Properties

Label 3344.2.a.t.1.2
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $5$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) 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 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.245526\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} +O(q^{10})\) \(q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} -1.00000 q^{11} +3.31182 q^{13} +7.40242 q^{15} +2.80637 q^{17} +1.00000 q^{19} +8.50056 q^{21} -6.88998 q^{23} +6.77018 q^{25} +2.90101 q^{27} +5.67979 q^{29} -2.51864 q^{31} +2.15766 q^{33} +13.5163 q^{35} -6.39893 q^{37} -7.14577 q^{39} +0.560629 q^{41} +9.40080 q^{43} -5.67958 q^{45} +12.1742 q^{47} +8.52137 q^{49} -6.05517 q^{51} +5.68316 q^{53} +3.43077 q^{55} -2.15766 q^{57} -4.35730 q^{59} -3.56412 q^{61} -6.52213 q^{63} -11.3621 q^{65} +9.95563 q^{67} +14.8662 q^{69} +11.4671 q^{71} -8.95834 q^{73} -14.6077 q^{75} +3.93972 q^{77} -8.49105 q^{79} -11.2258 q^{81} +5.21960 q^{83} -9.62799 q^{85} -12.2550 q^{87} -7.28423 q^{89} -13.0476 q^{91} +5.43436 q^{93} -3.43077 q^{95} +10.6574 q^{97} -1.65548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.15766 −1.24572 −0.622862 0.782332i \(-0.714030\pi\)
−0.622862 + 0.782332i \(0.714030\pi\)
\(4\) 0 0
\(5\) −3.43077 −1.53429 −0.767143 0.641476i \(-0.778322\pi\)
−0.767143 + 0.641476i \(0.778322\pi\)
\(6\) 0 0
\(7\) −3.93972 −1.48907 −0.744537 0.667582i \(-0.767329\pi\)
−0.744537 + 0.667582i \(0.767329\pi\)
\(8\) 0 0
\(9\) 1.65548 0.551827
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.31182 0.918534 0.459267 0.888298i \(-0.348112\pi\)
0.459267 + 0.888298i \(0.348112\pi\)
\(14\) 0 0
\(15\) 7.40242 1.91130
\(16\) 0 0
\(17\) 2.80637 0.680644 0.340322 0.940309i \(-0.389464\pi\)
0.340322 + 0.940309i \(0.389464\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 8.50056 1.85497
\(22\) 0 0
\(23\) −6.88998 −1.43666 −0.718330 0.695702i \(-0.755093\pi\)
−0.718330 + 0.695702i \(0.755093\pi\)
\(24\) 0 0
\(25\) 6.77018 1.35404
\(26\) 0 0
\(27\) 2.90101 0.558299
\(28\) 0 0
\(29\) 5.67979 1.05471 0.527355 0.849645i \(-0.323184\pi\)
0.527355 + 0.849645i \(0.323184\pi\)
\(30\) 0 0
\(31\) −2.51864 −0.452361 −0.226180 0.974085i \(-0.572624\pi\)
−0.226180 + 0.974085i \(0.572624\pi\)
\(32\) 0 0
\(33\) 2.15766 0.375600
\(34\) 0 0
\(35\) 13.5163 2.28466
\(36\) 0 0
\(37\) −6.39893 −1.05198 −0.525989 0.850491i \(-0.676305\pi\)
−0.525989 + 0.850491i \(0.676305\pi\)
\(38\) 0 0
\(39\) −7.14577 −1.14424
\(40\) 0 0
\(41\) 0.560629 0.0875555 0.0437778 0.999041i \(-0.486061\pi\)
0.0437778 + 0.999041i \(0.486061\pi\)
\(42\) 0 0
\(43\) 9.40080 1.43361 0.716805 0.697274i \(-0.245604\pi\)
0.716805 + 0.697274i \(0.245604\pi\)
\(44\) 0 0
\(45\) −5.67958 −0.846661
\(46\) 0 0
\(47\) 12.1742 1.77579 0.887896 0.460044i \(-0.152166\pi\)
0.887896 + 0.460044i \(0.152166\pi\)
\(48\) 0 0
\(49\) 8.52137 1.21734
\(50\) 0 0
\(51\) −6.05517 −0.847894
\(52\) 0 0
\(53\) 5.68316 0.780643 0.390321 0.920679i \(-0.372364\pi\)
0.390321 + 0.920679i \(0.372364\pi\)
\(54\) 0 0
\(55\) 3.43077 0.462605
\(56\) 0 0
\(57\) −2.15766 −0.285789
\(58\) 0 0
\(59\) −4.35730 −0.567273 −0.283636 0.958932i \(-0.591541\pi\)
−0.283636 + 0.958932i \(0.591541\pi\)
\(60\) 0 0
\(61\) −3.56412 −0.456339 −0.228169 0.973621i \(-0.573274\pi\)
−0.228169 + 0.973621i \(0.573274\pi\)
\(62\) 0 0
\(63\) −6.52213 −0.821711
\(64\) 0 0
\(65\) −11.3621 −1.40929
\(66\) 0 0
\(67\) 9.95563 1.21627 0.608137 0.793832i \(-0.291917\pi\)
0.608137 + 0.793832i \(0.291917\pi\)
\(68\) 0 0
\(69\) 14.8662 1.78968
\(70\) 0 0
\(71\) 11.4671 1.36089 0.680447 0.732797i \(-0.261786\pi\)
0.680447 + 0.732797i \(0.261786\pi\)
\(72\) 0 0
\(73\) −8.95834 −1.04849 −0.524247 0.851566i \(-0.675653\pi\)
−0.524247 + 0.851566i \(0.675653\pi\)
\(74\) 0 0
\(75\) −14.6077 −1.68675
\(76\) 0 0
\(77\) 3.93972 0.448972
\(78\) 0 0
\(79\) −8.49105 −0.955318 −0.477659 0.878545i \(-0.658515\pi\)
−0.477659 + 0.878545i \(0.658515\pi\)
\(80\) 0 0
\(81\) −11.2258 −1.24731
\(82\) 0 0
\(83\) 5.21960 0.572926 0.286463 0.958091i \(-0.407520\pi\)
0.286463 + 0.958091i \(0.407520\pi\)
\(84\) 0 0
