Properties

Label 1840.2.a.q.1.3
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
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] = [1840,2,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.93543 q^{3} +1.00000 q^{5} -4.93543 q^{7} +0.745898 q^{9} +O(q^{10})\) \(q+1.93543 q^{3} +1.00000 q^{5} -4.93543 q^{7} +0.745898 q^{9} -0.745898 q^{11} +1.74590 q^{13} +1.93543 q^{15} -6.10856 q^{17} -5.44364 q^{19} -9.55220 q^{21} +1.00000 q^{23} +1.00000 q^{25} -4.36266 q^{27} -1.66492 q^{29} +1.61676 q^{31} -1.44364 q^{33} -4.93543 q^{35} +4.34625 q^{37} +3.37907 q^{39} -6.95184 q^{41} -5.01641 q^{43} +0.745898 q^{45} -2.68133 q^{47} +17.3585 q^{49} -11.8227 q^{51} +13.7417 q^{53} -0.745898 q^{55} -10.5358 q^{57} -12.2171 q^{59} -13.9794 q^{61} -3.68133 q^{63} +1.74590 q^{65} -13.1044 q^{67} +1.93543 q^{69} +9.67716 q^{71} +5.69774 q^{73} +1.93543 q^{75} +3.68133 q^{77} -10.3791 q^{79} -10.6813 q^{81} -0.637339 q^{83} -6.10856 q^{85} -3.22235 q^{87} +2.72532 q^{89} -8.61676 q^{91} +3.12914 q^{93} -5.44364 q^{95} +7.12497 q^{97} -0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39} - 10 q^{41} - 12 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} - 9 q^{51} + 10 q^{53} - 3 q^{55} - 12 q^{57} - 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{65} + 6 q^{67} - 2 q^{69} - 10 q^{71} + 7 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} - 25 q^{81} - 16 q^{83} - 5 q^{85} + 5 q^{87} - 20 q^{89} - 11 q^{91} + 25 q^{93} - 7 q^{95} + 5 q^{97} - 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.93543 1.11742 0.558711 0.829362i \(-0.311296\pi\)
0.558711 + 0.829362i \(0.311296\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.93543 −1.86542 −0.932709 0.360630i \(-0.882562\pi\)
−0.932709 + 0.360630i \(0.882562\pi\)
\(8\) 0 0
\(9\) 0.745898 0.248633
\(10\) 0 0
\(11\) −0.745898 −0.224897 −0.112448 0.993658i \(-0.535869\pi\)
−0.112448 + 0.993658i \(0.535869\pi\)
\(12\) 0 0
\(13\) 1.74590 0.484225 0.242113 0.970248i \(-0.422160\pi\)
0.242113 + 0.970248i \(0.422160\pi\)
\(14\) 0 0
\(15\) 1.93543 0.499726
\(16\) 0 0
\(17\) −6.10856 −1.48154 −0.740772 0.671757i \(-0.765540\pi\)
−0.740772 + 0.671757i \(0.765540\pi\)
\(18\) 0 0
\(19\) −5.44364 −1.24886 −0.624428 0.781083i \(-0.714668\pi\)
−0.624428 + 0.781083i \(0.714668\pi\)
\(20\) 0 0
\(21\) −9.55220 −2.08446
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.36266 −0.839595
\(28\) 0 0
\(29\) −1.66492 −0.309169 −0.154584 0.987980i \(-0.549404\pi\)
−0.154584 + 0.987980i \(0.549404\pi\)
\(30\) 0 0
\(31\) 1.61676 0.290379 0.145190 0.989404i \(-0.453621\pi\)
0.145190 + 0.989404i \(0.453621\pi\)
\(32\) 0 0
\(33\) −1.44364 −0.251305
\(34\) 0 0
\(35\) −4.93543 −0.834240
\(36\) 0 0
\(37\) 4.34625 0.714520 0.357260 0.934005i \(-0.383711\pi\)
0.357260 + 0.934005i \(0.383711\pi\)
\(38\) 0 0
\(39\) 3.37907 0.541084
\(40\) 0 0
\(41\) −6.95184 −1.08569 −0.542847 0.839831i \(-0.682654\pi\)
−0.542847 + 0.839831i \(0.682654\pi\)
\(42\) 0 0
\(43\) −5.01641 −0.764995 −0.382497 0.923957i \(-0.624936\pi\)
−0.382497 + 0.923957i \(0.624936\pi\)
\(44\) 0 0
\(45\) 0.745898 0.111192
\(46\) 0 0
\(47\) −2.68133 −0.391112 −0.195556 0.980693i \(-0.562651\pi\)
−0.195556 + 0.980693i \(0.562651\pi\)
\(48\) 0 0
\(49\) 17.3585 2.47978
\(50\) 0 0
\(51\) −11.8227 −1.65551
\(52\) 0 0
\(53\) 13.7417 1.88757 0.943786 0.330558i \(-0.107237\pi\)
0.943786 + 0.330558i \(0.107237\pi\)
\(54\) 0 0
\(55\) −0.745898 −0.100577
\(56\) 0 0
\(57\) −10.5358 −1.39550
\(58\) 0 0
\(59\) −12.2171 −1.59053 −0.795267 0.606260i \(-0.792669\pi\)
−0.795267 + 0.606260i \(0.792669\pi\)
\(60\) 0 0
\(61\) −13.9794 −1.78988 −0.894941 0.446185i \(-0.852782\pi\)
−0.894941 + 0.446185i \(0.852782\pi\)
\(62\) 0 0
\(63\) −3.68133 −0.463804
\(64\) 0 0
\(65\) 1.74590 0.216552
\(66\) 0 0
\(67\) −13.1044 −1.60096 −0.800478 0.599362i \(-0.795421\pi\)
−0.800478 + 0.599362i \(0.795421\pi\)
\(68\) 0 0
\(69\) 1.93543 0.232999
\(70\) 0 0
\(71\) 9.67716 1.14847 0.574234 0.818691i \(-0.305300\pi\)
0.574234 + 0.818691i \(0.305300\pi\)
\(72\) 0 0
\(73\) 5.69774 0.666870 0.333435 0.942773i \(-0.391792\pi\)
0.333435 + 0.942773i \(0.391792\pi\)
\(74\) 0 0
\(75\) 1.93543 0.223484
