Properties

Label 3584.2.b.i.1793.6
Level $3584$
Weight $2$
Character 3584.1793
Analytic conductor $28.618$
Analytic rank $0$
Dimension $12$
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] = [3584,2,Mod(1793,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, 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("3584.1793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1793.6
Root \(0.271901 + 0.656426i\) of defining polynomial
Character \(\chi\) \(=\) 3584.1793
Dual form 3584.2.b.i.1793.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.101362i q^{3} +2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} +O(q^{10})\) \(q-0.101362i q^{3} +2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} -0.674013i q^{11} +5.30258i q^{13} +0.222631 q^{15} +3.87523 q^{17} +0.995183i q^{19} +0.101362i q^{21} +3.36590 q^{23} +0.175829 q^{25} -0.607128i q^{27} -0.134404i q^{29} -3.11512 q^{31} -0.0683190 q^{33} -2.19640i q^{35} +2.07795i q^{37} +0.537478 q^{39} +3.76710 q^{41} -3.61268i q^{43} +6.56663i q^{45} -6.68913 q^{47} +1.00000 q^{49} -0.392799i q^{51} +2.93656i q^{53} +1.48040 q^{55} +0.100873 q^{57} -6.92085i q^{59} -4.30265i q^{61} -2.98973 q^{63} -11.6466 q^{65} +13.9311i q^{67} -0.341173i q^{69} +10.6460 q^{71} -8.62368 q^{73} -0.0178223i q^{75} +0.674013i q^{77} +7.12273 q^{79} +8.90764 q^{81} +15.1833i q^{83} +8.51155i q^{85} -0.0136234 q^{87} -0.856726 q^{89} -5.30258i q^{91} +0.315754i q^{93} -2.18582 q^{95} -11.0125 q^{97} -2.01511i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 0.101362i − 0.0585211i −0.999572 0.0292606i \(-0.990685\pi\)
0.999572 0.0292606i \(-0.00931526\pi\)
\(4\) 0 0
\(5\) 2.19640i 0.982260i 0.871086 + 0.491130i \(0.163416\pi\)
−0.871086 + 0.491130i \(0.836584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 2.98973 0.996575
\(10\) 0 0
\(11\) − 0.674013i − 0.203223i −0.994824 0.101611i \(-0.967600\pi\)
0.994824 0.101611i \(-0.0323998\pi\)
\(12\) 0 0
\(13\) 5.30258i 1.47067i 0.677704 + 0.735335i \(0.262975\pi\)
−0.677704 + 0.735335i \(0.737025\pi\)
\(14\) 0 0
\(15\) 0.222631 0.0574830
\(16\) 0 0
\(17\) 3.87523 0.939881 0.469941 0.882698i \(-0.344275\pi\)
0.469941 + 0.882698i \(0.344275\pi\)
\(18\) 0 0
\(19\) 0.995183i 0.228311i 0.993463 + 0.114155i \(0.0364162\pi\)
−0.993463 + 0.114155i \(0.963584\pi\)
\(20\) 0 0
\(21\) 0.101362i 0.0221189i
\(22\) 0 0
\(23\) 3.36590 0.701840 0.350920 0.936406i \(-0.385869\pi\)
0.350920 + 0.936406i \(0.385869\pi\)
\(24\) 0 0
\(25\) 0.175829 0.0351657
\(26\) 0 0
\(27\) − 0.607128i − 0.116842i
\(28\) 0 0
\(29\) − 0.134404i − 0.0249582i −0.999922 0.0124791i \(-0.996028\pi\)
0.999922 0.0124791i \(-0.00397233\pi\)
\(30\) 0 0
\(31\) −3.11512 −0.559492 −0.279746 0.960074i \(-0.590250\pi\)
−0.279746 + 0.960074i \(0.590250\pi\)
\(32\) 0 0
\(33\) −0.0683190 −0.0118928
\(34\) 0 0
\(35\) − 2.19640i − 0.371259i
\(36\) 0 0
\(37\) 2.07795i 0.341613i 0.985305 + 0.170807i \(0.0546373\pi\)
−0.985305 + 0.170807i \(0.945363\pi\)
\(38\) 0 0
\(39\) 0.537478 0.0860653
\(40\) 0 0
\(41\) 3.76710 0.588321 0.294161 0.955756i \(-0.404960\pi\)
0.294161 + 0.955756i \(0.404960\pi\)
\(42\) 0 0
\(43\) − 3.61268i − 0.550929i −0.961311 0.275464i \(-0.911168\pi\)
0.961311 0.275464i \(-0.0888316\pi\)
\(44\) 0 0
\(45\) 6.56663i 0.978896i
\(46\) 0 0
\(47\) −6.68913 −0.975709 −0.487855 0.872925i \(-0.662220\pi\)
−0.487855 + 0.872925i \(0.662220\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 0.392799i − 0.0550029i
\(52\) 0 0
\(53\) 2.93656i 0.403368i 0.979451 + 0.201684i \(0.0646413\pi\)
−0.979451 + 0.201684i \(0.935359\pi\)
\(54\) 0 0
\(55\) 1.48040 0.199617
\(56\) 0 0
\(57\) 0.100873 0.0133610
\(58\) 0 0
\(59\) − 6.92085i − 0.901017i −0.892772 0.450509i \(-0.851243\pi\)
0.892772 0.450509i \(-0.148757\pi\)
\(60\) 0 0
\(61\) − 4.30265i − 0.550898i −0.961316 0.275449i \(-0.911173\pi\)
0.961316 0.275449i \(-0.0888265\pi\)
\(62\) 0 0
\(63\) −2.98973 −0.376670
\(64\) 0 0
\(65\) −11.6466 −1.44458
\(66\) 0 0
\(67\) 13.9311i 1.70196i 0.525198 + 0.850980i \(0.323991\pi\)
−0.525198 + 0.850980i \(0.676009\pi\)
\(68\) 0 0
\(69\) − 0.341173i − 0.0410724i
\(70\) 0 0
\(71\) 10.6460 1.26344 0.631721 0.775196i \(-0.282349\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(72\) 0 0
\(73\) −8.62368 −1.00932 −0.504662 0.863317i \(-0.668383\pi\)
−0.504662 + 0.863317i \(0.668383\pi\)
