Properties

Label 380.2.a.c.1.1
Level $380$
Weight $2$
Character 380.1
Self dual yes
Analytic conductor $3.034$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(1,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.a (trivial)

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: yes
Analytic conductor: \(3.03431527681\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 380.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.41421 q^{3} +1.00000 q^{5} +0.828427 q^{7} +8.65685 q^{9} +O(q^{10})\) \(q-3.41421 q^{3} +1.00000 q^{5} +0.828427 q^{7} +8.65685 q^{9} -2.00000 q^{11} -6.24264 q^{13} -3.41421 q^{15} +0.828427 q^{17} -1.00000 q^{19} -2.82843 q^{21} -6.00000 q^{23} +1.00000 q^{25} -19.3137 q^{27} -6.48528 q^{29} -6.82843 q^{31} +6.82843 q^{33} +0.828427 q^{35} -1.75736 q^{37} +21.3137 q^{39} +3.65685 q^{41} +4.82843 q^{43} +8.65685 q^{45} -4.82843 q^{47} -6.31371 q^{49} -2.82843 q^{51} +9.07107 q^{53} -2.00000 q^{55} +3.41421 q^{57} +13.6569 q^{59} -13.6569 q^{61} +7.17157 q^{63} -6.24264 q^{65} -3.41421 q^{67} +20.4853 q^{69} +5.17157 q^{71} -2.48528 q^{73} -3.41421 q^{75} -1.65685 q^{77} +1.65685 q^{79} +39.9706 q^{81} -13.3137 q^{83} +0.828427 q^{85} +22.1421 q^{87} -6.48528 q^{89} -5.17157 q^{91} +23.3137 q^{93} -1.00000 q^{95} -10.2426 q^{97} -17.3137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 2 q^{19} - 12 q^{23} + 2 q^{25} - 16 q^{27} + 4 q^{29} - 8 q^{31} + 8 q^{33} - 4 q^{35} - 12 q^{37} + 20 q^{39} - 4 q^{41} + 4 q^{43} + 6 q^{45} - 4 q^{47} + 10 q^{49} + 4 q^{53} - 4 q^{55} + 4 q^{57} + 16 q^{59} - 16 q^{61} + 20 q^{63} - 4 q^{65} - 4 q^{67} + 24 q^{69} + 16 q^{71} + 12 q^{73} - 4 q^{75} + 8 q^{77} - 8 q^{79} + 46 q^{81} - 4 q^{83} - 4 q^{85} + 16 q^{87} + 4 q^{89} - 16 q^{91} + 24 q^{93} - 2 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.41421 −1.97120 −0.985599 0.169102i \(-0.945913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 0 0
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −6.24264 −1.73140 −0.865699 0.500566i \(-0.833125\pi\)
−0.865699 + 0.500566i \(0.833125\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 0.828427 0.200923 0.100462 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −19.3137 −3.71692
\(28\) 0 0
\(29\) −6.48528 −1.20429 −0.602143 0.798388i \(-0.705686\pi\)
−0.602143 + 0.798388i \(0.705686\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) 6.82843 1.18868
\(34\) 0 0
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) −1.75736 −0.288908 −0.144454 0.989512i \(-0.546143\pi\)
−0.144454 + 0.989512i \(0.546143\pi\)
\(38\) 0 0
\(39\) 21.3137 3.41292
\(40\) 0 0
\(41\) 3.65685 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) 4.82843 0.736328 0.368164 0.929761i \(-0.379986\pi\)
0.368164 + 0.929761i \(0.379986\pi\)
\(44\) 0 0
\(45\) 8.65685 1.29049
\(46\) 0 0
\(47\) −4.82843 −0.704298 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) 9.07107 1.24601 0.623003 0.782219i \(-0.285912\pi\)
0.623003 + 0.782219i \(0.285912\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 3.41421 0.452224
\(58\) 0 0
\(59\) 13.6569 1.77797 0.888985 0.457935i \(-0.151411\pi\)
0.888985 + 0.457935i \(0.151411\pi\)
\(60\) 0 0
\(61\) −13.6569 −1.74858 −0.874291 0.485403i \(-0.838673\pi\)
−0.874291 + 0.485403i \(0.838673\pi\)
\(62\) 0 0
\(63\) 7.17157 0.903533
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) −3.41421 −0.417113 −0.208556 0.978010i \(-0.566876\pi\)
−0.208556 + 0.978010i \(0.566876\pi\)
\(68\) 0 0
\(69\) 20.4853 2.46614
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) −2.48528 −0.290880 −0.145440 0.989367i \(-0.546460\pi\)
−0.145440 + 0.989367i \(0.546460\pi\)
\(74\) 0 0
\(75\) −3.41421 −0.394239
\(76\) 0 0
\(77\) −1.65685 −0.188816
\(78\) 0 0
\(79\) 1.65685 0.186411 0.0932053 0.995647i \(-0.470289\pi\)
0.0932053 + 0.995647i \(0.470289\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 0 0
