Properties

Label 735.2.g.a.734.12
Level $735$
Weight $2$
Character 735.734
Analytic conductor $5.869$
Analytic rank $0$
Dimension $16$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(734,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.734");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.721389578983833600000000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 734.12
Root \(0.701515 - 1.98043i\) of defining polynomial
Character \(\chi\) \(=\) 735.734
Dual form 735.2.g.a.734.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.40303 q^{2} +1.73205i q^{3} -0.0315060 q^{4} +2.23607i q^{5} +2.43012i q^{6} -2.85026 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.40303 q^{2} +1.73205i q^{3} -0.0315060 q^{4} +2.23607i q^{5} +2.43012i q^{6} -2.85026 q^{8} -3.00000 q^{9} +3.13727i q^{10} -0.0545700i q^{12} -3.87298 q^{15} -3.93600 q^{16} +8.06126i q^{17} -4.20909 q^{18} -6.65977i q^{19} -0.0704496i q^{20} -0.222412 q^{23} -4.93680i q^{24} -5.00000 q^{25} -5.19615i q^{27} -5.43391 q^{30} +10.2178i q^{31} +0.178208 q^{32} +11.3102i q^{34} +0.0945180 q^{36} -9.34386i q^{38} -6.37339i q^{40} -6.70820i q^{45} -0.312051 q^{46} -1.02391i q^{47} -6.81734i q^{48} -7.01515 q^{50} -13.9625 q^{51} +11.0902 q^{53} -7.29036i q^{54} +11.5351 q^{57} +0.122022 q^{60} +15.0780i q^{61} +14.3359i q^{62} +8.12202 q^{64} -0.253978i q^{68} -0.385229i q^{69} +8.55079 q^{72} -8.66025i q^{75} +0.209823i q^{76} +5.83648 q^{79} -8.80115i q^{80} +9.00000 q^{81} +15.0986i q^{83} -18.0255 q^{85} -9.41181i q^{90} +0.00700732 q^{92} -17.6978 q^{93} -1.43658i q^{94} +14.8917 q^{95} +0.308666i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40303 0.992092 0.496046 0.868296i \(-0.334785\pi\)
0.496046 + 0.868296i \(0.334785\pi\)
\(3\) 1.73205i 1.00000i
\(4\) −0.0315060 −0.0157530
\(5\) 2.23607i 1.00000i
\(6\) 2.43012i 0.992092i
\(7\) 0 0
\(8\) −2.85026 −1.00772
\(9\) −3.00000 −1.00000
\(10\) 3.13727i 0.992092i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 0.0545700i − 0.0157530i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.87298 −1.00000
\(16\) −3.93600 −0.983999
\(17\) 8.06126i 1.95514i 0.210606 + 0.977571i \(0.432456\pi\)
−0.210606 + 0.977571i \(0.567544\pi\)
\(18\) −4.20909 −0.992092
\(19\) − 6.65977i − 1.52786i −0.645301 0.763928i \(-0.723268\pi\)
0.645301 0.763928i \(-0.276732\pi\)
\(20\) − 0.0704496i − 0.0157530i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.222412 −0.0463761 −0.0231881 0.999731i \(-0.507382\pi\)
−0.0231881 + 0.999731i \(0.507382\pi\)
\(24\) − 4.93680i − 1.00772i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −5.43391 −0.992092
\(31\) 10.2178i 1.83517i 0.397537 + 0.917586i \(0.369865\pi\)
−0.397537 + 0.917586i \(0.630135\pi\)
\(32\) 0.178208 0.0315031
\(33\) 0 0
\(34\) 11.3102i 1.93968i
\(35\) 0 0
\(36\) 0.0945180 0.0157530
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 9.34386i − 1.51577i
\(39\) 0 0
\(40\) − 6.37339i − 1.00772i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) − 6.70820i − 1.00000i
\(46\) −0.312051 −0.0460094
\(47\) − 1.02391i − 0.149353i −0.997208 0.0746766i \(-0.976208\pi\)
0.997208 0.0746766i \(-0.0237924\pi\)
\(48\) − 6.81734i − 0.983999i
\(49\) 0 0
\(50\) −7.01515 −0.992092
\(51\) −13.9625 −1.95514
\(52\) 0 0
\(53\) 11.0902 1.52336 0.761681 0.647953i \(-0.224375\pi\)
0.761681 + 0.647953i \(0.224375\pi\)
\(54\) − 7.29036i − 0.992092i
\(55\) 0 0
\(56\) 0 0
\(57\) 11.5351 1.52786
