Properties

Label 3150.2.g.p.2899.2
Level $3150$
Weight $2$
Character 3150.2899
Analytic conductor $25.153$
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] = [3150,2,Mod(2899,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, 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("3150.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.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: \(25.1528766367\)
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 1050)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3150.2899
Dual form 3150.2.g.p.2899.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +2.00000 q^{11} +7.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.00000i q^{17} -8.00000 q^{19} +2.00000i q^{22} +5.00000i q^{23} -7.00000 q^{26} +1.00000i q^{28} +9.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +7.00000 q^{34} +2.00000i q^{37} -8.00000i q^{38} -11.0000 q^{41} +3.00000i q^{43} -2.00000 q^{44} -5.00000 q^{46} +4.00000i q^{47} -1.00000 q^{49} -7.00000i q^{52} -3.00000i q^{53} -1.00000 q^{56} +9.00000i q^{58} +7.00000 q^{59} -5.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} +7.00000i q^{68} +4.00000 q^{71} +10.0000i q^{73} -2.00000 q^{74} +8.00000 q^{76} -2.00000i q^{77} +6.00000 q^{79} -11.0000i q^{82} +9.00000i q^{83} -3.00000 q^{86} -2.00000i q^{88} -10.0000 q^{89} +7.00000 q^{91} -5.00000i q^{92} -4.00000 q^{94} -10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{11} + 2 q^{14} + 2 q^{16} - 16 q^{19} - 14 q^{26} + 18 q^{29} + 2 q^{31} + 14 q^{34} - 22 q^{41} - 4 q^{44} - 10 q^{46} - 2 q^{49} - 2 q^{56} + 14 q^{59} - 10 q^{61} - 2 q^{64} + 8 q^{71} - 4 q^{74} + 16 q^{76} + 12 q^{79} - 6 q^{86} - 20 q^{89} + 14 q^{91} - 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 7.00000i 1.94145i 0.240192 + 0.970725i \(0.422790\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 5.00000i 1.04257i 0.853382 + 0.521286i \(0.174548\pi\)
−0.853382 + 0.521286i \(0.825452\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −7.00000 −1.37281
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 8.00000i − 1.29777i
\(39\) 0 0
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) 3.00000i 0.457496i 0.973486 + 0.228748i \(0.0734631\pi\)
−0.973486 + 0.228748i \(0.926537\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 7.00000i − 0.970725i
\(53\) − 3.00000i − 0.412082i −0.978543 0.206041i \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 9.00000i 1.18176i
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 7.00000i 0.848875i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 11.0000i − 1.21475i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.00000 −0.323498
\(87\) 0 0
\(88\) − 2.00000i − 0.213201i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) − 5.00000i − 0.521286i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 5.00000i 0.492665i 0.969185 + 0.246332i \(0.0792255\pi\)
−0.969185 + 0.246332i \(0.920775\pi\)
\(104\) 7.00000 0.686406
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) − 2.00000i − 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 7.00000i 0.644402i
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 5.00000i − 0.452679i
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) 16.0000i 1.36697i 0.729964 + 0.683486i \(0.239537\pi\)
−0.729964 + 0.683486i \(0.760463\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.00000i 0.335673i
\(143\) 14.0000i 1.17074i
\(144\) 0 0
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 8.00000i 0.648886i
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 0 0
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) − 15.0000i − 1.17489i −0.809264 0.587445i \(-0.800134\pi\)
0.809264 0.587445i \(-0.199866\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) −36.0000 −2.76923
\(170\) 0 0
\(171\) 0 0
\(172\) − 3.00000i − 0.228748i
\(173\) 16.0000i 1.21646i 0.793762 + 0.608229i \(0.208120\pi\)
−0.793762 + 0.608229i \(0.791880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 7.00000i 0.518875i
\(183\) 0 0
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 0 0
\(187\) − 14.0000i − 1.02378i