\(83\) −13.3137 −1.46137 −0.730685 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 22.1421 2.37389
\(88\) 0 0
\(89\) −6.48528 −0.687438 −0.343719 0.939072i \(-0.611687\pi\)
−0.343719 + 0.939072i \(0.611687\pi\)
\(90\) 0 0
\(91\) −5.17157 −0.542128
\(92\) 0 0
\(93\) 23.3137 2.41752
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −10.2426 −1.03998 −0.519991 0.854172i \(-0.674065\pi\)
−0.519991 + 0.854172i \(0.674065\pi\)
\(98\) 0 0
\(99\) −17.3137 −1.74009
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 3.89949 0.384229 0.192114 0.981373i \(-0.438466\pi\)
0.192114 + 0.981373i \(0.438466\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 0 0
\(107\) 7.41421 0.716759 0.358380 0.933576i \(-0.383329\pi\)
0.358380 + 0.933576i \(0.383329\pi\)
\(108\) 0 0
\(109\) −3.17157 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −9.07107 −0.853334 −0.426667 0.904409i \(-0.640312\pi\)
−0.426667 + 0.904409i \(0.640312\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −54.0416 −4.99615
\(118\) 0 0
\(119\) 0.686292 0.0629122
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −12.4853 −1.12576
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.55635 0.847989 0.423994 0.905665i \(-0.360628\pi\)
0.423994 + 0.905665i \(0.360628\pi\)
\(128\) 0 0
\(129\) −16.4853 −1.45145
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) −0.828427 −0.0718337
\(134\) 0 0
\(135\) −19.3137 −1.66226
\(136\) 0 0
\(137\) 8.14214 0.695630 0.347815 0.937563i \(-0.386924\pi\)
0.347815 + 0.937563i \(0.386924\pi\)
\(138\) 0 0
\(139\) 5.31371 0.450703 0.225351 0.974278i \(-0.427647\pi\)
0.225351 + 0.974278i \(0.427647\pi\)
\(140\) 0 0
\(141\) 16.4853 1.38831
\(142\) 0 0
\(143\) 12.4853 1.04407
\(144\) 0 0
\(145\) −6.48528 −0.538573
\(146\) 0 0
\(147\) 21.5563 1.77794
\(148\) 0 0
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) −10.1421 −0.825355 −0.412678 0.910877i \(-0.635406\pi\)
−0.412678 + 0.910877i \(0.635406\pi\)
\(152\) 0 0
\(153\) 7.17157 0.579787
\(154\) 0 0
\(155\) −6.82843 −0.548472
\(156\) 0 0
\(157\) 8.82843 0.704585 0.352293 0.935890i \(-0.385402\pi\)
0.352293 + 0.935890i \(0.385402\pi\)
\(158\) 0 0
\(159\) −30.9706 −2.45613
\(160\) 0 0
\(161\) −4.97056 −0.391735
\(162\) 0 0
\(163\) 14.4853 1.13457 0.567287 0.823520i \(-0.307993\pi\)
0.567287 + 0.823520i \(0.307993\pi\)
\(164\) 0 0
\(165\) 6.82843 0.531592
\(166\) 0 0
\(167\) 14.7279 1.13968 0.569840 0.821755i \(-0.307005\pi\)
0.569840 + 0.821755i \(0.307005\pi\)
\(168\) 0 0
\(169\) 25.9706 1.99774
\(170\) 0 0
\(171\) −8.65685 −0.662006
\(172\) 0 0
\(173\) 15.8995 1.20882 0.604408 0.796675i \(-0.293410\pi\)
0.604408 + 0.796675i \(0.293410\pi\)
\(174\) 0 0
\(175\) 0.828427 0.0626232
\(176\) 0 0
\(177\) −46.6274 −3.50473
\(178\) 0 0
\(179\) 10.3431 0.773083 0.386542 0.922272i \(-0.373670\pi\)
0.386542 + 0.922272i \(0.373670\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 46.6274 3.44680
\(184\) 0 0
\(185\) −1.75736 −0.129204
\(186\) 0 0
\(187\) −1.65685 −0.121161
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 3.41421 0.245760 0.122880 0.992422i \(-0.460787\pi\)
0.122880 + 0.992422i \(0.460787\pi\)
\(194\) 0 0
\(195\) 21.3137 1.52631
\(196\) 0 0
\(197\) 17.3137 1.23355 0.616775 0.787139i \(-0.288439\pi\)
0.616775 + 0.787139i \(0.288439\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 11.6569 0.822211
\(202\) 0 0
\(203\) −5.37258 −0.377081
\(204\) 0 0
\(205\) 3.65685 0.255406
\(206\) 0 0
\(207\) −51.9411 −3.61016
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −28.4853 −1.96101 −0.980504 0.196500i \(-0.937042\pi\)
−0.980504 + 0.196500i \(0.937042\pi\)
\(212\) 0 0
\(213\) −17.6569 −1.20983
\(214\) 0 0
\(215\) 4.82843 0.329296
\(216\) 0 0
\(217\) −5.65685 −0.384012
\(218\) 0 0
\(219\) 8.48528 0.573382
\(220\) 0 0
\(221\) −5.17157 −0.347878
\(222\) 0 0
\(223\) −23.2132 −1.55447 −0.777236 0.629210i \(-0.783379\pi\)
−0.777236 + 0.629210i \(0.783379\pi\)