\(85\) −9.62799 −1.04430
\(86\) 0 0
\(87\) −12.2550 −1.31388
\(88\) 0 0
\(89\) −7.28423 −0.772127 −0.386064 0.922472i \(-0.626166\pi\)
−0.386064 + 0.922472i \(0.626166\pi\)
\(90\) 0 0
\(91\) −13.0476 −1.36776
\(92\) 0 0
\(93\) 5.43436 0.563517
\(94\) 0 0
\(95\) −3.43077 −0.351989
\(96\) 0 0
\(97\) 10.6574 1.08209 0.541045 0.840993i \(-0.318029\pi\)
0.541045 + 0.840993i \(0.318029\pi\)
\(98\) 0 0
\(99\) −1.65548 −0.166382
\(100\) 0 0
\(101\) −11.4716 −1.14147 −0.570735 0.821134i \(-0.693342\pi\)
−0.570735 + 0.821134i \(0.693342\pi\)
\(102\) 0 0
\(103\) −18.3034 −1.80349 −0.901745 0.432268i \(-0.857714\pi\)
−0.901745 + 0.432268i \(0.857714\pi\)
\(104\) 0 0
\(105\) −29.1634 −2.84606
\(106\) 0 0
\(107\) −1.38838 −0.134220 −0.0671100 0.997746i \(-0.521378\pi\)
−0.0671100 + 0.997746i \(0.521378\pi\)
\(108\) 0 0
\(109\) −0.412113 −0.0394732 −0.0197366 0.999805i \(-0.506283\pi\)
−0.0197366 + 0.999805i \(0.506283\pi\)
\(110\) 0 0
\(111\) 13.8067 1.31047
\(112\) 0 0
\(113\) −6.54003 −0.615234 −0.307617 0.951510i \(-0.599532\pi\)
−0.307617 + 0.951510i \(0.599532\pi\)
\(114\) 0 0
\(115\) 23.6379 2.20425
\(116\) 0 0
\(117\) 5.48266 0.506872
\(118\) 0 0
\(119\) −11.0563 −1.01353
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −1.20964 −0.109070
\(124\) 0 0
\(125\) −6.07307 −0.543192
\(126\) 0 0
\(127\) −9.08005 −0.805724 −0.402862 0.915261i \(-0.631985\pi\)
−0.402862 + 0.915261i \(0.631985\pi\)
\(128\) 0 0
\(129\) −20.2837 −1.78588
\(130\) 0 0
\(131\) 10.3876 0.907571 0.453785 0.891111i \(-0.350073\pi\)
0.453785 + 0.891111i \(0.350073\pi\)
\(132\) 0 0
\(133\) −3.93972 −0.341617
\(134\) 0 0
\(135\) −9.95269 −0.856591
\(136\) 0 0
\(137\) −0.798293 −0.0682028 −0.0341014 0.999418i \(-0.510857\pi\)
−0.0341014 + 0.999418i \(0.510857\pi\)
\(138\) 0 0
\(139\) −5.03184 −0.426795 −0.213398 0.976965i \(-0.568453\pi\)
−0.213398 + 0.976965i \(0.568453\pi\)
\(140\) 0 0
\(141\) −26.2678 −2.21215
\(142\) 0 0
\(143\) −3.31182 −0.276948
\(144\) 0 0
\(145\) −19.4860 −1.61823
\(146\) 0 0
\(147\) −18.3862 −1.51647
\(148\) 0 0
\(149\) 19.8351 1.62496 0.812479 0.582991i \(-0.198118\pi\)
0.812479 + 0.582991i \(0.198118\pi\)
\(150\) 0 0
\(151\) −22.5447 −1.83466 −0.917331 0.398125i \(-0.869661\pi\)
−0.917331 + 0.398125i \(0.869661\pi\)
\(152\) 0 0
\(153\) 4.64589 0.375598
\(154\) 0 0
\(155\) 8.64087 0.694051
\(156\) 0 0
\(157\) −11.8013 −0.941843 −0.470921 0.882175i \(-0.656078\pi\)
−0.470921 + 0.882175i \(0.656078\pi\)
\(158\) 0 0
\(159\) −12.2623 −0.972465
\(160\) 0 0
\(161\) 27.1446 2.13929
\(162\) 0 0
\(163\) 24.8395 1.94558 0.972789 0.231691i \(-0.0744259\pi\)
0.972789 + 0.231691i \(0.0744259\pi\)
\(164\) 0 0
\(165\) −7.40242 −0.576278
\(166\) 0 0
\(167\) 2.79938 0.216623 0.108311 0.994117i \(-0.465456\pi\)
0.108311 + 0.994117i \(0.465456\pi\)
\(168\) 0 0
\(169\) −2.03184 −0.156295
\(170\) 0 0
\(171\) 1.65548 0.126598
\(172\) 0 0
\(173\) −6.43926 −0.489568 −0.244784 0.969578i \(-0.578717\pi\)
−0.244784 + 0.969578i \(0.578717\pi\)
\(174\) 0 0
\(175\) −26.6726 −2.01626
\(176\) 0 0
\(177\) 9.40156 0.706665
\(178\) 0 0
\(179\) −12.5241 −0.936095 −0.468048 0.883703i \(-0.655042\pi\)
−0.468048 + 0.883703i \(0.655042\pi\)
\(180\) 0 0
\(181\) 13.7515 1.02214 0.511071 0.859538i \(-0.329249\pi\)
0.511071 + 0.859538i \(0.329249\pi\)
\(182\) 0 0
\(183\) 7.69015 0.568472
\(184\) 0 0
\(185\) 21.9533 1.61404
\(186\) 0 0
\(187\) −2.80637 −0.205222
\(188\) 0 0
\(189\) −11.4292 −0.831348
\(190\) 0 0
\(191\) −3.10678 −0.224799 −0.112399 0.993663i \(-0.535854\pi\)
−0.112399 + 0.993663i \(0.535854\pi\)
\(192\) 0 0
\(193\) −0.747815 −0.0538289 −0.0269144 0.999638i \(-0.508568\pi\)
−0.0269144 + 0.999638i \(0.508568\pi\)
\(194\) 0 0
\(195\) 24.5155 1.75559
\(196\) 0 0
\(197\) −3.41798 −0.243521 −0.121761 0.992559i \(-0.538854\pi\)
−0.121761 + 0.992559i \(0.538854\pi\)
\(198\) 0 0
\(199\) −5.36785 −0.380517 −0.190258 0.981734i \(-0.560933\pi\)
−0.190258 + 0.981734i \(0.560933\pi\)
\(200\) 0 0
\(201\) −21.4808 −1.51514
\(202\) 0 0
\(203\) −22.3768 −1.57054
\(204\) 0 0
\(205\) −1.92339 −0.134335
\(206\) 0 0
\(207\) −11.4062 −0.792789
