
\(609\) 0 0
\(610\) −8.32051 −0.336888
\(611\) 0 0
\(612\) 0 0
\(613\) 40.8564i 1.65017i 0.565005 + 0.825087i \(0.308874\pi\)
−0.565005 + 0.825087i \(0.691126\pi\)
\(614\) 10.7321 0.433110
\(615\) 0 0
\(616\) 22.3923i 0.902212i
\(617\) 10.6077i 0.427050i 0.976938 + 0.213525i \(0.0684944\pi\)
−0.976938 + 0.213525i \(0.931506\pi\)
\(618\) 0 0
\(619\) − 7.60770i − 0.305779i −0.988243 0.152890i \(-0.951142\pi\)
0.988243 0.152890i \(-0.0488579\pi\)
\(620\) 4.39230 0.176399
\(621\) 0 0
\(622\) − 2.19615i − 0.0880577i
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 24.3923i 0.974913i
\(627\) 0 0
\(628\) 3.19615 0.127540
\(629\) − 15.5885i − 0.621552i
\(630\) 0 0
\(631\) 25.8564i 1.02933i 0.857392 + 0.514664i \(0.172083\pi\)
−0.857392 + 0.514664i \(0.827917\pi\)
\(632\) − 12.3923i − 0.492939i
\(633\) 0 0
\(634\) 6.12436 0.243229
\(635\) − 6.92820i − 0.274937i
\(636\) 0 0
\(637\) 0 0
\(638\) −14.1962 −0.562031
\(639\) 0 0
\(640\) 1.73205 0.0684653
\(641\) −30.8038 −1.21668 −0.608339 0.793677i \(-0.708164\pi\)
−0.608339 + 0.793677i \(0.708164\pi\)
\(642\) 0 0
\(643\) 27.7128i 1.09289i 0.837496 + 0.546443i \(0.184019\pi\)
−0.837496 + 0.546443i \(0.815981\pi\)
\(644\) − 10.3923i − 0.409514i
\(645\) 0 0
\(646\) 6.58846 0.259219
\(647\) 13.1769 0.518038 0.259019 0.965872i \(-0.416601\pi\)
0.259019 + 0.965872i \(0.416601\pi\)
\(648\) 0 0
\(649\) −65.5692 −2.57382
\(650\) 0 0
\(651\) 0 0
\(652\) − 9.46410i − 0.370643i
\(653\) 49.1769 1.92444 0.962221 0.272271i \(-0.0877746\pi\)
0.962221 + 0.272271i \(0.0877746\pi\)
\(654\) 0 0
\(655\) 28.3923i 1.10938i
\(656\) − 0.464102i − 0.0181201i
\(657\) 0 0
\(658\) 6.00000i 0.233904i
\(659\) −25.1769 −0.980753 −0.490377 0.871511i \(-0.663141\pi\)
−0.490377 + 0.871511i \(0.663141\pi\)
\(660\) 0 0
\(661\) 9.00000i 0.350059i 0.984563 + 0.175030i \(0.0560022\pi\)
−0.984563 + 0.175030i \(0.943998\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 11.6603 0.452506
\(665\) − 10.3923i − 0.402996i
\(666\) 0 0
\(667\) 6.58846 0.255106
\(668\) − 2.53590i − 0.0981169i
\(669\) 0 0
\(670\) 18.5885i 0.718135i
\(671\) − 22.7321i − 0.877561i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) − 31.0000i − 1.19408i
\(675\) 0 0
\(676\) 0 0
\(677\) −4.39230 −0.168810 −0.0844050 0.996432i \(-0.526899\pi\)
−0.0844050 + 0.996432i \(0.526899\pi\)
\(678\) 0 0
\(679\) 28.3923 1.08960
\(680\) −9.00000 −0.345134
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) − 27.7128i − 1.06040i −0.847872 0.530201i \(-0.822117\pi\)
0.847872 0.530201i \(-0.177883\pi\)
\(684\) 0 0
\(685\) 15.5885 0.595604
\(686\) −39.7128 −1.51624
\(687\) 0 0
\(688\) −6.19615 −0.236226
\(689\) 0 0
\(690\) 0 0
\(691\) 19.5167i 0.742449i 0.928543 + 0.371224i \(0.121062\pi\)
−0.928543 + 0.371224i \(0.878938\pi\)
\(692\) −16.3923 −0.623142
\(693\) 0 0
\(694\) − 12.5885i − 0.477851i
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) 2.41154i 0.0913437i