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0.122022 0.0157530
\(61\) 15.0780i 1.93055i 0.261243 + 0.965273i \(0.415868\pi\)
−0.261243 + 0.965273i \(0.584132\pi\)
\(62\) 14.3359i 1.82066i
\(63\) 0 0
\(64\) 8.12202 1.01525
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 0.253978i − 0.0307994i
\(69\) − 0.385229i − 0.0463761i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 8.55079 1.00772
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) − 8.66025i − 1.00000i
\(76\) 0.209823i 0.0240683i
\(77\) 0 0
\(78\) 0 0
\(79\) 5.83648 0.656656 0.328328 0.944564i \(-0.393515\pi\)
0.328328 + 0.944564i \(0.393515\pi\)
\(80\) − 8.80115i − 0.983999i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 15.0986i 1.65729i 0.559777 + 0.828643i \(0.310887\pi\)
−0.559777 + 0.828643i \(0.689113\pi\)
\(84\) 0 0
\(85\) −18.0255 −1.95514
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) − 9.41181i − 0.992092i
\(91\) 0 0
\(92\) 0.00700732 0.000730563 0
\(93\) −17.6978 −1.83517
\(94\) − 1.43658i − 0.148172i
\(95\) 14.8917 1.52786
\(96\) 0.308666i 0.0315031i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.157530 0.0157530
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −19.5898 −1.93968
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 15.5599 1.51131
\(107\) 20.5038 1.98218 0.991090 0.133196i \(-0.0425241\pi\)
0.991090 + 0.133196i \(0.0425241\pi\)
\(108\) 0.163710i 0.0157530i
\(109\) −15.4919 −1.48386 −0.741929 0.670478i \(-0.766089\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0856 1.13692 0.568461 0.822710i \(-0.307539\pi\)
0.568461 + 0.822710i \(0.307539\pi\)
\(114\) 16.1840 1.51577
\(115\) − 0.497329i − 0.0463761i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 11.0390 1.00772
\(121\) 11.0000 1.00000
\(122\) 21.1550i 1.91528i
\(123\) 0 0
\(124\) − 0.321922i − 0.0289095i
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.0390 0.975721
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 11.6190 1.00000
\(136\) − 22.9767i − 1.97024i
\(137\) −23.3099 −1.99150 −0.995749 0.0921099i \(-0.970639\pi\)
−0.995749 + 0.0921099i \(0.970639\pi\)
\(138\) − 0.540488i − 0.0460094i
\(139\) − 19.9383i − 1.69114i −0.533862 0.845572i \(-0.679260\pi\)
0.533862 0.845572i \(-0.320740\pi\)
\(140\) 0 0
\(141\) 1.77347 0.149353
\(142\) 0 0
\(143\) 0 0
\(144\) 11.8080 0.983999
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) − 12.1506i − 0.992092i
\(151\) 22.0885 1.79754 0.898770 0.438421i \(-0.144462\pi\)
0.898770 + 0.438421i \(0.144462\pi\)
\(152\) 18.9821i 1.53965i
\(153\) − 24.1838i − 1.95514i
\(154\) 0 0
\(155\) −22.8477 −1.83517
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 8.18876 0.651463
\(159\) 19.2089i 1.52336i
\(160\) 0.398486i 0.0315031i
\(161\) 0 0
\(162\) 12.6273 0.992092
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 21.1838i 1.64418i
\(167\) 8.94427i 0.692129i 0.938211 + 0.346064i \(0.112482\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −25.2903 −1.93968
\(171\) 19.9793i 1.52786i
\(172\) 0 0
\(173\) − 6.01343i − 0.457193i −0.973521 0.228596i \(-0.926586\pi\)
0.973521 0.228596i \(-0.0734136\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0.211349i 0.0157530i
\(181\) − 26.2539i − 1.95143i −0.219036 0.975717i \(-0.570291\pi\)
0.219036 0.975717i \(-0.429709\pi\)
\(182\) 0 0
\(183\) −26.1159 −1.93055
\(184\) 0.633933 0.0467342
\(185\) 0 0
\(186\) −24.8305 −1.82066
\(187\) 0 0
\(188\) 0.0322594i 0.00235276i