\(74\) 0 0
\(75\) − 0.0178223i − 0.00205794i
\(76\) 0 0
\(77\) 0.674013i 0.0768109i
\(78\) 0 0
\(79\) 7.12273 0.801369 0.400685 0.916216i \(-0.368772\pi\)
0.400685 + 0.916216i \(0.368772\pi\)
\(80\) 0 0
\(81\) 8.90764 0.989738
\(82\) 0 0
\(83\) 15.1833i 1.66658i 0.552834 + 0.833291i \(0.313546\pi\)
−0.552834 + 0.833291i \(0.686454\pi\)
\(84\) 0 0
\(85\) 8.51155i 0.923207i
\(86\) 0 0
\(87\) −0.0136234 −0.00146058
\(88\) 0 0
\(89\) −0.856726 −0.0908128 −0.0454064 0.998969i \(-0.514458\pi\)
−0.0454064 + 0.998969i \(0.514458\pi\)
\(90\) 0 0
\(91\) − 5.30258i − 0.555861i
\(92\) 0 0
\(93\) 0.315754i 0.0327421i
\(94\) 0 0
\(95\) −2.18582 −0.224260
\(96\) 0 0
\(97\) −11.0125 −1.11815 −0.559074 0.829118i \(-0.688843\pi\)
−0.559074 + 0.829118i \(0.688843\pi\)
\(98\) 0 0
\(99\) − 2.01511i − 0.202527i
\(100\) 0 0
\(101\) 0.354276i 0.0352518i 0.999845 + 0.0176259i \(0.00561079\pi\)
−0.999845 + 0.0176259i \(0.994389\pi\)
\(102\) 0 0
\(103\) −16.5750 −1.63318 −0.816590 0.577218i \(-0.804138\pi\)
−0.816590 + 0.577218i \(0.804138\pi\)
\(104\) 0 0
\(105\) −0.222631 −0.0217265
\(106\) 0 0
\(107\) − 17.7782i − 1.71869i −0.511400 0.859343i \(-0.670873\pi\)
0.511400 0.859343i \(-0.329127\pi\)
\(108\) 0 0
\(109\) 10.1967i 0.976667i 0.872657 + 0.488334i \(0.162395\pi\)
−0.872657 + 0.488334i \(0.837605\pi\)
\(110\) 0 0
\(111\) 0.210624 0.0199916
\(112\) 0 0
\(113\) 9.69262 0.911805 0.455902 0.890030i \(-0.349317\pi\)
0.455902 + 0.890030i \(0.349317\pi\)
\(114\) 0 0
\(115\) 7.39287i 0.689389i
\(116\) 0 0
\(117\) 15.8533i 1.46563i
\(118\) 0 0
\(119\) −3.87523 −0.355242
\(120\) 0 0
\(121\) 10.5457 0.958701
\(122\) 0 0
\(123\) − 0.381839i − 0.0344292i
\(124\) 0 0
\(125\) 11.3682i 1.01680i
\(126\) 0 0
\(127\) 20.2674 1.79844 0.899220 0.437497i \(-0.144135\pi\)
0.899220 + 0.437497i \(0.144135\pi\)
\(128\) 0 0
\(129\) −0.366187 −0.0322410
\(130\) 0 0
\(131\) 17.8643i 1.56081i 0.625272 + 0.780406i \(0.284988\pi\)
−0.625272 + 0.780406i \(0.715012\pi\)
\(132\) 0 0
\(133\) − 0.995183i − 0.0862933i
\(134\) 0 0
\(135\) 1.33350 0.114769
\(136\) 0 0
\(137\) −11.4894 −0.981606 −0.490803 0.871271i \(-0.663296\pi\)
−0.490803 + 0.871271i \(0.663296\pi\)
\(138\) 0 0
\(139\) − 0.209585i − 0.0177768i −0.999960 0.00888838i \(-0.997171\pi\)
0.999960 0.00888838i \(-0.00282929\pi\)
\(140\) 0 0
\(141\) 0.678020i 0.0570996i
\(142\) 0 0
\(143\) 3.57401 0.298873
\(144\) 0 0
\(145\) 0.295205 0.0245155
\(146\) 0 0
\(147\) − 0.101362i − 0.00836016i
\(148\) 0 0
\(149\) 10.5895i 0.867526i 0.901027 + 0.433763i \(0.142814\pi\)
−0.901027 + 0.433763i \(0.857186\pi\)
\(150\) 0 0
\(151\) −11.4820 −0.934391 −0.467196 0.884154i \(-0.654736\pi\)
−0.467196 + 0.884154i \(0.654736\pi\)
\(152\) 0 0
\(153\) 11.5859 0.936662
\(154\) 0 0
\(155\) − 6.84205i − 0.549567i
\(156\) 0 0
\(157\) 9.98207i 0.796656i 0.917243 + 0.398328i \(0.130409\pi\)
−0.917243 + 0.398328i \(0.869591\pi\)
\(158\) 0 0
\(159\) 0.297654 0.0236055
\(160\) 0 0
\(161\) −3.36590 −0.265270
\(162\) 0 0
\(163\) − 3.76859i − 0.295178i −0.989049 0.147589i \(-0.952849\pi\)
0.989049 0.147589i \(-0.0471513\pi\)
\(164\) 0 0
\(165\) − 0.150056i − 0.0116818i
\(166\) 0 0
\(167\) −8.45888 −0.654568 −0.327284 0.944926i \(-0.606133\pi\)
−0.327284 + 0.944926i \(0.606133\pi\)
\(168\) 0 0
\(169\) −15.1173 −1.16287
\(170\) 0 0
\(171\) 2.97533i 0.227529i
\(172\) 0 0
\(173\) − 1.41548i − 0.107617i −0.998551 0.0538085i \(-0.982864\pi\)
0.998551 0.0538085i \(-0.0171361\pi\)
\(174\) 0 0
\(175\) −0.175829 −0.0132914
\(176\) 0 0
\(177\) −0.701508 −0.0527286
\(178\) 0 0
\(179\) − 17.2449i − 1.28894i −0.764629 0.644470i \(-0.777078\pi\)
0.764629 0.644470i \(-0.222922\pi\)
\(180\) 0 0
\(181\) − 4.77705i − 0.355075i −0.984114 0.177538i \(-0.943187\pi\)
0.984114 0.177538i \(-0.0568132\pi\)
\(182\) 0 0
\(183\) −0.436123 −0.0322392
\(184\) 0 0
\(185\) −4.56401 −0.335553
\(186\) 0 0
\(187\) − 2.61195i − 0.191005i
\(188\) 0 0
\(189\) 0.607128i 0.0441621i
\(190\) 0 0
\(191\) −8.90148 −0.644089 −0.322044 0.946725i \(-0.604370\pi\)
−0.322044 + 0.946725i \(0.604370\pi\)
\(192\) 0 0
\(193\) −11.9910 −0.863129 −0.431565 0.902082i \(-0.642038\pi\)
−0.431565 + 0.902082i \(0.642038\pi\)
\(194\) 0 0
\(195\) 1.18052i 0.0845385i
\(196\) 0 0
\(197\) 18.5079i 1.31863i 0.751865 + 0.659317i \(0.229154\pi\)
−0.751865 + 0.659317i \(0.770846\pi\)
\(198\) 0 0