\(224\) 0 0
\(225\) 8.65685 0.577124
\(226\) 0 0
\(227\) 23.4142 1.55406 0.777028 0.629466i \(-0.216726\pi\)
0.777028 + 0.629466i \(0.216726\pi\)
\(228\) 0 0
\(229\) −10.3431 −0.683494 −0.341747 0.939792i \(-0.611019\pi\)
−0.341747 + 0.939792i \(0.611019\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −4.82843 −0.314972
\(236\) 0 0
\(237\) −5.65685 −0.367452
\(238\) 0 0
\(239\) 21.6569 1.40087 0.700433 0.713718i \(-0.252990\pi\)
0.700433 + 0.713718i \(0.252990\pi\)
\(240\) 0 0
\(241\) 26.2843 1.69312 0.846559 0.532294i \(-0.178670\pi\)
0.846559 + 0.532294i \(0.178670\pi\)
\(242\) 0 0
\(243\) −78.5269 −5.03750
\(244\) 0 0
\(245\) −6.31371 −0.403368
\(246\) 0 0
\(247\) 6.24264 0.397210
\(248\) 0 0
\(249\) 45.4558 2.88065
\(250\) 0 0
\(251\) −2.34315 −0.147898 −0.0739490 0.997262i \(-0.523560\pi\)
−0.0739490 + 0.997262i \(0.523560\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) −2.82843 −0.177123
\(256\) 0 0
\(257\) 6.24264 0.389405 0.194703 0.980862i \(-0.437626\pi\)
0.194703 + 0.980862i \(0.437626\pi\)
\(258\) 0 0
\(259\) −1.45584 −0.0904618
\(260\) 0 0
\(261\) −56.1421 −3.47511
\(262\) 0 0
\(263\) −7.65685 −0.472142 −0.236071 0.971736i \(-0.575860\pi\)
−0.236071 + 0.971736i \(0.575860\pi\)
\(264\) 0 0
\(265\) 9.07107 0.557231
\(266\) 0 0
\(267\) 22.1421 1.35508
\(268\) 0 0
\(269\) −5.51472 −0.336238 −0.168119 0.985767i \(-0.553769\pi\)
−0.168119 + 0.985767i \(0.553769\pi\)
\(270\) 0 0
\(271\) 12.3431 0.749793 0.374896 0.927067i \(-0.377678\pi\)
0.374896 + 0.927067i \(0.377678\pi\)
\(272\) 0 0
\(273\) 17.6569 1.06864
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 11.1716 0.671235 0.335617 0.941998i \(-0.391055\pi\)
0.335617 + 0.941998i \(0.391055\pi\)
\(278\) 0 0
\(279\) −59.1127 −3.53898
\(280\) 0 0
\(281\) −14.9706 −0.893069 −0.446534 0.894766i \(-0.647342\pi\)
−0.446534 + 0.894766i \(0.647342\pi\)
\(282\) 0 0
\(283\) −25.3137 −1.50474 −0.752372 0.658739i \(-0.771090\pi\)
−0.752372 + 0.658739i \(0.771090\pi\)
\(284\) 0 0
\(285\) 3.41421 0.202241
\(286\) 0 0
\(287\) 3.02944 0.178822
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 34.9706 2.05001
\(292\) 0 0
\(293\) −20.5858 −1.20263 −0.601317 0.799010i \(-0.705357\pi\)
−0.601317 + 0.799010i \(0.705357\pi\)
\(294\) 0 0
\(295\) 13.6569 0.795133
\(296\) 0 0
\(297\) 38.6274 2.24139
\(298\) 0 0
\(299\) 37.4558 2.16613
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −13.6569 −0.784566
\(304\) 0 0
\(305\) −13.6569 −0.781989
\(306\) 0 0
\(307\) −11.8995 −0.679140 −0.339570 0.940581i \(-0.610282\pi\)
−0.339570 + 0.940581i \(0.610282\pi\)
\(308\) 0 0
\(309\) −13.3137 −0.757390
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 0 0
\(315\) 7.17157 0.404072
\(316\) 0 0
\(317\) −14.2426 −0.799946 −0.399973 0.916527i \(-0.630981\pi\)
−0.399973 + 0.916527i \(0.630981\pi\)
\(318\) 0 0
\(319\) 12.9706 0.726212
\(320\) 0 0
\(321\) −25.3137 −1.41287
\(322\) 0 0
\(323\) −0.828427 −0.0460949
\(324\) 0 0
\(325\) −6.24264 −0.346279
\(326\) 0 0
\(327\) 10.8284 0.598813
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 6.82843 0.375324 0.187662 0.982234i \(-0.439909\pi\)
0.187662 + 0.982234i \(0.439909\pi\)
\(332\) 0 0
\(333\) −15.2132 −0.833678
\(334\) 0 0
\(335\) −3.41421 −0.186538
\(336\) 0 0
\(337\) 26.2426 1.42953 0.714764 0.699366i \(-0.246534\pi\)
0.714764 + 0.699366i \(0.246534\pi\)
\(338\) 0 0
\(339\) 30.9706 1.68209
\(340\) 0 0
\(341\) 13.6569 0.739560
\(342\) 0 0
\(343\) −11.0294 −0.595534
\(344\) 0 0
\(345\) 20.4853 1.10289
\(346\) 0 0
\(347\) −10.9706 −0.588931 −0.294465 0.955662i \(-0.595142\pi\)
−0.294465 + 0.955662i \(0.595142\pi\)
\(348\) 0 0
\(349\) −11.6569 −0.623977 −0.311989 0.950086i \(-0.600995\pi\)
−0.311989 + 0.950086i \(0.600995\pi\)
\(350\) 0 0
\(351\) 120.569 6.43547
\(352\) 0 0
\(353\) −32.1421 −1.71075 −0.855377 0.518007i \(-0.826674\pi\)
−0.855377 + 0.518007i \(0.826674\pi\)
\(354\) 0 0