\(189\) 0 0
\(190\) 20.8935 1.51577
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 14.0678i 1.01525i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.47352 −0.461219 −0.230610 0.973046i \(-0.574072\pi\)
−0.230610 + 0.973046i \(0.574072\pi\)
\(198\) 0 0
\(199\) − 5.88931i − 0.417482i −0.977971 0.208741i \(-0.933063\pi\)
0.977971 0.208741i \(-0.0669366\pi\)
\(200\) 14.2513 1.00772
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0.439903 0.0307994
\(205\) 0 0
\(206\) 0 0
\(207\) 0.667236 0.0463761
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −7.74597 −0.533254 −0.266627 0.963800i \(-0.585909\pi\)
−0.266627 + 0.963800i \(0.585909\pi\)
\(212\) −0.349409 −0.0239975
\(213\) 0 0
\(214\) 28.7675 1.96650
\(215\) 0 0
\(216\) 14.8104i 1.00772i
\(217\) 0 0
\(218\) −21.7357 −1.47212
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 16.9565 1.12793
\(227\) 17.8885i 1.18730i 0.804722 + 0.593652i \(0.202314\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) −0.363424 −0.0240683
\(229\) 4.36291i 0.288309i 0.989555 + 0.144155i \(0.0460463\pi\)
−0.989555 + 0.144155i \(0.953954\pi\)
\(230\) − 0.697767i − 0.0460094i
\(231\) 0 0
\(232\) 0 0
\(233\) −10.6454 −0.697404 −0.348702 0.937234i \(-0.613378\pi\)
−0.348702 + 0.937234i \(0.613378\pi\)
\(234\) 0 0
\(235\) 2.28954 0.149353
\(236\) 0 0
\(237\) 10.1091i 0.656656i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 15.2440 0.983999
\(241\) − 13.7048i − 0.882802i −0.897310 0.441401i \(-0.854482\pi\)
0.897310 0.441401i \(-0.145518\pi\)
\(242\) 15.4333 0.992092
\(243\) 15.5885i 1.00000i
\(244\) − 0.475049i − 0.0304119i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) − 29.1235i − 1.84934i
\(249\) −26.1515 −1.65729
\(250\) − 15.6864i − 0.992092i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) − 31.2211i − 1.95514i
\(256\) −0.755956 −0.0472472
\(257\) 6.92820i 0.432169i 0.976375 + 0.216085i \(0.0693287\pi\)
−0.976375 + 0.216085i \(0.930671\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.66746 −0.226145 −0.113073 0.993587i \(-0.536069\pi\)
−0.113073 + 0.993587i \(0.536069\pi\)
\(264\) 0 0
\(265\) 24.7985i 1.52336i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 16.3017 0.992092
\(271\) − 32.5284i − 1.97596i −0.154583 0.987980i \(-0.549404\pi\)
0.154583 0.987980i \(-0.450596\pi\)
\(272\) − 31.7291i − 1.92386i
\(273\) 0 0
\(274\) −32.7045 −1.97575
\(275\) 0 0
\(276\) 0.0121370i 0 0.000730563i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) − 27.9740i − 1.67777i
\(279\) − 30.6534i − 1.83517i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 2.48823 0.148172
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 25.7932i 1.52786i
\(286\) 0 0
\(287\) 0 0
\(288\) −0.534625 −0.0315031
\(289\) −47.9839 −2.82258
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.272850i 0.0157530i
\(301\) 0 0
\(302\) 30.9909 1.78332
\(303\) 0 0
\(304\) 26.2128i 1.50341i
\(305\) −33.7155 −1.93055
\(306\) − 33.9306i − 1.93968i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −32.0560 −1.82066
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.183884 −0.0103443
\(317\) 34.1604 1.91864 0.959319 0.282326i \(-0.0911060\pi\)
0.959319 + 0.282326i \(0.0911060\pi\)
\(318\) 26.9506i 1.51131i
\(319\) 0 0
\(320\) 18.1614i 1.01525i
\(321\) 35.5137i 1.98218i
\(322\) 0 0
\(323\) 53.6861 2.98718
\(324\) −0.283554 −0.0157530
\(325\) 0 0
\(326\) 0 0
