Properties

Label 3344.2.a.x.1.4
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.24102\) 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)
 
\(f(q)\) \(=\) \(q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} +O(q^{10})\) \(q+1.24102 q^{3} +2.83235 q^{5} +4.46564 q^{7} -1.45986 q^{9} -1.00000 q^{11} +2.14252 q^{13} +3.51500 q^{15} +7.81594 q^{17} -1.00000 q^{19} +5.54196 q^{21} -3.81016 q^{23} +3.02218 q^{25} -5.53479 q^{27} +1.86854 q^{29} +1.22996 q^{31} -1.24102 q^{33} +12.6482 q^{35} -8.10815 q^{37} +2.65892 q^{39} +10.9225 q^{41} -2.97782 q^{43} -4.13484 q^{45} -6.28504 q^{47} +12.9419 q^{49} +9.69975 q^{51} +6.43768 q^{53} -2.83235 q^{55} -1.24102 q^{57} -7.40754 q^{59} -9.83806 q^{61} -6.51922 q^{63} +6.06836 q^{65} +13.7013 q^{67} -4.72850 q^{69} +2.27693 q^{71} +2.77299 q^{73} +3.75060 q^{75} -4.46564 q^{77} -15.9183 q^{79} -2.48921 q^{81} -6.37820 q^{83} +22.1374 q^{85} +2.31890 q^{87} +1.65892 q^{89} +9.56772 q^{91} +1.52641 q^{93} -2.83235 q^{95} -5.99423 q^{97} +1.45986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 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
\(3\) 1.24102 0.716505 0.358252 0.933625i \(-0.383373\pi\)
0.358252 + 0.933625i \(0.383373\pi\)
\(4\) 0 0
\(5\) 2.83235 1.26666 0.633332 0.773880i \(-0.281687\pi\)
0.633332 + 0.773880i \(0.281687\pi\)
\(6\) 0 0
\(7\) 4.46564 1.68785 0.843926 0.536460i \(-0.180239\pi\)
0.843926 + 0.536460i \(0.180239\pi\)
\(8\) 0 0
\(9\) −1.45986 −0.486621
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.14252 0.594229 0.297114 0.954842i \(-0.403976\pi\)
0.297114 + 0.954842i \(0.403976\pi\)
\(14\) 0 0
\(15\) 3.51500 0.907570
\(16\) 0 0
\(17\) 7.81594 1.89564 0.947822 0.318801i \(-0.103280\pi\)
0.947822 + 0.318801i \(0.103280\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 5.54196 1.20935
\(22\) 0 0
\(23\) −3.81016 −0.794474 −0.397237 0.917716i \(-0.630031\pi\)
−0.397237 + 0.917716i \(0.630031\pi\)
\(24\) 0 0
\(25\) 3.02218 0.604436
\(26\) 0 0
\(27\) −5.53479 −1.06517
\(28\) 0 0
\(29\) 1.86854 0.346979 0.173490 0.984836i \(-0.444496\pi\)
0.173490 + 0.984836i \(0.444496\pi\)
\(30\) 0 0
\(31\) 1.22996 0.220907 0.110454 0.993881i \(-0.464770\pi\)
0.110454 + 0.993881i \(0.464770\pi\)
\(32\) 0 0
\(33\) −1.24102 −0.216034
\(34\) 0 0
\(35\) 12.6482 2.13794
\(36\) 0 0
\(37\) −8.10815 −1.33297 −0.666485 0.745518i \(-0.732202\pi\)
−0.666485 + 0.745518i \(0.732202\pi\)
\(38\) 0 0
\(39\) 2.65892 0.425768
\(40\) 0 0
\(41\) 10.9225 1.70582 0.852908 0.522062i \(-0.174837\pi\)
0.852908 + 0.522062i \(0.174837\pi\)
\(42\) 0 0
\(43\) −2.97782 −0.454113 −0.227057 0.973882i \(-0.572910\pi\)
−0.227057 + 0.973882i \(0.572910\pi\)
\(44\) 0 0
\(45\) −4.13484 −0.616385
\(46\) 0 0
\(47\) −6.28504 −0.916768 −0.458384 0.888754i \(-0.651572\pi\)
−0.458384 + 0.888754i \(0.651572\pi\)
\(48\) 0 0
\(49\) 12.9419 1.84884
\(50\) 0 0
\(51\) 9.69975 1.35824
\(52\) 0 0
\(53\) 6.43768 0.884284 0.442142 0.896945i \(-0.354219\pi\)
0.442142 + 0.896945i \(0.354219\pi\)
\(54\) 0 0
\(55\) −2.83235 −0.381913
\(56\) 0 0
\(57\) −1.24102 −0.164377
\(58\) 0 0
\(59\) −7.40754 −0.964380 −0.482190 0.876067i \(-0.660159\pi\)
−0.482190 + 0.876067i \(0.660159\pi\)
\(60\) 0 0
\(61\) −9.83806 −1.25963 −0.629817 0.776743i \(-0.716870\pi\)
−0.629817 + 0.776743i \(0.716870\pi\)
\(62\) 0 0
\(63\) −6.51922 −0.821344
\(64\) 0 0
\(65\) 6.06836 0.752688
\(66\) 0 0
\(67\) 13.7013 1.67388 0.836941 0.547293i \(-0.184342\pi\)
0.836941 + 0.547293i \(0.184342\pi\)
\(68\) 0 0
\(69\) −4.72850 −0.569244
\(70\) 0 0
\(71\) 2.27693 0.270222 0.135111 0.990830i \(-0.456861\pi\)
0.135111 + 0.990830i \(0.456861\pi\)
\(72\) 0 0
\(73\) 2.77299 0.324554 0.162277 0.986745i \(-0.448116\pi\)
0.162277 + 0.986745i \(0.448116\pi\)
\(74\) 0 0
\(75\) 3.75060 0.433081
\(76\) 0 0
\(77\) −4.46564 −0.508906
\(78\) 0 0
\(79\) −15.9183 −1.79095 −0.895475 0.445112i \(-0.853164\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(80\) 0 0
\(81\) −2.48921 −0.276579
\(82\) 0 0
\(83\) −6.37820 −0.700098 −0.350049 0.936731i \(-0.613835\pi\)
−0.350049 + 0.936731i \(0.613835\pi\)