\(199\) 2.13252 0.151170 0.0755852 0.997139i \(-0.475918\pi\)
0.0755852 + 0.997139i \(0.475918\pi\)
\(200\) 0 0
\(201\) 1.41208 0.0996006
\(202\) 0 0
\(203\) 0.134404i 0.00943332i
\(204\) 0 0
\(205\) 8.27405i 0.577884i
\(206\) 0 0
\(207\) 10.0631 0.699436
\(208\) 0 0
\(209\) 0.670766 0.0463979
\(210\) 0 0
\(211\) 22.7237i 1.56436i 0.623051 + 0.782181i \(0.285893\pi\)
−0.623051 + 0.782181i \(0.714107\pi\)
\(212\) 0 0
\(213\) − 1.07909i − 0.0739381i
\(214\) 0 0
\(215\) 7.93489 0.541155
\(216\) 0 0
\(217\) 3.11512 0.211468
\(218\) 0 0
\(219\) 0.874109i 0.0590668i
\(220\) 0 0
\(221\) 20.5487i 1.38226i
\(222\) 0 0
\(223\) −1.10025 −0.0736781 −0.0368391 0.999321i \(-0.511729\pi\)
−0.0368391 + 0.999321i \(0.511729\pi\)
\(224\) 0 0
\(225\) 0.525679 0.0350453
\(226\) 0 0
\(227\) − 2.63939i − 0.175183i −0.996156 0.0875913i \(-0.972083\pi\)
0.996156 0.0875913i \(-0.0279169\pi\)
\(228\) 0 0
\(229\) 19.2282i 1.27064i 0.772250 + 0.635319i \(0.219131\pi\)
−0.772250 + 0.635319i \(0.780869\pi\)
\(230\) 0 0
\(231\) 0.0683190 0.00449506
\(232\) 0 0
\(233\) −8.69975 −0.569939 −0.284970 0.958537i \(-0.591984\pi\)
−0.284970 + 0.958537i \(0.591984\pi\)
\(234\) 0 0
\(235\) − 14.6920i − 0.958400i
\(236\) 0 0
\(237\) − 0.721971i − 0.0468970i
\(238\) 0 0
\(239\) −1.72852 −0.111809 −0.0559044 0.998436i \(-0.517804\pi\)
−0.0559044 + 0.998436i \(0.517804\pi\)
\(240\) 0 0
\(241\) 17.6482 1.13682 0.568409 0.822746i \(-0.307559\pi\)
0.568409 + 0.822746i \(0.307559\pi\)
\(242\) 0 0
\(243\) − 2.72428i − 0.174762i
\(244\) 0 0
\(245\) 2.19640i 0.140323i
\(246\) 0 0
\(247\) −5.27704 −0.335770
\(248\) 0 0
\(249\) 1.53900 0.0975303
\(250\) 0 0
\(251\) 14.6099i 0.922169i 0.887356 + 0.461084i \(0.152539\pi\)
−0.887356 + 0.461084i \(0.847461\pi\)
\(252\) 0 0
\(253\) − 2.26866i − 0.142630i
\(254\) 0 0
\(255\) 0.862744 0.0540271
\(256\) 0 0
\(257\) −24.1929 −1.50911 −0.754556 0.656235i \(-0.772148\pi\)
−0.754556 + 0.656235i \(0.772148\pi\)
\(258\) 0 0
\(259\) − 2.07795i − 0.129118i
\(260\) 0 0
\(261\) − 0.401832i − 0.0248727i
\(262\) 0 0
\(263\) 23.8158 1.46854 0.734271 0.678856i \(-0.237524\pi\)
0.734271 + 0.678856i \(0.237524\pi\)
\(264\) 0 0
\(265\) −6.44986 −0.396212
\(266\) 0 0
\(267\) 0.0868391i 0.00531446i
\(268\) 0 0
\(269\) 0.304478i 0.0185643i 0.999957 + 0.00928216i \(0.00295465\pi\)
−0.999957 + 0.00928216i \(0.997045\pi\)
\(270\) 0 0
\(271\) −17.2018 −1.04493 −0.522467 0.852659i \(-0.674988\pi\)
−0.522467 + 0.852659i \(0.674988\pi\)
\(272\) 0 0
\(273\) −0.537478 −0.0325296
\(274\) 0 0
\(275\) − 0.118511i − 0.00714646i
\(276\) 0 0
\(277\) − 21.1489i − 1.27072i −0.772218 0.635358i \(-0.780853\pi\)
0.772218 0.635358i \(-0.219147\pi\)
\(278\) 0 0
\(279\) −9.31336 −0.557576
\(280\) 0 0
\(281\) −11.0974 −0.662018 −0.331009 0.943628i \(-0.607389\pi\)
−0.331009 + 0.943628i \(0.607389\pi\)
\(282\) 0 0
\(283\) − 23.8050i − 1.41506i −0.706683 0.707530i \(-0.749809\pi\)
0.706683 0.707530i \(-0.250191\pi\)
\(284\) 0 0
\(285\) 0.221558i 0.0131240i
\(286\) 0 0
\(287\) −3.76710 −0.222365
\(288\) 0 0
\(289\) −1.98260 −0.116624
\(290\) 0 0
\(291\) 1.11624i 0.0654353i
\(292\) 0 0
\(293\) 13.2713i 0.775316i 0.921803 + 0.387658i \(0.126716\pi\)
−0.921803 + 0.387658i \(0.873284\pi\)
\(294\) 0 0
\(295\) 15.2009 0.885033
\(296\) 0 0
\(297\) −0.409212 −0.0237449
\(298\) 0 0
\(299\) 17.8480i 1.03217i
\(300\) 0 0
\(301\) 3.61268i 0.208231i
\(302\) 0 0
\(303\) 0.0359100 0.00206298
\(304\) 0 0
\(305\) 9.45034 0.541125
\(306\) 0 0
\(307\) 14.4325i 0.823705i 0.911251 + 0.411852i \(0.135118\pi\)
−0.911251 + 0.411852i \(0.864882\pi\)
\(308\) 0 0
\(309\) 1.68007i 0.0955756i
\(310\) 0 0
\(311\) 22.4304 1.27191 0.635956 0.771725i \(-0.280606\pi\)
0.635956 + 0.771725i \(0.280606\pi\)
\(312\) 0 0
\(313\) 24.6010 1.39053 0.695266 0.718753i \(-0.255287\pi\)
0.695266 + 0.718753i \(0.255287\pi\)
\(314\) 0 0
\(315\) − 6.56663i − 0.369988i
\(316\) 0 0
\(317\) 30.9518i 1.73842i 0.494439 + 0.869212i \(0.335374\pi\)
−0.494439 + 0.869212i \(0.664626\pi\)
\(318\) 0 0
\(319\) −0.0905901 −0.00507207
\(320\) 0 0
\(321\) −1.80203 −0.100579
\(322\) 0 0
\(323\) 3.85656i 0.214585i
\(324\) 0 0
\(325\) 0.932345i 0.0517172i
\(326\) 0 0
\(327\) 1.03355 0.0571557
\(328\) 0 0
\(329\) 6.68913 0.368783
\(330\) 0 0
\(331\) − 33.4297i − 1.83746i −0.394884 0.918731i \(-0.629215\pi\)
0.394884 0.918731i \(-0.370785\pi\)
\(332\) 0 0