\(355\) 5.17157 0.274479
\(356\) 0 0
\(357\) −2.34315 −0.124012
\(358\) 0 0
\(359\) 1.31371 0.0693349 0.0346674 0.999399i \(-0.488963\pi\)
0.0346674 + 0.999399i \(0.488963\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 23.8995 1.25440
\(364\) 0 0
\(365\) −2.48528 −0.130086
\(366\) 0 0
\(367\) 0.142136 0.00741942 0.00370971 0.999993i \(-0.498819\pi\)
0.00370971 + 0.999993i \(0.498819\pi\)
\(368\) 0 0
\(369\) 31.6569 1.64799
\(370\) 0 0
\(371\) 7.51472 0.390145
\(372\) 0 0
\(373\) −21.5563 −1.11615 −0.558073 0.829792i \(-0.688459\pi\)
−0.558073 + 0.829792i \(0.688459\pi\)
\(374\) 0 0
\(375\) −3.41421 −0.176309
\(376\) 0 0
\(377\) 40.4853 2.08510
\(378\) 0 0
\(379\) 33.6569 1.72884 0.864418 0.502773i \(-0.167687\pi\)
0.864418 + 0.502773i \(0.167687\pi\)
\(380\) 0 0
\(381\) −32.6274 −1.67155
\(382\) 0 0
\(383\) 17.0711 0.872291 0.436145 0.899876i \(-0.356343\pi\)
0.436145 + 0.899876i \(0.356343\pi\)
\(384\) 0 0
\(385\) −1.65685 −0.0844411
\(386\) 0 0
\(387\) 41.7990 2.12476
\(388\) 0 0
\(389\) 16.6274 0.843044 0.421522 0.906818i \(-0.361496\pi\)
0.421522 + 0.906818i \(0.361496\pi\)
\(390\) 0 0
\(391\) −4.97056 −0.251372
\(392\) 0 0
\(393\) −19.3137 −0.974248
\(394\) 0 0
\(395\) 1.65685 0.0833654
\(396\) 0 0
\(397\) −3.85786 −0.193621 −0.0968103 0.995303i \(-0.530864\pi\)
−0.0968103 + 0.995303i \(0.530864\pi\)
\(398\) 0 0
\(399\) 2.82843 0.141598
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 0 0
\(403\) 42.6274 2.12342
\(404\) 0 0
\(405\) 39.9706 1.98615
\(406\) 0 0
\(407\) 3.51472 0.174218
\(408\) 0 0
\(409\) 26.4853 1.30961 0.654806 0.755797i \(-0.272750\pi\)
0.654806 + 0.755797i \(0.272750\pi\)
\(410\) 0 0
\(411\) −27.7990 −1.37122
\(412\) 0 0
\(413\) 11.3137 0.556711
\(414\) 0 0
\(415\) −13.3137 −0.653544
\(416\) 0 0
\(417\) −18.1421 −0.888424
\(418\) 0 0
\(419\) −1.65685 −0.0809426 −0.0404713 0.999181i \(-0.512886\pi\)
−0.0404713 + 0.999181i \(0.512886\pi\)
\(420\) 0 0
\(421\) −19.6569 −0.958016 −0.479008 0.877810i \(-0.659004\pi\)
−0.479008 + 0.877810i \(0.659004\pi\)
\(422\) 0 0
\(423\) −41.7990 −2.03234
\(424\) 0 0
\(425\) 0.828427 0.0401846
\(426\) 0 0
\(427\) −11.3137 −0.547509
\(428\) 0 0
\(429\) −42.6274 −2.05807
\(430\) 0 0
\(431\) −17.4558 −0.840818 −0.420409 0.907335i \(-0.638113\pi\)
−0.420409 + 0.907335i \(0.638113\pi\)
\(432\) 0 0
\(433\) 27.2132 1.30778 0.653892 0.756588i \(-0.273135\pi\)
0.653892 + 0.756588i \(0.273135\pi\)
\(434\) 0 0
\(435\) 22.1421 1.06163
\(436\) 0 0
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) −18.3431 −0.875471 −0.437735 0.899104i \(-0.644219\pi\)
−0.437735 + 0.899104i \(0.644219\pi\)
\(440\) 0 0
\(441\) −54.6569 −2.60271
\(442\) 0 0
\(443\) −22.2843 −1.05876 −0.529379 0.848386i \(-0.677575\pi\)
−0.529379 + 0.848386i \(0.677575\pi\)
\(444\) 0 0
\(445\) −6.48528 −0.307432
\(446\) 0 0
\(447\) 27.3137 1.29189
\(448\) 0 0
\(449\) −30.4853 −1.43869 −0.719345 0.694653i \(-0.755558\pi\)
−0.719345 + 0.694653i \(0.755558\pi\)
\(450\) 0 0
\(451\) −7.31371 −0.344389
\(452\) 0 0
\(453\) 34.6274 1.62694
\(454\) 0 0
\(455\) −5.17157 −0.242447
\(456\) 0 0
\(457\) −14.9706 −0.700293 −0.350147 0.936695i \(-0.613868\pi\)
−0.350147 + 0.936695i \(0.613868\pi\)
\(458\) 0 0
\(459\) −16.0000 −0.746816
\(460\) 0 0
\(461\) 17.3137 0.806380 0.403190 0.915116i \(-0.367901\pi\)
0.403190 + 0.915116i \(0.367901\pi\)
\(462\) 0 0
\(463\) 13.3137 0.618741 0.309370 0.950942i \(-0.399882\pi\)
0.309370 + 0.950942i \(0.399882\pi\)
\(464\) 0 0
\(465\) 23.3137 1.08115
\(466\) 0 0
\(467\) −13.3137 −0.616085 −0.308042 0.951373i \(-0.599674\pi\)
−0.308042 + 0.951373i \(0.599674\pi\)
\(468\) 0 0
\(469\) −2.82843 −0.130605
\(470\) 0 0
\(471\) −30.1421 −1.38888
\(472\) 0 0
\(473\) −9.65685 −0.444023
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 78.5269 3.59550
\(478\) 0 0
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 10.9706 0.500215
\(482\) 0 0
\(483\) 16.9706 0.772187