\(84\) 0 0
\(85\) 22.1374 2.40114
\(86\) 0 0
\(87\) 2.31890 0.248612
\(88\) 0 0
\(89\) 1.65892 0.175845 0.0879225 0.996127i \(-0.471977\pi\)
0.0879225 + 0.996127i \(0.471977\pi\)
\(90\) 0 0
\(91\) 9.56772 1.00297
\(92\) 0 0
\(93\) 1.52641 0.158281
\(94\) 0 0
\(95\) −2.83235 −0.290593
\(96\) 0 0
\(97\) −5.99423 −0.608622 −0.304311 0.952573i \(-0.598426\pi\)
−0.304311 + 0.952573i \(0.598426\pi\)
\(98\) 0 0
\(99\) 1.45986 0.146722
\(100\) 0 0
\(101\) 11.9627 1.19033 0.595165 0.803603i \(-0.297087\pi\)
0.595165 + 0.803603i \(0.297087\pi\)
\(102\) 0 0
\(103\) 0.753204 0.0742154 0.0371077 0.999311i \(-0.488186\pi\)
0.0371077 + 0.999311i \(0.488186\pi\)
\(104\) 0 0
\(105\) 15.6967 1.53184
\(106\) 0 0
\(107\) 1.05803 0.102284 0.0511420 0.998691i \(-0.483714\pi\)
0.0511420 + 0.998691i \(0.483714\pi\)
\(108\) 0 0
\(109\) 7.10240 0.680286 0.340143 0.940374i \(-0.389524\pi\)
0.340143 + 0.940374i \(0.389524\pi\)
\(110\) 0 0
\(111\) −10.0624 −0.955080
\(112\) 0 0
\(113\) −20.8752 −1.96378 −0.981888 0.189460i \(-0.939326\pi\)
−0.981888 + 0.189460i \(0.939326\pi\)
\(114\) 0 0
\(115\) −10.7917 −1.00633
\(116\) 0 0
\(117\) −3.12779 −0.289164
\(118\) 0 0
\(119\) 34.9031 3.19956
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 13.5551 1.22222
\(124\) 0 0
\(125\) −5.60187 −0.501046
\(126\) 0 0
\(127\) 19.9848 1.77336 0.886682 0.462380i \(-0.153004\pi\)
0.886682 + 0.462380i \(0.153004\pi\)
\(128\) 0 0
\(129\) −3.69554 −0.325374
\(130\) 0 0
\(131\) −14.5666 −1.27269 −0.636346 0.771404i \(-0.719555\pi\)
−0.636346 + 0.771404i \(0.719555\pi\)
\(132\) 0 0
\(133\) −4.46564 −0.387220
\(134\) 0 0
\(135\) −15.6764 −1.34921
\(136\) 0 0
\(137\) 2.89754 0.247554 0.123777 0.992310i \(-0.460499\pi\)
0.123777 + 0.992310i \(0.460499\pi\)
\(138\) 0 0
\(139\) −18.2399 −1.54709 −0.773544 0.633743i \(-0.781518\pi\)
−0.773544 + 0.633743i \(0.781518\pi\)
\(140\) 0 0
\(141\) −7.79988 −0.656868
\(142\) 0 0
\(143\) −2.14252 −0.179167
\(144\) 0 0
\(145\) 5.29235 0.439506
\(146\) 0 0
\(147\) 16.0612 1.32471
\(148\) 0 0
\(149\) −18.3109 −1.50009 −0.750044 0.661387i \(-0.769968\pi\)
−0.750044 + 0.661387i \(0.769968\pi\)
\(150\) 0 0
\(151\) 15.3574 1.24977 0.624884 0.780718i \(-0.285146\pi\)
0.624884 + 0.780718i \(0.285146\pi\)
\(152\) 0 0
\(153\) −11.4102 −0.922460
\(154\) 0 0
\(155\) 3.48367 0.279815
\(156\) 0 0
\(157\) −2.81022 −0.224280 −0.112140 0.993692i \(-0.535771\pi\)
−0.112140 + 0.993692i \(0.535771\pi\)
\(158\) 0 0
\(159\) 7.98931 0.633593
\(160\) 0 0
\(161\) −17.0148 −1.34095
\(162\) 0 0
\(163\) 6.73427 0.527469 0.263734 0.964595i \(-0.415046\pi\)
0.263734 + 0.964595i \(0.415046\pi\)
\(164\) 0 0
\(165\) −3.51500 −0.273643
\(166\) 0 0
\(167\) 8.27586 0.640405 0.320203 0.947349i \(-0.396249\pi\)
0.320203 + 0.947349i \(0.396249\pi\)
\(168\) 0 0
\(169\) −8.40960 −0.646892
\(170\) 0 0
\(171\) 1.45986 0.111639
\(172\) 0 0
\(173\) −4.83263 −0.367418 −0.183709 0.982981i \(-0.558810\pi\)
−0.183709 + 0.982981i \(0.558810\pi\)
\(174\) 0 0
\(175\) 13.4960 1.02020
\(176\) 0 0
\(177\) −9.19293 −0.690983
\(178\) 0 0
\(179\) 7.19714 0.537940 0.268970 0.963149i \(-0.413317\pi\)
0.268970 + 0.963149i \(0.413317\pi\)
\(180\) 0 0
\(181\) 2.75664 0.204899 0.102450 0.994738i \(-0.467332\pi\)
0.102450 + 0.994738i \(0.467332\pi\)
\(182\) 0 0
\(183\) −12.2093 −0.902534
\(184\) 0 0
\(185\) −22.9651 −1.68843
\(186\) 0 0
\(187\) −7.81594 −0.571558
\(188\) 0 0
\(189\) −24.7164 −1.79785
\(190\) 0 0
\(191\) −13.9906 −1.01232 −0.506161 0.862439i \(-0.668936\pi\)
−0.506161 + 0.862439i \(0.668936\pi\)
\(192\) 0 0
\(193\) −6.78688 −0.488530 −0.244265 0.969709i \(-0.578547\pi\)
−0.244265 + 0.969709i \(0.578547\pi\)
\(194\) 0 0
\(195\) 7.53098 0.539304
\(196\) 0 0
\(197\) −1.99861 −0.142395 −0.0711975 0.997462i \(-0.522682\pi\)
−0.0711975 + 0.997462i \(0.522682\pi\)
\(198\) 0 0
\(199\) −1.31094 −0.0929304 −0.0464652 0.998920i \(-0.514796\pi\)
−0.0464652 + 0.998920i \(0.514796\pi\)
\(200\) 0 0
\(201\) 17.0036 1.19934
\(202\) 0 0
\(203\) 8.34422 0.585649
\(204\) 0 0
\(205\) 30.9364 2.16069
\(206\) 0 0
\(207\) 5.56232 0.386608