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −2.55492 −0.175888 −0.0879441 0.996125i \(-0.528030\pi\)
−0.0879441 + 0.996125i \(0.528030\pi\)
\(212\) 0 0
\(213\) −24.7421 −1.69530
\(214\) 0 0
\(215\) −32.2520 −2.19957
\(216\) 0 0
\(217\) 9.92272 0.673598
\(218\) 0 0
\(219\) 19.3290 1.30613
\(220\) 0 0
\(221\) 9.29418 0.625194
\(222\) 0 0
\(223\) 24.9404 1.67013 0.835066 0.550149i \(-0.185429\pi\)
0.835066 + 0.550149i \(0.185429\pi\)
\(224\) 0 0
\(225\) 11.2079 0.747194
\(226\) 0 0
\(227\) 22.8254 1.51497 0.757487 0.652851i \(-0.226427\pi\)
0.757487 + 0.652851i \(0.226427\pi\)
\(228\) 0 0
\(229\) −0.603546 −0.0398834 −0.0199417 0.999801i \(-0.506348\pi\)
−0.0199417 + 0.999801i \(0.506348\pi\)
\(230\) 0 0
\(231\) −8.50056 −0.559296
\(232\) 0 0
\(233\) 17.3705 1.13798 0.568988 0.822346i \(-0.307335\pi\)
0.568988 + 0.822346i \(0.307335\pi\)
\(234\) 0 0
\(235\) −41.7669 −2.72457
\(236\) 0 0
\(237\) 18.3208 1.19006
\(238\) 0 0
\(239\) −7.23486 −0.467984 −0.233992 0.972238i \(-0.575179\pi\)
−0.233992 + 0.972238i \(0.575179\pi\)
\(240\) 0 0
\(241\) −12.2034 −0.786090 −0.393045 0.919519i \(-0.628578\pi\)
−0.393045 + 0.919519i \(0.628578\pi\)
\(242\) 0 0
\(243\) 15.5185 0.995509
\(244\) 0 0
\(245\) −29.2349 −1.86775
\(246\) 0 0
\(247\) 3.31182 0.210726
\(248\) 0 0
\(249\) −11.2621 −0.713707
\(250\) 0 0
\(251\) 14.0923 0.889499 0.444750 0.895655i \(-0.353293\pi\)
0.444750 + 0.895655i \(0.353293\pi\)
\(252\) 0 0
\(253\) 6.88998 0.433169
\(254\) 0 0
\(255\) 20.7739 1.30091
\(256\) 0 0
\(257\) 0.440920 0.0275038 0.0137519 0.999905i \(-0.495622\pi\)
0.0137519 + 0.999905i \(0.495622\pi\)
\(258\) 0 0
\(259\) 25.2100 1.56647
\(260\) 0 0
\(261\) 9.40279 0.582018
\(262\) 0 0
\(263\) 15.0661 0.929016 0.464508 0.885569i \(-0.346231\pi\)
0.464508 + 0.885569i \(0.346231\pi\)
\(264\) 0 0
\(265\) −19.4976 −1.19773
\(266\) 0 0
\(267\) 15.7169 0.961857
\(268\) 0 0
\(269\) −7.25751 −0.442498 −0.221249 0.975217i \(-0.571013\pi\)
−0.221249 + 0.975217i \(0.571013\pi\)
\(270\) 0 0
\(271\) 16.8878 1.02586 0.512931 0.858430i \(-0.328560\pi\)
0.512931 + 0.858430i \(0.328560\pi\)
\(272\) 0 0
\(273\) 28.1523 1.70386
\(274\) 0 0
\(275\) −6.77018 −0.408257
\(276\) 0 0
\(277\) −15.3818 −0.924204 −0.462102 0.886827i \(-0.652905\pi\)
−0.462102 + 0.886827i \(0.652905\pi\)
\(278\) 0 0
\(279\) −4.16956 −0.249625
\(280\) 0 0
\(281\) −25.8974 −1.54491 −0.772456 0.635069i \(-0.780972\pi\)
−0.772456 + 0.635069i \(0.780972\pi\)
\(282\) 0 0
\(283\) −18.6882 −1.11090 −0.555450 0.831550i \(-0.687454\pi\)
−0.555450 + 0.831550i \(0.687454\pi\)
\(284\) 0 0
\(285\) 7.40242 0.438482
\(286\) 0 0
\(287\) −2.20872 −0.130377
\(288\) 0 0
\(289\) −9.12431 −0.536724
\(290\) 0 0
\(291\) −22.9949 −1.34799
\(292\) 0 0
\(293\) 26.7471 1.56258 0.781291 0.624167i \(-0.214561\pi\)
0.781291 + 0.624167i \(0.214561\pi\)
\(294\) 0 0
\(295\) 14.9489 0.870359
\(296\) 0 0
\(297\) −2.90101 −0.168334
\(298\) 0 0
\(299\) −22.8184 −1.31962
\(300\) 0 0
\(301\) −37.0365 −2.13475
\(302\) 0 0
\(303\) 24.7518 1.42196
\(304\) 0 0
\(305\) 12.2277 0.700155
\(306\) 0 0
\(307\) −22.6415 −1.29222 −0.646109 0.763245i \(-0.723605\pi\)
−0.646109 + 0.763245i \(0.723605\pi\)
\(308\) 0 0
\(309\) 39.4925 2.24665
\(310\) 0 0
\(311\) 1.38723 0.0786628 0.0393314 0.999226i \(-0.487477\pi\)
0.0393314 + 0.999226i \(0.487477\pi\)
\(312\) 0 0
\(313\) 12.9018 0.729255 0.364627 0.931153i \(-0.381196\pi\)
0.364627 + 0.931153i \(0.381196\pi\)
\(314\) 0 0
\(315\) 22.3759 1.26074
\(316\) 0 0
\(317\) −19.0712 −1.07114 −0.535572 0.844489i \(-0.679904\pi\)
−0.535572 + 0.844489i \(0.679904\pi\)
\(318\) 0 0
\(319\) −5.67979 −0.318007
\(320\) 0 0
\(321\) 2.99565 0.167201
\(322\) 0 0
\(323\) 2.80637 0.156150
\(324\) 0 0
\(325\) 22.4216 1.24373
\(326\) 0 0
\(327\) 0.889197 0.0491727
\(328\) 0 0
\(329\) −47.9630 −2.64428
\(330\) 0 0
\(331\) 12.4616 0.684952 0.342476 0.939527i \(-0.388735\pi\)
0.342476 + 0.939527i \(0.388735\pi\)
\(332\) 0 0
\(333\) −10.5933 −0.580510
\(334\) 0 0
\(335\) −34.1555 −1.86611
\(336\) 0 0
\(337\) −0.401035 −0.0218458 −0.0109229 0.999940i \(-0.503477\pi\)
−0.0109229 + 0.999940i \(0.503477\pi\)