\(698\) 2.53590 0.0959852
\(699\) 0 0
\(700\) − 9.46410i − 0.357709i
\(701\) 4.39230 0.165895 0.0829475 0.996554i \(-0.473567\pi\)
0.0829475 + 0.996554i \(0.473567\pi\)
\(702\) 0 0
\(703\) 3.80385 0.143465
\(704\) 4.73205i 0.178346i
\(705\) 0 0
\(706\) 5.78461 0.217707
\(707\) − 6.58846i − 0.247784i
\(708\) 0 0
\(709\) 3.24871i 0.122008i 0.998138 + 0.0610040i \(0.0194302\pi\)
−0.998138 + 0.0610040i \(0.980570\pi\)
\(710\) − 14.1962i − 0.532772i
\(711\) 0 0
\(712\) 2.53590 0.0950368
\(713\) − 5.56922i − 0.208569i
\(714\) 0 0
\(715\) 0 0
\(716\) 8.19615 0.306305
\(717\) 0 0
\(718\) −22.0526 −0.822994
\(719\) 52.3923 1.95390 0.976952 0.213461i \(-0.0684736\pi\)
0.976952 + 0.213461i \(0.0684736\pi\)
\(720\) 0 0
\(721\) 19.8564i 0.739491i
\(722\) − 17.3923i − 0.647275i
\(723\) 0 0
\(724\) −11.5885 −0.430682
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 24.1962 0.897386 0.448693 0.893686i \(-0.351890\pi\)
0.448693 + 0.893686i \(0.351890\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 21.0000i 0.777245i
\(731\) 32.1962 1.19082
\(732\) 0 0
\(733\) 14.3205i 0.528940i 0.964394 + 0.264470i \(0.0851970\pi\)
−0.964394 + 0.264470i \(0.914803\pi\)
\(734\) 24.1962i 0.893096i
\(735\) 0 0
\(736\) − 2.19615i − 0.0809513i
\(737\) −50.7846 −1.87068
\(738\) 0 0
\(739\) − 18.9282i − 0.696285i −0.937442 0.348143i \(-0.886812\pi\)
0.937442 0.348143i \(-0.113188\pi\)
\(740\) −5.19615 −0.191014
\(741\) 0 0
\(742\) −14.1962 −0.521157
\(743\) 4.39230i 0.161138i 0.996749 + 0.0805690i \(0.0256737\pi\)
−0.996749 + 0.0805690i \(0.974326\pi\)
\(744\) 0 0
\(745\) −31.3923 −1.15013
\(746\) − 23.9808i − 0.877998i
\(747\) 0 0
\(748\) − 24.5885i − 0.899043i
\(749\) − 38.7846i − 1.41716i
\(750\) 0 0
\(751\) 24.9808 0.911561 0.455780 0.890092i \(-0.349360\pi\)
0.455780 + 0.890092i \(0.349360\pi\)
\(752\) 1.26795i 0.0462373i
\(753\) 0 0
\(754\) 0 0
\(755\) 12.5885 0.458141
\(756\) 0 0
\(757\) 18.7846 0.682738 0.341369 0.939929i \(-0.389109\pi\)
0.341369 + 0.939929i \(0.389109\pi\)
\(758\) 18.2487 0.662823
\(759\) 0 0
\(760\) − 2.19615i − 0.0796628i
\(761\) 4.39230i 0.159221i 0.996826 + 0.0796105i \(0.0253676\pi\)
−0.996826 + 0.0796105i \(0.974632\pi\)
\(762\) 0 0
\(763\) 77.5692 2.80819
\(764\) 20.7846 0.751961
\(765\) 0 0
\(766\) 11.3205 0.409027
\(767\) 0 0
\(768\) 0 0
\(769\) − 33.7128i − 1.21572i −0.794046 0.607858i \(-0.792029\pi\)
0.794046 0.607858i \(-0.207971\pi\)
\(770\) −38.7846 −1.39770
\(771\) 0 0
\(772\) 12.8038i 0.460821i
\(773\) − 50.7846i − 1.82660i −0.407293 0.913298i \(-0.633527\pi\)
0.407293 0.913298i \(-0.366473\pi\)
\(774\) 0 0
\(775\) − 5.07180i − 0.182184i
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 13.3923i 0.480137i
\(779\) −0.588457 −0.0210837
\(780\) 0 0
\(781\) 38.7846 1.38782
\(782\) 11.4115i 0.408076i
\(783\) 0 0
\(784\) −15.3923 −0.549725
\(785\) 5.53590i 0.197585i
\(786\) 0 0