\(76\) 0 0
\(77\) 3.68133 0.419527
\(78\) 0 0
\(79\) −10.3791 −1.16774 −0.583868 0.811848i \(-0.698462\pi\)
−0.583868 + 0.811848i \(0.698462\pi\)
\(80\) 0 0
\(81\) −10.6813 −1.18681
\(82\) 0 0
\(83\) −0.637339 −0.0699570 −0.0349785 0.999388i \(-0.511136\pi\)
−0.0349785 + 0.999388i \(0.511136\pi\)
\(84\) 0 0
\(85\) −6.10856 −0.662566
\(86\) 0 0
\(87\) −3.22235 −0.345472
\(88\) 0 0
\(89\) 2.72532 0.288884 0.144442 0.989513i \(-0.453861\pi\)
0.144442 + 0.989513i \(0.453861\pi\)
\(90\) 0 0
\(91\) −8.61676 −0.903282
\(92\) 0 0
\(93\) 3.12914 0.324476
\(94\) 0 0
\(95\) −5.44364 −0.558505
\(96\) 0 0
\(97\) 7.12497 0.723431 0.361715 0.932289i \(-0.382191\pi\)
0.361715 + 0.932289i \(0.382191\pi\)
\(98\) 0 0
\(99\) −0.556364 −0.0559167
\(100\) 0 0
\(101\) 14.9753 1.49009 0.745047 0.667012i \(-0.232427\pi\)
0.745047 + 0.667012i \(0.232427\pi\)
\(102\) 0 0
\(103\) 10.8667 1.07073 0.535364 0.844622i \(-0.320175\pi\)
0.535364 + 0.844622i \(0.320175\pi\)
\(104\) 0 0
\(105\) −9.55220 −0.932199
\(106\) 0 0
\(107\) 7.49180 0.724259 0.362130 0.932128i \(-0.382050\pi\)
0.362130 + 0.932128i \(0.382050\pi\)
\(108\) 0 0
\(109\) −18.3309 −1.75578 −0.877891 0.478860i \(-0.841050\pi\)
−0.877891 + 0.478860i \(0.841050\pi\)
\(110\) 0 0
\(111\) 8.41188 0.798420
\(112\) 0 0
\(113\) 0.379068 0.0356597 0.0178299 0.999841i \(-0.494324\pi\)
0.0178299 + 0.999841i \(0.494324\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 1.30226 0.120394
\(118\) 0 0
\(119\) 30.1484 2.76370
\(120\) 0 0
\(121\) −10.4436 −0.949421
\(122\) 0 0
\(123\) −13.4548 −1.21318
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.64852 −0.235018 −0.117509 0.993072i \(-0.537491\pi\)
−0.117509 + 0.993072i \(0.537491\pi\)
\(128\) 0 0
\(129\) −9.70892 −0.854822
\(130\) 0 0
\(131\) 5.53579 0.483664 0.241832 0.970318i \(-0.422252\pi\)
0.241832 + 0.970318i \(0.422252\pi\)
\(132\) 0 0
\(133\) 26.8667 2.32964
\(134\) 0 0
\(135\) −4.36266 −0.375478
\(136\) 0 0
\(137\) 6.16896 0.527050 0.263525 0.964653i \(-0.415115\pi\)
0.263525 + 0.964653i \(0.415115\pi\)
\(138\) 0 0
\(139\) 20.7693 1.76163 0.880815 0.473460i \(-0.156995\pi\)
0.880815 + 0.473460i \(0.156995\pi\)
\(140\) 0 0
\(141\) −5.18953 −0.437038
\(142\) 0 0
\(143\) −1.30226 −0.108901
\(144\) 0 0
\(145\) −1.66492 −0.138264
\(146\) 0 0
\(147\) 33.5962 2.77097
\(148\) 0 0
\(149\) −1.06457 −0.0872128 −0.0436064 0.999049i \(-0.513885\pi\)
−0.0436064 + 0.999049i \(0.513885\pi\)
\(150\) 0 0
\(151\) 12.6608 1.03032 0.515159 0.857095i \(-0.327733\pi\)
0.515159 + 0.857095i \(0.327733\pi\)
\(152\) 0 0
\(153\) −4.55636 −0.368360
\(154\) 0 0
\(155\) 1.61676 0.129862
\(156\) 0 0
\(157\) 14.8873 1.18813 0.594067 0.804416i \(-0.297521\pi\)
0.594067 + 0.804416i \(0.297521\pi\)
\(158\) 0 0
\(159\) 26.5962 2.10921
\(160\) 0 0
\(161\) −4.93543 −0.388967
\(162\) 0 0
\(163\) −6.72949 −0.527094 −0.263547 0.964646i \(-0.584892\pi\)
−0.263547 + 0.964646i \(0.584892\pi\)
\(164\) 0 0
\(165\) −1.44364 −0.112387
\(166\) 0 0
\(167\) 2.98359 0.230877 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(168\) 0 0
\(169\) −9.95184 −0.765526
\(170\) 0 0
\(171\) −4.06040 −0.310506
\(172\) 0 0
\(173\) −15.8503 −1.20508 −0.602538 0.798091i \(-0.705844\pi\)
−0.602538 + 0.798091i \(0.705844\pi\)
\(174\) 0 0
\(175\) −4.93543 −0.373084
\(176\) 0 0
\(177\) −23.6454 −1.77730
\(178\) 0 0
\(179\) −3.02759 −0.226292 −0.113146 0.993578i \(-0.536093\pi\)
−0.113146 + 0.993578i \(0.536093\pi\)
\(180\) 0 0
\(181\) 1.82270 0.135481 0.0677403 0.997703i \(-0.478421\pi\)
0.0677403 + 0.997703i \(0.478421\pi\)
\(182\) 0 0
\(183\) −27.0562 −2.00005
\(184\) 0 0
\(185\) 4.34625 0.319543
\(186\) 0 0
\(187\) 4.55636 0.333194
\(188\) 0 0
\(189\) 21.5316 1.56619
\(190\) 0 0
\(191\) −19.6126 −1.41912 −0.709559 0.704646i \(-0.751106\pi\)
−0.709559 + 0.704646i \(0.751106\pi\)
\(192\) 0 0
\(193\) −0.335076 −0.0241193 −0.0120597 0.999927i \(-0.503839\pi\)
−0.0120597 + 0.999927i \(0.503839\pi\)
\(194\) 0 0
\(195\) 3.37907 0.241980
\(196\) 0 0
\(197\) −20.5316 −1.46282 −0.731409 0.681939i \(-0.761137\pi\)
−0.731409 + 0.681939i \(0.761137\pi\)
\(198\) 0 0
\(199\) −16.7253 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(200\) 0 0