\(338\) 0 0
\(339\) 14.1111 0.766411
\(340\) 0 0
\(341\) 2.51864 0.136392
\(342\) 0 0
\(343\) −5.99377 −0.323633
\(344\) 0 0
\(345\) −51.0026 −2.74589
\(346\) 0 0
\(347\) −1.06608 −0.0572304 −0.0286152 0.999591i \(-0.509110\pi\)
−0.0286152 + 0.999591i \(0.509110\pi\)
\(348\) 0 0
\(349\) 22.2695 1.19206 0.596029 0.802963i \(-0.296744\pi\)
0.596029 + 0.802963i \(0.296744\pi\)
\(350\) 0 0
\(351\) 9.60762 0.512817
\(352\) 0 0
\(353\) 12.3631 0.658023 0.329012 0.944326i \(-0.393284\pi\)
0.329012 + 0.944326i \(0.393284\pi\)
\(354\) 0 0
\(355\) −39.3410 −2.08800
\(356\) 0 0
\(357\) 23.8557 1.26258
\(358\) 0 0
\(359\) −32.1914 −1.69899 −0.849497 0.527593i \(-0.823095\pi\)
−0.849497 + 0.527593i \(0.823095\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.15766 −0.113248
\(364\) 0 0
\(365\) 30.7340 1.60869
\(366\) 0 0
\(367\) 18.9777 0.990628 0.495314 0.868714i \(-0.335053\pi\)
0.495314 + 0.868714i \(0.335053\pi\)
\(368\) 0 0
\(369\) 0.928111 0.0483155
\(370\) 0 0
\(371\) −22.3901 −1.16243
\(372\) 0 0
\(373\) 1.15238 0.0596678 0.0298339 0.999555i \(-0.490502\pi\)
0.0298339 + 0.999555i \(0.490502\pi\)
\(374\) 0 0
\(375\) 13.1036 0.676667
\(376\) 0 0
\(377\) 18.8104 0.968787
\(378\) 0 0
\(379\) −13.5365 −0.695325 −0.347663 0.937620i \(-0.613025\pi\)
−0.347663 + 0.937620i \(0.613025\pi\)
\(380\) 0 0
\(381\) 19.5916 1.00371
\(382\) 0 0
\(383\) −13.1671 −0.672809 −0.336404 0.941718i \(-0.609211\pi\)
−0.336404 + 0.941718i \(0.609211\pi\)
\(384\) 0 0
\(385\) −13.5163 −0.688852
\(386\) 0 0
\(387\) 15.5629 0.791105
\(388\) 0 0
\(389\) 0.223588 0.0113364 0.00566819 0.999984i \(-0.498196\pi\)
0.00566819 + 0.999984i \(0.498196\pi\)
\(390\) 0 0
\(391\) −19.3358 −0.977854
\(392\) 0 0
\(393\) −22.4129 −1.13058
\(394\) 0 0
\(395\) 29.1308 1.46573
\(396\) 0 0
\(397\) −21.6504 −1.08660 −0.543301 0.839538i \(-0.682826\pi\)
−0.543301 + 0.839538i \(0.682826\pi\)
\(398\) 0 0
\(399\) 8.50056 0.425560
\(400\) 0 0
\(401\) 1.30180 0.0650089 0.0325045 0.999472i \(-0.489652\pi\)
0.0325045 + 0.999472i \(0.489652\pi\)
\(402\) 0 0
\(403\) −8.34128 −0.415509
\(404\) 0 0
\(405\) 38.5132 1.91374
\(406\) 0 0
\(407\) 6.39893 0.317183
\(408\) 0 0
\(409\) −2.15631 −0.106622 −0.0533112 0.998578i \(-0.516978\pi\)
−0.0533112 + 0.998578i \(0.516978\pi\)
\(410\) 0 0
\(411\) 1.72244 0.0849618
\(412\) 0 0
\(413\) 17.1665 0.844710
\(414\) 0 0
\(415\) −17.9073 −0.879032
\(416\) 0 0
\(417\) 10.8570 0.531669
\(418\) 0 0
\(419\) 29.6335 1.44769 0.723844 0.689963i \(-0.242373\pi\)
0.723844 + 0.689963i \(0.242373\pi\)
\(420\) 0 0
\(421\) −5.25385 −0.256057 −0.128028 0.991770i \(-0.540865\pi\)
−0.128028 + 0.991770i \(0.540865\pi\)
\(422\) 0 0
\(423\) 20.1542 0.979931
\(424\) 0 0
\(425\) 18.9996 0.921616
\(426\) 0 0
\(427\) 14.0416 0.679522
\(428\) 0 0
\(429\) 7.14577 0.345001
\(430\) 0 0
\(431\) −17.8489 −0.859752 −0.429876 0.902888i \(-0.641443\pi\)
−0.429876 + 0.902888i \(0.641443\pi\)
\(432\) 0 0
\(433\) −0.696383 −0.0334660 −0.0167330 0.999860i \(-0.505327\pi\)
−0.0167330 + 0.999860i \(0.505327\pi\)
\(434\) 0 0
\(435\) 42.0442 2.01586
\(436\) 0 0
\(437\) −6.88998 −0.329593
\(438\) 0 0
\(439\) 24.8210 1.18464 0.592321 0.805702i \(-0.298212\pi\)
0.592321 + 0.805702i \(0.298212\pi\)
\(440\) 0 0
\(441\) 14.1070 0.671761
\(442\) 0 0
\(443\) 3.93424 0.186922 0.0934608 0.995623i \(-0.470207\pi\)
0.0934608 + 0.995623i \(0.470207\pi\)
\(444\) 0 0
\(445\) 24.9905 1.18466
\(446\) 0 0
\(447\) −42.7974 −2.02425
\(448\) 0 0
\(449\) 6.13394 0.289479 0.144739 0.989470i \(-0.453766\pi\)
0.144739 + 0.989470i \(0.453766\pi\)
\(450\) 0 0
\(451\) −0.560629 −0.0263990
\(452\) 0 0
\(453\) 48.6437 2.28548
\(454\) 0 0
\(455\) 44.7634 2.09854
\(456\) 0 0
\(457\) 8.40189 0.393024 0.196512 0.980501i \(-0.437039\pi\)
0.196512 + 0.980501i \(0.437039\pi\)
\(458\) 0 0
\(459\) 8.14129 0.380003
\(460\) 0 0
\(461\) 25.9903 1.21049 0.605246 0.796039i \(-0.293075\pi\)
0.605246 + 0.796039i \(0.293075\pi\)
\(462\) 0 0
\(463\) −19.3724 −0.900311 −0.450155 0.892950i \(-0.648631\pi\)
−0.450155 + 0.892950i \(0.648631\pi\)
\(464\) 0 0
\(465\) −18.6440 −0.864596
\(466\) 0 0