\(327\) − 26.8328i − 1.48386i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 23.2379 1.27727 0.638635 0.769510i \(-0.279499\pi\)
0.638635 + 0.769510i \(0.279499\pi\)
\(332\) − 0.475697i − 0.0261072i
\(333\) 0 0
\(334\) 12.5491i 0.686655i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −18.2394 −0.992092
\(339\) 20.9329i 1.13692i
\(340\) 0.567912 0.0307994
\(341\) 0 0
\(342\) 28.0316i 1.51577i
\(343\) 0 0
\(344\) 0 0
\(345\) 0.861398 0.0463761
\(346\) − 8.43702i − 0.453577i
\(347\) −23.2925 −1.25041 −0.625204 0.780461i \(-0.714984\pi\)
−0.625204 + 0.780461i \(0.714984\pi\)
\(348\) 0 0
\(349\) 25.4834i 1.36410i 0.731308 + 0.682048i \(0.238910\pi\)
−0.731308 + 0.682048i \(0.761090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 20.7846i − 1.10625i −0.833097 0.553127i \(-0.813435\pi\)
0.833097 0.553127i \(-0.186565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 19.1202i 1.00772i
\(361\) −25.3526 −1.33434
\(362\) − 36.8350i − 1.93600i
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) −36.6415 −1.91528
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0.875413 0.0456341
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.557586 0.0289095
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 19.3649 1.00000
\(376\) 2.91843i 0.150506i
\(377\) 0 0
\(378\) 0 0
\(379\) 30.2146 1.55202 0.776009 0.630722i \(-0.217241\pi\)
0.776009 + 0.630722i \(0.217241\pi\)
\(380\) −0.469178 −0.0240683
\(381\) 0 0
\(382\) 0 0
\(383\) 33.2689i 1.69996i 0.526812 + 0.849982i \(0.323387\pi\)
−0.526812 + 0.849982i \(0.676613\pi\)
\(384\) 19.1202i 0.975721i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) − 1.79292i − 0.0906719i
\(392\) 0 0
\(393\) 0 0
\(394\) −9.08254 −0.457572
\(395\) 13.0508i 0.656656i
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) − 8.26288i − 0.414181i
\(399\) 0 0
\(400\) 19.6800 0.983999
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 20.1246i 1.00000i
\(406\) 0 0
\(407\) 0 0
\(408\) 39.7968 1.97024
\(409\) 12.1639i 0.601464i 0.953709 + 0.300732i \(0.0972310\pi\)
−0.953709 + 0.300732i \(0.902769\pi\)
\(410\) 0 0
\(411\) − 40.3739i − 1.99150i
\(412\) 0 0
\(413\) 0 0
\(414\) 0.936153 0.0460094
\(415\) −33.7615 −1.65729
\(416\) 0 0
\(417\) 34.5341 1.69114
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −37.8245 −1.84345 −0.921727 0.387840i \(-0.873221\pi\)
−0.921727 + 0.387840i \(0.873221\pi\)
\(422\) −10.8678 −0.529037
\(423\) 3.07174i 0.149353i
\(424\) −31.6101 −1.53512
\(425\) − 40.3063i − 1.95514i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.645994 −0.0312253
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 20.4520i 0.983999i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.488089 0.0233752
\(437\) 1.48121i 0.0708561i
\(438\) 0 0
\(439\) − 29.6588i − 1.41554i −0.706445 0.707768i \(-0.749702\pi\)
0.706445 0.707768i \(-0.250298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.94466 0.0923937 0.0461968 0.998932i \(-0.485290\pi\)
0.0461968 + 0.998932i \(0.485290\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 21.0455 0.992092
\(451\) 0 0
\(452\) −0.380770 −0.0179099
\(453\) 38.2585i 1.79754i
\(454\) 25.0982i 1.17792i
\(455\) 0 0
\(456\) −32.8780 −1.53965
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 6.12130i 0.286029i
\(459\) 41.8875 1.95514
\(460\) 0.0156688i 0 0.000730563i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) − 39.5734i − 1.83517i
\(466\) −14.9358 −0.691889
\(467\) 35.7771i 1.65557i 0.561048 + 0.827783i \(0.310398\pi\)