\(201\) −25.3627 −1.78894
\(202\) 0 0
\(203\) 8.21712 0.576729
\(204\) 0 0
\(205\) −6.95184 −0.485538
\(206\) 0 0
\(207\) 0.745898 0.0518435
\(208\) 0 0
\(209\) 4.06040 0.280864
\(210\) 0 0
\(211\) 13.3955 0.922183 0.461091 0.887353i \(-0.347458\pi\)
0.461091 + 0.887353i \(0.347458\pi\)
\(212\) 0 0
\(213\) 18.7295 1.28332
\(214\) 0 0
\(215\) −5.01641 −0.342116
\(216\) 0 0
\(217\) −7.97942 −0.541679
\(218\) 0 0
\(219\) 11.0276 0.745175
\(220\) 0 0
\(221\) −10.6649 −0.717400
\(222\) 0 0
\(223\) −1.27468 −0.0853587 −0.0426794 0.999089i \(-0.513589\pi\)
−0.0426794 + 0.999089i \(0.513589\pi\)
\(224\) 0 0
\(225\) 0.745898 0.0497266
\(226\) 0 0
\(227\) −3.57978 −0.237598 −0.118799 0.992918i \(-0.537904\pi\)
−0.118799 + 0.992918i \(0.537904\pi\)
\(228\) 0 0
\(229\) 10.4119 0.688036 0.344018 0.938963i \(-0.388212\pi\)
0.344018 + 0.938963i \(0.388212\pi\)
\(230\) 0 0
\(231\) 7.12497 0.468788
\(232\) 0 0
\(233\) 3.06040 0.200493 0.100247 0.994963i \(-0.468037\pi\)
0.100247 + 0.994963i \(0.468037\pi\)
\(234\) 0 0
\(235\) −2.68133 −0.174911
\(236\) 0 0
\(237\) −20.0880 −1.30485
\(238\) 0 0
\(239\) 19.1895 1.24127 0.620634 0.784100i \(-0.286875\pi\)
0.620634 + 0.784100i \(0.286875\pi\)
\(240\) 0 0
\(241\) 0.346255 0.0223042 0.0111521 0.999938i \(-0.496450\pi\)
0.0111521 + 0.999938i \(0.496450\pi\)
\(242\) 0 0
\(243\) −7.58501 −0.486579
\(244\) 0 0
\(245\) 17.3585 1.10899
\(246\) 0 0
\(247\) −9.50403 −0.604727
\(248\) 0 0
\(249\) −1.23353 −0.0781715
\(250\) 0 0
\(251\) −14.3861 −0.908041 −0.454021 0.890991i \(-0.650011\pi\)
−0.454021 + 0.890991i \(0.650011\pi\)
\(252\) 0 0
\(253\) −0.745898 −0.0468942
\(254\) 0 0
\(255\) −11.8227 −0.740366
\(256\) 0 0
\(257\) 1.28586 0.0802095 0.0401047 0.999195i \(-0.487231\pi\)
0.0401047 + 0.999195i \(0.487231\pi\)
\(258\) 0 0
\(259\) −21.4506 −1.33288
\(260\) 0 0
\(261\) −1.24186 −0.0768694
\(262\) 0 0
\(263\) −28.2294 −1.74070 −0.870348 0.492437i \(-0.836106\pi\)
−0.870348 + 0.492437i \(0.836106\pi\)
\(264\) 0 0
\(265\) 13.7417 0.844148
\(266\) 0 0
\(267\) 5.27468 0.322805
\(268\) 0 0
\(269\) −19.4395 −1.18525 −0.592623 0.805480i \(-0.701907\pi\)
−0.592623 + 0.805480i \(0.701907\pi\)
\(270\) 0 0
\(271\) 6.42723 0.390426 0.195213 0.980761i \(-0.437460\pi\)
0.195213 + 0.980761i \(0.437460\pi\)
\(272\) 0 0
\(273\) −16.6772 −1.00935
\(274\) 0 0
\(275\) −0.745898 −0.0449794
\(276\) 0 0
\(277\) 4.08514 0.245452 0.122726 0.992441i \(-0.460836\pi\)
0.122726 + 0.992441i \(0.460836\pi\)
\(278\) 0 0
\(279\) 1.20594 0.0721978
\(280\) 0 0
\(281\) −6.38741 −0.381041 −0.190520 0.981683i \(-0.561018\pi\)
−0.190520 + 0.981683i \(0.561018\pi\)
\(282\) 0 0
\(283\) 21.4835 1.27706 0.638530 0.769597i \(-0.279543\pi\)
0.638530 + 0.769597i \(0.279543\pi\)
\(284\) 0 0
\(285\) −10.5358 −0.624086
\(286\) 0 0
\(287\) 34.3103 2.02527
\(288\) 0 0
\(289\) 20.3145 1.19497
\(290\) 0 0
\(291\) 13.7899 0.808378
\(292\) 0 0
\(293\) −16.1208 −0.941787 −0.470894 0.882190i \(-0.656068\pi\)
−0.470894 + 0.882190i \(0.656068\pi\)
\(294\) 0 0
\(295\) −12.2171 −0.711308
\(296\) 0 0
\(297\) 3.25410 0.188822
\(298\) 0 0
\(299\) 1.74590 0.100968
\(300\) 0 0
\(301\) 24.7581 1.42704
\(302\) 0 0
\(303\) 28.9836 1.66506
\(304\) 0 0
\(305\) −13.9794 −0.800460
\(306\) 0 0
\(307\) 6.96302 0.397400 0.198700 0.980060i \(-0.436328\pi\)
0.198700 + 0.980060i \(0.436328\pi\)
\(308\) 0 0
\(309\) 21.0318 1.19645
\(310\) 0 0
\(311\) 4.14031 0.234776 0.117388 0.993086i \(-0.462548\pi\)
0.117388 + 0.993086i \(0.462548\pi\)
\(312\) 0 0
\(313\) 6.26528 0.354135 0.177067 0.984199i \(-0.443339\pi\)
0.177067 + 0.984199i \(0.443339\pi\)
\(314\) 0 0
\(315\) −3.68133 −0.207419
\(316\) 0 0
\(317\) 9.53996 0.535817 0.267909 0.963444i \(-0.413667\pi\)
0.267909 + 0.963444i \(0.413667\pi\)
\(318\) 0 0
\(319\) 1.24186 0.0695310
\(320\) 0 0
\(321\) 14.4999 0.809304
\(322\) 0 0
\(323\) 33.2528 1.85023
\(324\) 0 0
\(325\) 1.74590 0.0968450
\(326\) 0 0
\(327\) −35.4782 −1.96195
\(328\) 0 0
\(329\) 13.2335 0.729588
\(330\) 0 0
\(331\) 14.3023 0.786123 0.393062 0.919512i \(-0.371416\pi\)
0.393062 + 0.919512i \(0.371416\pi\)
\(332\) 0 0
\(333\) 3.24186 0.177653
\(334\) 0 0
\(335\) −13.1044 −0.715969
\(336\) 0 0