\(188\) − 4.00000i − 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 0.0723575 0.0361787 0.999345i \(-0.488481\pi\)
0.0361787 + 0.999345i \(0.488481\pi\)
\(192\) 0 0
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 21.0000i − 1.49619i −0.663593 0.748094i \(-0.730969\pi\)
0.663593 0.748094i \(-0.269031\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 9.00000i − 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.00000 −0.348367
\(207\) 0 0
\(208\) 7.00000i 0.485363i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 3.00000i 0.206041i
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.00000i − 0.0678844i
\(218\) − 4.00000i − 0.270914i
\(219\) 0 0
\(220\) 0 0
\(221\) 49.0000 3.29610
\(222\) 0 0
\(223\) 9.00000i 0.602685i 0.953516 + 0.301342i \(0.0974347\pi\)
−0.953516 + 0.301342i \(0.902565\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 3.00000i − 0.199117i −0.995032 0.0995585i \(-0.968257\pi\)
0.995032 0.0995585i \(-0.0317430\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 9.00000i − 0.590879i
\(233\) − 16.0000i − 1.04819i −0.851658 0.524097i \(-0.824403\pi\)
0.851658 0.524097i \(-0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 0 0
\(238\) − 7.00000i − 0.453743i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) − 56.0000i − 3.56319i
\(248\) − 1.00000i − 0.0635001i
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) 10.0000i 0.628695i
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.00000i 0.311891i 0.987766 + 0.155946i \(0.0498425\pi\)
−0.987766 + 0.155946i \(0.950158\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000i 0.554964i 0.960731 + 0.277482i \(0.0894999\pi\)
−0.960731 + 0.277482i \(0.910500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) − 7.00000i − 0.424437i
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −14.0000 −0.827837
\(287\) 11.0000i 0.649309i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) − 10.0000i − 0.585206i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) − 3.00000i − 0.173785i
\(299\) −35.0000 −2.02410
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) − 22.0000i − 1.26596i
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 23.0000i 1.29181i 0.763418 + 0.645904i \(0.223520\pi\)
−0.763418 + 0.645904i \(0.776480\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 5.00000i 0.278639i
\(323\) 56.0000i 3.11592i
\(324\) 0 0
\(325\) 0 0
\(326\) 15.0000 0.830773
\(327\) 0 0
\(328\) 11.0000i 0.607373i
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) − 36.0000i − 1.95814i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 3.00000 0.161749
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) − 6.00000i − 0.322097i −0.986947 0.161048i \(-0.948512\pi\)
0.986947 0.161048i \(-0.0514875\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000i 0.106600i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) − 2.00000i − 0.105703i
\(359\) 19.0000 1.00278 0.501391 0.865221i \(-0.332822\pi\)
0.501391 + 0.865221i \(0.332822\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 2.00000i 0.105118i
\(363\) 0 0
\(364\) −7.00000 −0.366900
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 5.00000i 0.260643i
\(369\) 0 0
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 14.0000 0.723923
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 63.0000i 3.24467i
\(378\) 0 0
\(379\) −7.00000 −0.359566 −0.179783 0.983706i \(-0.557540\pi\)
−0.179783 + 0.983706i \(0.557540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.00000i 0.0511645i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 10.0000i 0.507673i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 35.0000 1.77003
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 21.0000 1.05796
\(395\) 0 0
\(396\) 0 0
\(397\) − 37.0000i − 1.85698i −0.371361 0.928488i \(-0.621109\pi\)
0.371361 0.928488i \(-0.378891\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 7.00000i 0.348695i
\(404\) 0 0
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 5.00000i − 0.246332i
\(413\) − 7.00000i − 0.344447i
\(414\) 0 0
\(415\) 0 0
\(416\) −7.00000 −0.343203
\(417\) 0 0
\(418\) − 16.0000i − 0.782586i