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −7.31644 −0.503685 −0.251842 0.967768i \(-0.581036\pi\)
−0.251842 + 0.967768i \(0.581036\pi\)
\(212\) 0 0
\(213\) 2.82572 0.193615
\(214\) 0 0
\(215\) −8.43421 −0.575208
\(216\) 0 0
\(217\) 5.49255 0.372859
\(218\) 0 0
\(219\) 3.44134 0.232544
\(220\) 0 0
\(221\) 16.7458 1.12645
\(222\) 0 0
\(223\) 7.69849 0.515529 0.257765 0.966208i \(-0.417014\pi\)
0.257765 + 0.966208i \(0.417014\pi\)
\(224\) 0 0
\(225\) −4.41197 −0.294131
\(226\) 0 0
\(227\) −9.61894 −0.638431 −0.319216 0.947682i \(-0.603419\pi\)
−0.319216 + 0.947682i \(0.603419\pi\)
\(228\) 0 0
\(229\) 12.1361 0.801976 0.400988 0.916083i \(-0.368667\pi\)
0.400988 + 0.916083i \(0.368667\pi\)
\(230\) 0 0
\(231\) −5.54196 −0.364634
\(232\) 0 0
\(233\) −13.5356 −0.886745 −0.443372 0.896338i \(-0.646218\pi\)
−0.443372 + 0.896338i \(0.646218\pi\)
\(234\) 0 0
\(235\) −17.8014 −1.16124
\(236\) 0 0
\(237\) −19.7550 −1.28322
\(238\) 0 0
\(239\) −18.0084 −1.16487 −0.582433 0.812879i \(-0.697899\pi\)
−0.582433 + 0.812879i \(0.697899\pi\)
\(240\) 0 0
\(241\) 24.4331 1.57387 0.786936 0.617034i \(-0.211666\pi\)
0.786936 + 0.617034i \(0.211666\pi\)
\(242\) 0 0
\(243\) 13.5152 0.867001
\(244\) 0 0
\(245\) 36.6560 2.34186
\(246\) 0 0
\(247\) −2.14252 −0.136325
\(248\) 0 0
\(249\) −7.91549 −0.501624
\(250\) 0 0
\(251\) 27.0408 1.70680 0.853400 0.521256i \(-0.174536\pi\)
0.853400 + 0.521256i \(0.174536\pi\)
\(252\) 0 0
\(253\) 3.81016 0.239543
\(254\) 0 0
\(255\) 27.4731 1.72043
\(256\) 0 0
\(257\) −26.1689 −1.63237 −0.816184 0.577792i \(-0.803915\pi\)
−0.816184 + 0.577792i \(0.803915\pi\)
\(258\) 0 0
\(259\) −36.2080 −2.24986
\(260\) 0 0
\(261\) −2.72781 −0.168847
\(262\) 0 0
\(263\) −12.1061 −0.746494 −0.373247 0.927732i \(-0.621756\pi\)
−0.373247 + 0.927732i \(0.621756\pi\)
\(264\) 0 0
\(265\) 18.2337 1.12009
\(266\) 0 0
\(267\) 2.05875 0.125994
\(268\) 0 0
\(269\) 27.5041 1.67696 0.838479 0.544934i \(-0.183445\pi\)
0.838479 + 0.544934i \(0.183445\pi\)
\(270\) 0 0
\(271\) 4.45270 0.270482 0.135241 0.990813i \(-0.456819\pi\)
0.135241 + 0.990813i \(0.456819\pi\)
\(272\) 0 0
\(273\) 11.8738 0.718633
\(274\) 0 0
\(275\) −3.02218 −0.182244
\(276\) 0 0
\(277\) 14.0559 0.844538 0.422269 0.906471i \(-0.361234\pi\)
0.422269 + 0.906471i \(0.361234\pi\)
\(278\) 0 0
\(279\) −1.79557 −0.107498
\(280\) 0 0
\(281\) −12.0282 −0.717543 −0.358772 0.933425i \(-0.616804\pi\)
−0.358772 + 0.933425i \(0.616804\pi\)
\(282\) 0 0
\(283\) 19.1496 1.13833 0.569164 0.822224i \(-0.307267\pi\)
0.569164 + 0.822224i \(0.307267\pi\)
\(284\) 0 0
\(285\) −3.51500 −0.208211
\(286\) 0 0
\(287\) 48.7761 2.87916
\(288\) 0 0
\(289\) 44.0889 2.59346
\(290\) 0 0
\(291\) −7.43897 −0.436080
\(292\) 0 0
\(293\) 17.3732 1.01495 0.507476 0.861666i \(-0.330579\pi\)
0.507476 + 0.861666i \(0.330579\pi\)
\(294\) 0 0
\(295\) −20.9807 −1.22155
\(296\) 0 0
\(297\) 5.53479 0.321161
\(298\) 0 0
\(299\) −8.16336 −0.472099
\(300\) 0 0
\(301\) −13.2979 −0.766476
\(302\) 0 0
\(303\) 14.8459 0.852877
\(304\) 0 0
\(305\) −27.8648 −1.59553
\(306\) 0 0
\(307\) 25.4419 1.45205 0.726023 0.687670i \(-0.241366\pi\)
0.726023 + 0.687670i \(0.241366\pi\)
\(308\) 0 0
\(309\) 0.934744 0.0531757
\(310\) 0 0
\(311\) 6.22913 0.353222 0.176611 0.984281i \(-0.443487\pi\)
0.176611 + 0.984281i \(0.443487\pi\)
\(312\) 0 0
\(313\) 23.4024 1.32278 0.661392 0.750041i \(-0.269966\pi\)
0.661392 + 0.750041i \(0.269966\pi\)
\(314\) 0 0
\(315\) −18.4647 −1.04037
\(316\) 0 0
\(317\) −13.9440 −0.783171 −0.391585 0.920142i \(-0.628073\pi\)
−0.391585 + 0.920142i \(0.628073\pi\)
\(318\) 0 0
\(319\) −1.86854 −0.104618
\(320\) 0 0
\(321\) 1.31304 0.0732870
\(322\) 0 0
\(323\) −7.81594 −0.434890
\(324\) 0 0
\(325\) 6.47509 0.359173
\(326\) 0 0
\(327\) 8.81424 0.487428
\(328\) 0 0
\(329\) −28.0667 −1.54737
\(330\) 0 0
\(331\) −8.96372 −0.492691 −0.246345 0.969182i \(-0.579230\pi\)
−0.246345 + 0.969182i \(0.579230\pi\)
\(332\) 0 0
\(333\) 11.8368 0.648652
\(334\) 0 0
\(335\) 38.8069 2.12024
\(336\) 0 0
\(337\) 7.52217 0.409759 0.204879 0.978787i \(-0.434320\pi\)
0.204879 + 0.978787i \(0.434320\pi\)