\(337\) −3.34731 −0.182340 −0.0911699 0.995835i \(-0.529061\pi\)
−0.0911699 + 0.995835i \(0.529061\pi\)
\(338\) 0 0
\(339\) 0.733661 0.0398470
\(340\) 0 0
\(341\) −1.20594 −0.0653054
\(342\) 0 0
\(343\) −51.1236 −2.76042
\(344\) 0 0
\(345\) 1.93543 0.104200
\(346\) 0 0
\(347\) 23.9383 1.28507 0.642537 0.766255i \(-0.277882\pi\)
0.642537 + 0.766255i \(0.277882\pi\)
\(348\) 0 0
\(349\) −11.9149 −0.637788 −0.318894 0.947790i \(-0.603311\pi\)
−0.318894 + 0.947790i \(0.603311\pi\)
\(350\) 0 0
\(351\) −7.61676 −0.406553
\(352\) 0 0
\(353\) −25.2447 −1.34364 −0.671820 0.740714i \(-0.734487\pi\)
−0.671820 + 0.740714i \(0.734487\pi\)
\(354\) 0 0
\(355\) 9.67716 0.513610
\(356\) 0 0
\(357\) 58.3502 3.08822
\(358\) 0 0
\(359\) −20.4671 −1.08021 −0.540105 0.841598i \(-0.681615\pi\)
−0.540105 + 0.841598i \(0.681615\pi\)
\(360\) 0 0
\(361\) 10.6332 0.559641
\(362\) 0 0
\(363\) −20.2130 −1.06090
\(364\) 0 0
\(365\) 5.69774 0.298233
\(366\) 0 0
\(367\) 36.4999 1.90528 0.952639 0.304104i \(-0.0983571\pi\)
0.952639 + 0.304104i \(0.0983571\pi\)
\(368\) 0 0
\(369\) −5.18537 −0.269939
\(370\) 0 0
\(371\) −67.8214 −3.52111
\(372\) 0 0
\(373\) −5.57978 −0.288910 −0.144455 0.989511i \(-0.546143\pi\)
−0.144455 + 0.989511i \(0.546143\pi\)
\(374\) 0 0
\(375\) 1.93543 0.0999453
\(376\) 0 0
\(377\) −2.90679 −0.149707
\(378\) 0 0
\(379\) −36.6168 −1.88088 −0.940438 0.339964i \(-0.889585\pi\)
−0.940438 + 0.339964i \(0.889585\pi\)
\(380\) 0 0
\(381\) −5.12603 −0.262614
\(382\) 0 0
\(383\) 28.7581 1.46947 0.734736 0.678353i \(-0.237306\pi\)
0.734736 + 0.678353i \(0.237306\pi\)
\(384\) 0 0
\(385\) 3.68133 0.187618
\(386\) 0 0
\(387\) −3.74173 −0.190203
\(388\) 0 0
\(389\) 26.3913 1.33809 0.669046 0.743221i \(-0.266703\pi\)
0.669046 + 0.743221i \(0.266703\pi\)
\(390\) 0 0
\(391\) −6.10856 −0.308923
\(392\) 0 0
\(393\) 10.7141 0.540457
\(394\) 0 0
\(395\) −10.3791 −0.522228
\(396\) 0 0
\(397\) 1.08931 0.0546710 0.0273355 0.999626i \(-0.491298\pi\)
0.0273355 + 0.999626i \(0.491298\pi\)
\(398\) 0 0
\(399\) 51.9987 2.60319
\(400\) 0 0
\(401\) −22.5962 −1.12840 −0.564200 0.825638i \(-0.690815\pi\)
−0.564200 + 0.825638i \(0.690815\pi\)
\(402\) 0 0
\(403\) 2.82270 0.140609
\(404\) 0 0
\(405\) −10.6813 −0.530760
\(406\) 0 0
\(407\) −3.24186 −0.160693
\(408\) 0 0
\(409\) −16.0510 −0.793671 −0.396835 0.917890i \(-0.629892\pi\)
−0.396835 + 0.917890i \(0.629892\pi\)
\(410\) 0 0
\(411\) 11.9396 0.588937
\(412\) 0 0
\(413\) 60.2968 2.96701
\(414\) 0 0
\(415\) −0.637339 −0.0312857
\(416\) 0 0
\(417\) 40.1976 1.96849
\(418\) 0 0
\(419\) 24.4342 1.19369 0.596845 0.802356i \(-0.296421\pi\)
0.596845 + 0.802356i \(0.296421\pi\)
\(420\) 0 0
\(421\) −11.3473 −0.553034 −0.276517 0.961009i \(-0.589180\pi\)
−0.276517 + 0.961009i \(0.589180\pi\)
\(422\) 0 0
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) −6.10856 −0.296309
\(426\) 0 0
\(427\) 68.9945 3.33888
\(428\) 0 0
\(429\) −2.52044 −0.121688
\(430\) 0 0
\(431\) −7.83805 −0.377546 −0.188773 0.982021i \(-0.560451\pi\)
−0.188773 + 0.982021i \(0.560451\pi\)
\(432\) 0 0
\(433\) −37.9023 −1.82147 −0.910735 0.412990i \(-0.864484\pi\)
−0.910735 + 0.412990i \(0.864484\pi\)
\(434\) 0 0
\(435\) −3.22235 −0.154500
\(436\) 0 0
\(437\) −5.44364 −0.260404
\(438\) 0 0
\(439\) 17.7704 0.848134 0.424067 0.905631i \(-0.360602\pi\)
0.424067 + 0.905631i \(0.360602\pi\)
\(440\) 0 0
\(441\) 12.9477 0.616556
\(442\) 0 0
\(443\) −16.9466 −0.805158 −0.402579 0.915385i \(-0.631886\pi\)
−0.402579 + 0.915385i \(0.631886\pi\)
\(444\) 0 0
\(445\) 2.72532 0.129193
\(446\) 0 0
\(447\) −2.06040 −0.0974535
\(448\) 0 0
\(449\) −21.0890 −0.995253 −0.497627 0.867391i \(-0.665795\pi\)
−0.497627 + 0.867391i \(0.665795\pi\)
\(450\) 0 0
\(451\) 5.18537 0.244169
\(452\) 0 0
\(453\) 24.5040 1.15130
\(454\) 0 0
\(455\) −8.61676 −0.403960
\(456\) 0 0
\(457\) 13.1372 0.614532 0.307266 0.951624i \(-0.400586\pi\)
0.307266 + 0.951624i \(0.400586\pi\)
\(458\) 0 0
\(459\) 26.6496 1.24390
\(460\) 0 0
\(461\) −14.5439 −0.677375 −0.338687 0.940899i \(-0.609983\pi\)
−0.338687 + 0.940899i \(0.609983\pi\)
\(462\) 0 0
\(463\) −19.4283 −0.902909 −0.451455 0.892294i \(-0.649095\pi\)
−0.451455 + 0.892294i \(0.649095\pi\)
\(464\) 0 0