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) − 13.0000i − 0.632830i
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000i 0.241967i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) −39.0000 −1.87856 −0.939282 0.343146i \(-0.888507\pi\)
−0.939282 + 0.343146i \(0.888507\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i 0.989553 + 0.144171i \(0.0460515\pi\)
−0.989553 + 0.144171i \(0.953949\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) − 40.0000i − 1.91346i
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 49.0000i 2.33069i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −9.00000 −0.426162
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −22.0000 −1.03594
\(452\) − 14.0000i − 0.658505i
\(453\) 0 0
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.00000i − 0.327446i −0.986506 0.163723i \(-0.947650\pi\)
0.986506 0.163723i \(-0.0523504\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) − 29.0000i − 1.34196i −0.741475 0.670980i \(-0.765874\pi\)
0.741475 0.670980i \(-0.234126\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) − 7.00000i − 0.322201i
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 7.00000 0.320844
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) − 30.0000i − 1.36646i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) − 6.00000i − 0.271886i −0.990717 0.135943i \(-0.956594\pi\)
0.990717 0.135943i \(-0.0434064\pi\)
\(488\) 5.00000i 0.226339i
\(489\) 0 0
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) 0 0
\(493\) − 63.0000i − 2.83738i
\(494\) 56.0000 2.51956
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 4.00000i − 0.179425i
\(498\) 0 0
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.00000i 0.223161i
\(503\) 26.0000i 1.15928i 0.814872 + 0.579641i \(0.196807\pi\)
−0.814872 + 0.579641i \(0.803193\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −10.0000 −0.444554
\(507\) 0 0
\(508\) − 18.0000i − 0.798621i
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −5.00000 −0.220541
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000i 0.351840i
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) − 7.00000i − 0.304925i
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) − 8.00000i − 0.346844i
\(533\) − 77.0000i − 3.33524i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) − 6.00000i − 0.258678i
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 12.0000i 0.515444i
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) 35.0000i 1.49649i 0.663421 + 0.748246i \(0.269104\pi\)
−0.663421 + 0.748246i \(0.730896\pi\)
\(548\) − 16.0000i − 0.683486i
\(549\) 0 0
\(550\) 0 0
\(551\) −72.0000 −3.06730
\(552\) 0 0
\(553\) − 6.00000i − 0.255146i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 0 0
\(559\) −21.0000 −0.888205
\(560\) 0 0
\(561\) 0 0
\(562\) 8.00000i 0.337460i
\(563\) − 27.0000i − 1.13791i −0.822367 0.568957i \(-0.807347\pi\)
0.822367 0.568957i \(-0.192653\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) − 4.00000i − 0.167836i
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) − 14.0000i − 0.585369i
\(573\) 0 0
\(574\) −11.0000 −0.459131
\(575\) 0 0
\(576\) 0 0
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) − 32.0000i − 1.33102i
\(579\) 0 0
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) − 6.00000i − 0.248495i
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 39.0000i 1.60970i 0.593477 + 0.804851i \(0.297755\pi\)
−0.593477 + 0.804851i \(0.702245\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) 0 0
\(598\) − 35.0000i − 1.43126i
\(599\) −1.00000 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 3.00000i 0.122271i
\(603\) 0 0
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) 36.0000i 1.46119i 0.682808 + 0.730597i \(0.260758\pi\)
−0.682808 + 0.730597i \(0.739242\pi\)
\(608\) − 8.00000i − 0.324443i
\(609\) 0 0
\(610\) 0 0
\(611\) −28.0000 −1.13276
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) − 38.0000i − 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 2.00000i − 0.0801927i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) 42.0000 1.67199 0.835997 0.548734i \(-0.184890\pi\)
0.835997 + 0.548734i \(0.184890\pi\)