−0.561048 + 0.827783i \(0.689602\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.21230 0.148172
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 14.1834i 0.651463i
\(475\) 33.2989i 1.52786i
\(476\) 0 0
\(477\) −33.2707 −1.52336
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −0.690197 −0.0315031
\(481\) 0 0
\(482\) − 19.2282i − 0.875821i
\(483\) 0 0
\(484\) −0.346566 −0.0157530
\(485\) 0 0
\(486\) 21.8711i 0.992092i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) − 42.9764i − 1.94545i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) − 40.2172i − 1.80581i
\(497\) 0 0
\(498\) −36.6914 −1.64418
\(499\) 25.6355 1.14760 0.573801 0.818995i \(-0.305468\pi\)
0.573801 + 0.818995i \(0.305468\pi\)
\(500\) 0.352248i 0.0157530i
\(501\) −15.4919 −0.692129
\(502\) 0 0
\(503\) − 29.1733i − 1.30077i −0.759604 0.650386i \(-0.774607\pi\)
0.759604 0.650386i \(-0.225393\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 22.5167i − 1.00000i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) − 43.8042i − 1.93968i
\(511\) 0 0
\(512\) −23.1387 −1.02259
\(513\) −34.6052 −1.52786
\(514\) 9.72048i 0.428752i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.4156 0.457193
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −5.14556 −0.224357
\(527\) −82.3684 −3.58802
\(528\) 0 0
\(529\) −22.9505 −0.997849
\(530\) 34.7931i 1.51131i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 45.8479i 1.98218i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −0.366067 −0.0157530
\(541\) 34.2776 1.47371 0.736854 0.676052i \(-0.236311\pi\)
0.736854 + 0.676052i \(0.236311\pi\)
\(542\) − 45.6383i − 1.96033i
\(543\) 45.4730 1.95143
\(544\) 1.43658i 0.0615930i
\(545\) − 34.6410i − 1.48386i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0.734401 0.0313721
\(549\) − 45.2341i − 1.93055i
\(550\) 0 0
\(551\) 0 0
\(552\) 1.09800i 0.0467342i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.628176i 0.0266406i
\(557\) −4.75072 −0.201295 −0.100647 0.994922i \(-0.532091\pi\)
−0.100647 + 0.994922i \(0.532091\pi\)
\(558\) − 43.0077i − 1.82066i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 47.3436i 1.99530i 0.0685449 + 0.997648i \(0.478164\pi\)
−0.0685449 + 0.997648i \(0.521836\pi\)
\(564\) −0.0558750 −0.00235276
\(565\) 27.0243i 1.13692i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 36.1886i 1.51577i
\(571\) 38.7298 1.62079 0.810397 0.585882i \(-0.199252\pi\)
0.810397 + 0.585882i \(0.199252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.11206 0.0463761
\(576\) −24.3661 −1.01525
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −67.3228 −2.80026
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 19.4410i 0.803099i
\(587\) − 19.1943i − 0.792232i −0.918201 0.396116i \(-0.870358\pi\)
0.918201 0.396116i \(-0.129642\pi\)
\(588\) 0 0
\(589\) 68.0483 2.80388
\(590\) 0 0
\(591\) − 11.2125i − 0.461219i
\(592\) 0 0
\(593\) 4.47214i 0.183649i 0.995775 + 0.0918243i \(0.0292698\pi\)
−0.995775 + 0.0918243i \(0.970730\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.2006 0.417482
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 24.6840i 1.00772i
\(601\) 14.0834i 0.574473i 0.957860 + 0.287237i \(0.0927366\pi\)
−0.957860 + 0.287237i \(0.907263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.695921 −0.0283166
\(605\) 24.5967i 1.00000i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) − 1.18683i − 0.0481321i
\(609\) 0 0
\(610\) −47.3039 −1.91528