\(338\) 0 0
\(339\) −25.9066 −1.40706
\(340\) 0 0
\(341\) −1.22996 −0.0666061
\(342\) 0 0
\(343\) 26.5344 1.43272
\(344\) 0 0
\(345\) −13.3927 −0.721041
\(346\) 0 0
\(347\) 7.09789 0.381035 0.190517 0.981684i \(-0.438983\pi\)
0.190517 + 0.981684i \(0.438983\pi\)
\(348\) 0 0
\(349\) −30.8496 −1.65134 −0.825670 0.564153i \(-0.809203\pi\)
−0.825670 + 0.564153i \(0.809203\pi\)
\(350\) 0 0
\(351\) −11.8584 −0.632955
\(352\) 0 0
\(353\) 32.7894 1.74521 0.872603 0.488430i \(-0.162430\pi\)
0.872603 + 0.488430i \(0.162430\pi\)
\(354\) 0 0
\(355\) 6.44906 0.342281
\(356\) 0 0
\(357\) 43.3156 2.29250
\(358\) 0 0
\(359\) −9.72206 −0.513111 −0.256555 0.966530i \(-0.582588\pi\)
−0.256555 + 0.966530i \(0.582588\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.24102 0.0651368
\(364\) 0 0
\(365\) 7.85407 0.411101
\(366\) 0 0
\(367\) −13.3115 −0.694856 −0.347428 0.937707i \(-0.612945\pi\)
−0.347428 + 0.937707i \(0.612945\pi\)
\(368\) 0 0
\(369\) −15.9454 −0.830086
\(370\) 0 0
\(371\) 28.7483 1.49254
\(372\) 0 0
\(373\) −6.88925 −0.356712 −0.178356 0.983966i \(-0.557078\pi\)
−0.178356 + 0.983966i \(0.557078\pi\)
\(374\) 0 0
\(375\) −6.95204 −0.359002
\(376\) 0 0
\(377\) 4.00339 0.206185
\(378\) 0 0
\(379\) −15.6685 −0.804837 −0.402418 0.915456i \(-0.631830\pi\)
−0.402418 + 0.915456i \(0.631830\pi\)
\(380\) 0 0
\(381\) 24.8016 1.27062
\(382\) 0 0
\(383\) 38.2914 1.95660 0.978299 0.207198i \(-0.0664346\pi\)
0.978299 + 0.207198i \(0.0664346\pi\)
\(384\) 0 0
\(385\) −12.6482 −0.644613
\(386\) 0 0
\(387\) 4.34721 0.220981
\(388\) 0 0
\(389\) 13.4533 0.682108 0.341054 0.940044i \(-0.389216\pi\)
0.341054 + 0.940044i \(0.389216\pi\)
\(390\) 0 0
\(391\) −29.7800 −1.50604
\(392\) 0 0
\(393\) −18.0775 −0.911889
\(394\) 0 0
\(395\) −45.0862 −2.26853
\(396\) 0 0
\(397\) −30.5925 −1.53539 −0.767697 0.640813i \(-0.778597\pi\)
−0.767697 + 0.640813i \(0.778597\pi\)
\(398\) 0 0
\(399\) −5.54196 −0.277445
\(400\) 0 0
\(401\) −15.2750 −0.762798 −0.381399 0.924411i \(-0.624558\pi\)
−0.381399 + 0.924411i \(0.624558\pi\)
\(402\) 0 0
\(403\) 2.63522 0.131270
\(404\) 0 0
\(405\) −7.05030 −0.350332
\(406\) 0 0
\(407\) 8.10815 0.401906
\(408\) 0 0
\(409\) −15.3255 −0.757799 −0.378900 0.925438i \(-0.623697\pi\)
−0.378900 + 0.925438i \(0.623697\pi\)
\(410\) 0 0
\(411\) 3.59592 0.177374
\(412\) 0 0
\(413\) −33.0794 −1.62773
\(414\) 0 0
\(415\) −18.0653 −0.886789
\(416\) 0 0
\(417\) −22.6361 −1.10850
\(418\) 0 0
\(419\) −5.42923 −0.265235 −0.132618 0.991167i \(-0.542338\pi\)
−0.132618 + 0.991167i \(0.542338\pi\)
\(420\) 0 0
\(421\) −13.4074 −0.653437 −0.326719 0.945122i \(-0.605943\pi\)
−0.326719 + 0.945122i \(0.605943\pi\)
\(422\) 0 0
\(423\) 9.17531 0.446119
\(424\) 0 0
\(425\) 23.6212 1.14580
\(426\) 0 0
\(427\) −43.9332 −2.12608
\(428\) 0 0
\(429\) −2.65892 −0.128374
\(430\) 0 0
\(431\) 9.96718 0.480102 0.240051 0.970760i \(-0.422836\pi\)
0.240051 + 0.970760i \(0.422836\pi\)
\(432\) 0 0
\(433\) 23.3082 1.12012 0.560061 0.828451i \(-0.310778\pi\)
0.560061 + 0.828451i \(0.310778\pi\)
\(434\) 0 0
\(435\) 6.56793 0.314908
\(436\) 0 0
\(437\) 3.81016 0.182265
\(438\) 0 0
\(439\) 11.5107 0.549374 0.274687 0.961534i \(-0.411426\pi\)
0.274687 + 0.961534i \(0.411426\pi\)
\(440\) 0 0
\(441\) −18.8934 −0.899686
\(442\) 0 0
\(443\) −31.2930 −1.48678 −0.743389 0.668860i \(-0.766783\pi\)
−0.743389 + 0.668860i \(0.766783\pi\)
\(444\) 0 0
\(445\) 4.69863 0.222736
\(446\) 0 0
\(447\) −22.7243 −1.07482
\(448\) 0 0
\(449\) −34.4750 −1.62697 −0.813487 0.581583i \(-0.802434\pi\)
−0.813487 + 0.581583i \(0.802434\pi\)
\(450\) 0 0
\(451\) −10.9225 −0.514323
\(452\) 0 0
\(453\) 19.0589 0.895465
\(454\) 0 0
\(455\) 27.0991 1.27043
\(456\) 0 0
\(457\) 29.2398 1.36778 0.683891 0.729584i \(-0.260286\pi\)
0.683891 + 0.729584i \(0.260286\pi\)
\(458\) 0 0
\(459\) −43.2596 −2.01918
\(460\) 0 0
\(461\) 9.66554 0.450169 0.225085 0.974339i \(-0.427734\pi\)
0.225085 + 0.974339i \(0.427734\pi\)
\(462\) 0 0
\(463\) 7.41991 0.344833 0.172416 0.985024i \(-0.444843\pi\)
0.172416 + 0.985024i \(0.444843\pi\)
\(464\) 0 0
\(465\) 4.32332 0.200489
\(466\) 0 0