\(787\) − 14.5359i − 0.518149i −0.965857 0.259074i \(-0.916583\pi\)
0.965857 0.259074i \(-0.0834175\pi\)
\(788\) − 6.92820i − 0.246807i
\(789\) 0 0
\(790\) 21.4641 0.763658
\(791\) − 52.9808i − 1.88378i
\(792\) 0 0
\(793\) 0 0
\(794\) −16.3923 −0.581741
\(795\) 0 0
\(796\) 8.58846 0.304410
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) − 6.58846i − 0.233083i
\(800\) − 2.00000i − 0.0707107i
\(801\) 0 0
\(802\) −21.0000 −0.741536
\(803\) −57.3731 −2.02465
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 0 0
\(807\) 0 0
\(808\) − 1.39230i − 0.0489811i
\(809\) 47.1962 1.65933 0.829664 0.558263i \(-0.188532\pi\)
0.829664 + 0.558263i \(0.188532\pi\)
\(810\) 0 0
\(811\) − 4.39230i − 0.154235i −0.997022 0.0771173i \(-0.975428\pi\)
0.997022 0.0771173i \(-0.0245716\pi\)
\(812\) − 14.1962i − 0.498187i
\(813\) 0 0
\(814\) − 14.1962i − 0.497575i
\(815\) 16.3923 0.574197
\(816\) 0 0
\(817\) 7.85641i 0.274861i
\(818\) 3.33975 0.116771
\(819\) 0 0
\(820\) 0.803848 0.0280716
\(821\) − 40.6410i − 1.41838i −0.705017 0.709191i \(-0.749061\pi\)
0.705017 0.709191i \(-0.250939\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 4.19615i 0.146180i
\(825\) 0 0
\(826\) − 65.5692i − 2.28144i
\(827\) 32.1051i 1.11640i 0.829705 + 0.558202i \(0.188509\pi\)
−0.829705 + 0.558202i \(0.811491\pi\)
\(828\) 0 0
\(829\) 11.9808 0.416109 0.208055 0.978117i \(-0.433287\pi\)
0.208055 + 0.978117i \(0.433287\pi\)
\(830\) 20.1962i 0.701019i
\(831\) 0 0
\(832\) 0 0
\(833\) 79.9808 2.77117
\(834\) 0 0
\(835\) 4.39230 0.152002
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 16.3923i 0.566263i
\(839\) 12.0000i 0.414286i 0.978311 + 0.207143i \(0.0664165\pi\)
−0.978311 + 0.207143i \(0.933583\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −0.464102 −0.0159940
\(843\) 0 0
\(844\) 3.60770 0.124182
\(845\) 0 0
\(846\) 0 0
\(847\) − 53.9090i − 1.85233i
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) 10.3923i 0.356453i
\(851\) 6.58846i 0.225849i
\(852\) 0 0
\(853\) 9.00000i 0.308154i 0.988059 + 0.154077i \(0.0492404\pi\)
−0.988059 + 0.154077i \(0.950760\pi\)
\(854\) 22.7321 0.777875
\(855\) 0 0
\(856\) − 8.19615i − 0.280139i
\(857\) −54.3731 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(858\) 0 0
\(859\) 10.5885 0.361273 0.180637 0.983550i \(-0.442184\pi\)
0.180637 + 0.983550i \(0.442184\pi\)
\(860\) − 10.7321i − 0.365960i
\(861\) 0 0
\(862\) 27.8038 0.947003
\(863\) 4.48334i 0.152615i 0.997084 + 0.0763073i \(0.0243130\pi\)
−0.997084 + 0.0763073i \(0.975687\pi\)
\(864\) 0 0
\(865\) − 28.3923i − 0.965367i
\(866\) − 33.7846i − 1.14805i
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 58.6410i 1.98926i
\(870\) 0 0
\(871\) 0 0
\(872\) 16.3923 0.555113
\(873\) 0 0
\(874\) −2.78461 −0.0941908
\(875\) 57.3731 1.93956
\(876\) 0 0
\(877\) − 43.3923i − 1.46525i −0.680630 0.732627i \(-0.738294\pi\)
0.680630 0.732627i \(-0.261706\pi\)