\(632\) − 6.00000i − 0.238667i
\(633\) 0 0
\(634\) −23.0000 −0.913447
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.00000i − 0.277350i
\(638\) 18.0000i 0.712627i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 10.0000i 0.394362i 0.980367 + 0.197181i \(0.0631786\pi\)
−0.980367 + 0.197181i \(0.936821\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) −56.0000 −2.20329
\(647\) 46.0000i 1.80845i 0.427060 + 0.904223i \(0.359549\pi\)
−0.427060 + 0.904223i \(0.640451\pi\)
\(648\) 0 0
\(649\) 14.0000 0.549548
\(650\) 0 0
\(651\) 0 0
\(652\) 15.0000i 0.587445i
\(653\) 34.0000i 1.33052i 0.746611 + 0.665261i \(0.231680\pi\)
−0.746611 + 0.665261i \(0.768320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −11.0000 −0.429478
\(657\) 0 0
\(658\) 4.00000i 0.155936i
\(659\) 34.0000 1.32445 0.662226 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 25.0000i 0.971653i
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 45.0000i 1.74241i
\(668\) − 2.00000i − 0.0773823i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) − 13.0000i − 0.501113i −0.968102 0.250557i \(-0.919386\pi\)
0.968102 0.250557i \(-0.0806136\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) 2.00000i 0.0765840i
\(683\) − 38.0000i − 1.45403i −0.686622 0.727015i \(-0.740907\pi\)
0.686622 0.727015i \(-0.259093\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 3.00000i 0.114374i
\(689\) 21.0000 0.800036
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) − 16.0000i − 0.608229i
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 77.0000i 2.91658i
\(698\) 7.00000i 0.264954i
\(699\) 0 0
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 0 0
\(703\) − 16.0000i − 0.603451i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) 5.00000i 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 19.0000i 0.709074i
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) 45.0000i 1.67473i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 21.0000i − 0.778847i −0.921059 0.389423i \(-0.872674\pi\)
0.921059 0.389423i \(-0.127326\pi\)
\(728\) − 7.00000i − 0.259437i
\(729\) 0 0
\(730\) 0 0
\(731\) 21.0000 0.776713
\(732\) 0 0
\(733\) 45.0000i 1.66211i 0.556188 + 0.831056i \(0.312263\pi\)
−0.556188 + 0.831056i \(0.687737\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 24.0000i 0.884051i
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 3.00000i − 0.110133i
\(743\) 39.0000i 1.43077i 0.698730 + 0.715386i \(0.253749\pi\)
−0.698730 + 0.715386i \(0.746251\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 14.0000i 0.511891i
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 0 0
\(754\) −63.0000 −2.29432
\(755\) 0 0
\(756\) 0 0
\(757\) − 26.0000i − 0.944986i −0.881334 0.472493i \(-0.843354\pi\)
0.881334 0.472493i \(-0.156646\pi\)
\(758\) − 7.00000i − 0.254251i
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) −1.00000 −0.0361787
\(765\) 0 0
\(766\) 0 0
\(767\) 49.0000i 1.76929i
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) − 28.0000i − 1.00709i −0.863969 0.503545i \(-0.832029\pi\)
0.863969 0.503545i \(-0.167971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) 88.0000 3.15293
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 35.0000i 1.25160i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 26.0000i − 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) 21.0000i 0.748094i
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) − 35.0000i − 1.24289i
\(794\) 37.0000 1.31308
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) 12.0000i 0.423735i
\(803\) 20.0000i 0.705785i
\(804\) 0 0
\(805\) 0 0
\(806\) −7.00000 −0.246564
\(807\) 0 0
\(808\) 0 0
\(809\) −44.0000 −1.54696 −0.773479 0.633822i \(-0.781485\pi\)
−0.773479 + 0.633822i \(0.781485\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) 9.00000i 0.315838i
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) − 24.0000i − 0.839654i
\(818\) − 10.0000i − 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) 7.00000 0.243561
\(827\) − 6.00000i − 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 7.00000i − 0.242681i
\(833\) 7.00000i 0.242536i
\(834\) 0 0