\(467\) 3.09212 0.143086 0.0715430 0.997438i \(-0.477208\pi\)
0.0715430 + 0.997438i \(0.477208\pi\)
\(468\) 0 0
\(469\) 61.1851 2.82526
\(470\) 0 0
\(471\) −3.48755 −0.160698
\(472\) 0 0
\(473\) 2.97782 0.136920
\(474\) 0 0
\(475\) −3.02218 −0.138667
\(476\) 0 0
\(477\) −9.39813 −0.430311
\(478\) 0 0
\(479\) −30.3725 −1.38776 −0.693879 0.720092i \(-0.744100\pi\)
−0.693879 + 0.720092i \(0.744100\pi\)
\(480\) 0 0
\(481\) −17.3719 −0.792090
\(482\) 0 0
\(483\) −21.1158 −0.960800
\(484\) 0 0
\(485\) −16.9777 −0.770919
\(486\) 0 0
\(487\) 36.6521 1.66086 0.830432 0.557120i \(-0.188094\pi\)
0.830432 + 0.557120i \(0.188094\pi\)
\(488\) 0 0
\(489\) 8.35738 0.377934
\(490\) 0 0
\(491\) −14.8522 −0.670270 −0.335135 0.942170i \(-0.608782\pi\)
−0.335135 + 0.942170i \(0.608782\pi\)
\(492\) 0 0
\(493\) 14.6044 0.657749
\(494\) 0 0
\(495\) 4.13484 0.185847
\(496\) 0 0
\(497\) 10.1680 0.456095
\(498\) 0 0
\(499\) −27.9369 −1.25063 −0.625314 0.780374i \(-0.715029\pi\)
−0.625314 + 0.780374i \(0.715029\pi\)
\(500\) 0 0
\(501\) 10.2705 0.458853
\(502\) 0 0
\(503\) −8.65406 −0.385865 −0.192933 0.981212i \(-0.561800\pi\)
−0.192933 + 0.981212i \(0.561800\pi\)
\(504\) 0 0
\(505\) 33.8824 1.50775
\(506\) 0 0
\(507\) −10.4365 −0.463501
\(508\) 0 0
\(509\) −26.6016 −1.17910 −0.589548 0.807733i \(-0.700694\pi\)
−0.589548 + 0.807733i \(0.700694\pi\)
\(510\) 0 0
\(511\) 12.3832 0.547799
\(512\) 0 0
\(513\) 5.53479 0.244367
\(514\) 0 0
\(515\) 2.13334 0.0940060
\(516\) 0 0
\(517\) 6.28504 0.276416
\(518\) 0 0
\(519\) −5.99740 −0.263257
\(520\) 0 0
\(521\) 4.17685 0.182991 0.0914955 0.995805i \(-0.470835\pi\)
0.0914955 + 0.995805i \(0.470835\pi\)
\(522\) 0 0
\(523\) −3.07716 −0.134555 −0.0672773 0.997734i \(-0.521431\pi\)
−0.0672773 + 0.997734i \(0.521431\pi\)
\(524\) 0 0
\(525\) 16.7488 0.730977
\(526\) 0 0
\(527\) 9.61329 0.418762
\(528\) 0 0
\(529\) −8.48265 −0.368811
\(530\) 0 0
\(531\) 10.8140 0.469288
\(532\) 0 0
\(533\) 23.4018 1.01364
\(534\) 0 0
\(535\) 2.99672 0.129560
\(536\) 0 0
\(537\) 8.93182 0.385436
\(538\) 0 0
\(539\) −12.9419 −0.557447
\(540\) 0 0
\(541\) −39.9766 −1.71873 −0.859365 0.511363i \(-0.829141\pi\)
−0.859365 + 0.511363i \(0.829141\pi\)
\(542\) 0 0
\(543\) 3.42105 0.146811
\(544\) 0 0
\(545\) 20.1164 0.861694
\(546\) 0 0
\(547\) −13.7551 −0.588125 −0.294063 0.955786i \(-0.595007\pi\)
−0.294063 + 0.955786i \(0.595007\pi\)
\(548\) 0 0
\(549\) 14.3622 0.612965
\(550\) 0 0
\(551\) −1.86854 −0.0796025
\(552\) 0 0
\(553\) −71.0854 −3.02286
\(554\) 0 0
\(555\) −28.5002 −1.20976
\(556\) 0 0
\(557\) 3.37375 0.142950 0.0714751 0.997442i \(-0.477229\pi\)
0.0714751 + 0.997442i \(0.477229\pi\)
\(558\) 0 0
\(559\) −6.38004 −0.269847
\(560\) 0 0
\(561\) −9.69975 −0.409524
\(562\) 0 0
\(563\) −32.4119 −1.36600 −0.682999 0.730419i \(-0.739325\pi\)
−0.682999 + 0.730419i \(0.739325\pi\)
\(564\) 0 0
\(565\) −59.1259 −2.48744
\(566\) 0 0
\(567\) −11.1159 −0.466824
\(568\) 0 0
\(569\) 29.7569 1.24748 0.623738 0.781633i \(-0.285613\pi\)
0.623738 + 0.781633i \(0.285613\pi\)
\(570\) 0 0
\(571\) −11.9446 −0.499865 −0.249933 0.968263i \(-0.580409\pi\)
−0.249933 + 0.968263i \(0.580409\pi\)
\(572\) 0 0
\(573\) −17.3626 −0.725334
\(574\) 0 0
\(575\) −11.5150 −0.480209
\(576\) 0 0
\(577\) −4.05443 −0.168788 −0.0843941 0.996432i \(-0.526895\pi\)
−0.0843941 + 0.996432i \(0.526895\pi\)
\(578\) 0 0
\(579\) −8.42266 −0.350034
\(580\) 0 0
\(581\) −28.4827 −1.18166
\(582\) 0 0
\(583\) −6.43768 −0.266622
\(584\) 0 0
\(585\) −8.85898 −0.366274
\(586\) 0 0
\(587\) −17.7020 −0.730640 −0.365320 0.930882i \(-0.619040\pi\)
−0.365320 + 0.930882i \(0.619040\pi\)
\(588\) 0 0
\(589\) −1.22996 −0.0506796
\(590\) 0 0
\(591\) −2.48032 −0.102027
\(592\) 0 0
\(593\) 9.42897 0.387201 0.193601 0.981080i \(-0.437983\pi\)
0.193601 + 0.981080i \(0.437983\pi\)
\(594\) 0 0
\(595\) 98.8577 4.05277
\(596\) 0 0
\(597\) −1.62691 −0.0665851
\(598\) 0 0
\(599\) −2.75320 −0.112493 −0.0562464 0.998417i \(-0.517913\pi\)
−0.0562464 + 0.998417i \(0.517913\pi\)
\(600\) 0 0
\(601\) −45.2029 −1.84386 −0.921932 0.387351i \(-0.873390\pi\)