\(333\) 6.21251i 0.340443i
\(334\) 0 0
\(335\) −30.5984 −1.67177
\(336\) 0 0
\(337\) 27.1717 1.48014 0.740069 0.672530i \(-0.234793\pi\)
0.740069 + 0.672530i \(0.234793\pi\)
\(338\) 0 0
\(339\) − 0.982459i − 0.0533599i
\(340\) 0 0
\(341\) 2.09963i 0.113701i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.749353 0.0403438
\(346\) 0 0
\(347\) − 25.0784i − 1.34628i −0.739516 0.673139i \(-0.764946\pi\)
0.739516 0.673139i \(-0.235054\pi\)
\(348\) 0 0
\(349\) − 23.7113i − 1.26924i −0.772826 0.634618i \(-0.781157\pi\)
0.772826 0.634618i \(-0.218843\pi\)
\(350\) 0 0
\(351\) 3.21934 0.171836
\(352\) 0 0
\(353\) −20.6382 −1.09846 −0.549230 0.835672i \(-0.685079\pi\)
−0.549230 + 0.835672i \(0.685079\pi\)
\(354\) 0 0
\(355\) 23.3828i 1.24103i
\(356\) 0 0
\(357\) 0.392799i 0.0207891i
\(358\) 0 0
\(359\) −27.2976 −1.44071 −0.720355 0.693605i \(-0.756021\pi\)
−0.720355 + 0.693605i \(0.756021\pi\)
\(360\) 0 0
\(361\) 18.0096 0.947874
\(362\) 0 0
\(363\) − 1.06893i − 0.0561042i
\(364\) 0 0
\(365\) − 18.9410i − 0.991419i
\(366\) 0 0
\(367\) 28.7125 1.49878 0.749390 0.662128i \(-0.230347\pi\)
0.749390 + 0.662128i \(0.230347\pi\)
\(368\) 0 0
\(369\) 11.2626 0.586307
\(370\) 0 0
\(371\) − 2.93656i − 0.152459i
\(372\) 0 0
\(373\) − 29.1078i − 1.50715i −0.657364 0.753573i \(-0.728329\pi\)
0.657364 0.753573i \(-0.271671\pi\)
\(374\) 0 0
\(375\) 1.15230 0.0595044
\(376\) 0 0
\(377\) 0.712688 0.0367053
\(378\) 0 0
\(379\) 19.5975i 1.00666i 0.864096 + 0.503328i \(0.167891\pi\)
−0.864096 + 0.503328i \(0.832109\pi\)
\(380\) 0 0
\(381\) − 2.05433i − 0.105247i
\(382\) 0 0
\(383\) 19.5193 0.997389 0.498694 0.866778i \(-0.333813\pi\)
0.498694 + 0.866778i \(0.333813\pi\)
\(384\) 0 0
\(385\) −1.48040 −0.0754482
\(386\) 0 0
\(387\) − 10.8009i − 0.549042i
\(388\) 0 0
\(389\) − 10.7701i − 0.546065i −0.962005 0.273033i \(-0.911973\pi\)
0.962005 0.273033i \(-0.0880267\pi\)
\(390\) 0 0
\(391\) 13.0437 0.659646
\(392\) 0 0
\(393\) 1.81076 0.0913405
\(394\) 0 0
\(395\) 15.6444i 0.787153i
\(396\) 0 0
\(397\) − 29.9845i − 1.50488i −0.658661 0.752440i \(-0.728877\pi\)
0.658661 0.752440i \(-0.271123\pi\)
\(398\) 0 0
\(399\) −0.100873 −0.00504998
\(400\) 0 0
\(401\) 34.4739 1.72155 0.860773 0.508989i \(-0.169981\pi\)
0.860773 + 0.508989i \(0.169981\pi\)
\(402\) 0 0
\(403\) − 16.5182i − 0.822829i
\(404\) 0 0
\(405\) 19.5647i 0.972179i
\(406\) 0 0
\(407\) 1.40057 0.0694235
\(408\) 0 0
\(409\) 14.7405 0.728869 0.364435 0.931229i \(-0.381262\pi\)
0.364435 + 0.931229i \(0.381262\pi\)
\(410\) 0 0
\(411\) 1.16458i 0.0574447i
\(412\) 0 0
\(413\) 6.92085i 0.340553i
\(414\) 0 0
\(415\) −33.3486 −1.63702
\(416\) 0 0
\(417\) −0.0212438 −0.00104032
\(418\) 0 0
\(419\) − 5.42033i − 0.264800i −0.991196 0.132400i \(-0.957732\pi\)
0.991196 0.132400i \(-0.0422684\pi\)
\(420\) 0 0
\(421\) − 9.72201i − 0.473822i −0.971531 0.236911i \(-0.923865\pi\)
0.971531 0.236911i \(-0.0761350\pi\)
\(422\) 0 0
\(423\) −19.9987 −0.972368
\(424\) 0 0
\(425\) 0.681376 0.0330516
\(426\) 0 0
\(427\) 4.30265i 0.208220i
\(428\) 0 0
\(429\) − 0.362267i − 0.0174904i
\(430\) 0 0
\(431\) 12.1520 0.585341 0.292670 0.956213i \(-0.405456\pi\)
0.292670 + 0.956213i \(0.405456\pi\)
\(432\) 0 0
\(433\) 30.0836 1.44573 0.722864 0.690991i \(-0.242825\pi\)
0.722864 + 0.690991i \(0.242825\pi\)
\(434\) 0 0
\(435\) − 0.0299225i − 0.00143467i
\(436\) 0 0
\(437\) 3.34969i 0.160238i
\(438\) 0 0
\(439\) 0.195719 0.00934117 0.00467058 0.999989i \(-0.498513\pi\)
0.00467058 + 0.999989i \(0.498513\pi\)
\(440\) 0 0
\(441\) 2.98973 0.142368
\(442\) 0 0
\(443\) 28.1256i 1.33629i 0.744032 + 0.668144i \(0.232911\pi\)
−0.744032 + 0.668144i \(0.767089\pi\)
\(444\) 0 0
\(445\) − 1.88171i − 0.0892017i
\(446\) 0 0
\(447\) 1.07337 0.0507686
\(448\) 0 0
\(449\) −12.7288 −0.600709 −0.300355 0.953828i \(-0.597105\pi\)
−0.300355 + 0.953828i \(0.597105\pi\)
\(450\) 0 0
\(451\) − 2.53907i − 0.119560i
\(452\) 0 0
\(453\) 1.16383i 0.0546816i
\(454\) 0 0
\(455\) 11.6466 0.546000
\(456\) 0 0
\(457\) −3.64289 −0.170407 −0.0852035 0.996364i \(-0.527154\pi\)
−0.0852035 + 0.996364i \(0.527154\pi\)
\(458\) 0 0
\(459\) − 2.35276i − 0.109817i
\(460\) 0 0
\(461\) − 11.4883i − 0.535064i −0.963549 0.267532i \(-0.913792\pi\)
0.963549 0.267532i \(-0.0862081\pi\)
\(462\) 0 0
\(463\) −7.98371 −0.371035 −0.185517 0.982641i \(-0.559396\pi\)