\(878\) 16.5885i 0.559833i
\(879\) 0 0
\(880\) −8.19615 −0.276292
\(881\) 37.9808 1.27960 0.639802 0.768540i \(-0.279016\pi\)
0.639802 + 0.768540i \(0.279016\pi\)
\(882\) 0 0
\(883\) 24.7846 0.834069 0.417034 0.908891i \(-0.363070\pi\)
0.417034 + 0.908891i \(0.363070\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 4.39230i − 0.147562i
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 18.9282i 0.634832i
\(890\) 4.39230i 0.147230i
\(891\) 0 0
\(892\) − 18.9282i − 0.633763i
\(893\) 1.60770 0.0537995
\(894\) 0 0
\(895\) 14.1962i 0.474525i
\(896\) −4.73205 −0.158087
\(897\) 0 0
\(898\) −33.4641 −1.11671
\(899\) − 7.60770i − 0.253731i
\(900\) 0 0
\(901\) 15.5885 0.519327
\(902\) 2.19615i 0.0731239i
\(903\) 0 0
\(904\) − 11.1962i − 0.372378i
\(905\) − 20.0718i − 0.667209i
\(906\) 0 0
\(907\) 41.1769 1.36726 0.683629 0.729830i \(-0.260401\pi\)
0.683629 + 0.729830i \(0.260401\pi\)
\(908\) 9.80385i 0.325352i
\(909\) 0 0
\(910\) 0 0
\(911\) −37.1769 −1.23173 −0.615863 0.787853i \(-0.711193\pi\)
−0.615863 + 0.787853i \(0.711193\pi\)
\(912\) 0 0
\(913\) −55.1769 −1.82609
\(914\) −19.9808 −0.660905
\(915\) 0 0
\(916\) − 19.8564i − 0.656074i
\(917\) − 77.5692i − 2.56156i
\(918\) 0 0
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) 3.80385 0.125409
\(921\) 0 0
\(922\) −19.9808 −0.658031
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000i 0.197279i
\(926\) −26.1962 −0.860859
\(927\) 0 0
\(928\) − 3.00000i − 0.0984798i
\(929\) − 34.6077i − 1.13544i −0.823221 0.567721i \(-0.807825\pi\)
0.823221 0.567721i \(-0.192175\pi\)
\(930\) 0 0
\(931\) 19.5167i 0.639633i
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) − 36.5885i − 1.19721i
\(935\) 42.5885 1.39279
\(936\) 0 0
\(937\) 5.39230 0.176159 0.0880795 0.996113i \(-0.471927\pi\)
0.0880795 + 0.996113i \(0.471927\pi\)
\(938\) − 50.7846i − 1.65818i
\(939\) 0 0
\(940\) −2.19615 −0.0716306
\(941\) 2.78461i 0.0907757i 0.998969 + 0.0453878i \(0.0144524\pi\)
−0.998969 + 0.0453878i \(0.985548\pi\)
\(942\) 0 0
\(943\) − 1.01924i − 0.0331910i
\(944\) − 13.8564i − 0.450988i
\(945\) 0 0
\(946\) 29.3205 0.953292
\(947\) 42.9282i 1.39498i 0.716595 + 0.697490i \(0.245700\pi\)
−0.716595 + 0.697490i \(0.754300\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.53590 −0.0822754
\(951\) 0 0
\(952\) 24.5885 0.796916
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) 36.0000i 1.16493i
\(956\) 24.5885i 0.795248i
\(957\) 0 0
\(958\) 35.3205 1.14115
\(959\) −42.5885 −1.37525
\(960\) 0 0
\(961\) 24.5692 0.792555
\(962\) 0 0
\(963\) 0 0
\(964\) − 0.803848i − 0.0258902i
\(965\) −22.1769 −0.713900
\(966\) 0 0
\(967\) 14.8756i 0.478368i 0.970974 + 0.239184i \(0.0768800\pi\)
−0.970974 + 0.239184i \(0.923120\pi\)
\(968\) − 11.3923i − 0.366163i
\(969\) 0 0
\(970\) 10.3923i 0.333677i
\(971\) 13.1769 0.422867 0.211434 0.977392i \(-0.432187\pi\)
0.211434 + 0.977392i \(0.432187\pi\)
\(972\) 0 0