\(465\) 3.12914 0.145110
\(466\) 0 0
\(467\) 3.61259 0.167171 0.0835855 0.996501i \(-0.473363\pi\)
0.0835855 + 0.996501i \(0.473363\pi\)
\(468\) 0 0
\(469\) 64.6758 2.98645
\(470\) 0 0
\(471\) 28.8133 1.32765
\(472\) 0 0
\(473\) 3.74173 0.172045
\(474\) 0 0
\(475\) −5.44364 −0.249771
\(476\) 0 0
\(477\) 10.2499 0.469312
\(478\) 0 0
\(479\) −5.13720 −0.234725 −0.117362 0.993089i \(-0.537444\pi\)
−0.117362 + 0.993089i \(0.537444\pi\)
\(480\) 0 0
\(481\) 7.58812 0.345988
\(482\) 0 0
\(483\) −9.55220 −0.434640
\(484\) 0 0
\(485\) 7.12497 0.323528
\(486\) 0 0
\(487\) 24.7693 1.12240 0.561202 0.827679i \(-0.310339\pi\)
0.561202 + 0.827679i \(0.310339\pi\)
\(488\) 0 0
\(489\) −13.0245 −0.588987
\(490\) 0 0
\(491\) −3.54413 −0.159944 −0.0799721 0.996797i \(-0.525483\pi\)
−0.0799721 + 0.996797i \(0.525483\pi\)
\(492\) 0 0
\(493\) 10.1703 0.458047
\(494\) 0 0
\(495\) −0.556364 −0.0250067
\(496\) 0 0
\(497\) −47.7610 −2.14237
\(498\) 0 0
\(499\) −31.5358 −1.41174 −0.705868 0.708344i \(-0.749443\pi\)
−0.705868 + 0.708344i \(0.749443\pi\)
\(500\) 0 0
\(501\) 5.77454 0.257988
\(502\) 0 0
\(503\) 32.2346 1.43727 0.718635 0.695388i \(-0.244767\pi\)
0.718635 + 0.695388i \(0.244767\pi\)
\(504\) 0 0
\(505\) 14.9753 0.666390
\(506\) 0 0
\(507\) −19.2611 −0.855416
\(508\) 0 0
\(509\) 0.302263 0.0133976 0.00669878 0.999978i \(-0.497868\pi\)
0.00669878 + 0.999978i \(0.497868\pi\)
\(510\) 0 0
\(511\) −28.1208 −1.24399
\(512\) 0 0
\(513\) 23.7487 1.04853
\(514\) 0 0
\(515\) 10.8667 0.478844
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) −30.6772 −1.34658
\(520\) 0 0
\(521\) −30.3463 −1.32949 −0.664747 0.747069i \(-0.731461\pi\)
−0.664747 + 0.747069i \(0.731461\pi\)
\(522\) 0 0
\(523\) 42.5878 1.86224 0.931118 0.364717i \(-0.118834\pi\)
0.931118 + 0.364717i \(0.118834\pi\)
\(524\) 0 0
\(525\) −9.55220 −0.416892
\(526\) 0 0
\(527\) −9.87609 −0.430209
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.11273 −0.395459
\(532\) 0 0
\(533\) −12.1372 −0.525721
\(534\) 0 0
\(535\) 7.49180 0.323899
\(536\) 0 0
\(537\) −5.85969 −0.252864
\(538\) 0 0
\(539\) −12.9477 −0.557696
\(540\) 0 0
\(541\) 38.9313 1.67379 0.836893 0.547367i \(-0.184370\pi\)
0.836893 + 0.547367i \(0.184370\pi\)
\(542\) 0 0
\(543\) 3.52772 0.151389
\(544\) 0 0
\(545\) −18.3309 −0.785210
\(546\) 0 0
\(547\) −16.6004 −0.709780 −0.354890 0.934908i \(-0.615482\pi\)
−0.354890 + 0.934908i \(0.615482\pi\)
\(548\) 0 0
\(549\) −10.4272 −0.445023
\(550\) 0 0
\(551\) 9.06324 0.386107
\(552\) 0 0
\(553\) 51.2252 2.17832
\(554\) 0 0
\(555\) 8.41188 0.357064
\(556\) 0 0
\(557\) 17.4506 0.739408 0.369704 0.929150i \(-0.379459\pi\)
0.369704 + 0.929150i \(0.379459\pi\)
\(558\) 0 0
\(559\) −8.75814 −0.370430
\(560\) 0 0
\(561\) 8.81853 0.372319
\(562\) 0 0
\(563\) −8.25827 −0.348045 −0.174022 0.984742i \(-0.555676\pi\)
−0.174022 + 0.984742i \(0.555676\pi\)
\(564\) 0 0
\(565\) 0.379068 0.0159475
\(566\) 0 0
\(567\) 52.7170 2.21391
\(568\) 0 0
\(569\) −41.4178 −1.73633 −0.868163 0.496279i \(-0.834699\pi\)
−0.868163 + 0.496279i \(0.834699\pi\)
\(570\) 0 0
\(571\) −7.19059 −0.300917 −0.150458 0.988616i \(-0.548075\pi\)
−0.150458 + 0.988616i \(0.548075\pi\)
\(572\) 0 0
\(573\) −37.9588 −1.58575
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 39.3103 1.63651 0.818255 0.574855i \(-0.194942\pi\)
0.818255 + 0.574855i \(0.194942\pi\)
\(578\) 0 0
\(579\) −0.648517 −0.0269515
\(580\) 0 0
\(581\) 3.14554 0.130499
\(582\) 0 0
\(583\) −10.2499 −0.424509
\(584\) 0 0
\(585\) 1.30226 0.0538419
\(586\) 0 0
\(587\) −28.7899 −1.18829 −0.594143 0.804359i \(-0.702509\pi\)
−0.594143 + 0.804359i \(0.702509\pi\)
\(588\) 0 0
\(589\) −8.80107 −0.362642
\(590\) 0 0
\(591\) −39.7376 −1.63458
\(592\) 0 0
\(593\) −40.8873 −1.67904 −0.839519 0.543330i \(-0.817163\pi\)
−0.839519 + 0.543330i \(0.817163\pi\)
\(594\) 0 0
\(595\) 30.1484 1.23596
\(596\) 0 0
\(597\) −32.3707 −1.32485
\(598\) 0 0
\(599\) −19.1801 −0.783679 −0.391840 0.920034i \(-0.628161\pi\)
−0.391840 + 0.920034i \(0.628161\pi\)
\(600\) 0 0
\(601\) 3.36683 0.137336 0.0686679 0.997640i \(-0.478125\pi\)
0.0686679 + 0.997640i \(0.478125\pi\)
\(602\) 0 0
\(603\) −9.77454 −0.398050
\(604\) 0 0