−0.185517 + 0.982641i \(0.559396\pi\)
\(464\) 0 0
\(465\) −0.693521 −0.0321613
\(466\) 0 0
\(467\) 27.5689i 1.27573i 0.770146 + 0.637867i \(0.220183\pi\)
−0.770146 + 0.637867i \(0.779817\pi\)
\(468\) 0 0
\(469\) − 13.9311i − 0.643280i
\(470\) 0 0
\(471\) 1.01180 0.0466212
\(472\) 0 0
\(473\) −2.43499 −0.111961
\(474\) 0 0
\(475\) 0.174982i 0.00802871i
\(476\) 0 0
\(477\) 8.77951i 0.401986i
\(478\) 0 0
\(479\) −12.9612 −0.592210 −0.296105 0.955155i \(-0.595688\pi\)
−0.296105 + 0.955155i \(0.595688\pi\)
\(480\) 0 0
\(481\) −11.0185 −0.502401
\(482\) 0 0
\(483\) 0.341173i 0.0155239i
\(484\) 0 0
\(485\) − 24.1878i − 1.09831i
\(486\) 0 0
\(487\) −8.86309 −0.401625 −0.200812 0.979630i \(-0.564358\pi\)
−0.200812 + 0.979630i \(0.564358\pi\)
\(488\) 0 0
\(489\) −0.381990 −0.0172742
\(490\) 0 0
\(491\) 17.5466i 0.791865i 0.918280 + 0.395933i \(0.129579\pi\)
−0.918280 + 0.395933i \(0.870421\pi\)
\(492\) 0 0
\(493\) − 0.520847i − 0.0234578i
\(494\) 0 0
\(495\) 4.42599 0.198934
\(496\) 0 0
\(497\) −10.6460 −0.477536
\(498\) 0 0
\(499\) − 0.153293i − 0.00686231i −0.999994 0.00343116i \(-0.998908\pi\)
0.999994 0.00343116i \(-0.00109217\pi\)
\(500\) 0 0
\(501\) 0.857406i 0.0383061i
\(502\) 0 0
\(503\) −15.5136 −0.691719 −0.345860 0.938286i \(-0.612413\pi\)
−0.345860 + 0.938286i \(0.612413\pi\)
\(504\) 0 0
\(505\) −0.778133 −0.0346264
\(506\) 0 0
\(507\) 1.53232i 0.0680526i
\(508\) 0 0
\(509\) − 27.0386i − 1.19846i −0.800575 0.599232i \(-0.795473\pi\)
0.800575 0.599232i \(-0.204527\pi\)
\(510\) 0 0
\(511\) 8.62368 0.381489
\(512\) 0 0
\(513\) 0.604204 0.0266762
\(514\) 0 0
\(515\) − 36.4053i − 1.60421i
\(516\) 0 0
\(517\) 4.50856i 0.198286i
\(518\) 0 0
\(519\) −0.143475 −0.00629787
\(520\) 0 0
\(521\) 15.7310 0.689190 0.344595 0.938752i \(-0.388016\pi\)
0.344595 + 0.938752i \(0.388016\pi\)
\(522\) 0 0
\(523\) 43.4999i 1.90212i 0.309009 + 0.951059i \(0.400003\pi\)
−0.309009 + 0.951059i \(0.599997\pi\)
\(524\) 0 0
\(525\) 0.0178223i 0 0.000777827i
\(526\) 0 0
\(527\) −12.0718 −0.525856
\(528\) 0 0
\(529\) −11.6707 −0.507421
\(530\) 0 0
\(531\) − 20.6914i − 0.897932i
\(532\) 0 0
\(533\) 19.9753i 0.865227i
\(534\) 0 0
\(535\) 39.0481 1.68820
\(536\) 0 0
\(537\) −1.74797 −0.0754303
\(538\) 0 0
\(539\) − 0.674013i − 0.0290318i
\(540\) 0 0
\(541\) − 30.1374i − 1.29571i −0.761765 0.647853i \(-0.775667\pi\)
0.761765 0.647853i \(-0.224333\pi\)
\(542\) 0 0
\(543\) −0.484209 −0.0207794
\(544\) 0 0
\(545\) −22.3960 −0.959341
\(546\) 0 0
\(547\) 22.1918i 0.948852i 0.880295 + 0.474426i \(0.157344\pi\)
−0.880295 + 0.474426i \(0.842656\pi\)
\(548\) 0 0
\(549\) − 12.8637i − 0.549011i
\(550\) 0 0
\(551\) 0.133757 0.00569823
\(552\) 0 0
\(553\) −7.12273 −0.302889
\(554\) 0 0
\(555\) 0.462616i 0.0196369i
\(556\) 0 0
\(557\) 4.64011i 0.196608i 0.995156 + 0.0983038i \(0.0313417\pi\)
−0.995156 + 0.0983038i \(0.968658\pi\)
\(558\) 0 0
\(559\) 19.1565 0.810235
\(560\) 0 0
\(561\) −0.264752 −0.0111778
\(562\) 0 0
\(563\) − 42.9470i − 1.81000i −0.425411 0.905000i \(-0.639870\pi\)
0.425411 0.905000i \(-0.360130\pi\)
\(564\) 0 0
\(565\) 21.2889i 0.895629i
\(566\) 0 0
\(567\) −8.90764 −0.374086
\(568\) 0 0
\(569\) −28.2672 −1.18502 −0.592512 0.805561i \(-0.701864\pi\)
−0.592512 + 0.805561i \(0.701864\pi\)
\(570\) 0 0
\(571\) − 22.6294i − 0.947012i −0.880791 0.473506i \(-0.842988\pi\)
0.880791 0.473506i \(-0.157012\pi\)
\(572\) 0 0
\(573\) 0.902268i 0.0376928i
\(574\) 0 0
\(575\) 0.591822 0.0246807
\(576\) 0 0
\(577\) 12.2701 0.510809 0.255405 0.966834i \(-0.417791\pi\)
0.255405 + 0.966834i \(0.417791\pi\)
\(578\) 0 0
\(579\) 1.21542i 0.0505113i
\(580\) 0 0
\(581\) − 15.1833i − 0.629909i
\(582\) 0 0
\(583\) 1.97928 0.0819734
\(584\) 0 0
\(585\) −34.8201 −1.43963
\(586\) 0 0
\(587\) − 21.0007i − 0.866793i −0.901203 0.433396i \(-0.857315\pi\)
0.901203 0.433396i \(-0.142685\pi\)
\(588\) 0 0
\(589\) − 3.10012i − 0.127738i
\(590\) 0 0
\(591\) 1.87599 0.0771679
\(592\) 0 0
\(593\) 4.30581 0.176819 0.0884093 0.996084i \(-0.471822\pi\)
0.0884093 + 0.996084i \(0.471822\pi\)
\(594\) 0 0
\(595\) − 8.51155i − 0.348940i
\(596\) 0 0
\(597\) − 0.216156i − 0.00884666i
\(598\) 0 0
\(599\) −2.42319 −0.0990087 −0.0495044 0.998774i \(-0.515764\pi\)
−0.0495044 + 0.998774i \(0.515764\pi\)
\(600\) 0 0
\(601\) −6.18973 −0.252485 −0.126242 0.991999i \(-0.540292\pi\)
−0.126242 + 0.991999i \(0.540292\pi\)