\(467\) 34.6720 1.60443 0.802214 0.597037i \(-0.203655\pi\)
0.802214 + 0.597037i \(0.203655\pi\)
\(468\) 0 0
\(469\) −39.2224 −1.81112
\(470\) 0 0
\(471\) 25.4631 1.17328
\(472\) 0 0
\(473\) −9.40080 −0.432249
\(474\) 0 0
\(475\) 6.77018 0.310637
\(476\) 0 0
\(477\) 9.40838 0.430780
\(478\) 0 0
\(479\) −23.2352 −1.06164 −0.530821 0.847484i \(-0.678117\pi\)
−0.530821 + 0.847484i \(0.678117\pi\)
\(480\) 0 0
\(481\) −21.1921 −0.966277
\(482\) 0 0
\(483\) −58.5687 −2.66497
\(484\) 0 0
\(485\) −36.5629 −1.66024
\(486\) 0 0
\(487\) −14.2680 −0.646545 −0.323272 0.946306i \(-0.604783\pi\)
−0.323272 + 0.946306i \(0.604783\pi\)
\(488\) 0 0
\(489\) −53.5951 −2.42365
\(490\) 0 0
\(491\) −41.6168 −1.87814 −0.939071 0.343724i \(-0.888311\pi\)
−0.939071 + 0.343724i \(0.888311\pi\)
\(492\) 0 0
\(493\) 15.9396 0.717882
\(494\) 0 0
\(495\) 5.67958 0.255278
\(496\) 0 0
\(497\) −45.1771 −2.02647
\(498\) 0 0
\(499\) −19.9664 −0.893820 −0.446910 0.894579i \(-0.647476\pi\)
−0.446910 + 0.894579i \(0.647476\pi\)
\(500\) 0 0
\(501\) −6.04010 −0.269852
\(502\) 0 0
\(503\) −2.31362 −0.103159 −0.0515797 0.998669i \(-0.516426\pi\)
−0.0515797 + 0.998669i \(0.516426\pi\)
\(504\) 0 0
\(505\) 39.3565 1.75134
\(506\) 0 0
\(507\) 4.38401 0.194701
\(508\) 0 0
\(509\) −13.2009 −0.585120 −0.292560 0.956247i \(-0.594507\pi\)
−0.292560 + 0.956247i \(0.594507\pi\)
\(510\) 0 0
\(511\) 35.2933 1.56128
\(512\) 0 0
\(513\) 2.90101 0.128083
\(514\) 0 0
\(515\) 62.7948 2.76707
\(516\) 0 0
\(517\) −12.1742 −0.535421
\(518\) 0 0
\(519\) 13.8937 0.609866
\(520\) 0 0
\(521\) −15.6498 −0.685629 −0.342814 0.939403i \(-0.611380\pi\)
−0.342814 + 0.939403i \(0.611380\pi\)
\(522\) 0 0
\(523\) 17.7350 0.775498 0.387749 0.921765i \(-0.373253\pi\)
0.387749 + 0.921765i \(0.373253\pi\)
\(524\) 0 0
\(525\) 57.5503 2.51170
\(526\) 0 0
\(527\) −7.06822 −0.307897
\(528\) 0 0
\(529\) 24.4719 1.06399
\(530\) 0 0
\(531\) −7.21344 −0.313037
\(532\) 0 0
\(533\) 1.85670 0.0804227
\(534\) 0 0
\(535\) 4.76322 0.205932
\(536\) 0 0
\(537\) 27.0227 1.16612
\(538\) 0 0
\(539\) −8.52137 −0.367041
\(540\) 0 0
\(541\) −34.5196 −1.48412 −0.742058 0.670336i \(-0.766150\pi\)
−0.742058 + 0.670336i \(0.766150\pi\)
\(542\) 0 0
\(543\) −29.6711 −1.27331
\(544\) 0 0
\(545\) 1.41386 0.0605632
\(546\) 0 0
\(547\) −1.99561 −0.0853263 −0.0426632 0.999090i \(-0.513584\pi\)
−0.0426632 + 0.999090i \(0.513584\pi\)
\(548\) 0 0
\(549\) −5.90034 −0.251820
\(550\) 0 0
\(551\) 5.67979 0.241967
\(552\) 0 0
\(553\) 33.4523 1.42254
\(554\) 0 0
\(555\) −47.3676 −2.01064
\(556\) 0 0
\(557\) −23.5272 −0.996881 −0.498440 0.866924i \(-0.666094\pi\)
−0.498440 + 0.866924i \(0.666094\pi\)
\(558\) 0 0
\(559\) 31.1338 1.31682
\(560\) 0 0
\(561\) 6.05517 0.255650
\(562\) 0 0
\(563\) −14.2122 −0.598972 −0.299486 0.954101i \(-0.596815\pi\)
−0.299486 + 0.954101i \(0.596815\pi\)
\(564\) 0 0
\(565\) 22.4373 0.943945
\(566\) 0 0
\(567\) 44.2266 1.85734
\(568\) 0 0
\(569\) 22.1906 0.930277 0.465138 0.885238i \(-0.346005\pi\)
0.465138 + 0.885238i \(0.346005\pi\)
\(570\) 0 0
\(571\) −8.67954 −0.363227 −0.181614 0.983370i \(-0.558132\pi\)
−0.181614 + 0.983370i \(0.558132\pi\)
\(572\) 0 0
\(573\) 6.70337 0.280037
\(574\) 0 0
\(575\) −46.6464 −1.94529
\(576\) 0 0
\(577\) 40.9150 1.70331 0.851657 0.524100i \(-0.175598\pi\)
0.851657 + 0.524100i \(0.175598\pi\)
\(578\) 0 0
\(579\) 1.61353 0.0670559
\(580\) 0 0
\(581\) −20.5638 −0.853128
\(582\) 0 0
\(583\) −5.68316 −0.235373
\(584\) 0 0
\(585\) −18.8097 −0.777687
\(586\) 0 0
\(587\) 21.2684 0.877840 0.438920 0.898526i \(-0.355361\pi\)
0.438920 + 0.898526i \(0.355361\pi\)
\(588\) 0 0
\(589\) −2.51864 −0.103779
\(590\) 0 0
\(591\) 7.37484 0.303360
\(592\) 0 0
\(593\) −23.0212 −0.945368 −0.472684 0.881232i \(-0.656715\pi\)
−0.472684 + 0.881232i \(0.656715\pi\)
\(594\) 0 0
\(595\) 37.9316 1.55504
\(596\) 0 0
\(597\) 11.5820 0.474019
\(598\) 0 0
\(599\) 41.6249 1.70075 0.850374 0.526179i \(-0.176376\pi\)
0.850374 + 0.526179i \(0.176376\pi\)
\(600\) 0 0
\(601\) −20.8629 −0.851016 −0.425508 0.904955i \(-0.639905\pi\)
−0.425508 + 0.904955i \(0.639905\pi\)
\(602\) 0 0
\(603\) 16.4814 0.671173
\(604\) 0 0