\(484\) 0 0
\(485\) −10.2426 −0.465094
\(486\) 0 0
\(487\) −12.3848 −0.561208 −0.280604 0.959824i \(-0.590535\pi\)
−0.280604 + 0.959824i \(0.590535\pi\)
\(488\) 0 0
\(489\) −49.4558 −2.23647
\(490\) 0 0
\(491\) 12.6863 0.572524 0.286262 0.958151i \(-0.407587\pi\)
0.286262 + 0.958151i \(0.407587\pi\)
\(492\) 0 0
\(493\) −5.37258 −0.241969
\(494\) 0 0
\(495\) −17.3137 −0.778193
\(496\) 0 0
\(497\) 4.28427 0.192176
\(498\) 0 0
\(499\) 6.97056 0.312045 0.156023 0.987753i \(-0.450133\pi\)
0.156023 + 0.987753i \(0.450133\pi\)
\(500\) 0 0
\(501\) −50.2843 −2.24654
\(502\) 0 0
\(503\) −30.9706 −1.38091 −0.690455 0.723376i \(-0.742589\pi\)
−0.690455 + 0.723376i \(0.742589\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) −88.6690 −3.93793
\(508\) 0 0
\(509\) 5.79899 0.257036 0.128518 0.991707i \(-0.458978\pi\)
0.128518 + 0.991707i \(0.458978\pi\)
\(510\) 0 0
\(511\) −2.05887 −0.0910792
\(512\) 0 0
\(513\) 19.3137 0.852721
\(514\) 0 0
\(515\) 3.89949 0.171832
\(516\) 0 0
\(517\) 9.65685 0.424708
\(518\) 0 0
\(519\) −54.2843 −2.38282
\(520\) 0 0
\(521\) 39.6569 1.73740 0.868699 0.495340i \(-0.164957\pi\)
0.868699 + 0.495340i \(0.164957\pi\)
\(522\) 0 0
\(523\) 18.7279 0.818915 0.409457 0.912329i \(-0.365718\pi\)
0.409457 + 0.912329i \(0.365718\pi\)
\(524\) 0 0
\(525\) −2.82843 −0.123443
\(526\) 0 0
\(527\) −5.65685 −0.246416
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 118.225 5.13055
\(532\) 0 0
\(533\) −22.8284 −0.988809
\(534\) 0 0
\(535\) 7.41421 0.320544
\(536\) 0 0
\(537\) −35.3137 −1.52390
\(538\) 0 0
\(539\) 12.6274 0.543901
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) 0 0
\(543\) 61.4558 2.63732
\(544\) 0 0
\(545\) −3.17157 −0.135855
\(546\) 0 0
\(547\) −0.384776 −0.0164518 −0.00822592 0.999966i \(-0.502618\pi\)
−0.00822592 + 0.999966i \(0.502618\pi\)
\(548\) 0 0
\(549\) −118.225 −5.04574
\(550\) 0 0
\(551\) 6.48528 0.276282
\(552\) 0 0
\(553\) 1.37258 0.0583682
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) 2.20101 0.0932598 0.0466299 0.998912i \(-0.485152\pi\)
0.0466299 + 0.998912i \(0.485152\pi\)
\(558\) 0 0
\(559\) −30.1421 −1.27488
\(560\) 0 0
\(561\) 5.65685 0.238833
\(562\) 0 0
\(563\) −8.58579 −0.361848 −0.180924 0.983497i \(-0.557909\pi\)
−0.180924 + 0.983497i \(0.557909\pi\)
\(564\) 0 0
\(565\) −9.07107 −0.381623
\(566\) 0 0
\(567\) 33.1127 1.39060
\(568\) 0 0
\(569\) −4.82843 −0.202418 −0.101209 0.994865i \(-0.532271\pi\)
−0.101209 + 0.994865i \(0.532271\pi\)
\(570\) 0 0
\(571\) −37.3137 −1.56153 −0.780765 0.624825i \(-0.785170\pi\)
−0.780765 + 0.624825i \(0.785170\pi\)
\(572\) 0 0
\(573\) −13.6569 −0.570523
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) −11.6569 −0.484442
\(580\) 0 0
\(581\) −11.0294 −0.457578
\(582\) 0 0
\(583\) −18.1421 −0.751370
\(584\) 0 0
\(585\) −54.0416 −2.23435
\(586\) 0 0
\(587\) 18.9706 0.782999 0.391499 0.920178i \(-0.371956\pi\)
0.391499 + 0.920178i \(0.371956\pi\)
\(588\) 0 0
\(589\) 6.82843 0.281360
\(590\) 0 0
\(591\) −59.1127 −2.43157
\(592\) 0 0
\(593\) −36.6274 −1.50411 −0.752054 0.659102i \(-0.770937\pi\)
−0.752054 + 0.659102i \(0.770937\pi\)
\(594\) 0 0
\(595\) 0.686292 0.0281352
\(596\) 0 0
\(597\) 73.9411 3.02621
\(598\) 0 0
\(599\) −27.3137 −1.11601 −0.558004 0.829838i \(-0.688433\pi\)
−0.558004 + 0.829838i \(0.688433\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) −29.5563 −1.20363
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −9.07107 −0.368183 −0.184092 0.982909i \(-0.558934\pi\)
−0.184092 + 0.982909i \(0.558934\pi\)
\(608\) 0 0
\(609\) 18.3431 0.743302
\(610\) 0 0
\(611\) 30.1421 1.21942
\(612\) 0 0
\(613\) 34.4853 1.39285 0.696424 0.717631i \(-0.254773\pi\)
0.696424 + 0.717631i \(0.254773\pi\)
\(614\) 0 0
\(615\) −12.4853 −0.503455
\(616\) 0 0
\(617\) −45.7990 −1.84380 −0.921899 0.387430i \(-0.873363\pi\)
−0.921899 + 0.387430i \(0.873363\pi\)
\(618\) 0 0
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 0 0