\(602\) 0 0
\(603\) 41.6503i 1.69613i
\(604\) 0 0
\(605\) 23.1626i 0.941693i
\(606\) 0 0
\(607\) −8.80892 −0.357543 −0.178772 0.983891i \(-0.557212\pi\)
−0.178772 + 0.983891i \(0.557212\pi\)
\(608\) 0 0
\(609\) 0.0136234 0.000552049 0
\(610\) 0 0
\(611\) − 35.4696i − 1.43495i
\(612\) 0 0
\(613\) − 19.7718i − 0.798577i −0.916825 0.399289i \(-0.869257\pi\)
0.916825 0.399289i \(-0.130743\pi\)
\(614\) 0 0
\(615\) 0.838670 0.0338185
\(616\) 0 0
\(617\) −12.9420 −0.521026 −0.260513 0.965470i \(-0.583892\pi\)
−0.260513 + 0.965470i \(0.583892\pi\)
\(618\) 0 0
\(619\) − 17.1723i − 0.690213i −0.938563 0.345107i \(-0.887843\pi\)
0.938563 0.345107i \(-0.112157\pi\)
\(620\) 0 0
\(621\) − 2.04354i − 0.0820042i
\(622\) 0 0
\(623\) 0.856726 0.0343240
\(624\) 0 0
\(625\) −24.0899 −0.963598
\(626\) 0 0
\(627\) − 0.0679899i − 0.00271526i
\(628\) 0 0
\(629\) 8.05254i 0.321076i
\(630\) 0 0
\(631\) −13.7312 −0.546629 −0.273314 0.961925i \(-0.588120\pi\)
−0.273314 + 0.961925i \(0.588120\pi\)
\(632\) 0 0
\(633\) 2.30331 0.0915482
\(634\) 0 0
\(635\) 44.5153i 1.76654i
\(636\) 0 0
\(637\) 5.30258i 0.210096i
\(638\) 0 0
\(639\) 31.8285 1.25912
\(640\) 0 0
\(641\) −11.7414 −0.463759 −0.231879 0.972745i \(-0.574487\pi\)
−0.231879 + 0.972745i \(0.574487\pi\)
\(642\) 0 0
\(643\) − 2.59560i − 0.102361i −0.998689 0.0511803i \(-0.983702\pi\)
0.998689 0.0511803i \(-0.0162983\pi\)
\(644\) 0 0
\(645\) − 0.804293i − 0.0316690i
\(646\) 0 0
\(647\) −29.0272 −1.14118 −0.570588 0.821236i \(-0.693285\pi\)
−0.570588 + 0.821236i \(0.693285\pi\)
\(648\) 0 0
\(649\) −4.66474 −0.183107
\(650\) 0 0
\(651\) − 0.315754i − 0.0123754i
\(652\) 0 0
\(653\) 39.2320i 1.53527i 0.640890 + 0.767633i \(0.278565\pi\)
−0.640890 + 0.767633i \(0.721435\pi\)
\(654\) 0 0
\(655\) −39.2372 −1.53312
\(656\) 0 0
\(657\) −25.7824 −1.00587
\(658\) 0 0
\(659\) − 18.2689i − 0.711655i −0.934552 0.355828i \(-0.884199\pi\)
0.934552 0.355828i \(-0.115801\pi\)
\(660\) 0 0
\(661\) − 39.5936i − 1.54001i −0.638036 0.770007i \(-0.720253\pi\)
0.638036 0.770007i \(-0.279747\pi\)
\(662\) 0 0
\(663\) 2.08285 0.0808911
\(664\) 0 0
\(665\) 2.18582 0.0847625
\(666\) 0 0
\(667\) − 0.452392i − 0.0175167i
\(668\) 0 0
\(669\) 0.111523i 0.00431173i
\(670\) 0 0
\(671\) −2.90004 −0.111955
\(672\) 0 0
\(673\) −8.38015 −0.323031 −0.161516 0.986870i \(-0.551638\pi\)
−0.161516 + 0.986870i \(0.551638\pi\)
\(674\) 0 0
\(675\) − 0.106750i − 0.00410883i
\(676\) 0 0
\(677\) 20.5355i 0.789243i 0.918844 + 0.394621i \(0.129124\pi\)
−0.918844 + 0.394621i \(0.870876\pi\)
\(678\) 0 0
\(679\) 11.0125 0.422620
\(680\) 0 0
\(681\) −0.267533 −0.0102519
\(682\) 0 0
\(683\) 24.2522i 0.927983i 0.885840 + 0.463991i \(0.153583\pi\)
−0.885840 + 0.463991i \(0.846417\pi\)
\(684\) 0 0
\(685\) − 25.2353i − 0.964192i
\(686\) 0 0
\(687\) 1.94900 0.0743592
\(688\) 0 0
\(689\) −15.5713 −0.593221
\(690\) 0 0
\(691\) 30.8579i 1.17389i 0.809627 + 0.586945i \(0.199669\pi\)
−0.809627 + 0.586945i \(0.800331\pi\)
\(692\) 0 0
\(693\) 2.01511i 0.0765478i
\(694\) 0 0
\(695\) 0.460332 0.0174614
\(696\) 0 0
\(697\) 14.5984 0.552952
\(698\) 0 0
\(699\) 0.881820i 0.0333535i
\(700\) 0 0
\(701\) − 49.9348i − 1.88601i −0.332775 0.943006i \(-0.607985\pi\)
0.332775 0.943006i \(-0.392015\pi\)
\(702\) 0 0
\(703\) −2.06794 −0.0779940
\(704\) 0 0
\(705\) −1.48920 −0.0560867
\(706\) 0 0
\(707\) − 0.354276i − 0.0133239i
\(708\) 0 0
\(709\) − 30.1773i − 1.13333i −0.823948 0.566665i \(-0.808233\pi\)
0.823948 0.566665i \(-0.191767\pi\)
\(710\) 0 0
\(711\) 21.2950 0.798625
\(712\) 0 0
\(713\) −10.4852 −0.392674
\(714\) 0 0
\(715\) 7.84994i 0.293571i
\(716\) 0 0
\(717\) 0.175206i 0.00654318i
\(718\) 0 0
\(719\) 25.5563 0.953089 0.476544 0.879150i \(-0.341889\pi\)
0.476544 + 0.879150i \(0.341889\pi\)
\(720\) 0 0
\(721\) 16.5750 0.617284
\(722\) 0 0
\(723\) − 1.78885i − 0.0665279i
\(724\) 0 0
\(725\) − 0.0236321i 0 0.000877674i
\(726\) 0 0
\(727\) −11.4845 −0.425937 −0.212969 0.977059i \(-0.568313\pi\)
−0.212969 + 0.977059i \(0.568313\pi\)
\(728\) 0 0
\(729\) 26.4468 0.979510
\(730\) 0 0
\(731\) − 14.0000i − 0.517807i
\(732\) 0 0
\(733\) − 52.0835i − 1.92375i −0.273495 0.961873i \(-0.588180\pi\)
0.273495 0.961873i \(-0.411820\pi\)
\(734\) 0 0
\(735\) 0.222631 0.00821185
\(736\) 0 0
\(737\) 9.38977 0.345877
\(738\) 0 0
\(739\) − 25.2269i − 0.927986i −0.885839 0.463993i \(-0.846416\pi\)