\(605\) −3.43077 −0.139481
\(606\) 0 0
\(607\) 0.146703 0.00595450 0.00297725 0.999996i \(-0.499052\pi\)
0.00297725 + 0.999996i \(0.499052\pi\)
\(608\) 0 0
\(609\) 48.2813 1.95646
\(610\) 0 0
\(611\) 40.3188 1.63113
\(612\) 0 0
\(613\) −18.3393 −0.740716 −0.370358 0.928889i \(-0.620765\pi\)
−0.370358 + 0.928889i \(0.620765\pi\)
\(614\) 0 0
\(615\) 4.15001 0.167345
\(616\) 0 0
\(617\) 4.12050 0.165885 0.0829425 0.996554i \(-0.473568\pi\)
0.0829425 + 0.996554i \(0.473568\pi\)
\(618\) 0 0
\(619\) −28.5699 −1.14832 −0.574161 0.818743i \(-0.694672\pi\)
−0.574161 + 0.818743i \(0.694672\pi\)
\(620\) 0 0
\(621\) −19.9879 −0.802087
\(622\) 0 0
\(623\) 28.6978 1.14975
\(624\) 0 0
\(625\) −13.0156 −0.520624
\(626\) 0 0
\(627\) 2.15766 0.0861685
\(628\) 0 0
\(629\) −17.9577 −0.716022
\(630\) 0 0
\(631\) −33.9323 −1.35082 −0.675412 0.737441i \(-0.736034\pi\)
−0.675412 + 0.737441i \(0.736034\pi\)
\(632\) 0 0
\(633\) 5.51265 0.219108
\(634\) 0 0
\(635\) 31.1516 1.23621
\(636\) 0 0
\(637\) 28.2213 1.11817
\(638\) 0 0
\(639\) 18.9836 0.750979
\(640\) 0 0
\(641\) 19.7694 0.780844 0.390422 0.920636i \(-0.372329\pi\)
0.390422 + 0.920636i \(0.372329\pi\)
\(642\) 0 0
\(643\) −36.7642 −1.44984 −0.724920 0.688833i \(-0.758123\pi\)
−0.724920 + 0.688833i \(0.758123\pi\)
\(644\) 0 0
\(645\) 69.5887 2.74005
\(646\) 0 0
\(647\) −24.9430 −0.980610 −0.490305 0.871551i \(-0.663115\pi\)
−0.490305 + 0.871551i \(0.663115\pi\)
\(648\) 0 0
\(649\) 4.35730 0.171039
\(650\) 0 0
\(651\) −21.4098 −0.839117
\(652\) 0 0
\(653\) 8.60802 0.336858 0.168429 0.985714i \(-0.446131\pi\)
0.168429 + 0.985714i \(0.446131\pi\)
\(654\) 0 0
\(655\) −35.6375 −1.39247
\(656\) 0 0
\(657\) −14.8304 −0.578588
\(658\) 0 0
\(659\) −27.7805 −1.08217 −0.541087 0.840967i \(-0.681987\pi\)
−0.541087 + 0.840967i \(0.681987\pi\)
\(660\) 0 0
\(661\) 30.8037 1.19812 0.599062 0.800702i \(-0.295540\pi\)
0.599062 + 0.800702i \(0.295540\pi\)
\(662\) 0 0
\(663\) −20.0537 −0.778819
\(664\) 0 0
\(665\) 13.5163 0.524138
\(666\) 0 0
\(667\) −39.1336 −1.51526
\(668\) 0 0
\(669\) −53.8128 −2.08052
\(670\) 0 0
\(671\) 3.56412 0.137591
\(672\) 0 0
\(673\) 15.2015 0.585974 0.292987 0.956116i \(-0.405351\pi\)
0.292987 + 0.956116i \(0.405351\pi\)
\(674\) 0 0
\(675\) 19.6403 0.755957
\(676\) 0 0
\(677\) −9.49027 −0.364741 −0.182370 0.983230i \(-0.558377\pi\)
−0.182370 + 0.983230i \(0.558377\pi\)
\(678\) 0 0
\(679\) −41.9870 −1.61131
\(680\) 0 0
\(681\) −49.2493 −1.88724
\(682\) 0 0
\(683\) −15.0960 −0.577631 −0.288816 0.957385i \(-0.593261\pi\)
−0.288816 + 0.957385i \(0.593261\pi\)
\(684\) 0 0
\(685\) 2.73876 0.104643
\(686\) 0 0
\(687\) 1.30224 0.0496837
\(688\) 0 0
\(689\) 18.8216 0.717047
\(690\) 0 0
\(691\) −3.46673 −0.131881 −0.0659403 0.997824i \(-0.521005\pi\)
−0.0659403 + 0.997824i \(0.521005\pi\)
\(692\) 0 0
\(693\) 6.52213 0.247755
\(694\) 0 0
\(695\) 17.2631 0.654826
\(696\) 0 0
\(697\) 1.57333 0.0595941
\(698\) 0 0
\(699\) −37.4795 −1.41760
\(700\) 0 0
\(701\) −26.2612 −0.991872 −0.495936 0.868359i \(-0.665175\pi\)
−0.495936 + 0.868359i \(0.665175\pi\)
\(702\) 0 0
\(703\) −6.39893 −0.241340
\(704\) 0 0
\(705\) 90.1187 3.39407
\(706\) 0 0
\(707\) 45.1950 1.69973
\(708\) 0 0
\(709\) −20.1488 −0.756704 −0.378352 0.925662i \(-0.623509\pi\)
−0.378352 + 0.925662i \(0.623509\pi\)
\(710\) 0 0
\(711\) −14.0568 −0.527171
\(712\) 0 0
\(713\) 17.3534 0.649889
\(714\) 0 0
\(715\) 11.3621 0.424918
\(716\) 0 0
\(717\) 15.6104 0.582979
\(718\) 0 0
\(719\) 43.4738 1.62130 0.810650 0.585532i \(-0.199114\pi\)
0.810650 + 0.585532i \(0.199114\pi\)
\(720\) 0 0
\(721\) 72.1103 2.68553
\(722\) 0 0
\(723\) 26.3307 0.979250
\(724\) 0 0
\(725\) 38.4532 1.42811
\(726\) 0 0
\(727\) −9.55640 −0.354427 −0.177214 0.984172i \(-0.556708\pi\)
−0.177214 + 0.984172i \(0.556708\pi\)
\(728\) 0 0
\(729\) 0.193986 0.00718467
\(730\) 0 0
\(731\) 26.3821 0.975777
\(732\) 0 0
\(733\) −20.3072 −0.750065 −0.375032 0.927012i \(-0.622368\pi\)
−0.375032 + 0.927012i \(0.622368\pi\)
\(734\) 0 0
\(735\) 63.0788 2.32670
\(736\) 0 0
\(737\) −9.95563 −0.366720
\(738\) 0 0
\(739\) 6.60855 0.243099 0.121550 0.992585i \(-0.461214\pi\)