\(621\) 115.882 4.65019
\(622\) 0 0
\(623\) −5.37258 −0.215248
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.82843 −0.272701
\(628\) 0 0
\(629\) −1.45584 −0.0580483
\(630\) 0 0
\(631\) 7.65685 0.304815 0.152407 0.988318i \(-0.451297\pi\)
0.152407 + 0.988318i \(0.451297\pi\)
\(632\) 0 0
\(633\) 97.2548 3.86553
\(634\) 0 0
\(635\) 9.55635 0.379232
\(636\) 0 0
\(637\) 39.4142 1.56165
\(638\) 0 0
\(639\) 44.7696 1.77106
\(640\) 0 0
\(641\) 1.31371 0.0518884 0.0259442 0.999663i \(-0.491741\pi\)
0.0259442 + 0.999663i \(0.491741\pi\)
\(642\) 0 0
\(643\) −20.6274 −0.813466 −0.406733 0.913547i \(-0.633332\pi\)
−0.406733 + 0.913547i \(0.633332\pi\)
\(644\) 0 0
\(645\) −16.4853 −0.649107
\(646\) 0 0
\(647\) 29.3137 1.15244 0.576220 0.817294i \(-0.304527\pi\)
0.576220 + 0.817294i \(0.304527\pi\)
\(648\) 0 0
\(649\) −27.3137 −1.07216
\(650\) 0 0
\(651\) 19.3137 0.756964
\(652\) 0 0
\(653\) −25.7990 −1.00959 −0.504796 0.863239i \(-0.668432\pi\)
−0.504796 + 0.863239i \(0.668432\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) −21.5147 −0.839369
\(658\) 0 0
\(659\) 10.6274 0.413985 0.206993 0.978342i \(-0.433632\pi\)
0.206993 + 0.978342i \(0.433632\pi\)
\(660\) 0 0
\(661\) 32.6274 1.26906 0.634530 0.772898i \(-0.281194\pi\)
0.634530 + 0.772898i \(0.281194\pi\)
\(662\) 0 0
\(663\) 17.6569 0.685735
\(664\) 0 0
\(665\) −0.828427 −0.0321250
\(666\) 0 0
\(667\) 38.9117 1.50667
\(668\) 0 0
\(669\) 79.2548 3.06417
\(670\) 0 0
\(671\) 27.3137 1.05443
\(672\) 0 0
\(673\) 1.75736 0.0677412 0.0338706 0.999426i \(-0.489217\pi\)
0.0338706 + 0.999426i \(0.489217\pi\)
\(674\) 0 0
\(675\) −19.3137 −0.743385
\(676\) 0 0
\(677\) −5.55635 −0.213548 −0.106774 0.994283i \(-0.534052\pi\)
−0.106774 + 0.994283i \(0.534052\pi\)
\(678\) 0 0
\(679\) −8.48528 −0.325635
\(680\) 0 0
\(681\) −79.9411 −3.06335
\(682\) 0 0
\(683\) −13.2721 −0.507842 −0.253921 0.967225i \(-0.581720\pi\)
−0.253921 + 0.967225i \(0.581720\pi\)
\(684\) 0 0
\(685\) 8.14214 0.311095
\(686\) 0 0
\(687\) 35.3137 1.34730
\(688\) 0 0
\(689\) −56.6274 −2.15733
\(690\) 0 0
\(691\) −4.34315 −0.165221 −0.0826105 0.996582i \(-0.526326\pi\)
−0.0826105 + 0.996582i \(0.526326\pi\)
\(692\) 0 0
\(693\) −14.3431 −0.544851
\(694\) 0 0
\(695\) 5.31371 0.201560
\(696\) 0 0
\(697\) 3.02944 0.114748
\(698\) 0 0
\(699\) −34.1421 −1.29137
\(700\) 0 0
\(701\) 38.3431 1.44820 0.724100 0.689695i \(-0.242255\pi\)
0.724100 + 0.689695i \(0.242255\pi\)
\(702\) 0 0
\(703\) 1.75736 0.0662801
\(704\) 0 0
\(705\) 16.4853 0.620872
\(706\) 0 0
\(707\) 3.31371 0.124625
\(708\) 0 0
\(709\) −14.9706 −0.562231 −0.281116 0.959674i \(-0.590704\pi\)
−0.281116 + 0.959674i \(0.590704\pi\)
\(710\) 0 0
\(711\) 14.3431 0.537910
\(712\) 0 0
\(713\) 40.9706 1.53436
\(714\) 0 0
\(715\) 12.4853 0.466923
\(716\) 0 0
\(717\) −73.9411 −2.76138
\(718\) 0 0
\(719\) −18.2843 −0.681888 −0.340944 0.940084i \(-0.610747\pi\)
−0.340944 + 0.940084i \(0.610747\pi\)
\(720\) 0 0
\(721\) 3.23045 0.120308
\(722\) 0 0
\(723\) −89.7401 −3.33747
\(724\) 0 0
\(725\) −6.48528 −0.240857
\(726\) 0 0
\(727\) −46.4853 −1.72404 −0.862022 0.506871i \(-0.830802\pi\)
−0.862022 + 0.506871i \(0.830802\pi\)
\(728\) 0 0
\(729\) 148.196 5.48874
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 28.3431 1.04688 0.523439 0.852063i \(-0.324649\pi\)
0.523439 + 0.852063i \(0.324649\pi\)
\(734\) 0 0
\(735\) 21.5563 0.795118
\(736\) 0 0
\(737\) 6.82843 0.251528
\(738\) 0 0
\(739\) −10.6274 −0.390936 −0.195468 0.980710i \(-0.562623\pi\)
−0.195468 + 0.980710i \(0.562623\pi\)
\(740\) 0 0
\(741\) −21.3137 −0.782979
\(742\) 0 0
\(743\) −34.0416 −1.24887 −0.624433 0.781078i \(-0.714670\pi\)
−0.624433 + 0.781078i \(0.714670\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) −115.255 −4.21695
\(748\) 0 0
\(749\) 6.14214 0.224429
\(750\) 0 0
\(751\) −14.1421 −0.516054 −0.258027 0.966138i \(-0.583072\pi\)