0.885839 0.463993i \(-0.153584\pi\)
\(740\) 0 0
\(741\) 0.534889i 0.0196496i
\(742\) 0 0
\(743\) −23.8950 −0.876623 −0.438311 0.898823i \(-0.644423\pi\)
−0.438311 + 0.898823i \(0.644423\pi\)
\(744\) 0 0
\(745\) −23.2588 −0.852136
\(746\) 0 0
\(747\) 45.3939i 1.66087i
\(748\) 0 0
\(749\) 17.7782i 0.649602i
\(750\) 0 0
\(751\) 29.7013 1.08381 0.541907 0.840438i \(-0.317702\pi\)
0.541907 + 0.840438i \(0.317702\pi\)
\(752\) 0 0
\(753\) 1.48088 0.0539664
\(754\) 0 0
\(755\) − 25.2190i − 0.917815i
\(756\) 0 0
\(757\) − 47.1188i − 1.71256i −0.516511 0.856280i \(-0.672770\pi\)
0.516511 0.856280i \(-0.327230\pi\)
\(758\) 0 0
\(759\) −0.229955 −0.00834685
\(760\) 0 0
\(761\) −11.5153 −0.417430 −0.208715 0.977977i \(-0.566928\pi\)
−0.208715 + 0.977977i \(0.566928\pi\)
\(762\) 0 0
\(763\) − 10.1967i − 0.369145i
\(764\) 0 0
\(765\) 25.4472i 0.920046i
\(766\) 0 0
\(767\) 36.6983 1.32510
\(768\) 0 0
\(769\) 27.5195 0.992378 0.496189 0.868215i \(-0.334732\pi\)
0.496189 + 0.868215i \(0.334732\pi\)
\(770\) 0 0
\(771\) 2.45223i 0.0883150i
\(772\) 0 0
\(773\) − 13.6192i − 0.489850i −0.969542 0.244925i \(-0.921237\pi\)
0.969542 0.244925i \(-0.0787634\pi\)
\(774\) 0 0
\(775\) −0.547727 −0.0196749
\(776\) 0 0
\(777\) −0.210624 −0.00755611
\(778\) 0 0
\(779\) 3.74895i 0.134320i
\(780\) 0 0
\(781\) − 7.17551i − 0.256760i
\(782\) 0 0
\(783\) −0.0816005 −0.00291616
\(784\) 0 0
\(785\) −21.9246 −0.782523
\(786\) 0 0
\(787\) − 34.6258i − 1.23428i −0.786855 0.617138i \(-0.788292\pi\)
0.786855 0.617138i \(-0.211708\pi\)
\(788\) 0 0
\(789\) − 2.41400i − 0.0859408i
\(790\) 0 0
\(791\) −9.69262 −0.344630
\(792\) 0 0
\(793\) 22.8151 0.810189
\(794\) 0 0
\(795\) 0.653768i 0.0231868i
\(796\) 0 0
\(797\) − 35.0863i − 1.24282i −0.783486 0.621410i \(-0.786560\pi\)
0.783486 0.621410i \(-0.213440\pi\)
\(798\) 0 0
\(799\) −25.9219 −0.917051
\(800\) 0 0
\(801\) −2.56138 −0.0905017
\(802\) 0 0
\(803\) 5.81247i 0.205118i
\(804\) 0 0
\(805\) − 7.39287i − 0.260565i
\(806\) 0 0
\(807\) 0.0308623 0.00108641
\(808\) 0 0
\(809\) 28.3561 0.996947 0.498473 0.866905i \(-0.333894\pi\)
0.498473 + 0.866905i \(0.333894\pi\)
\(810\) 0 0
\(811\) − 2.29432i − 0.0805646i −0.999188 0.0402823i \(-0.987174\pi\)
0.999188 0.0402823i \(-0.0128257\pi\)
\(812\) 0 0
\(813\) 1.74360i 0.0611508i
\(814\) 0 0
\(815\) 8.27732 0.289942
\(816\) 0 0
\(817\) 3.59528 0.125783
\(818\) 0 0
\(819\) − 15.8533i − 0.553958i
\(820\) 0 0
\(821\) − 41.7766i − 1.45801i −0.684506 0.729007i \(-0.739982\pi\)
0.684506 0.729007i \(-0.260018\pi\)
\(822\) 0 0
\(823\) −45.2110 −1.57596 −0.787979 0.615702i \(-0.788872\pi\)
−0.787979 + 0.615702i \(0.788872\pi\)
\(824\) 0 0
\(825\) −0.0120124 −0.000418219 0
\(826\) 0 0
\(827\) − 5.06276i − 0.176049i −0.996118 0.0880246i \(-0.971945\pi\)
0.996118 0.0880246i \(-0.0280554\pi\)
\(828\) 0 0
\(829\) − 51.1701i − 1.77721i −0.458671 0.888606i \(-0.651674\pi\)
0.458671 0.888606i \(-0.348326\pi\)
\(830\) 0 0
\(831\) −2.14369 −0.0743637
\(832\) 0 0
\(833\) 3.87523 0.134269
\(834\) 0 0
\(835\) − 18.5791i − 0.642956i
\(836\) 0 0
\(837\) 1.89128i 0.0653721i
\(838\) 0 0
\(839\) 40.5224 1.39899 0.699495 0.714638i \(-0.253408\pi\)
0.699495 + 0.714638i \(0.253408\pi\)
\(840\) 0 0
\(841\) 28.9819 0.999377
\(842\) 0 0
\(843\) 1.12485i 0.0387420i
\(844\) 0 0
\(845\) − 33.2037i − 1.14224i
\(846\) 0 0
\(847\) −10.5457 −0.362355
\(848\) 0 0
\(849\) −2.41291 −0.0828109
\(850\) 0 0
\(851\) 6.99419i 0.239758i
\(852\) 0 0
\(853\) − 8.02166i − 0.274656i −0.990526 0.137328i \(-0.956148\pi\)
0.990526 0.137328i \(-0.0438515\pi\)
\(854\) 0 0
\(855\) −6.53500 −0.223492
\(856\) 0 0
\(857\) 0.785744 0.0268405 0.0134203 0.999910i \(-0.495728\pi\)
0.0134203 + 0.999910i \(0.495728\pi\)
\(858\) 0 0
\(859\) 17.1625i 0.585577i 0.956177 + 0.292789i \(0.0945833\pi\)
−0.956177 + 0.292789i \(0.905417\pi\)
\(860\) 0 0
\(861\) 0.381839i 0.0130130i
\(862\) 0 0
\(863\) 8.01096 0.272696 0.136348 0.990661i \(-0.456463\pi\)
0.136348 + 0.990661i \(0.456463\pi\)
\(864\) 0 0
\(865\) 3.10896 0.105708
\(866\) 0 0
\(867\) 0.200959i 0.00682494i
\(868\) 0 0
\(869\) − 4.80081i − 0.162856i
\(870\) 0 0
\(871\) −73.8710 −2.50302
\(872\) 0 0
\(873\) −32.9243 −1.11432
\(874\) 0 0
\(875\) − 11.3682i − 0.384315i
\(876\) 0 0
\(877\) − 7.20234i − 0.243206i −0.992579 0.121603i \(-0.961197\pi\)
0.992579 0.121603i \(-0.0388034\pi\)