−0.921932 + 0.387351i \(0.873390\pi\)
\(602\) 0 0
\(603\) −20.0020 −0.814546
\(604\) 0 0
\(605\) 2.83235 0.115151
\(606\) 0 0
\(607\) 38.2093 1.55087 0.775433 0.631430i \(-0.217532\pi\)
0.775433 + 0.631430i \(0.217532\pi\)
\(608\) 0 0
\(609\) 10.3554 0.419621
\(610\) 0 0
\(611\) −13.4658 −0.544770
\(612\) 0 0
\(613\) −2.78003 −0.112284 −0.0561421 0.998423i \(-0.517880\pi\)
−0.0561421 + 0.998423i \(0.517880\pi\)
\(614\) 0 0
\(615\) 38.3928 1.54815
\(616\) 0 0
\(617\) 21.4168 0.862207 0.431104 0.902302i \(-0.358124\pi\)
0.431104 + 0.902302i \(0.358124\pi\)
\(618\) 0 0
\(619\) −5.63033 −0.226302 −0.113151 0.993578i \(-0.536094\pi\)
−0.113151 + 0.993578i \(0.536094\pi\)
\(620\) 0 0
\(621\) 21.0885 0.846251
\(622\) 0 0
\(623\) 7.40813 0.296800
\(624\) 0 0
\(625\) −30.9773 −1.23909
\(626\) 0 0
\(627\) 1.24102 0.0495617
\(628\) 0 0
\(629\) −63.3728 −2.52684
\(630\) 0 0
\(631\) 47.0442 1.87280 0.936400 0.350935i \(-0.114136\pi\)
0.936400 + 0.350935i \(0.114136\pi\)
\(632\) 0 0
\(633\) −9.07987 −0.360893
\(634\) 0 0
\(635\) 56.6039 2.24626
\(636\) 0 0
\(637\) 27.7283 1.09864
\(638\) 0 0
\(639\) −3.32401 −0.131496
\(640\) 0 0
\(641\) −4.26926 −0.168626 −0.0843128 0.996439i \(-0.526870\pi\)
−0.0843128 + 0.996439i \(0.526870\pi\)
\(642\) 0 0
\(643\) −9.72241 −0.383414 −0.191707 0.981452i \(-0.561402\pi\)
−0.191707 + 0.981452i \(0.561402\pi\)
\(644\) 0 0
\(645\) −10.4670 −0.412139
\(646\) 0 0
\(647\) −30.8738 −1.21377 −0.606887 0.794788i \(-0.707582\pi\)
−0.606887 + 0.794788i \(0.707582\pi\)
\(648\) 0 0
\(649\) 7.40754 0.290772
\(650\) 0 0
\(651\) 6.81638 0.267155
\(652\) 0 0
\(653\) −2.21765 −0.0867832 −0.0433916 0.999058i \(-0.513816\pi\)
−0.0433916 + 0.999058i \(0.513816\pi\)
\(654\) 0 0
\(655\) −41.2577 −1.61207
\(656\) 0 0
\(657\) −4.04819 −0.157935
\(658\) 0 0
\(659\) −8.82521 −0.343781 −0.171891 0.985116i \(-0.554988\pi\)
−0.171891 + 0.985116i \(0.554988\pi\)
\(660\) 0 0
\(661\) 47.0328 1.82937 0.914683 0.404173i \(-0.132441\pi\)
0.914683 + 0.404173i \(0.132441\pi\)
\(662\) 0 0
\(663\) 20.7819 0.807104
\(664\) 0 0
\(665\) −12.6482 −0.490477
\(666\) 0 0
\(667\) −7.11944 −0.275666
\(668\) 0 0
\(669\) 9.55400 0.369379
\(670\) 0 0
\(671\) 9.83806 0.379794
\(672\) 0 0
\(673\) 23.0771 0.889556 0.444778 0.895641i \(-0.353283\pi\)
0.444778 + 0.895641i \(0.353283\pi\)
\(674\) 0 0
\(675\) −16.7271 −0.643828
\(676\) 0 0
\(677\) 6.34225 0.243753 0.121876 0.992545i \(-0.461109\pi\)
0.121876 + 0.992545i \(0.461109\pi\)
\(678\) 0 0
\(679\) −26.7680 −1.02726
\(680\) 0 0
\(681\) −11.9373 −0.457439
\(682\) 0 0
\(683\) −3.29657 −0.126140 −0.0630698 0.998009i \(-0.520089\pi\)
−0.0630698 + 0.998009i \(0.520089\pi\)
\(684\) 0 0
\(685\) 8.20685 0.313568
\(686\) 0 0
\(687\) 15.0612 0.574620
\(688\) 0 0
\(689\) 13.7929 0.525467
\(690\) 0 0
\(691\) 15.7705 0.599937 0.299969 0.953949i \(-0.403024\pi\)
0.299969 + 0.953949i \(0.403024\pi\)
\(692\) 0 0
\(693\) 6.51922 0.247645
\(694\) 0 0
\(695\) −51.6617 −1.95964
\(696\) 0 0
\(697\) 85.3700 3.23362
\(698\) 0 0
\(699\) −16.7979 −0.635357
\(700\) 0 0
\(701\) −18.1060 −0.683855 −0.341927 0.939726i \(-0.611080\pi\)
−0.341927 + 0.939726i \(0.611080\pi\)
\(702\) 0 0
\(703\) 8.10815 0.305805
\(704\) 0 0
\(705\) −22.0920 −0.832031
\(706\) 0 0
\(707\) 53.4209 2.00910
\(708\) 0 0
\(709\) −25.6798 −0.964426 −0.482213 0.876054i \(-0.660167\pi\)
−0.482213 + 0.876054i \(0.660167\pi\)
\(710\) 0 0
\(711\) 23.2386 0.871514
\(712\) 0 0
\(713\) −4.68635 −0.175505
\(714\) 0 0
\(715\) −6.06836 −0.226944
\(716\) 0 0
\(717\) −22.3488 −0.834632
\(718\) 0 0
\(719\) 27.8775 1.03966 0.519828 0.854271i \(-0.325996\pi\)
0.519828 + 0.854271i \(0.325996\pi\)
\(720\) 0 0
\(721\) 3.36354 0.125265
\(722\) 0 0
\(723\) 30.3220 1.12769
\(724\) 0 0
\(725\) 5.64707 0.209727
\(726\) 0 0
\(727\) 41.7941 1.55006 0.775029 0.631925i \(-0.217735\pi\)
0.775029 + 0.631925i \(0.217735\pi\)
\(728\) 0 0
\(729\) 24.2403 0.897789
\(730\) 0 0
\(731\) −23.2744 −0.860836
\(732\) 0 0
\(733\) 9.36881 0.346045 0.173022 0.984918i \(-0.444647\pi\)
0.173022 + 0.984918i \(0.444647\pi\)
\(734\) 0 0
\(735\) 45.4909 1.67796
\(736\) 0 0