\(973\) − 18.9282i − 0.606810i
\(974\) −9.12436 −0.292363
\(975\) 0 0
\(976\) 4.80385 0.153767
\(977\) − 16.8564i − 0.539284i −0.962961 0.269642i \(-0.913095\pi\)
0.962961 0.269642i \(-0.0869054\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) − 26.6603i − 0.851631i
\(981\) 0 0
\(982\) − 0.588457i − 0.0187784i
\(983\) 20.7846i 0.662926i 0.943468 + 0.331463i \(0.107542\pi\)
−0.943468 + 0.331463i \(0.892458\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 15.5885i 0.496438i
\(987\) 0 0
\(988\) 0 0
\(989\) −13.6077 −0.432700
\(990\) 0 0
\(991\) 29.3731 0.933066 0.466533 0.884504i \(-0.345503\pi\)
0.466533 + 0.884504i \(0.345503\pi\)
\(992\) −2.53590 −0.0805149
\(993\) 0 0
\(994\) 38.7846i 1.23017i
\(995\) 14.8756i 0.471590i
\(996\) 0 0
\(997\) 13.1962 0.417926 0.208963 0.977924i \(-0.432991\pi\)
0.208963 + 0.977924i \(0.432991\pi\)
\(998\) 0 0
\(999\) 0 0
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 3042.2.b.l.1351.1 4
3.2 odd 2 1014.2.b.d.337.4 4
13.5 odd 4 3042.2.a.s.1.1 2
13.8 odd 4 3042.2.a.v.1.2 2
13.9 even 3 234.2.l.a.127.1 4
13.10 even 6 234.2.l.a.199.1 4
13.12 even 2 inner 3042.2.b.l.1351.4 4
39.2 even 12 1014.2.e.h.529.2 4
39.5 even 4 1014.2.a.j.1.2 2
39.8 even 4 1014.2.a.h.1.1 2
39.11 even 12 1014.2.e.j.529.1 4
39.17 odd 6 1014.2.i.f.361.1 4
39.20 even 12 1014.2.e.j.991.1 4
39.23 odd 6 78.2.i.b.43.2 4
39.29 odd 6 1014.2.i.f.823.1 4
39.32 even 12 1014.2.e.h.991.2 4
39.35 odd 6 78.2.i.b.49.2 yes 4
39.38 odd 2 1014.2.b.d.337.1 4
52.23 odd 6 1872.2.by.k.433.2 4
52.35 odd 6 1872.2.by.k.1297.2 4
156.23 even 6 624.2.bv.d.433.2 4
156.35 even 6 624.2.bv.d.49.2 4
156.47 odd 4 8112.2.a.bq.1.1 2
156.83 odd 4 8112.2.a.bx.1.2 2
195.23 even 12 1950.2.y.h.199.2 4
195.62 even 12 1950.2.y.a.199.1 4
195.74 odd 6 1950.2.bc.c.751.1 4
195.113 even 12 1950.2.y.a.49.1 4
195.152 even 12 1950.2.y.h.49.2 4
195.179 odd 6 1950.2.bc.c.901.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.b.43.2 4 39.23 odd 6
78.2.i.b.49.2 yes 4 39.35 odd 6
234.2.l.a.127.1 4 13.9 even 3
234.2.l.a.199.1 4 13.10 even 6
624.2.bv.d.49.2 4 156.35 even 6
624.2.bv.d.433.2 4 156.23 even 6
1014.2.a.h.1.1 2 39.8 even 4
1014.2.a.j.1.2 2 39.5 even 4
1014.2.b.d.337.1 4 39.38 odd 2
1014.2.b.d.337.4 4 3.2 odd 2
1014.2.e.h.529.2 4 39.2 even 12
1014.2.e.h.991.2 4 39.32 even 12
1014.2.e.j.529.1 4 39.11 even 12
1014.2.e.j.991.1 4 39.20 even 12
1014.2.i.f.361.1 4 39.17 odd 6
1014.2.i.f.823.1 4 39.29 odd 6
1872.2.by.k.433.2 4 52.23 odd 6
1872.2.by.k.1297.2 4 52.35 odd 6
1950.2.y.a.49.1 4 195.113 even 12
1950.2.y.a.199.1 4 195.62 even 12
1950.2.y.h.49.2 4 195.152 even 12
1950.2.y.h.199.2 4 195.23 even 12
1950.2.bc.c.751.1 4 195.74 odd 6
1950.2.bc.c.901.1 4 195.179 odd 6
3042.2.a.s.1.1 2 13.5 odd 4
3042.2.a.v.1.2 2 13.8 odd 4
3042.2.b.l.1351.1 4 1.1 even 1 trivial
3042.2.b.l.1351.4 4 13.12 even 2 inner
8112.2.a.bq.1.1 2 156.47 odd 4
8112.2.a.bx.1.2 2 156.83 odd 4