−0.258027 + 0.966138i \(0.583072\pi\)
\(752\) 0 0
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) −10.1421 −0.369110
\(756\) 0 0
\(757\) 8.34315 0.303237 0.151618 0.988439i \(-0.451552\pi\)
0.151618 + 0.988439i \(0.451552\pi\)
\(758\) 0 0
\(759\) −40.9706 −1.48714
\(760\) 0 0
\(761\) −14.6274 −0.530243 −0.265122 0.964215i \(-0.585412\pi\)
−0.265122 + 0.964215i \(0.585412\pi\)
\(762\) 0 0
\(763\) −2.62742 −0.0951189
\(764\) 0 0
\(765\) 7.17157 0.259289
\(766\) 0 0
\(767\) −85.2548 −3.07837
\(768\) 0 0
\(769\) −6.62742 −0.238991 −0.119495 0.992835i \(-0.538128\pi\)
−0.119495 + 0.992835i \(0.538128\pi\)
\(770\) 0 0
\(771\) −21.3137 −0.767594
\(772\) 0 0
\(773\) −19.8995 −0.715735 −0.357868 0.933772i \(-0.616496\pi\)
−0.357868 + 0.933772i \(0.616496\pi\)
\(774\) 0 0
\(775\) −6.82843 −0.245284
\(776\) 0 0
\(777\) 4.97056 0.178318
\(778\) 0 0
\(779\) −3.65685 −0.131020
\(780\) 0 0
\(781\) −10.3431 −0.370107
\(782\) 0 0
\(783\) 125.255 4.47624
\(784\) 0 0
\(785\) 8.82843 0.315100
\(786\) 0 0
\(787\) 45.8406 1.63404 0.817021 0.576608i \(-0.195624\pi\)
0.817021 + 0.576608i \(0.195624\pi\)
\(788\) 0 0
\(789\) 26.1421 0.930685
\(790\) 0 0
\(791\) −7.51472 −0.267193
\(792\) 0 0
\(793\) 85.2548 3.02749
\(794\) 0 0
\(795\) −30.9706 −1.09841
\(796\) 0 0
\(797\) −7.89949 −0.279814 −0.139907 0.990165i \(-0.544680\pi\)
−0.139907 + 0.990165i \(0.544680\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) −56.1421 −1.98368
\(802\) 0 0
\(803\) 4.97056 0.175407
\(804\) 0 0
\(805\) −4.97056 −0.175189
\(806\) 0 0
\(807\) 18.8284 0.662792
\(808\) 0 0
\(809\) 51.2548 1.80202 0.901012 0.433794i \(-0.142825\pi\)
0.901012 + 0.433794i \(0.142825\pi\)
\(810\) 0 0
\(811\) −19.7990 −0.695237 −0.347618 0.937636i \(-0.613009\pi\)
−0.347618 + 0.937636i \(0.613009\pi\)
\(812\) 0 0
\(813\) −42.1421 −1.47799
\(814\) 0 0
\(815\) 14.4853 0.507397
\(816\) 0 0
\(817\) −4.82843 −0.168925
\(818\) 0 0
\(819\) −44.7696 −1.56437
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −20.8284 −0.726033 −0.363017 0.931783i \(-0.618253\pi\)
−0.363017 + 0.931783i \(0.618253\pi\)
\(824\) 0 0
\(825\) 6.82843 0.237735
\(826\) 0 0
\(827\) −14.9289 −0.519130 −0.259565 0.965726i \(-0.583579\pi\)
−0.259565 + 0.965726i \(0.583579\pi\)
\(828\) 0 0
\(829\) 44.4264 1.54299 0.771496 0.636234i \(-0.219509\pi\)
0.771496 + 0.636234i \(0.219509\pi\)
\(830\) 0 0
\(831\) −38.1421 −1.32314
\(832\) 0 0
\(833\) −5.23045 −0.181224
\(834\) 0 0
\(835\) 14.7279 0.509681
\(836\) 0 0
\(837\) 131.882 4.55852
\(838\) 0 0
\(839\) −35.5980 −1.22898 −0.614489 0.788925i \(-0.710638\pi\)
−0.614489 + 0.788925i \(0.710638\pi\)
\(840\) 0 0
\(841\) 13.0589 0.450306
\(842\) 0 0
\(843\) 51.1127 1.76041
\(844\) 0 0
\(845\) 25.9706 0.893415
\(846\) 0 0
\(847\) −5.79899 −0.199256
\(848\) 0 0
\(849\) 86.4264 2.96615
\(850\) 0 0
\(851\) 10.5442 0.361449
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) −8.65685 −0.296058
\(856\) 0 0
\(857\) 15.2132 0.519673 0.259837 0.965653i \(-0.416331\pi\)
0.259837 + 0.965653i \(0.416331\pi\)
\(858\) 0 0
\(859\) 32.2843 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(860\) 0 0
\(861\) −10.3431 −0.352493
\(862\) 0 0
\(863\) 15.8995 0.541225 0.270613 0.962688i \(-0.412774\pi\)
0.270613 + 0.962688i \(0.412774\pi\)
\(864\) 0 0
\(865\) 15.8995 0.540599
\(866\) 0 0
\(867\) 55.6985 1.89162
\(868\) 0 0
\(869\) −3.31371 −0.112410
\(870\) 0 0
\(871\) 21.3137 0.722187
\(872\) 0 0
\(873\) −88.6690 −3.00099
\(874\) 0 0
\(875\) 0.828427 0.0280059
\(876\) 0 0
\(877\) −22.9289 −0.774255 −0.387128 0.922026i \(-0.626533\pi\)
−0.387128 + 0.922026i \(0.626533\pi\)
\(878\) 0 0
\(879\) 70.2843 2.37063
\(880\) 0 0
\(881\) −12.2843 −0.413868 −0.206934 0.978355i \(-0.566348\pi\)
−0.206934 + 0.978355i \(0.566348\pi\)
\(882\) 0 0
\(883\) −44.8284 −1.50860 −0.754298 0.656532i \(-0.772023\pi\)
−0.754298 + 0.656532i \(0.772023\pi\)
\(884\) 0 0
\(885\) −46.6274 −1.56736
\(886\) 0 0
