Properties

Label 1050.2.g.d.799.1
Level $1050$
Weight $2$
Character 1050.799
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
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] = [1050,2,Mod(799,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, 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("1050.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.g (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.799
Dual form 1050.2.g.d.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -1.00000 q^{21} -4.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} +1.00000i q^{42} -12.0000i q^{43} -4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +2.00000i q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} +12.0000i q^{83} +1.00000 q^{84} -12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -2.00000 q^{89} -2.00000 q^{91} +8.00000i q^{92} +8.00000i q^{93} +8.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} - 4 q^{39} + 4 q^{41} - 8 q^{44} - 16 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} + 2 q^{56} - 8 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} - 16 q^{69} + 16 q^{71} + 4 q^{74} - 8 q^{76} + 2 q^{81} + 2 q^{84} - 24 q^{86} - 4 q^{89} - 4 q^{91} + 16 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) − 4.00000i − 0.852803i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000i 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000i 0.154303i
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 4.00000i − 0.455842i
\(78\) 2.00000i 0.226455i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 6.00000i 0.643268i
\(88\) 4.00000i 0.426401i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 8.00000i 0.834058i
\(93\) 8.00000i 0.829561i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 20.0000i 1.93347i 0.255774 + 0.966736i \(0.417670\pi\)
−0.255774 + 0.966736i \(0.582330\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000i 0.184900i
\(118\) 4.00000i 0.368230i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) − 2.00000i − 0.180334i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) − 8.00000i − 0.671345i
\(143\) − 8.00000i − 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 1.00000i 0.0824786i
\(148\) − 2.00000i − 0.164399i
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 2.00000i 0.161690i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 12.0000i 0.914991i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000i 0.300658i
\(178\) 2.00000i 0.149906i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 2.00000i 0.147844i
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) − 8.00000i − 0.585018i
\(188\) − 8.00000i − 0.583460i
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) − 6.00000i − 0.422159i
\(203\) 6.00000i 0.421117i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 8.00000i 0.556038i
\(208\) − 2.00000i − 0.138675i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 8.00000i − 0.548151i
\(214\) 20.0000 1.36717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 8.00000i 0.543075i
\(218\) − 2.00000i − 0.135457i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) − 2.00000i − 0.134231i
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) − 6.00000i − 0.393919i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 8.00000i − 0.509028i
\(248\) − 8.00000i − 0.508001i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 32.0000i − 2.01182i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 2.00000i 0.122398i
\(268\) 12.0000i 0.733017i
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 2.00000i 0.121046i
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 14.0000i − 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) − 2.00000i − 0.118056i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 14.0000i 0.819288i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000i 0.232104i
\(298\) − 18.0000i − 1.04271i
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) − 8.00000i − 0.460348i
\(303\) − 6.00000i − 0.344691i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 4.00000i 0.227921i
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 34.0000i 1.92179i 0.276907 + 0.960897i \(0.410691\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 20.0000 1.11629
\(322\) 8.00000i 0.445823i
\(323\) − 8.00000i − 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) − 2.00000i − 0.110600i
\(328\) 2.00000i 0.110432i
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 4.00000i 0.216295i
\(343\) 1.00000i 0.0539949i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) − 4.00000i − 0.213201i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 2.00000i 0.105851i
\(358\) − 20.0000i − 1.05703i
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 10.0000i 0.525588i
\(363\) − 5.00000i − 0.262432i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) − 8.00000i − 0.414781i
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 12.0000i 0.618031i
\(378\) − 1.00000i − 0.0514344i
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000i 0.818631i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000i 0.609994i
\(388\) 10.0000i 0.507673i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 1.00000i − 0.0505076i
\(393\) − 12.0000i − 0.605320i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 12.0000i 0.598506i
\(403\) 16.0000i 0.797017i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 8.00000i 0.396545i
\(408\) − 2.00000i − 0.0990148i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) − 8.00000i − 0.394132i
\(413\) 4.00000i 0.196827i
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 20.0000i 0.979404i