\(737\) −13.7013 −0.504694
\(738\) 0 0
\(739\) −34.2879 −1.26130 −0.630650 0.776067i \(-0.717212\pi\)
−0.630650 + 0.776067i \(0.717212\pi\)
\(740\) 0 0
\(741\) −2.65892 −0.0976778
\(742\) 0 0
\(743\) −25.3459 −0.929851 −0.464925 0.885350i \(-0.653919\pi\)
−0.464925 + 0.885350i \(0.653919\pi\)
\(744\) 0 0
\(745\) −51.8629 −1.90011
\(746\) 0 0
\(747\) 9.31130 0.340683
\(748\) 0 0
\(749\) 4.72480 0.172640
\(750\) 0 0
\(751\) −36.2198 −1.32168 −0.660840 0.750527i \(-0.729800\pi\)
−0.660840 + 0.750527i \(0.729800\pi\)
\(752\) 0 0
\(753\) 33.5583 1.22293
\(754\) 0 0
\(755\) 43.4975 1.58304
\(756\) 0 0
\(757\) −33.7822 −1.22784 −0.613918 0.789370i \(-0.710407\pi\)
−0.613918 + 0.789370i \(0.710407\pi\)
\(758\) 0 0
\(759\) 4.72850 0.171634
\(760\) 0 0
\(761\) −24.9369 −0.903963 −0.451981 0.892027i \(-0.649283\pi\)
−0.451981 + 0.892027i \(0.649283\pi\)
\(762\) 0 0
\(763\) 31.7167 1.14822
\(764\) 0 0
\(765\) −32.3176 −1.16845
\(766\) 0 0
\(767\) −15.8708 −0.573062
\(768\) 0 0
\(769\) 13.4936 0.486592 0.243296 0.969952i \(-0.421771\pi\)
0.243296 + 0.969952i \(0.421771\pi\)
\(770\) 0 0
\(771\) −32.4761 −1.16960
\(772\) 0 0
\(773\) −11.4133 −0.410509 −0.205254 0.978709i \(-0.565802\pi\)
−0.205254 + 0.978709i \(0.565802\pi\)
\(774\) 0 0
\(775\) 3.71716 0.133524
\(776\) 0 0
\(777\) −44.9350 −1.61203
\(778\) 0 0
\(779\) −10.9225 −0.391341
\(780\) 0 0
\(781\) −2.27693 −0.0814751
\(782\) 0 0
\(783\) −10.3420 −0.369592
\(784\) 0 0
\(785\) −7.95952 −0.284087
\(786\) 0 0
\(787\) −20.7206 −0.738609 −0.369304 0.929308i \(-0.620404\pi\)
−0.369304 + 0.929308i \(0.620404\pi\)
\(788\) 0 0
\(789\) −15.0239 −0.534866
\(790\) 0 0
\(791\) −93.2212 −3.31456
\(792\) 0 0
\(793\) −21.0783 −0.748511
\(794\) 0 0
\(795\) 22.6285 0.802549
\(796\) 0 0
\(797\) 22.4629 0.795675 0.397838 0.917456i \(-0.369761\pi\)
0.397838 + 0.917456i \(0.369761\pi\)
\(798\) 0 0
\(799\) −49.1235 −1.73786
\(800\) 0 0
\(801\) −2.42179 −0.0855699
\(802\) 0 0
\(803\) −2.77299 −0.0978567
\(804\) 0 0
\(805\) −48.1918 −1.69854
\(806\) 0 0
\(807\) 34.1333 1.20155
\(808\) 0 0
\(809\) 1.49572 0.0525866 0.0262933 0.999654i \(-0.491630\pi\)
0.0262933 + 0.999654i \(0.491630\pi\)
\(810\) 0 0
\(811\) 47.8558 1.68044 0.840222 0.542243i \(-0.182425\pi\)
0.840222 + 0.542243i \(0.182425\pi\)
\(812\) 0 0
\(813\) 5.52590 0.193802
\(814\) 0 0
\(815\) 19.0738 0.668126
\(816\) 0 0
\(817\) 2.97782 0.104181
\(818\) 0 0
\(819\) −13.9676 −0.488066
\(820\) 0 0
\(821\) −43.4124 −1.51510 −0.757552 0.652774i \(-0.773605\pi\)
−0.757552 + 0.652774i \(0.773605\pi\)
\(822\) 0 0
\(823\) −11.7858 −0.410826 −0.205413 0.978675i \(-0.565854\pi\)
−0.205413 + 0.978675i \(0.565854\pi\)
\(824\) 0 0
\(825\) −3.75060 −0.130579
\(826\) 0 0
\(827\) −42.6416 −1.48279 −0.741397 0.671067i \(-0.765836\pi\)
−0.741397 + 0.671067i \(0.765836\pi\)
\(828\) 0 0
\(829\) −56.4118 −1.95926 −0.979631 0.200808i \(-0.935643\pi\)
−0.979631 + 0.200808i \(0.935643\pi\)
\(830\) 0 0
\(831\) 17.4437 0.605115
\(832\) 0 0
\(833\) 101.153 3.50475
\(834\) 0 0
\(835\) 23.4401 0.811178
\(836\) 0 0
\(837\) −6.80757 −0.235304
\(838\) 0 0
\(839\) 27.6179 0.953474 0.476737 0.879046i \(-0.341819\pi\)
0.476737 + 0.879046i \(0.341819\pi\)
\(840\) 0 0
\(841\) −25.5086 −0.879605
\(842\) 0 0
\(843\) −14.9273 −0.514123
\(844\) 0 0
\(845\) −23.8189 −0.819395
\(846\) 0 0
\(847\) 4.46564 0.153441
\(848\) 0 0
\(849\) 23.7651 0.815617
\(850\) 0 0
\(851\) 30.8934 1.05901
\(852\) 0 0
\(853\) 22.2378 0.761409 0.380705 0.924697i \(-0.375681\pi\)
0.380705 + 0.924697i \(0.375681\pi\)
\(854\) 0 0
\(855\) 4.13484 0.141408
\(856\) 0 0
\(857\) −15.2312 −0.520289 −0.260144 0.965570i \(-0.583770\pi\)
−0.260144 + 0.965570i \(0.583770\pi\)
\(858\) 0 0
\(859\) −40.9212 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(860\) 0 0
\(861\) 60.5323 2.06293
\(862\) 0 0
\(863\) 13.2896 0.452384 0.226192 0.974083i \(-0.427372\pi\)
0.226192 + 0.974083i \(0.427372\pi\)
\(864\) 0 0
\(865\) −13.6877 −0.465395
\(866\) 0 0
\(867\) 54.7153 1.85823
\(868\) 0 0
\(869\) 15.9183 0.539992
\(870\) 0 0
\(871\) 29.3554 0.994669
\(872\) 0 0
\(873\) 8.75075 0.296168