\(887\) −34.9289 −1.17280 −0.586399 0.810022i \(-0.699455\pi\)
−0.586399 + 0.810022i \(0.699455\pi\)
\(888\) 0 0
\(889\) 7.91674 0.265519
\(890\) 0 0
\(891\) −79.9411 −2.67813
\(892\) 0 0
\(893\) 4.82843 0.161577
\(894\) 0 0
\(895\) 10.3431 0.345733
\(896\) 0 0
\(897\) −127.882 −4.26986
\(898\) 0 0
\(899\) 44.2843 1.47696
\(900\) 0 0
\(901\) 7.51472 0.250352
\(902\) 0 0
\(903\) −13.6569 −0.454472
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 16.3848 0.544048 0.272024 0.962291i \(-0.412307\pi\)
0.272024 + 0.962291i \(0.412307\pi\)
\(908\) 0 0
\(909\) 34.6274 1.14852
\(910\) 0 0
\(911\) 43.7990 1.45113 0.725563 0.688156i \(-0.241580\pi\)
0.725563 + 0.688156i \(0.241580\pi\)
\(912\) 0 0
\(913\) 26.6274 0.881239
\(914\) 0 0
\(915\) 46.6274 1.54145
\(916\) 0 0
\(917\) 4.68629 0.154755
\(918\) 0 0
\(919\) 12.6863 0.418482 0.209241 0.977864i \(-0.432901\pi\)
0.209241 + 0.977864i \(0.432901\pi\)
\(920\) 0 0
\(921\) 40.6274 1.33872
\(922\) 0 0
\(923\) −32.2843 −1.06265
\(924\) 0 0
\(925\) −1.75736 −0.0577816
\(926\) 0 0
\(927\) 33.7574 1.10874
\(928\) 0 0
\(929\) 29.3137 0.961752 0.480876 0.876789i \(-0.340319\pi\)
0.480876 + 0.876789i \(0.340319\pi\)
\(930\) 0 0
\(931\) 6.31371 0.206923
\(932\) 0 0
\(933\) 47.7990 1.56487
\(934\) 0 0
\(935\) −1.65685 −0.0541849
\(936\) 0 0
\(937\) −11.8579 −0.387380 −0.193690 0.981063i \(-0.562046\pi\)
−0.193690 + 0.981063i \(0.562046\pi\)
\(938\) 0 0
\(939\) −61.4558 −2.00554
\(940\) 0 0
\(941\) 26.2843 0.856843 0.428421 0.903579i \(-0.359070\pi\)
0.428421 + 0.903579i \(0.359070\pi\)
\(942\) 0 0
\(943\) −21.9411 −0.714501
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) −19.6569 −0.638762 −0.319381 0.947626i \(-0.603475\pi\)
−0.319381 + 0.947626i \(0.603475\pi\)
\(948\) 0 0
\(949\) 15.5147 0.503629
\(950\) 0 0
\(951\) 48.6274 1.57685
\(952\) 0 0
\(953\) 14.2426 0.461364 0.230682 0.973029i \(-0.425904\pi\)
0.230682 + 0.973029i \(0.425904\pi\)
\(954\) 0 0
\(955\) 4.00000 0.129437
\(956\) 0 0
\(957\) −44.2843 −1.43151
\(958\) 0 0
\(959\) 6.74517 0.217813
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) 64.1838 2.06829
\(964\) 0 0
\(965\) 3.41421 0.109907
\(966\) 0 0
\(967\) −24.6274 −0.791964 −0.395982 0.918258i \(-0.629596\pi\)
−0.395982 + 0.918258i \(0.629596\pi\)
\(968\) 0 0
\(969\) 2.82843 0.0908622
\(970\) 0 0
\(971\) 27.1127 0.870088 0.435044 0.900409i \(-0.356733\pi\)
0.435044 + 0.900409i \(0.356733\pi\)
\(972\) 0 0
\(973\) 4.40202 0.141122
\(974\) 0 0
\(975\) 21.3137 0.682585
\(976\) 0 0
\(977\) 34.7279 1.11104 0.555522 0.831502i \(-0.312518\pi\)
0.555522 + 0.831502i \(0.312518\pi\)
\(978\) 0 0
\(979\) 12.9706 0.414541
\(980\) 0 0
\(981\) −27.4558 −0.876598
\(982\) 0 0
\(983\) −20.5858 −0.656585 −0.328292 0.944576i \(-0.606473\pi\)
−0.328292 + 0.944576i \(0.606473\pi\)
\(984\) 0 0
\(985\) 17.3137 0.551661
\(986\) 0 0
\(987\) 13.6569 0.434702
\(988\) 0 0
\(989\) −28.9706 −0.921210
\(990\) 0 0
\(991\) 11.1127 0.353006 0.176503 0.984300i \(-0.443521\pi\)
0.176503 + 0.984300i \(0.443521\pi\)
\(992\) 0 0
\(993\) −23.3137 −0.739838
\(994\) 0 0
\(995\) −21.6569 −0.686568
\(996\) 0 0
\(997\) −30.4853 −0.965479 −0.482739 0.875764i \(-0.660358\pi\)
−0.482739 + 0.875764i \(0.660358\pi\)
\(998\) 0 0
\(999\) 33.9411 1.07385
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 380.2.a.c.1.1 2
3.2 odd 2 3420.2.a.g.1.2 2
4.3 odd 2 1520.2.a.o.1.2 2
5.2 odd 4 1900.2.c.d.1749.4 4
5.3 odd 4 1900.2.c.d.1749.1 4
5.4 even 2 1900.2.a.e.1.2 2
8.3 odd 2 6080.2.a.y.1.1 2
8.5 even 2 6080.2.a.bl.1.2 2
19.18 odd 2 7220.2.a.m.1.2 2
20.19 odd 2 7600.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.1 2 1.1 even 1 trivial
1520.2.a.o.1.2 2 4.3 odd 2
1900.2.a.e.1.2 2 5.4 even 2
1900.2.c.d.1749.1 4 5.3 odd 4
1900.2.c.d.1749.4 4 5.2 odd 4
3420.2.a.g.1.2 2 3.2 odd 2
6080.2.a.y.1.1 2 8.3 odd 2
6080.2.a.bl.1.2 2 8.5 even 2
7220.2.a.m.1.2 2 19.18 odd 2
7600.2.a.u.1.1 2 20.19 odd 2