\(418\) − 16.0000i − 0.782586i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) − 8.00000i − 0.388973i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 2.00000i 0.0967868i
\(428\) − 20.0000i − 0.966736i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 6.00000i − 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) − 32.0000i − 1.53077i
\(438\) 14.0000i 0.668946i
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000i 0.190261i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 0 0
\(447\) − 18.0000i − 0.851371i
\(448\) 1.00000i 0.0472456i
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 14.0000i 0.658505i
\(453\) − 8.00000i − 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 38.0000i 1.77757i 0.458329 + 0.888783i \(0.348448\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 20.0000i 0.925490i 0.886492 + 0.462745i \(0.153135\pi\)
−0.886492 + 0.462745i \(0.846865\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) − 4.00000i − 0.184115i
\(473\) − 48.0000i − 2.20704i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) − 6.00000i − 0.274721i
\(478\) − 16.0000i − 0.731823i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) − 18.0000i − 0.819878i
\(483\) 8.00000i 0.364013i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 24.0000i 1.08754i 0.839233 + 0.543772i \(0.183004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 12.0000i 0.540453i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 8.00000i − 0.358849i
\(498\) − 12.0000i − 0.537733i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) − 4.00000i − 0.178529i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 32.0000i 1.40736i
\(518\) − 2.00000i − 0.0878750i
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 16.0000i 0.696971i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 4.00000i 0.173422i
\(533\) − 4.00000i − 0.173259i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 20.0000i − 0.863064i
\(538\) 30.0000i 1.29339i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 24.0000i 1.03089i
\(543\) 10.0000i 0.429141i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 10.0000i 0.427179i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) − 8.00000i − 0.340503i
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) − 10.0000i − 0.421825i
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 1.00000i − 0.0419961i
\(568\) 8.00000i 0.335673i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 16.0000i 0.668410i
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 26.0000i − 1.08239i −0.840896 0.541197i \(-0.817971\pi\)
0.840896 0.541197i \(-0.182029\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 10.0000i 0.414513i
\(583\) 24.0000i 0.993978i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) − 1.00000i − 0.0412393i
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 2.00000i 0.0821995i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) − 16.0000i − 0.654836i
\(598\) 16.0000i 0.654289i
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 12.0000i 0.489083i
\(603\) 12.0000i 0.488678i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) − 2.00000i − 0.0808452i
\(613\) − 34.0000i − 1.37325i −0.727013 0.686624i \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) − 10.0000i − 0.402585i −0.979531 0.201292i \(-0.935486\pi\)
0.979531 0.201292i \(-0.0645141\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) − 8.00000i − 0.320771i
\(623\) 2.00000i 0.0801283i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 34.0000 1.35891
\(627\) − 16.0000i − 0.638978i
\(628\) 14.0000i 0.558661i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) − 20.0000i − 0.794929i
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 2.00000i 0.0792429i
\(638\) 24.0000i 0.950169i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) − 20.0000i − 0.789337i
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) − 12.0000i − 0.469956i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 14.0000i 0.546192i
\(658\) − 8.00000i − 0.311872i
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 4.00000i 0.155347i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 48.0000i 1.85857i
\(668\) − 16.0000i − 0.619059i
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 1.00000i 0.0385758i
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 14.0000i 0.537667i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 32.0000i 1.22534i
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 26.0000i − 0.991962i
\(688\) − 12.0000i − 0.457496i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 4.00000i 0.151947i
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 4.00000i − 0.151511i
\(698\) − 34.0000i − 1.28692i
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 8.00000i 0.301726i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) − 6.00000i − 0.225653i
\(708\) − 4.00000i − 0.150329i
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 2.00000i − 0.0749532i
\(713\) 64.0000i 2.39682i
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) − 16.0000i − 0.597531i
\(718\) − 8.00000i − 0.298557i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 3.00000i 0.111648i
\(723\) − 18.0000i − 0.669427i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 24.0000i 0.890111i 0.895503 + 0.445055i \(0.146816\pi\)
−0.895503 + 0.445055i \(0.853184\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) − 2.00000i − 0.0739221i
\(733\) 30.0000i 1.10808i 0.832492 + 0.554038i \(0.186914\pi\)
−0.832492 + 0.554038i \(0.813086\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) − 48.0000i − 1.76810i
\(738\) 2.00000i 0.0736210i