0.121550 + 0.992585i \(0.461214\pi\)
\(740\) 0 0
\(741\) −7.14577 −0.262507
\(742\) 0 0
\(743\) 39.1431 1.43602 0.718010 0.696033i \(-0.245053\pi\)
0.718010 + 0.696033i \(0.245053\pi\)
\(744\) 0 0
\(745\) −68.0498 −2.49315
\(746\) 0 0
\(747\) 8.64096 0.316156
\(748\) 0 0
\(749\) 5.46983 0.199863
\(750\) 0 0
\(751\) −43.2173 −1.57702 −0.788511 0.615020i \(-0.789148\pi\)
−0.788511 + 0.615020i \(0.789148\pi\)
\(752\) 0 0
\(753\) −30.4064 −1.10807
\(754\) 0 0
\(755\) 77.3457 2.81490
\(756\) 0 0
\(757\) 9.93714 0.361171 0.180586 0.983559i \(-0.442201\pi\)
0.180586 + 0.983559i \(0.442201\pi\)
\(758\) 0 0
\(759\) −14.8662 −0.539609
\(760\) 0 0
\(761\) 14.5829 0.528630 0.264315 0.964436i \(-0.414854\pi\)
0.264315 + 0.964436i \(0.414854\pi\)
\(762\) 0 0
\(763\) 1.62361 0.0587785
\(764\) 0 0
\(765\) −15.9390 −0.576275
\(766\) 0 0
\(767\) −14.4306 −0.521059
\(768\) 0 0
\(769\) −11.8924 −0.428853 −0.214426 0.976740i \(-0.568788\pi\)
−0.214426 + 0.976740i \(0.568788\pi\)
\(770\) 0 0
\(771\) −0.951354 −0.0342622
\(772\) 0 0
\(773\) 15.0935 0.542876 0.271438 0.962456i \(-0.412501\pi\)
0.271438 + 0.962456i \(0.412501\pi\)
\(774\) 0 0
\(775\) −17.0516 −0.612513
\(776\) 0 0
\(777\) −54.3945 −1.95139
\(778\) 0 0
\(779\) 0.560629 0.0200866
\(780\) 0 0
\(781\) −11.4671 −0.410325
\(782\) 0 0
\(783\) 16.4771 0.588844
\(784\) 0 0
\(785\) 40.4874 1.44506
\(786\) 0 0
\(787\) 5.98214 0.213240 0.106620 0.994300i \(-0.465997\pi\)
0.106620 + 0.994300i \(0.465997\pi\)
\(788\) 0 0
\(789\) −32.5075 −1.15730
\(790\) 0 0
\(791\) 25.7659 0.916128
\(792\) 0 0
\(793\) −11.8037 −0.419163
\(794\) 0 0
\(795\) 42.0692 1.49204
\(796\) 0 0
\(797\) 24.2718 0.859751 0.429876 0.902888i \(-0.358557\pi\)
0.429876 + 0.902888i \(0.358557\pi\)
\(798\) 0 0
\(799\) 34.1653 1.20868
\(800\) 0 0
\(801\) −12.0589 −0.426081
\(802\) 0 0
\(803\) 8.95834 0.316133
\(804\) 0 0
\(805\) −93.1268 −3.28229
\(806\) 0 0
\(807\) 15.6592 0.551230
\(808\) 0 0
\(809\) −17.9557 −0.631289 −0.315644 0.948878i \(-0.602221\pi\)
−0.315644 + 0.948878i \(0.602221\pi\)
\(810\) 0 0
\(811\) −5.48300 −0.192534 −0.0962671 0.995356i \(-0.530690\pi\)
−0.0962671 + 0.995356i \(0.530690\pi\)
\(812\) 0 0
\(813\) −36.4381 −1.27794
\(814\) 0 0
\(815\) −85.2185 −2.98508
\(816\) 0 0
\(817\) 9.40080 0.328892
\(818\) 0 0
\(819\) −21.6001 −0.754770
\(820\) 0 0
\(821\) 52.6324 1.83688 0.918441 0.395558i \(-0.129449\pi\)
0.918441 + 0.395558i \(0.129449\pi\)
\(822\) 0 0
\(823\) 20.4784 0.713831 0.356916 0.934137i \(-0.383828\pi\)
0.356916 + 0.934137i \(0.383828\pi\)
\(824\) 0 0
\(825\) 14.6077 0.508575
\(826\) 0 0
\(827\) 0.359755 0.0125099 0.00625496 0.999980i \(-0.498009\pi\)
0.00625496 + 0.999980i \(0.498009\pi\)
\(828\) 0 0
\(829\) 22.7422 0.789870 0.394935 0.918709i \(-0.370767\pi\)
0.394935 + 0.918709i \(0.370767\pi\)
\(830\) 0 0
\(831\) 33.1887 1.15130
\(832\) 0 0
\(833\) 23.9141 0.828574
\(834\) 0 0
\(835\) −9.60403 −0.332361
\(836\) 0 0
\(837\) −7.30659 −0.252553
\(838\) 0 0
\(839\) −13.1616 −0.454388 −0.227194 0.973849i \(-0.572955\pi\)
−0.227194 + 0.973849i \(0.572955\pi\)
\(840\) 0 0
\(841\) 3.25998 0.112413
\(842\) 0 0
\(843\) 55.8778 1.92453
\(844\) 0 0
\(845\) 6.97077 0.239802
\(846\) 0 0
\(847\) −3.93972 −0.135370
\(848\) 0 0
\(849\) 40.3228 1.38387
\(850\) 0 0
\(851\) 44.0885 1.51133
\(852\) 0 0
\(853\) −36.9102 −1.26378 −0.631891 0.775057i \(-0.717721\pi\)
−0.631891 + 0.775057i \(0.717721\pi\)
\(854\) 0 0
\(855\) −5.67958 −0.194237
\(856\) 0 0
\(857\) 24.3615 0.832174 0.416087 0.909325i \(-0.363401\pi\)
0.416087 + 0.909325i \(0.363401\pi\)
\(858\) 0 0
\(859\) 7.69750 0.262635 0.131318 0.991340i \(-0.458079\pi\)
0.131318 + 0.991340i \(0.458079\pi\)
\(860\) 0 0
\(861\) 4.76566 0.162413
\(862\) 0 0
\(863\) 29.0225 0.987937 0.493969 0.869480i \(-0.335546\pi\)
0.493969 + 0.869480i \(0.335546\pi\)
\(864\) 0 0
\(865\) 22.0916 0.751137
\(866\) 0 0
\(867\) 19.6871 0.668610
\(868\) 0 0
\(869\) 8.49105 0.288039
\(870\) 0 0
\(871\) 32.9713 1.11719
\(872\) 0 0
\(873\) 17.6431 0.597127
\(874\) 0 0
\(875\) 23.9262 0.808852
\(876\) 0 0
\(877\) −36.0010 −1.21567 −0.607834 0.794064i \(-0.707961\pi\)