\(605\) −10.4436 −0.424594
\(606\) 0 0
\(607\) −13.1924 −0.535462 −0.267731 0.963494i \(-0.586274\pi\)
−0.267731 + 0.963494i \(0.586274\pi\)
\(608\) 0 0
\(609\) 15.9037 0.644450
\(610\) 0 0
\(611\) −4.68133 −0.189386
\(612\) 0 0
\(613\) 32.6207 1.31754 0.658768 0.752346i \(-0.271078\pi\)
0.658768 + 0.752346i \(0.271078\pi\)
\(614\) 0 0
\(615\) −13.4548 −0.542550
\(616\) 0 0
\(617\) 41.7212 1.67963 0.839815 0.542872i \(-0.182663\pi\)
0.839815 + 0.542872i \(0.182663\pi\)
\(618\) 0 0
\(619\) 0.339245 0.0136354 0.00681771 0.999977i \(-0.497830\pi\)
0.00681771 + 0.999977i \(0.497830\pi\)
\(620\) 0 0
\(621\) −4.36266 −0.175068
\(622\) 0 0
\(623\) −13.4506 −0.538889
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.85863 0.313843
\(628\) 0 0
\(629\) −26.5494 −1.05859
\(630\) 0 0
\(631\) −3.71725 −0.147982 −0.0739908 0.997259i \(-0.523574\pi\)
−0.0739908 + 0.997259i \(0.523574\pi\)
\(632\) 0 0
\(633\) 25.9260 1.03047
\(634\) 0 0
\(635\) −2.64852 −0.105103
\(636\) 0 0
\(637\) 30.3062 1.20077
\(638\) 0 0
\(639\) 7.21818 0.285547
\(640\) 0 0
\(641\) −22.0328 −0.870244 −0.435122 0.900372i \(-0.643295\pi\)
−0.435122 + 0.900372i \(0.643295\pi\)
\(642\) 0 0
\(643\) 24.6842 0.973449 0.486724 0.873556i \(-0.338192\pi\)
0.486724 + 0.873556i \(0.338192\pi\)
\(644\) 0 0
\(645\) −9.70892 −0.382288
\(646\) 0 0
\(647\) −32.1731 −1.26486 −0.632428 0.774619i \(-0.717942\pi\)
−0.632428 + 0.774619i \(0.717942\pi\)
\(648\) 0 0
\(649\) 9.11273 0.357706
\(650\) 0 0
\(651\) −15.4436 −0.605284
\(652\) 0 0
\(653\) −26.9466 −1.05450 −0.527251 0.849709i \(-0.676777\pi\)
−0.527251 + 0.849709i \(0.676777\pi\)
\(654\) 0 0
\(655\) 5.53579 0.216301
\(656\) 0 0
\(657\) 4.24993 0.165806
\(658\) 0 0
\(659\) 19.7417 0.769029 0.384514 0.923119i \(-0.374369\pi\)
0.384514 + 0.923119i \(0.374369\pi\)
\(660\) 0 0
\(661\) −1.28692 −0.0500552 −0.0250276 0.999687i \(-0.507967\pi\)
−0.0250276 + 0.999687i \(0.507967\pi\)
\(662\) 0 0
\(663\) −20.6412 −0.801639
\(664\) 0 0
\(665\) 26.8667 1.04185
\(666\) 0 0
\(667\) −1.66492 −0.0644661
\(668\) 0 0
\(669\) −2.46705 −0.0953817
\(670\) 0 0
\(671\) 10.4272 0.402539
\(672\) 0 0
\(673\) 12.4970 0.481725 0.240862 0.970559i \(-0.422570\pi\)
0.240862 + 0.970559i \(0.422570\pi\)
\(674\) 0 0
\(675\) −4.36266 −0.167919
\(676\) 0 0
\(677\) −2.69251 −0.103482 −0.0517408 0.998661i \(-0.516477\pi\)
−0.0517408 + 0.998661i \(0.516477\pi\)
\(678\) 0 0
\(679\) −35.1648 −1.34950
\(680\) 0 0
\(681\) −6.92842 −0.265498
\(682\) 0 0
\(683\) 2.72115 0.104122 0.0520610 0.998644i \(-0.483421\pi\)
0.0520610 + 0.998644i \(0.483421\pi\)
\(684\) 0 0
\(685\) 6.16896 0.235704
\(686\) 0 0
\(687\) 20.1515 0.768827
\(688\) 0 0
\(689\) 23.9917 0.914010
\(690\) 0 0
\(691\) −38.3051 −1.45719 −0.728597 0.684942i \(-0.759827\pi\)
−0.728597 + 0.684942i \(0.759827\pi\)
\(692\) 0 0
\(693\) 2.74590 0.104308
\(694\) 0 0
\(695\) 20.7693 0.787825
\(696\) 0 0
\(697\) 42.4657 1.60850
\(698\) 0 0
\(699\) 5.92319 0.224036
\(700\) 0 0
\(701\) 39.8472 1.50501 0.752504 0.658588i \(-0.228846\pi\)
0.752504 + 0.658588i \(0.228846\pi\)
\(702\) 0 0
\(703\) −23.6594 −0.892332
\(704\) 0 0
\(705\) −5.18953 −0.195449
\(706\) 0 0
\(707\) −73.9094 −2.77965
\(708\) 0 0
\(709\) −44.8995 −1.68624 −0.843118 0.537728i \(-0.819283\pi\)
−0.843118 + 0.537728i \(0.819283\pi\)
\(710\) 0 0
\(711\) −7.74173 −0.290338
\(712\) 0 0
\(713\) 1.61676 0.0605482
\(714\) 0 0
\(715\) −1.30226 −0.0487019
\(716\) 0 0
\(717\) 37.1400 1.38702
\(718\) 0 0
\(719\) 31.5400 1.17624 0.588121 0.808773i \(-0.299868\pi\)
0.588121 + 0.808773i \(0.299868\pi\)
\(720\) 0 0
\(721\) −53.6318 −1.99735
\(722\) 0 0
\(723\) 0.670152 0.0249232
\(724\) 0 0
\(725\) −1.66492 −0.0618337
\(726\) 0 0
\(727\) 48.9219 1.81441 0.907206 0.420687i \(-0.138211\pi\)
0.907206 + 0.420687i \(0.138211\pi\)
\(728\) 0 0
\(729\) 17.3637 0.643101
\(730\) 0 0
\(731\) 30.6430 1.13337
\(732\) 0 0
\(733\) −14.1536 −0.522776 −0.261388 0.965234i \(-0.584180\pi\)
−0.261388 + 0.965234i \(0.584180\pi\)
\(734\) 0 0
\(735\) 33.5962 1.23921
\(736\) 0 0
\(737\) 9.77454 0.360050
\(738\) 0 0
\(739\) −4.46421 −0.164219 −0.0821093 0.996623i \(-0.526166\pi\)
−0.0821093 + 0.996623i \(0.526166\pi\)
\(740\) 0 0