\(611\) 0 0
\(612\) 0.761934i 0.0307994i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.4247 −0.500200 −0.250100 0.968220i \(-0.580464\pi\)
−0.250100 + 0.968220i \(0.580464\pi\)
\(618\) 0 0
\(619\) − 31.7579i − 1.27646i −0.769846 0.638230i \(-0.779667\pi\)
0.769846 0.638230i \(-0.220333\pi\)
\(620\) 0.719840 0.0289095
\(621\) 1.15569i 0.0463761i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 50.0135 1.99101 0.995504 0.0947206i \(-0.0301958\pi\)
0.995504 + 0.0947206i \(0.0301958\pi\)
\(632\) −16.6355 −0.661725
\(633\) − 13.4164i − 0.533254i
\(634\) 47.9280 1.90346
\(635\) 0 0
\(636\) − 0.605194i − 0.0239975i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 24.6840i 0.975721i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 49.8267i 1.96650i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 75.3233 2.96355
\(647\) − 44.7214i − 1.75818i −0.476658 0.879089i \(-0.658152\pi\)
0.476658 0.879089i \(-0.341848\pi\)
\(648\) −25.6524 −1.00772
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 40.1462 1.57104 0.785522 0.618834i \(-0.212395\pi\)
0.785522 + 0.618834i \(0.212395\pi\)
\(654\) − 37.6473i − 1.47212i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) − 34.5190i − 1.34263i −0.741170 0.671317i \(-0.765729\pi\)
0.741170 0.671317i \(-0.234271\pi\)
\(662\) 32.6035 1.26717
\(663\) 0 0
\(664\) − 43.0350i − 1.67008i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 0.281798i − 0.0109031i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 0.409578 0.0157530
\(677\) − 41.5692i − 1.59763i −0.601574 0.798817i \(-0.705459\pi\)
0.601574 0.798817i \(-0.294541\pi\)
\(678\) 29.3696i 1.12793i
\(679\) 0 0
\(680\) 51.3775 1.97024
\(681\) −30.9839 −1.18730
\(682\) 0 0
\(683\) 21.5132 0.823181 0.411591 0.911369i \(-0.364973\pi\)
0.411591 + 0.911369i \(0.364973\pi\)
\(684\) − 0.629468i − 0.0240683i
\(685\) − 52.1225i − 1.99150i
\(686\) 0 0
\(687\) −7.55678 −0.288309
\(688\) 0 0
\(689\) 0 0
\(690\) 1.20857 0.0460094
\(691\) 45.0775i 1.71483i 0.514627 + 0.857414i \(0.327930\pi\)
−0.514627 + 0.857414i \(0.672070\pi\)
\(692\) 0.189459i 0.00720216i
\(693\) 0 0
\(694\) −32.6801 −1.24052
\(695\) 44.5834 1.69114
\(696\) 0 0
\(697\) 0 0
\(698\) 35.7540i 1.35331i
\(699\) − 18.4384i − 0.697404i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 3.96560i 0.149353i
\(706\) − 29.1614i − 1.09751i
\(707\) 0 0
\(708\) 0 0
\(709\) 14.4786 0.543754 0.271877 0.962332i \(-0.412356\pi\)
0.271877 + 0.962332i \(0.412356\pi\)
\(710\) 0 0
\(711\) −17.5094 −0.656656
\(712\) 0 0
\(713\) − 2.27256i − 0.0851082i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 26.4035i 0.983999i
\(721\) 0 0
\(722\) −35.5704 −1.32379
\(723\) 23.7374 0.882802
\(724\) 0.827154i 0.0307409i
\(725\) 0 0
\(726\) 26.7313i 0.992092i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0.822809 0.0304119
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −0.0396357 −0.00146099
\(737\) 0 0
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.9178 1.68456 0.842281 0.539039i \(-0.181212\pi\)
0.842281 + 0.539039i \(0.181212\pi\)
\(744\) 50.4433 1.84934
\(745\) 0 0
\(746\) 0 0
\(747\) − 45.2958i − 1.65729i
\(748\) 0 0
\(749\) 0 0
\(750\) 27.1696 0.992092
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 4.03012i 0.146963i
\(753\) 0 0
\(754\) 0 0
\(755\) 49.3915i 1.79754i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 42.3919 1.53974
\(759\) 0 0
\(760\) −42.4453 −1.53965