\(878\) 0 0
\(879\) 1.34520 0.0453724
\(880\) 0 0
\(881\) 28.3868 0.956377 0.478188 0.878257i \(-0.341294\pi\)
0.478188 + 0.878257i \(0.341294\pi\)
\(882\) 0 0
\(883\) 27.6390i 0.930126i 0.885278 + 0.465063i \(0.153968\pi\)
−0.885278 + 0.465063i \(0.846032\pi\)
\(884\) 0 0
\(885\) − 1.54079i − 0.0517931i
\(886\) 0 0
\(887\) 6.99105 0.234737 0.117368 0.993088i \(-0.462554\pi\)
0.117368 + 0.993088i \(0.462554\pi\)
\(888\) 0 0
\(889\) −20.2674 −0.679746
\(890\) 0 0
\(891\) − 6.00386i − 0.201137i
\(892\) 0 0
\(893\) − 6.65691i − 0.222765i
\(894\) 0 0
\(895\) 37.8766 1.26607
\(896\) 0 0
\(897\) 1.80910 0.0604040
\(898\) 0 0
\(899\) 0.418685i 0.0139639i
\(900\) 0 0
\(901\) 11.3798i 0.379118i
\(902\) 0 0
\(903\) 0.366187 0.0121859
\(904\) 0 0
\(905\) 10.4923 0.348776
\(906\) 0 0
\(907\) − 8.75000i − 0.290539i −0.989392 0.145270i \(-0.953595\pi\)
0.989392 0.145270i \(-0.0464049\pi\)
\(908\) 0 0
\(909\) 1.05919i 0.0351311i
\(910\) 0 0
\(911\) −23.5428 −0.780009 −0.390005 0.920813i \(-0.627527\pi\)
−0.390005 + 0.920813i \(0.627527\pi\)
\(912\) 0 0
\(913\) 10.2337 0.338687
\(914\) 0 0
\(915\) − 0.957901i − 0.0316672i
\(916\) 0 0
\(917\) − 17.8643i − 0.589932i
\(918\) 0 0
\(919\) 53.5349 1.76595 0.882976 0.469417i \(-0.155536\pi\)
0.882976 + 0.469417i \(0.155536\pi\)
\(920\) 0 0
\(921\) 1.46290 0.0482041
\(922\) 0 0
\(923\) 56.4510i 1.85811i
\(924\) 0 0
\(925\) 0.365363i 0.0120131i
\(926\) 0 0
\(927\) −49.5546 −1.62759
\(928\) 0 0
\(929\) 43.7405 1.43508 0.717540 0.696518i \(-0.245268\pi\)
0.717540 + 0.696518i \(0.245268\pi\)
\(930\) 0 0
\(931\) 0.995183i 0.0326158i
\(932\) 0 0
\(933\) − 2.27358i − 0.0744337i
\(934\) 0 0
\(935\) 5.73689 0.187617
\(936\) 0 0
\(937\) 8.90798 0.291011 0.145506 0.989357i \(-0.453519\pi\)
0.145506 + 0.989357i \(0.453519\pi\)
\(938\) 0 0
\(939\) − 2.49360i − 0.0813755i
\(940\) 0 0
\(941\) 24.1765i 0.788133i 0.919082 + 0.394066i \(0.128932\pi\)
−0.919082 + 0.394066i \(0.871068\pi\)
\(942\) 0 0
\(943\) 12.6797 0.412907
\(944\) 0 0
\(945\) −1.33350 −0.0433786
\(946\) 0 0
\(947\) − 8.55120i − 0.277877i −0.990301 0.138938i \(-0.955631\pi\)
0.990301 0.138938i \(-0.0443690\pi\)
\(948\) 0 0
\(949\) − 45.7277i − 1.48438i
\(950\) 0 0
\(951\) 3.13732 0.101735
\(952\) 0 0
\(953\) 23.7559 0.769529 0.384765 0.923015i \(-0.374283\pi\)
0.384765 + 0.923015i \(0.374283\pi\)
\(954\) 0 0
\(955\) − 19.5512i − 0.632662i
\(956\) 0 0
\(957\) 0.00918236i 0 0.000296823i
\(958\) 0 0
\(959\) 11.4894 0.371012
\(960\) 0 0
\(961\) −21.2960 −0.686968
\(962\) 0 0
\(963\) − 53.1520i − 1.71280i
\(964\) 0 0
\(965\) − 26.3370i − 0.847817i
\(966\) 0 0
\(967\) 56.9403 1.83108 0.915538 0.402231i \(-0.131765\pi\)
0.915538 + 0.402231i \(0.131765\pi\)
\(968\) 0 0
\(969\) 0.390907 0.0125578
\(970\) 0 0
\(971\) 25.9137i 0.831610i 0.909454 + 0.415805i \(0.136500\pi\)
−0.909454 + 0.415805i \(0.863500\pi\)
\(972\) 0 0
\(973\) 0.209585i 0.00671898i
\(974\) 0 0
\(975\) 0.0945039 0.00302655
\(976\) 0 0
\(977\) 13.2222 0.423014 0.211507 0.977376i \(-0.432163\pi\)
0.211507 + 0.977376i \(0.432163\pi\)
\(978\) 0 0
\(979\) 0.577444i 0.0184552i
\(980\) 0 0
\(981\) 30.4854i 0.973322i
\(982\) 0 0
\(983\) 39.0935 1.24689 0.623445 0.781867i \(-0.285732\pi\)
0.623445 + 0.781867i \(0.285732\pi\)
\(984\) 0 0
\(985\) −40.6508 −1.29524
\(986\) 0 0
\(987\) − 0.678020i − 0.0215816i
\(988\) 0 0
\(989\) − 12.1599i − 0.386664i
\(990\) 0 0
\(991\) −21.0291 −0.668012 −0.334006 0.942571i \(-0.608401\pi\)
−0.334006 + 0.942571i \(0.608401\pi\)
\(992\) 0 0
\(993\) −3.38849 −0.107530
\(994\) 0 0
\(995\) 4.68387i 0.148489i
\(996\) 0 0
\(997\) − 26.9937i − 0.854899i −0.904039 0.427449i \(-0.859412\pi\)
0.904039 0.427449i \(-0.140588\pi\)
\(998\) 0 0
\(999\) 1.26158 0.0399147
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 3584.2.b.i.1793.6 12
4.3 odd 2 3584.2.b.k.1793.7 12
8.3 odd 2 3584.2.b.k.1793.6 12
8.5 even 2 inner 3584.2.b.i.1793.7 12
16.3 odd 4 3584.2.a.k.1.3 yes 6
16.5 even 4 3584.2.a.l.1.3 yes 6
16.11 odd 4 3584.2.a.e.1.4 6
16.13 even 4 3584.2.a.f.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3584.2.a.e.1.4 6 16.11 odd 4
3584.2.a.f.1.4 yes 6 16.13 even 4
3584.2.a.k.1.3 yes 6 16.3 odd 4
3584.2.a.l.1.3 yes 6 16.5 even 4
3584.2.b.i.1793.6 12 1.1 even 1 trivial
3584.2.b.i.1793.7 12 8.5 even 2 inner
3584.2.b.k.1793.6 12 8.3 odd 2
3584.2.b.k.1793.7 12 4.3 odd 2