\(741\) −18.3944 −0.675736
\(742\) 0 0
\(743\) 34.2898 1.25797 0.628985 0.777418i \(-0.283471\pi\)
0.628985 + 0.777418i \(0.283471\pi\)
\(744\) 0 0
\(745\) −1.06457 −0.0390027
\(746\) 0 0
\(747\) −0.475390 −0.0173936
\(748\) 0 0
\(749\) −36.9753 −1.35105
\(750\) 0 0
\(751\) 8.28275 0.302242 0.151121 0.988515i \(-0.451712\pi\)
0.151121 + 0.988515i \(0.451712\pi\)
\(752\) 0 0
\(753\) −27.8433 −1.01467
\(754\) 0 0
\(755\) 12.6608 0.460772
\(756\) 0 0
\(757\) 3.32985 0.121025 0.0605127 0.998167i \(-0.480726\pi\)
0.0605127 + 0.998167i \(0.480726\pi\)
\(758\) 0 0
\(759\) −1.44364 −0.0524007
\(760\) 0 0
\(761\) −18.4077 −0.667279 −0.333640 0.942701i \(-0.608277\pi\)
−0.333640 + 0.942701i \(0.608277\pi\)
\(762\) 0 0
\(763\) 90.4710 3.27527
\(764\) 0 0
\(765\) −4.55636 −0.164736
\(766\) 0 0
\(767\) −21.3298 −0.770176
\(768\) 0 0
\(769\) 51.4506 1.85536 0.927679 0.373379i \(-0.121801\pi\)
0.927679 + 0.373379i \(0.121801\pi\)
\(770\) 0 0
\(771\) 2.48869 0.0896279
\(772\) 0 0
\(773\) 17.4506 0.627656 0.313828 0.949480i \(-0.398388\pi\)
0.313828 + 0.949480i \(0.398388\pi\)
\(774\) 0 0
\(775\) 1.61676 0.0580758
\(776\) 0 0
\(777\) −41.5163 −1.48939
\(778\) 0 0
\(779\) 37.8433 1.35588
\(780\) 0 0
\(781\) −7.21818 −0.258287
\(782\) 0 0
\(783\) 7.26350 0.259576
\(784\) 0 0
\(785\) 14.8873 0.531350
\(786\) 0 0
\(787\) 21.2335 0.756893 0.378447 0.925623i \(-0.376458\pi\)
0.378447 + 0.925623i \(0.376458\pi\)
\(788\) 0 0
\(789\) −54.6360 −1.94509
\(790\) 0 0
\(791\) −1.87086 −0.0665203
\(792\) 0 0
\(793\) −24.4067 −0.866706
\(794\) 0 0
\(795\) 26.5962 0.943270
\(796\) 0 0
\(797\) −25.3215 −0.896934 −0.448467 0.893799i \(-0.648030\pi\)
−0.448467 + 0.893799i \(0.648030\pi\)
\(798\) 0 0
\(799\) 16.3791 0.579450
\(800\) 0 0
\(801\) 2.03281 0.0718259
\(802\) 0 0
\(803\) −4.24993 −0.149977
\(804\) 0 0
\(805\) −4.93543 −0.173951
\(806\) 0 0
\(807\) −37.6238 −1.32442
\(808\) 0 0
\(809\) 6.14137 0.215919 0.107960 0.994155i \(-0.465568\pi\)
0.107960 + 0.994155i \(0.465568\pi\)
\(810\) 0 0
\(811\) −15.5051 −0.544457 −0.272229 0.962233i \(-0.587761\pi\)
−0.272229 + 0.962233i \(0.587761\pi\)
\(812\) 0 0
\(813\) 12.4395 0.436271
\(814\) 0 0
\(815\) −6.72949 −0.235724
\(816\) 0 0
\(817\) 27.3075 0.955368
\(818\) 0 0
\(819\) −6.42723 −0.224586
\(820\) 0 0
\(821\) −21.4835 −0.749778 −0.374889 0.927070i \(-0.622319\pi\)
−0.374889 + 0.927070i \(0.622319\pi\)
\(822\) 0 0
\(823\) −40.1565 −1.39977 −0.699883 0.714258i \(-0.746765\pi\)
−0.699883 + 0.714258i \(0.746765\pi\)
\(824\) 0 0
\(825\) −1.44364 −0.0502609
\(826\) 0 0
\(827\) −38.0573 −1.32338 −0.661691 0.749777i \(-0.730161\pi\)
−0.661691 + 0.749777i \(0.730161\pi\)
\(828\) 0 0
\(829\) −51.2580 −1.78026 −0.890132 0.455703i \(-0.849388\pi\)
−0.890132 + 0.455703i \(0.849388\pi\)
\(830\) 0 0
\(831\) 7.90652 0.274274
\(832\) 0 0
\(833\) −106.035 −3.67391
\(834\) 0 0
\(835\) 2.98359 0.103252
\(836\) 0 0
\(837\) −7.05339 −0.243801
\(838\) 0 0
\(839\) −45.4200 −1.56807 −0.784035 0.620716i \(-0.786842\pi\)
−0.784035 + 0.620716i \(0.786842\pi\)
\(840\) 0 0
\(841\) −26.2280 −0.904415
\(842\) 0 0
\(843\) −12.3624 −0.425783
\(844\) 0 0
\(845\) −9.95184 −0.342354
\(846\) 0 0
\(847\) 51.5439 1.77107
\(848\) 0 0
\(849\) 41.5798 1.42701
\(850\) 0 0
\(851\) 4.34625 0.148988
\(852\) 0 0
\(853\) 40.0867 1.37254 0.686270 0.727346i \(-0.259247\pi\)
0.686270 + 0.727346i \(0.259247\pi\)
\(854\) 0 0
\(855\) −4.06040 −0.138863
\(856\) 0 0
\(857\) −3.63423 −0.124143 −0.0620715 0.998072i \(-0.519771\pi\)
−0.0620715 + 0.998072i \(0.519771\pi\)
\(858\) 0 0
\(859\) 37.7529 1.28811 0.644056 0.764978i \(-0.277250\pi\)
0.644056 + 0.764978i \(0.277250\pi\)
\(860\) 0 0
\(861\) 66.4053 2.26309
\(862\) 0 0
\(863\) 3.94555 0.134308 0.0671541 0.997743i \(-0.478608\pi\)
0.0671541 + 0.997743i \(0.478608\pi\)
\(864\) 0 0
\(865\) −15.8503 −0.538926
\(866\) 0 0
\(867\) 39.3173 1.33529
\(868\) 0 0
\(869\) 7.74173 0.262620
\(870\) 0 0
\(871\) −22.8789 −0.775223
\(872\) 0 0
\(873\) 5.31450 0.179869
\(874\) 0 0
\(875\) −4.93543 −0.166848
\(876\) 0 0
\(877\) −8.64958 −0.292075 −0.146038 0.989279i \(-0.546652\pi\)
−0.146038 + 0.989279i \(0.546652\pi\)
\(878\) 0 0
\(879\) −31.2007 −1.05237
\(880\) 0 0