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 54.0766 1.95514
\(766\) 46.6773i 1.68652i
\(767\) 0 0
\(768\) − 1.30935i − 0.0472472i
\(769\) 46.2288i 1.66705i 0.552479 + 0.833527i \(0.313682\pi\)
−0.552479 + 0.833527i \(0.686318\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 42.3541i 1.52337i 0.647947 + 0.761686i \(0.275628\pi\)
−0.647947 + 0.761686i \(0.724372\pi\)
\(774\) 0 0
\(775\) − 51.0890i − 1.83517i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) − 2.51552i − 0.0899549i
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0.203955 0.00726559
\(789\) − 6.35223i − 0.226145i
\(790\) 18.3106i 0.651463i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −42.9523 −1.52336
\(796\) 0.185549i 0.00657660i
\(797\) 54.3810i 1.92627i 0.269013 + 0.963136i \(0.413302\pi\)
−0.269013 + 0.963136i \(0.586698\pi\)
\(798\) 0 0
\(799\) 8.25403 0.292007
\(800\) −0.891041 −0.0315031
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 28.2354i 0.992092i
\(811\) − 41.3686i − 1.45265i −0.687353 0.726323i \(-0.741228\pi\)
0.687353 0.726323i \(-0.258772\pi\)
\(812\) 0 0
\(813\) 56.3408 1.97596
\(814\) 0 0
\(815\) 0 0
\(816\) 54.9564 1.92386
\(817\) 0 0
\(818\) 17.0663i 0.596708i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) − 56.6458i − 1.97575i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 50.2872 1.74866 0.874329 0.485334i \(-0.161302\pi\)
0.874329 + 0.485334i \(0.161302\pi\)
\(828\) −0.0210220 −0.000730563 0
\(829\) − 55.9493i − 1.94320i −0.236632 0.971599i \(-0.576044\pi\)
0.236632 0.971599i \(-0.423956\pi\)
\(830\) −47.3684 −1.64418
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 48.4524 1.67777
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 53.0933 1.83517
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) −53.0689 −1.82888
\(843\) 0 0
\(844\) 0.244044 0.00840036
\(845\) − 29.0689i − 1.00000i
\(846\) 4.30975i 0.148172i
\(847\) 0 0
\(848\) −43.6511 −1.49899
\(849\) 0 0
\(850\) − 56.5509i − 1.93968i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) −44.6751 −1.52786
\(856\) −58.4413 −1.99748
\(857\) − 36.2106i − 1.23693i −0.785812 0.618466i \(-0.787755\pi\)
0.785812 0.618466i \(-0.212245\pi\)
\(858\) 0 0
\(859\) − 18.9436i − 0.646348i −0.946340 0.323174i \(-0.895250\pi\)
0.946340 0.323174i \(-0.104750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −55.8993 −1.90284 −0.951418 0.307902i \(-0.900373\pi\)
−0.951418 + 0.307902i \(0.900373\pi\)
\(864\) − 0.925997i − 0.0315031i
\(865\) 13.4464 0.457193
\(866\) 0 0
\(867\) − 83.1105i − 2.82258i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 44.1561 1.49531
\(873\) 0 0
\(874\) 2.07819i 0.0702957i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) − 41.6121i − 1.40434i
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.72842 0.0916631
\(887\) − 35.3168i − 1.18582i −0.805268 0.592911i \(-0.797979\pi\)
0.805268 0.592911i \(-0.202021\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.81903 −0.228190
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.472590 −0.0157530
\(901\) 89.4013i 2.97839i
\(902\) 0 0
\(903\) 0 0
\(904\) −34.4473 −1.14570
\(905\) 58.7054 1.95143
\(906\) 53.6778i 1.78332i
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) − 0.563597i − 0.0187036i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −45.4020 −1.50341
\(913\) 0 0
\(914\) 0 0
\(915\) − 58.3970i − 1.93055i
\(916\) − 0.137458i − 0.00454174i
\(917\) 0 0
\(918\) 58.7695 1.93968