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) − 3.00000i − 0.103633i
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 10.0000i 0.344623i
\(843\) 0 0
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) − 3.00000i − 0.103020i
\(849\) 0 0
\(850\) 0 0
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) − 35.0000i − 1.19838i −0.800608 0.599189i \(-0.795490\pi\)
0.800608 0.599189i \(-0.204510\pi\)
\(854\) −5.00000 −0.171096
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) 0 0
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 39.0000i − 1.32835i
\(863\) − 48.0000i − 1.63394i −0.576681 0.816970i \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −6.00000 −0.203888
\(867\) 0 0
\(868\) 1.00000i 0.0339422i
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −84.0000 −2.84623
\(872\) 4.00000i 0.135457i
\(873\) 0 0
\(874\) 40.0000 1.35302
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 19.0000i 0.641219i
\(879\) 0 0
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) − 23.0000i − 0.774012i −0.922077 0.387006i \(-0.873509\pi\)
0.922077 0.387006i \(-0.126491\pi\)
\(884\) −49.0000 −1.64805
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 14.0000i − 0.470074i −0.971986 0.235037i \(-0.924479\pi\)
0.971986 0.235037i \(-0.0755211\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) − 9.00000i − 0.301342i
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) − 20.0000i − 0.667409i
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) − 22.0000i − 0.732520i
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 39.0000i 1.29497i 0.762077 + 0.647487i \(0.224180\pi\)
−0.762077 + 0.647487i \(0.775820\pi\)
\(908\) 3.00000i 0.0995585i
\(909\) 0 0
\(910\) 0 0
\(911\) 39.0000 1.29213 0.646064 0.763283i \(-0.276414\pi\)
0.646064 + 0.763283i \(0.276414\pi\)
\(912\) 0 0
\(913\) 18.0000i 0.595713i
\(914\) 7.00000 0.231539
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 8.00000i − 0.263466i
\(923\) 28.0000i 0.921631i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) 9.00000i 0.295439i
\(929\) 7.00000 0.229663 0.114831 0.993385i \(-0.463367\pi\)
0.114831 + 0.993385i \(0.463367\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 16.0000i 0.524097i
\(933\) 0 0
\(934\) 29.0000 0.948909
\(935\) 0 0
\(936\) 0 0
\(937\) 6.00000i 0.196011i 0.995186 + 0.0980057i \(0.0312463\pi\)
−0.995186 + 0.0980057i \(0.968754\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) − 55.0000i − 1.79105i
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) −70.0000 −2.27230
\(950\) 0 0
\(951\) 0 0
\(952\) 7.00000i 0.226871i
\(953\) 48.0000i 1.55487i 0.628962 + 0.777436i \(0.283480\pi\)
−0.628962 + 0.777436i \(0.716520\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) − 12.0000i − 0.387702i
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) − 14.0000i − 0.451378i
\(963\) 0 0
\(964\) 30.0000 0.966235
\(965\) 0 0
\(966\) 0 0
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 6.00000 0.192252
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 42.0000i 1.34027i
\(983\) − 32.0000i − 1.02064i −0.859984 0.510321i \(-0.829527\pi\)
0.859984 0.510321i \(-0.170473\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 63.0000 2.00633
\(987\) 0 0
\(988\) 56.0000i 1.78160i
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) 23.0000i 0.728052i
\(999\) 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 3150.2.g.p.2899.2 2
3.2 odd 2 1050.2.g.b.799.1 2
5.2 odd 4 3150.2.a.r.1.1 1
5.3 odd 4 3150.2.a.bd.1.1 1
5.4 even 2 inner 3150.2.g.p.2899.1 2
15.2 even 4 1050.2.a.n.1.1 yes 1
15.8 even 4 1050.2.a.f.1.1 1
15.14 odd 2 1050.2.g.b.799.2 2
60.23 odd 4 8400.2.a.bb.1.1 1
60.47 odd 4 8400.2.a.cb.1.1 1
105.62 odd 4 7350.2.a.cm.1.1 1
105.83 odd 4 7350.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.f.1.1 1 15.8 even 4
1050.2.a.n.1.1 yes 1 15.2 even 4
1050.2.g.b.799.1 2 3.2 odd 2
1050.2.g.b.799.2 2 15.14 odd 2
3150.2.a.r.1.1 1 5.2 odd 4
3150.2.a.bd.1.1 1 5.3 odd 4
3150.2.g.p.2899.1 2 5.4 even 2 inner
3150.2.g.p.2899.2 2 1.1 even 1 trivial
7350.2.a.i.1.1 1 105.83 odd 4
7350.2.a.cm.1.1 1 105.62 odd 4
8400.2.a.bb.1.1 1 60.23 odd 4
8400.2.a.cb.1.1 1 60.47 odd 4