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 6.00000i − 0.220267i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) − 12.0000i − 0.439057i
\(748\) 8.00000i 0.292509i
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000i 0.291730i
\(753\) − 4.00000i − 0.145768i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 34.0000i 1.23575i 0.786276 + 0.617876i \(0.212006\pi\)
−0.786276 + 0.617876i \(0.787994\pi\)
\(758\) 28.0000i 1.01701i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) − 2.00000i − 0.0724049i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 8.00000i 0.288863i
\(768\) − 1.00000i − 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 14.0000i 0.503871i
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) − 2.00000i − 0.0717496i
\(778\) − 18.0000i − 0.645331i
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 16.0000i 0.572159i
\(783\) − 6.00000i − 0.214423i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) − 4.00000i − 0.142134i
\(793\) 4.00000i 0.142044i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) − 18.0000i − 0.635602i
\(803\) − 56.0000i − 1.97620i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 30.0000i 1.05605i
\(808\) 6.00000i 0.211079i
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 24.0000i 0.841717i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 48.0000i − 1.67931i
\(818\) 10.0000i 0.349642i
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 10.0000i 0.348790i
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 2.00000i 0.0693375i
\(833\) 2.00000i 0.0692959i
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) − 8.00000i − 0.276520i
\(838\) 12.0000i 0.414533i
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000i 0.896019i
\(843\) − 10.0000i − 0.344418i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 5.00000i − 0.171802i
\(848\) 6.00000i 0.206041i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 8.00000i 0.274075i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −6.00000 −0.203888
\(867\) − 13.0000i − 0.441503i
\(868\) − 8.00000i − 0.271538i
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 2.00000i 0.0677285i
\(873\) 10.0000i 0.338449i
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 10.0000i 0.337676i 0.985644 + 0.168838i \(0.0540015\pi\)
−0.985644 + 0.168838i \(0.945999\pi\)
\(878\) 32.0000i 1.07995i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 16.0000i − 0.537227i −0.963248 0.268614i \(-0.913434\pi\)
0.963248 0.268614i \(-0.0865655\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 16.0000i 0.534224i
\(898\) − 14.0000i − 0.467186i
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 8.00000i − 0.266371i
\(903\) 12.0000i 0.399335i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 48.0000i 1.58857i
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) − 12.0000i − 0.396275i
\(918\) − 2.00000i − 0.0660098i
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 34.0000i 1.11973i
\(923\) − 16.0000i − 0.526646i
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 8.00000i − 0.262754i
\(928\) 6.00000i 0.196960i
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) − 10.0000i − 0.327561i
\(933\) − 8.00000i − 0.261908i
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) − 18.0000i − 0.588034i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949923\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 34.0000 1.10955
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 14.0000i 0.456145i
\(943\) − 16.0000i − 0.521032i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) − 2.00000i − 0.0648204i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 24.0000i 0.775810i
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 4.00000i − 0.128965i
\(963\) − 20.0000i − 0.644491i
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 20.0000i 0.641171i
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) − 36.0000i − 1.14881i
\(983\) 32.0000i 1.02064i 0.859984 + 0.510321i \(0.170473\pi\)
−0.859984 + 0.510321i \(0.829527\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) − 8.00000i − 0.254643i
\(988\) 8.00000i 0.254514i
\(989\) −96.0000 −3.05262
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 28.0000i − 0.888553i
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) 4.00000i 0.126618i
\(999\) −2.00000 −0.0632772
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 1050.2.g.d.799.1 2
3.2 odd 2 3150.2.g.e.2899.2 2
5.2 odd 4 210.2.a.c.1.1 1
5.3 odd 4 1050.2.a.h.1.1 1
5.4 even 2 inner 1050.2.g.d.799.2 2
15.2 even 4 630.2.a.b.1.1 1
15.8 even 4 3150.2.a.w.1.1 1
15.14 odd 2 3150.2.g.e.2899.1 2
20.3 even 4 8400.2.a.p.1.1 1
20.7 even 4 1680.2.a.q.1.1 1
35.2 odd 12 1470.2.i.f.361.1 2
35.12 even 12 1470.2.i.b.361.1 2
35.13 even 4 7350.2.a.p.1.1 1
35.17 even 12 1470.2.i.b.961.1 2
35.27 even 4 1470.2.a.q.1.1 1
35.32 odd 12 1470.2.i.f.961.1 2
40.27 even 4 6720.2.a.k.1.1 1
40.37 odd 4 6720.2.a.bp.1.1 1
60.47 odd 4 5040.2.a.i.1.1 1
105.62 odd 4 4410.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 1 5.2 odd 4
630.2.a.b.1.1 1 15.2 even 4
1050.2.a.h.1.1 1 5.3 odd 4
1050.2.g.d.799.1 2 1.1 even 1 trivial
1050.2.g.d.799.2 2 5.4 even 2 inner
1470.2.a.q.1.1 1 35.27 even 4
1470.2.i.b.361.1 2 35.12 even 12
1470.2.i.b.961.1 2 35.17 even 12
1470.2.i.f.361.1 2 35.2 odd 12
1470.2.i.f.961.1 2 35.32 odd 12
1680.2.a.q.1.1 1 20.7 even 4
3150.2.a.w.1.1 1 15.8 even 4
3150.2.g.e.2899.1 2 15.14 odd 2
3150.2.g.e.2899.2 2 3.2 odd 2
4410.2.a.l.1.1 1 105.62 odd 4
5040.2.a.i.1.1 1 60.47 odd 4
6720.2.a.k.1.1 1 40.27 even 4
6720.2.a.bp.1.1 1 40.37 odd 4
7350.2.a.p.1.1 1 35.13 even 4
8400.2.a.p.1.1 1 20.3 even 4