\(919\) 23.2379 0.766548 0.383274 0.923635i \(-0.374797\pi\)
0.383274 + 0.923635i \(0.374797\pi\)
\(920\) 1.41752i 0.0467342i
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) − 55.5227i − 1.82066i
\(931\) 0 0
\(932\) 0.335394 0.0109862
\(933\) 0 0
\(934\) 50.1963i 1.64247i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.0721343 −0.00235276
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.0630 1.26938 0.634688 0.772768i \(-0.281129\pi\)
0.634688 + 0.772768i \(0.281129\pi\)
\(948\) − 0.318497i − 0.0103443i
\(949\) 0 0
\(950\) 46.7193i 1.51577i
\(951\) 59.1675i 1.91864i
\(952\) 0 0
\(953\) −45.7584 −1.48226 −0.741129 0.671362i \(-0.765710\pi\)
−0.741129 + 0.671362i \(0.765710\pi\)
\(954\) −46.6798 −1.51131
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −31.4565 −1.01525
\(961\) −73.4036 −2.36786
\(962\) 0 0
\(963\) −61.5115 −1.98218
\(964\) 0.431783i 0.0139068i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −31.3529 −1.00772
\(969\) 92.9871i 2.98718i
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) − 0.491130i − 0.0157530i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) − 59.3471i − 1.89966i
\(977\) 55.8960 1.78827 0.894136 0.447796i \(-0.147791\pi\)
0.894136 + 0.447796i \(0.147791\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 46.4758 1.48386
\(982\) 0 0
\(983\) 3.46410i 0.110488i 0.998473 + 0.0552438i \(0.0175936\pi\)
−0.998473 + 0.0552438i \(0.982406\pi\)
\(984\) 0 0
\(985\) − 14.4752i − 0.461219i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 1.82090i 0.0578135i
\(993\) 40.2492i 1.27727i
\(994\) 0 0
\(995\) 13.1689 0.417482
\(996\) 0.823931 0.0261072
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 35.9673 1.13853
\(999\) 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 735.2.g.a.734.12 yes 16
3.2 odd 2 inner 735.2.g.a.734.5 16
5.4 even 2 inner 735.2.g.a.734.5 16
7.2 even 3 735.2.p.d.374.3 16
7.3 odd 6 735.2.p.d.509.3 16
7.4 even 3 735.2.p.e.509.3 16
7.5 odd 6 735.2.p.e.374.3 16
7.6 odd 2 inner 735.2.g.a.734.11 yes 16
15.14 odd 2 CM 735.2.g.a.734.12 yes 16
21.2 odd 6 735.2.p.e.374.6 16
21.5 even 6 735.2.p.d.374.6 16
21.11 odd 6 735.2.p.d.509.6 16
21.17 even 6 735.2.p.e.509.6 16
21.20 even 2 inner 735.2.g.a.734.6 yes 16
35.4 even 6 735.2.p.d.509.6 16
35.9 even 6 735.2.p.e.374.6 16
35.19 odd 6 735.2.p.d.374.6 16
35.24 odd 6 735.2.p.e.509.6 16
35.34 odd 2 inner 735.2.g.a.734.6 yes 16
105.44 odd 6 735.2.p.d.374.3 16
105.59 even 6 735.2.p.d.509.3 16
105.74 odd 6 735.2.p.e.509.3 16
105.89 even 6 735.2.p.e.374.3 16
105.104 even 2 inner 735.2.g.a.734.11 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.2.g.a.734.5 16 3.2 odd 2 inner
735.2.g.a.734.5 16 5.4 even 2 inner
735.2.g.a.734.6 yes 16 21.20 even 2 inner
735.2.g.a.734.6 yes 16 35.34 odd 2 inner
735.2.g.a.734.11 yes 16 7.6 odd 2 inner
735.2.g.a.734.11 yes 16 105.104 even 2 inner
735.2.g.a.734.12 yes 16 1.1 even 1 trivial
735.2.g.a.734.12 yes 16 15.14 odd 2 CM
735.2.p.d.374.3 16 7.2 even 3
735.2.p.d.374.3 16 105.44 odd 6
735.2.p.d.374.6 16 21.5 even 6
735.2.p.d.374.6 16 35.19 odd 6
735.2.p.d.509.3 16 7.3 odd 6
735.2.p.d.509.3 16 105.59 even 6
735.2.p.d.509.6 16 21.11 odd 6
735.2.p.d.509.6 16 35.4 even 6
735.2.p.e.374.3 16 7.5 odd 6
735.2.p.e.374.3 16 105.89 even 6
735.2.p.e.374.6 16 21.2 odd 6
735.2.p.e.374.6 16 35.9 even 6
735.2.p.e.509.3 16 7.4 even 3
735.2.p.e.509.3 16 105.74 odd 6
735.2.p.e.509.6 16 21.17 even 6
735.2.p.e.509.6 16 35.24 odd 6