\(881\) −56.1760 −1.89262 −0.946308 0.323266i \(-0.895219\pi\)
−0.946308 + 0.323266i \(0.895219\pi\)
\(882\) 0 0
\(883\) 12.1054 0.407381 0.203690 0.979035i \(-0.434706\pi\)
0.203690 + 0.979035i \(0.434706\pi\)
\(884\) 0 0
\(885\) −23.6454 −0.794832
\(886\) 0 0
\(887\) 24.7058 0.829540 0.414770 0.909926i \(-0.363862\pi\)
0.414770 + 0.909926i \(0.363862\pi\)
\(888\) 0 0
\(889\) 13.0716 0.438407
\(890\) 0 0
\(891\) 7.96719 0.266911
\(892\) 0 0
\(893\) 14.5962 0.488443
\(894\) 0 0
\(895\) −3.02759 −0.101201
\(896\) 0 0
\(897\) 3.37907 0.112824
\(898\) 0 0
\(899\) −2.69179 −0.0897761
\(900\) 0 0
\(901\) −83.9422 −2.79652
\(902\) 0 0
\(903\) 47.9177 1.59460
\(904\) 0 0
\(905\) 1.82270 0.0605887
\(906\) 0 0
\(907\) −22.7170 −0.754305 −0.377153 0.926151i \(-0.623097\pi\)
−0.377153 + 0.926151i \(0.623097\pi\)
\(908\) 0 0
\(909\) 11.1700 0.370486
\(910\) 0 0
\(911\) 12.3156 0.408033 0.204016 0.978967i \(-0.434600\pi\)
0.204016 + 0.978967i \(0.434600\pi\)
\(912\) 0 0
\(913\) 0.475390 0.0157331
\(914\) 0 0
\(915\) −27.0562 −0.894451
\(916\) 0 0
\(917\) −27.3215 −0.902236
\(918\) 0 0
\(919\) −46.2088 −1.52429 −0.762144 0.647408i \(-0.775853\pi\)
−0.762144 + 0.647408i \(0.775853\pi\)
\(920\) 0 0
\(921\) 13.4764 0.444064
\(922\) 0 0
\(923\) 16.8953 0.556117
\(924\) 0 0
\(925\) 4.34625 0.142904
\(926\) 0 0
\(927\) 8.10545 0.266218
\(928\) 0 0
\(929\) 9.12391 0.299346 0.149673 0.988736i \(-0.452178\pi\)
0.149673 + 0.988736i \(0.452178\pi\)
\(930\) 0 0
\(931\) −94.4933 −3.09689
\(932\) 0 0
\(933\) 8.01330 0.262344
\(934\) 0 0
\(935\) 4.55636 0.149009
\(936\) 0 0
\(937\) −33.9190 −1.10809 −0.554043 0.832488i \(-0.686916\pi\)
−0.554043 + 0.832488i \(0.686916\pi\)
\(938\) 0 0
\(939\) 12.1260 0.395718
\(940\) 0 0
\(941\) −37.2926 −1.21570 −0.607852 0.794050i \(-0.707969\pi\)
−0.607852 + 0.794050i \(0.707969\pi\)
\(942\) 0 0
\(943\) −6.95184 −0.226383
\(944\) 0 0
\(945\) 21.5316 0.700424
\(946\) 0 0
\(947\) −33.7047 −1.09526 −0.547629 0.836722i \(-0.684469\pi\)
−0.547629 + 0.836722i \(0.684469\pi\)
\(948\) 0 0
\(949\) 9.94767 0.322915
\(950\) 0 0
\(951\) 18.4639 0.598734
\(952\) 0 0
\(953\) −48.4465 −1.56934 −0.784668 0.619917i \(-0.787166\pi\)
−0.784668 + 0.619917i \(0.787166\pi\)
\(954\) 0 0
\(955\) −19.6126 −0.634649
\(956\) 0 0
\(957\) 2.40354 0.0776955
\(958\) 0 0
\(959\) −30.4465 −0.983168
\(960\) 0 0
\(961\) −28.3861 −0.915680
\(962\) 0 0
\(963\) 5.58812 0.180075
\(964\) 0 0
\(965\) −0.335076 −0.0107865
\(966\) 0 0
\(967\) 28.2528 0.908548 0.454274 0.890862i \(-0.349899\pi\)
0.454274 + 0.890862i \(0.349899\pi\)
\(968\) 0 0
\(969\) 64.3585 2.06749
\(970\) 0 0
\(971\) −1.65553 −0.0531284 −0.0265642 0.999647i \(-0.508457\pi\)
−0.0265642 + 0.999647i \(0.508457\pi\)
\(972\) 0 0
\(973\) −102.506 −3.28618
\(974\) 0 0
\(975\) 3.37907 0.108217
\(976\) 0 0
\(977\) −11.1630 −0.357136 −0.178568 0.983928i \(-0.557146\pi\)
−0.178568 + 0.983928i \(0.557146\pi\)
\(978\) 0 0
\(979\) −2.03281 −0.0649690
\(980\) 0 0
\(981\) −13.6730 −0.436545
\(982\) 0 0
\(983\) −15.0615 −0.480386 −0.240193 0.970725i \(-0.577211\pi\)
−0.240193 + 0.970725i \(0.577211\pi\)
\(984\) 0 0
\(985\) −20.5316 −0.654192
\(986\) 0 0
\(987\) 25.6126 0.815258
\(988\) 0 0
\(989\) −5.01641 −0.159512
\(990\) 0 0
\(991\) 43.8971 1.39444 0.697219 0.716858i \(-0.254421\pi\)
0.697219 + 0.716858i \(0.254421\pi\)
\(992\) 0 0
\(993\) 27.6811 0.878432
\(994\) 0 0
\(995\) −16.7253 −0.530228
\(996\) 0 0
\(997\) 38.5655 1.22138 0.610691 0.791869i \(-0.290892\pi\)
0.610691 + 0.791869i \(0.290892\pi\)
\(998\) 0 0
\(999\) −18.9612 −0.599907
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 1840.2.a.q.1.3 3
4.3 odd 2 920.2.a.i.1.1 3
5.4 even 2 9200.2.a.ci.1.1 3
8.3 odd 2 7360.2.a.bw.1.3 3
8.5 even 2 7360.2.a.cf.1.1 3
12.11 even 2 8280.2.a.bl.1.3 3
20.3 even 4 4600.2.e.q.4049.2 6
20.7 even 4 4600.2.e.q.4049.5 6
20.19 odd 2 4600.2.a.v.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.1 3 4.3 odd 2
1840.2.a.q.1.3 3 1.1 even 1 trivial
4600.2.a.v.1.3 3 20.19 odd 2
4600.2.e.q.4049.2 6 20.3 even 4
4600.2.e.q.4049.5 6 20.7 even 4
7360.2.a.bw.1.3 3 8.3 odd 2
7360.2.a.cf.1.1 3 8.5 even 2
8280.2.a.bl.1.3 3 12.11 even 2
9200.2.a.ci.1.1 3 5.4 even 2