−0.607834 + 0.794064i \(0.707961\pi\)
\(878\) 0 0
\(879\) −57.7111 −1.94655
\(880\) 0 0
\(881\) −15.3587 −0.517448 −0.258724 0.965951i \(-0.583302\pi\)
−0.258724 + 0.965951i \(0.583302\pi\)
\(882\) 0 0
\(883\) −40.2870 −1.35577 −0.677883 0.735170i \(-0.737102\pi\)
−0.677883 + 0.735170i \(0.737102\pi\)
\(884\) 0 0
\(885\) −32.2546 −1.08423
\(886\) 0 0
\(887\) 7.09855 0.238346 0.119173 0.992874i \(-0.461976\pi\)
0.119173 + 0.992874i \(0.461976\pi\)
\(888\) 0 0
\(889\) 35.7728 1.19978
\(890\) 0 0
\(891\) 11.2258 0.376079
\(892\) 0 0
\(893\) 12.1742 0.407395
\(894\) 0 0
\(895\) 42.9673 1.43624
\(896\) 0 0
\(897\) 49.2342 1.64388
\(898\) 0 0
\(899\) −14.3053 −0.477110
\(900\) 0 0
\(901\) 15.9490 0.531339
\(902\) 0 0
\(903\) 79.9121 2.65931
\(904\) 0 0
\(905\) −47.1783 −1.56826
\(906\) 0 0
\(907\) −28.4435 −0.944452 −0.472226 0.881478i \(-0.656549\pi\)
−0.472226 + 0.881478i \(0.656549\pi\)
\(908\) 0 0
\(909\) −18.9911 −0.629895
\(910\) 0 0
\(911\) −49.5740 −1.64246 −0.821230 0.570598i \(-0.806712\pi\)
−0.821230 + 0.570598i \(0.806712\pi\)
\(912\) 0 0
\(913\) −5.21960 −0.172744
\(914\) 0 0
\(915\) −26.3831 −0.872199
\(916\) 0 0
\(917\) −40.9243 −1.35144
\(918\) 0 0
\(919\) −17.8421 −0.588556 −0.294278 0.955720i \(-0.595079\pi\)
−0.294278 + 0.955720i \(0.595079\pi\)
\(920\) 0 0
\(921\) 48.8526 1.60975
\(922\) 0 0
\(923\) 37.9770 1.25003
\(924\) 0 0
\(925\) −43.3219 −1.42441
\(926\) 0 0
\(927\) −30.3010 −0.995215
\(928\) 0 0
\(929\) 36.0244 1.18192 0.590961 0.806700i \(-0.298749\pi\)
0.590961 + 0.806700i \(0.298749\pi\)
\(930\) 0 0
\(931\) 8.52137 0.279277
\(932\) 0 0
\(933\) −2.99317 −0.0979921
\(934\) 0 0
\(935\) 9.62799 0.314869
\(936\) 0 0
\(937\) −3.50371 −0.114461 −0.0572307 0.998361i \(-0.518227\pi\)
−0.0572307 + 0.998361i \(0.518227\pi\)
\(938\) 0 0
\(939\) −27.8377 −0.908450
\(940\) 0 0
\(941\) 44.2074 1.44112 0.720560 0.693392i \(-0.243885\pi\)
0.720560 + 0.693392i \(0.243885\pi\)
\(942\) 0 0
\(943\) −3.86272 −0.125788
\(944\) 0 0
\(945\) 39.2108 1.27553
\(946\) 0 0
\(947\) −37.9515 −1.23326 −0.616629 0.787254i \(-0.711502\pi\)
−0.616629 + 0.787254i \(0.711502\pi\)
\(948\) 0 0
\(949\) −29.6684 −0.963077
\(950\) 0 0
\(951\) 41.1491 1.33435
\(952\) 0 0
\(953\) 13.0303 0.422094 0.211047 0.977476i \(-0.432313\pi\)
0.211047 + 0.977476i \(0.432313\pi\)
\(954\) 0 0
\(955\) 10.6586 0.344906
\(956\) 0 0
\(957\) 12.2550 0.396149
\(958\) 0 0
\(959\) 3.14505 0.101559
\(960\) 0 0
\(961\) −24.6565 −0.795370
\(962\) 0 0
\(963\) −2.29844 −0.0740662
\(964\) 0 0
\(965\) 2.56558 0.0825889
\(966\) 0 0
\(967\) −24.0504 −0.773409 −0.386705 0.922204i \(-0.626387\pi\)
−0.386705 + 0.922204i \(0.626387\pi\)
\(968\) 0 0
\(969\) −6.05517 −0.194520
\(970\) 0 0
\(971\) −54.1335 −1.73723 −0.868614 0.495490i \(-0.834988\pi\)
−0.868614 + 0.495490i \(0.834988\pi\)
\(972\) 0 0
\(973\) 19.8240 0.635529
\(974\) 0 0
\(975\) −48.3781 −1.54934
\(976\) 0 0
\(977\) −50.2507 −1.60766 −0.803832 0.594857i \(-0.797209\pi\)
−0.803832 + 0.594857i \(0.797209\pi\)
\(978\) 0 0
\(979\) 7.28423 0.232805
\(980\) 0 0
\(981\) −0.682245 −0.0217824
\(982\) 0 0
\(983\) −6.64633 −0.211985 −0.105993 0.994367i \(-0.533802\pi\)
−0.105993 + 0.994367i \(0.533802\pi\)
\(984\) 0 0
\(985\) 11.7263 0.373631
\(986\) 0 0
\(987\) 103.488 3.29405
\(988\) 0 0
\(989\) −64.7714 −2.05961
\(990\) 0 0
\(991\) −51.7993 −1.64546 −0.822729 0.568434i \(-0.807549\pi\)
−0.822729 + 0.568434i \(0.807549\pi\)
\(992\) 0 0
\(993\) −26.8879 −0.853260
\(994\) 0 0
\(995\) 18.4159 0.583822
\(996\) 0 0
\(997\) −14.2042 −0.449851 −0.224925 0.974376i \(-0.572214\pi\)
−0.224925 + 0.974376i \(0.572214\pi\)
\(998\) 0 0
\(999\) −18.5633 −0.587318
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 3344.2.a.t.1.2 5
4.3 odd 2 209.2.a.c.1.1 5
12.11 even 2 1881.2.a.k.1.5 5
20.19 odd 2 5225.2.a.h.1.5 5
44.43 even 2 2299.2.a.n.1.5 5
76.75 even 2 3971.2.a.h.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.1 5 4.3 odd 2
1881.2.a.k.1.5 5 12.11 even 2
2299.2.a.n.1.5 5 44.43 even 2
3344.2.a.t.1.2 5 1.1 even 1 trivial
3971.2.a.h.1.5 5 76.75 even 2
5225.2.a.h.1.5 5 20.19 odd 2