\(874\) 0 0
\(875\) −25.0159 −0.845692
\(876\) 0 0
\(877\) 17.3501 0.585872 0.292936 0.956132i \(-0.405368\pi\)
0.292936 + 0.956132i \(0.405368\pi\)
\(878\) 0 0
\(879\) 21.5605 0.727217
\(880\) 0 0
\(881\) −24.6924 −0.831907 −0.415954 0.909386i \(-0.636552\pi\)
−0.415954 + 0.909386i \(0.636552\pi\)
\(882\) 0 0
\(883\) −14.7443 −0.496185 −0.248092 0.968736i \(-0.579804\pi\)
−0.248092 + 0.968736i \(0.579804\pi\)
\(884\) 0 0
\(885\) −26.0376 −0.875243
\(886\) 0 0
\(887\) 22.0908 0.741737 0.370869 0.928685i \(-0.379060\pi\)
0.370869 + 0.928685i \(0.379060\pi\)
\(888\) 0 0
\(889\) 89.2448 2.99318
\(890\) 0 0
\(891\) 2.48921 0.0833917
\(892\) 0 0
\(893\) 6.28504 0.210321
\(894\) 0 0
\(895\) 20.3848 0.681389
\(896\) 0 0
\(897\) −10.1309 −0.338261
\(898\) 0 0
\(899\) 2.29823 0.0766503
\(900\) 0 0
\(901\) 50.3165 1.67629
\(902\) 0 0
\(903\) −16.5029 −0.549183
\(904\) 0 0
\(905\) 7.80775 0.259539
\(906\) 0 0
\(907\) 37.1803 1.23455 0.617275 0.786747i \(-0.288236\pi\)
0.617275 + 0.786747i \(0.288236\pi\)
\(908\) 0 0
\(909\) −17.4639 −0.579240
\(910\) 0 0
\(911\) −6.79211 −0.225033 −0.112516 0.993650i \(-0.535891\pi\)
−0.112516 + 0.993650i \(0.535891\pi\)
\(912\) 0 0
\(913\) 6.37820 0.211088
\(914\) 0 0
\(915\) −34.5808 −1.14321
\(916\) 0 0
\(917\) −65.0492 −2.14811
\(918\) 0 0
\(919\) 33.7646 1.11379 0.556896 0.830582i \(-0.311992\pi\)
0.556896 + 0.830582i \(0.311992\pi\)
\(920\) 0 0
\(921\) 31.5740 1.04040
\(922\) 0 0
\(923\) 4.87838 0.160574
\(924\) 0 0
\(925\) −24.5043 −0.805696
\(926\) 0 0
\(927\) −1.09958 −0.0361148
\(928\) 0 0
\(929\) 4.77431 0.156640 0.0783200 0.996928i \(-0.475044\pi\)
0.0783200 + 0.996928i \(0.475044\pi\)
\(930\) 0 0
\(931\) −12.9419 −0.424154
\(932\) 0 0
\(933\) 7.73050 0.253085
\(934\) 0 0
\(935\) −22.1374 −0.723972
\(936\) 0 0
\(937\) −52.6610 −1.72036 −0.860180 0.509990i \(-0.829649\pi\)
−0.860180 + 0.509990i \(0.829649\pi\)
\(938\) 0 0
\(939\) 29.0429 0.947781
\(940\) 0 0
\(941\) −26.6960 −0.870264 −0.435132 0.900367i \(-0.643298\pi\)
−0.435132 + 0.900367i \(0.643298\pi\)
\(942\) 0 0
\(943\) −41.6167 −1.35523
\(944\) 0 0
\(945\) −70.0053 −2.27727
\(946\) 0 0
\(947\) −49.0764 −1.59477 −0.797384 0.603472i \(-0.793783\pi\)
−0.797384 + 0.603472i \(0.793783\pi\)
\(948\) 0 0
\(949\) 5.94119 0.192859
\(950\) 0 0
\(951\) −17.3048 −0.561146
\(952\) 0 0
\(953\) −47.8969 −1.55153 −0.775767 0.631020i \(-0.782637\pi\)
−0.775767 + 0.631020i \(0.782637\pi\)
\(954\) 0 0
\(955\) −39.6261 −1.28227
\(956\) 0 0
\(957\) −2.31890 −0.0749594
\(958\) 0 0
\(959\) 12.9394 0.417834
\(960\) 0 0
\(961\) −29.4872 −0.951200
\(962\) 0 0
\(963\) −1.54459 −0.0497736
\(964\) 0 0
\(965\) −19.2228 −0.618803
\(966\) 0 0
\(967\) 33.0737 1.06358 0.531789 0.846877i \(-0.321520\pi\)
0.531789 + 0.846877i \(0.321520\pi\)
\(968\) 0 0
\(969\) −9.69975 −0.311601
\(970\) 0 0
\(971\) 2.21427 0.0710593 0.0355296 0.999369i \(-0.488688\pi\)
0.0355296 + 0.999369i \(0.488688\pi\)
\(972\) 0 0
\(973\) −81.4527 −2.61125
\(974\) 0 0
\(975\) 8.03573 0.257349
\(976\) 0 0
\(977\) −32.6454 −1.04442 −0.522210 0.852817i \(-0.674892\pi\)
−0.522210 + 0.852817i \(0.674892\pi\)
\(978\) 0 0
\(979\) −1.65892 −0.0530193
\(980\) 0 0
\(981\) −10.3685 −0.331042
\(982\) 0 0
\(983\) 47.4633 1.51385 0.756923 0.653505i \(-0.226702\pi\)
0.756923 + 0.653505i \(0.226702\pi\)
\(984\) 0 0
\(985\) −5.66075 −0.180366
\(986\) 0 0
\(987\) −34.8314 −1.10870
\(988\) 0 0
\(989\) 11.3460 0.360781
\(990\) 0 0
\(991\) −44.0157 −1.39821 −0.699103 0.715021i \(-0.746417\pi\)
−0.699103 + 0.715021i \(0.746417\pi\)
\(992\) 0 0
\(993\) −11.1242 −0.353015
\(994\) 0 0
\(995\) −3.71305 −0.117712
\(996\) 0 0
\(997\) 24.1273 0.764120 0.382060 0.924138i \(-0.375215\pi\)
0.382060 + 0.924138i \(0.375215\pi\)
\(998\) 0 0
\(999\) 44.8769 1.41984
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.x.1.4 6
4.3 odd 2 836.2.a.d.1.3 6
12.11 even 2 7524.2.a.r.1.2 6
44.43 even 2 9196.2.a.k.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.d.1.3 6 4.3 odd 2
3344.2.a.x.1.4 6 1.1 even 1 trivial
7524.2.a.r.1.2 6 12.11 even 2
9196.2.a.k.1.3 6 44.43 even 2